Uniqueness for the Brezis-Nirenberg type problems on spheres and hemispheres

TL;DR

This paper investigates Brezis-Nirenberg type nonlinear PDEs on spheres and hemispheres with Neumann boundary conditions, establishing conditions for solutions to be constant and extending results to systems.

Contribution

It provides new conditions ensuring only constant solutions for these PDEs on spheres and hemispheres, including extensions to systems.

Findings

Conditions for constant solutions on spheres and hemispheres

Extension of results to nonlinear PDE systems

Analysis of solutions under Neumann boundary conditions

Abstract

In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for some nonlinear partial differential systems.

Equations

$$\{\begin{array}{rlr}-L_{g}u=& f(u)& \text{mbox}onM,\\ u>& 0& \text{mbox}onM,\\ \frac{\partial u}{\partial \nu }=& 0& \text{mbox}on\partial M,\end{array}$$

$$\{\begin{array}{rlr}-{\Delta }_{g}u+\lambda u=& F(u)u^{\frac{n+2}{n-2}}& \text{mbox}onM,\\ u>& 0& \text{mbox}onM,\\ \frac{\partial u}{\partial \nu }=& 0& \text{mbox}on\partial M,\end{array}$$

$$\{\begin{array}{rlr}-L_g^{1}u& =f(u)& \text{ on }{\mathbb{S}}_{+}^n,\\ \frac{\partial u}{\partial \nu }& =b(u)& \text{ on }\partial {\mathbb{S}}_{+}^n,\end{array}$$

$$\{\begin{array}{rlr}& -L_g^1{u}_1=f_1(u_1,u_2)& \text{ on }{\mathbb{S}}_{+}^n,\\ & -L_g^1{u}_2=f_2(u_1,u_2)& \text{ on }{\mathbb{S}}_{+}^n,\\ & \frac{\partial u_1}{\partial \nu }=b_1(u_1,u_2),\frac{\partial u_2}{\partial \nu }=b_2(u_1,u_2)& \text{ on }\partial {\mathbb{S}}_{+}^n,\end{array}$$

$$\{\begin{array}{rl}& h_1(t):=t^{-\frac{n+2}{n-2}}\left(f(t)+\frac{n(n-2)}{4}t\right)\text{ is decreasing on }(0,+\infty ),\text{ and }\\ & k_1(t):=t^{-\frac{n}{n-2}}b(t)\text{ is nonincreasing nonnegative on }(0,+\infty ).\end{array}$$

$$h_{j1}(t_1,t_2)$$

$$k_{j1}(t_1,t_2)$$

$$\{\begin{array}{rlr}& h_{j1}(t_1,t_2),k_{j1}(t_1,t_2)\text{ are nondecreasing in }{t}_l>0,\text{ with }j\ne l,& \\ & h_{j1}(t_1,t_2)t_j^{\frac{n+2}{n-2}},b_{j1}(t_1,t_2)\text{ are nondecreasing in }{t}_j>0,\text{ and }& \\ & h_{j1}(a_{1}t,a_{2}t),k_{j1}(a_{1}t,a_{2}t)\text{ are decreasing in }t>0\text{ for any }{a}_j>0.& \end{array}$$

$$\{\begin{array}{rlrl}-{\Delta }_{g}u& =u^p-\lambda u& & \text{on }{\mathbb{S}}_{+}^n,\\ \frac{\partial u}{\partial \nu }& =0& & \text{on }\partial {\mathbb{S}}_{+}^n,\end{array}$$

$$\{\begin{array}{rlrl}-{\Delta }_g{u}_1& =u_1^{p_1}-{\lambda }_1{u}_1-{\lambda }_2{u}_2,& & \text{on }{\mathbb{S}}_{+}^n,\\ -{\Delta }_g{u}_2& =u_2^{p_2}-{\lambda }_3{u}_1-{\lambda }_4{u}_2& & \text{on }{\mathbb{S}}_{+}^n,\\ \frac{\partial u_1}{\partial \nu }& =\frac{\partial u_2}{\partial \nu }=0& & \text{on }\partial {\mathbb{S}}_{+}^n,\end{array}$$

$$P_s=\frac{\Gamma \left(B+\frac{1}{2}+s\right)}{\Gamma \left(B+\frac{1}{2}-s\right)},B=\sqrt{-{\Delta }_{g_{{\mathbb{S}}^n}}+{\left(\frac{n-1}{2}\right)}^2},\text{mbox}fors\notin \frac{n}{2}+\mathbb{N},$$

$$P_s^{{\phi }^{\frac{4}{n-2s}}{g}_{{\mathbb{S}}^n}}(u)={\phi }^{-\frac{n+2s}{n-2s}}{P}_s(\phi u)\text{ for all }\phi ,u\in C^{\infty }({\mathbb{S}}^n),\phi >0.$$

$$(P_s^{-1}\varphi )(\zeta )=c_{n,s}{\int }_{{\mathbb{S}}^n}\frac{\varphi (\omega )}{|\zeta -\omega {|}^{n-2s}}d{\upsilon }_{g_{{\mathbb{S}}^n}}^{(\omega )}\text{ for }\varphi \in L^p({\mathbb{S}}^n),$$

$$-L_{g_{{\mathbb{S}}^n}}^{s}u=P_{s}u-d_{n,s}u\text{ and }{d}_{n,s}=:\frac{\Gamma (\frac{n}{2}+s)}{\Gamma (\frac{n}{2}-s)}=P_s(1).$$

$$\{\begin{array}{rlrl}-L_{g_{{\mathbb{S}}^n}}^{s}u& =f(u)& & \text{on }{\mathbb{S}}^n,\\ u& >0& & \text{on }{\mathbb{S}}^n,\end{array}$$

$$\{\begin{array}{rlrl}-L_{g_{{\mathbb{S}}^n}}^s{u}_1& =f_1(u_1,u_2)& & \text{on }{\mathbb{S}}^n,\\ -L_{g_{{\mathbb{S}}^n}}^s{u}_2& =f_2(u_1,u_2)& & \text{on }{\mathbb{S}}^n,\\ u_1,u_2& >0& & \text{on }{\mathbb{S}}^n,\end{array}$$

$$h_s(t):=t^{-\frac{n+2s}{n-2s}}\left(f(t)+d_{n,s}t\right)\text{ is decreasing on }(0,+\infty )\text{ for }s\in (0,n/2)$$

$$h_s(t)t^{\frac{n+2s}{n-2s}}\text{ is nondecreasing on }(0,+\infty )\text{ if }s>1.$$

$$h_{is}(t_1,t_2):=t_i^{-\frac{n+2s}{n-2s}}\left(f_i(t_1,t_2)+d_{n,s}{t}_i\right),t_1>0,t_2>0,\text{ for }i=1,2,$$

$$\{\begin{array}{rlr}& h_{is}(t_1,t_2)\text{ is nondecreasing in }{t}_j>0,\text{ with }i\ne j,& \\ & h_{is}(t_1,t_2)t_i^{\frac{n+2s}{n-2s}}\text{ is nondecreasing in }{t}_i>0,& \\ & h_{is}(a_{1}t,a_{2}t)\text{ is decreasing in }t>0\text{ for any }{a}_i>0.& \end{array}$$

$$-L_g^{s}u$$

$$\frac{\partial u}{\partial \nu }$$

$$\{\begin{array}{rlrl}-L_g^s{u}_1& ={\tilde{f}}_1(u_1,u_2)-{\lambda }_1{u}_1-{\lambda }_2{u}_2& & \text{in }M,\\ -L_g^s{u}_2& ={\tilde{f}}_2(u_1,u_2)-{\lambda }_3{u}_1-{\lambda }_4{u}_2& & \text{in }M,\\ \frac{\partial u_1}{\partial \nu }& =\frac{\partial u_1}{\partial \nu }=0& & \text{on }\partial M.\end{array}$$

$$\left|\frac{{\tilde{f}}_j(r_1,t_1)-{\tilde{f}}_j(r_2,t_2)}{(r_1,t_1)-(r_2,t_2)}\right|\le C\left[(r_1+t_1{)}^{\mu }+(r_1+t_1{)}^{p-1}\right]$$

$$\underset{t\to +\infty }{\lim }\frac{{\tilde{f}}_j(a_{1}t,a_{2}t)}{t^p}={\psi }_j(a_1,a_2)\text{ for all }{a}_j\ge 0\text{ and }$$

$$A=\left(\begin{array}{cc}{\lambda }_1& {\lambda }_2\\ {\lambda }_3& {\lambda }_4\end{array}\right),\text{mbox}and\Vert A\Vert \text{mbox}itsnorm.$$

$${\lambda }_{1,s,g}=\underset{u\in H^s({\mathbb{S}}^n,g)}{\min }\frac{{\int }_{{\mathbb{S}}^n}(u-\overline{u})(P_s^g(u-\overline{u})-R_s^g(u-\overline{u}))d{\upsilon }_g}{{\int }_{{\mathbb{S}}^n}(u-\overline{u}{)}^{2}d{\upsilon }_g}>0,$$

$$\Vert u{\Vert }_{s,g}^2:={\int }_{{\mathbb{S}}^n}u{P}_s^{g}ud{\upsilon }_g,u\in C^{\infty }({\mathbb{S}}^n),$$

$$-L_g^{s}u=u^p-\lambda u\text{ in }{\mathbb{S}}^n,$$

$$\{\begin{array}{rl}-L_g^s{u}_1=u_2^{p_1}-{\lambda }_1{u}_1-{\lambda }_2{u}_2& \text{ in }{\mathbb{S}}^n,\\ -L_g^s{u}_2=u_1^{p_2}-{\lambda }_3{u}_1-{\lambda }_4{u}_2& \text{ in }{\mathbb{S}}^n,\end{array}$$

$$u(\zeta )=c_{n,s}{\int }_{{\mathbb{S}}^n}\frac{\tilde{f}(u)(\omega )+(R_s^g(\omega )-\lambda )u(\omega )}{\mathcal{K}(\zeta ,\omega )}d{\upsilon }_g^{(\omega )},\zeta \in {\mathbb{S}}^n,$$

Full text

This study was financed by the Brazilian agencies: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG)

1. Introduction and main results

Let $(M^{n},g)$, $n\geq 3$, be a compact Riemannian manifold (possibly with non-empty boundary). We consider the following problem

[TABLE]

where $L_{g}$ is a second order partial differential operator on $M^{n}$ with respect to the metric $g$, and $\frac{\partial u}{\partial\nu}$ is the normal derivative of $u$ with respect to the unit exterior normal vector field $\nu$ of the boundary $\partial M$, and $f:(0,\infty)\rightarrow\mathbb{R}$ is a smooth function. The problem will be assumed without boundary conditions if the boundary of $M$ is empty. Our main interest here is to find conditions on $f$, and on the geometry, or on the topology of $M$ that imply the existence of only constant solutions to the problem (P). A particular case of this problem is the following:

[TABLE]

where $\Delta_{g}$ is the Laplace-Beltrami operator on $M^{n}$ with respect to the metric $g$, $\lambda$ is a real smooth function on $M$ and $F:(0,\infty)\rightarrow\mathbb{R}$ is a real smooth function. Note that when $\lambda>0$ is a constant and $F(u)u^{\frac{n+2}{n-2}}=u^{p}$, $p>1$, then $u=\lambda^{\frac{1}{p-1}}$ is a solution of the problem (Q). In the case where $F$ is a constant and $\lambda=\frac{(n-2)}{4(n-1)}R_{g}$, where $R_{g}$ denotes the scalar curvature of the Riemannian manifold $(M,g)$, which is just the Yamabe problem in the conformal geometry for the closed case. Otherwise, when $\partial M$ is not empty, we are in the case of minimal boundary. See [14] for more information on this subject. When $M^{n}=\mathbb{S}^{n}$ is the standard unit $n$-sphere and $g$ is the standard metric, then there are infinitely many solutions to the Yamabe problem with respect to the metric $g$ since the conformal group of the standard unit $n$-sphere is also infinite, see [37]. If $(M^{n},g)$ is an Einstein manifold which is conformally distinct from the standard $n$-sphere, it was shown by Obata that the Yamabe problem has a unique solution. However, R. Schoen has proved that there are at least three solutions for the Yamabe problem on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$ with respect to the standard product metric. In this direction and a good bibliography of literature, see the survey [6], to this and other related questions.

In a more specific situation, the problem (Q) were studied by Lin, Ni and Takagi for the case when $f(u)u^{\frac{n+2}{n-2}}=u^{p}$, $p>1$, $\lambda>0$ is a constant function and $M$ is a bounded convex domain with smooth boundary in the Euclidean space $\mathbb{R}^{n}$ (see [32], [36] and references therein). When $p$ is a subcritical exponent, that is, $p<\frac{n+2}{n-2}$, it was shown in [32] that exists a unique solution if $\lambda$ is sufficiently small. This kind of uniqueness of radially symmetric solution of (P) were also obtained by Lin and Ni in [31], when $\Omega$ is an annulus and $p>1$, or when $\Omega$ is a ball, and $p>\frac{n+2}{n-2}$. They conjectured that, for any $p>1$ there exists a $\bar{\lambda}>0$ such that the problem (P) has only the constant solution, for $0<\lambda<\bar{\lambda}$. However, this conjecture is not true in general, especially when $p$ is the critical Sobolev exponent, that is, $p=\frac{n+2}{n-2}$. In this case, when $\Omega$ is a unit ball and $n=4,5,6$, it was shown by Adimurthi and Yadava [2] that the problem (P) has at least two radial solutions if $\lambda>0$ is close enough to 0 (see also [8]). On the other hand, if $\Omega$ is not a ball or $n$ is different from 4, 5 or 6, that conjecture is open.

In [7], Brezis and Li studied the problem (P) for the case of the standard unit sphere $(\mathbb{S}^{n},g)$, where $g$ is the standard metric, and so, by using results of symmetry due to Gidas, Ni and Nirenberg [16], they showed that this problem has only constant solutions provided that $L_{g}=\Delta_{g}$, and $f$ is a function with the property that $h(t)=t^{-\frac{n+2}{n-2}}(f(t)+\frac{n(n-2)}{4}t)$ is a decreasing function on $(0,+\infty)$. As a consequence of this result, they proved that the problem (Q) has only constant solutions if $0<\lambda<\frac{n(n-2)}{4}$, and $F$ is a decreasing function on $(0,+\infty)$.

Motivated by these results, we will carry out a study on the nonexistence of positive solutions for a wide class of problems, which include equations, systems, and some types of operators, on spheres and hemispheres. Our first attempt is the following nonlinear elliptic equations and systems:

[TABLE]

and

[TABLE]

where $L^{1}_{g}=\Delta_{g}$, and $g$ is a metric in the conformal class of the standard metric $g_{\mathbb{S}^{n}_{+}}$, $n>2$, and $f:[0,+\infty)\rightarrow\mathbb{R}$, $f_{1}$, $f_{2}:[0,+\infty)\times[0,+\infty)\rightarrow\mathbb{R}$ are continuous functions. The notation $L^{1}_{g}$ will make sense in Section 4.

Thus we have the following result:

Theorem 1.1**.**

Let $g$ be the standard metric on $\mathbb{S}^{n}_{+}$. Suppose that

[TABLE]

Then

1. $(i)$

any continuous positive solution $u$ on $\overline{\mathbb{S}^{n}_{+}}$ of (1.1) is constant in each circle parallel to the equator. Moreover, if $b(u(\zeta_{0}))=0$ for some $\zeta_{0}\in\partial\mathbb{S}^{n}_{+}$, then $u$ is constant in $\mathbb{S}^{n}_{+}$; 2. $(ii)$

if $h_{1}(t)t^{\frac{n+2}{n-2}}$ is nonnegative in $t\geq 0$, then any nonnegative continuous solution of (1.1) is zero or positive on $\overline{\mathbb{S}^{n}_{+}}$, and so, it follows the same conclusion of (i). Moreover, if $b(t)\geq\frac{n-2}{2}t$ for $t>0$, then there are no positive solutions of (1.1).

We can extend Theorem 1.1 for systems. Define for $j=1,2$:

[TABLE]

Theorem 1.2**.**

Let $g$ be the standard metric on $\mathbb{S}^{n}_{+}$. Suppose that for $j,l=1,2$:

[TABLE]

Then

1. $(i)$

for any pair $(u_{1},u_{2})$ of positive continuous solutions on $\overline{\mathbb{S}^{n}_{+}}$ of (1.2), $u_{1}$ and $u_{2}$ are constant in each circle parallel to the equator. Moreover, if $b_{1}(u_{1}(\zeta_{1}),u_{2}(\zeta_{2}))=b_{2}(u_{1}(\zeta_{3}),u_{2}(\zeta_{4}))=0$ for some $\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{4}\in\partial\mathbb{S}^{n}_{+}$, then $u_{1}$ and $u_{2}$ are constant; 2. $(ii)$

for any pair $(u_{1},u_{2})$ of nonnegative continuous solutions of (1.2), $u_{j}$ is zero or positive, and so, it follows the same conclusion of (i). Moreover, if $b_{j}(t_{1},t_{2})\geq\frac{n-2}{2}t_{j}$ for some $j=1,2$, then there are no positive solutions of (1.2).

To get a better understanding of the above results, let us consider the following problems:

[TABLE]

where $p>1$, $\lambda>0$, and

[TABLE]

for $\lambda_{1},-\lambda_{2},-\lambda_{3},\lambda_{4}$ being nonnegative values, and $p_{i}>1$ for $i=1,2$. The above problems are related to Brezis-Nirenberg type problems. Then, in the spirit of Lin-Ni’s conjecture, we obtain the following uniqueness results:

Corollary 1.1**.**

Let $g$ be the standard metric on $\mathbb{S}^{n}_{+}$. Assume that $p\leq\frac{n+2}{n-2}$ and $\lambda\leq\frac{n(n-2)}{4}$, and at least one of these inequalities is strict. Then the only positive solution of (1.5) is the constant $u\equiv\lambda^{\frac{1}{p-1}}$.

A similar result on a smooth domain in the Euclidean space with zero Neumann boundary data was established in [46].

Corollary 1.2**.**

Let $g$ be the standard metric on $\mathbb{S}^{n}_{+}$. Assume that $\lambda_{1},-\lambda_{2},-\lambda_{3},\lambda_{4}$ are nonnegative values,

1. $(i)$

$p_{1}\leq\frac{n+2}{n-2}$, $\lambda_{1}\leq\frac{n(n-2)}{4}$, 2. $(ii)$

$p_{2}\leq\frac{n+2}{n-2}$, $\lambda_{4}\leq\frac{n(n-2)}{4}$,

and in each above item at least one of these inequalities is strict. Then for any pair $(u_{1},u_{2})$ of positive solutions of (1.6), $u_{1}$ and $u_{2}$ are constant.

In order to generalize the results of Brezis and Li, let us recall a little bit on conformal operators. In [20], Graham, Jenne, Mason and Sparling constructed a sequence of conformally covariant elliptic operators $P_{k}^{g}$, on Riemannian manifolds $(M,g)$ for all positive integers $k$ if $n$ is odd, and for $k\in\{1,..,n/2\}$ if $n$ is even. Moreover, $P_{1}^{g}$ is the well known conformal Laplacian $L^{1}_{g}:=-\Delta_{g}+d_{n,1}R_{g}$, where $d_{n,1}=(n-2)/[4(n-1)]$, $n\geq 3$, and $P_{2}^{g}$ is the Paneitz operator.

After that, by using a generalized Dirichlet to Neumann map, Graham and Zworski [21] introduced a meromorphic family of conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, see Mazzeo and Melrose [33]. Chang and González [11] established a connection with the method of Graham and Zworski, and the localization method of Caffarelli and Silvestre [9], for fractional Laplacian $(-\Delta)^{s}$ on the Euclidean space $\mathbb{R}^{n}$, and used it to define conformally invariant operators $P_{s}^{g}$ of non-integer order $s\in(0,n/2)$. These lead naturally to a fractional order curvature $R_{s}^{g}=P_{s}^{g}(1)$, which is called $s$-curvature. There are several works on these conformally invariant equations of fractional order and prescribing $s$-curvature problems, which are related to fractional Yamabe and Nirenberg problems, see, e.g., [19, 25, 27, 28] and references therein.

Let $({\mathbb{S}^{n}},g_{\mathbb{S}^{n}})$ be the $n$-sphere, $n\geq 3$ and $g_{\mathbb{S}^{n}}$ the standard metric. The operator $P_{s}:=P_{s}^{g_{\mathbb{S}^{n}}}$ has the formula (see [5])

[TABLE]

where $\Gamma$ is the Gamma function. Moreover, from [11, 19],

[TABLE]

The Green function of $P_{s}^{-1}$ is the spherical Riesz potential,

[TABLE]

where $c_{n,s}=\frac{\Gamma(\frac{n}{2}-s)}{\pi^{\frac{n}{2}}2^{2s}\Gamma(s)}$, $p>1$ and $|\cdot|$ is the Euclidean distance in $\mathbb{R}^{n+1}$.

Denote by

[TABLE]

In Section 3, we will establish uniqueness results for the following problems:

[TABLE]

and

[TABLE]

where $n>2s$, and $f:(0,+\infty)\rightarrow\mathbb{R}$, $f_{1}$, $f_{2}:(0,+\infty)\times(0,+\infty)\rightarrow\mathbb{R}$ are continuous functions. Then, we have the following.

Theorem 1.3**.**

Assume that

[TABLE]

and

[TABLE]

Any continuous solution of (1.10) is constant. In particular, if$f(t)=t^{p}-\lambda t$ with $p\leq\frac{n+2s}{n-2s}$ and $\lambda\leq d_{n,s}$, and at least one of these inequalities is strict, then the only positive solution of (1.10) is the constant $u\equiv\lambda^{\frac{1}{p-1}}$ provided that $p>1$ and $\lambda>0$.

For systems, following our approach, we define

[TABLE]

the result is:

Theorem 1.4**.**

Let $s\in(0,n/2)$. Assume that for $i,j=1,2$:

[TABLE]

Any pair $(u_{1},u_{2})$ of continuous solutions of (1.11) is given by constant solutions.

Now we consider the above problems on the conformal hemisphere $(\mathbb{S}^{n}_{+},g)$ and sphere $(\mathbb{S}^{n},g)$, where $g$ is a conformal metric in the conformal class $[g_{\mathbb{S}^{n}_{+}}]$ or $[g_{\mathbb{S}^{n}}]$ . We consider the following problems:

[TABLE]

and

[TABLE]

where:

for $M=\mathbb{S}^{n}_{+}$, we have that $n\geq 3$, $s=1$, $g$ belongs to the conformal class $[g_{\mathbb{S}^{n}_{+}}]$, $L^{1}_{g}=\Delta_{g}$, $R^{1}_{g}$ is the scalar curvature on $(\mathbb{S}^{n}_{+},g)$; 2. -

for $M=\mathbb{S}^{n}$, we have that $\partial M=\emptyset$, $n\geq 3$, $0<2s<n$, $g$ belongs to the conformal class $[g_{\mathbb{S}^{n}}]$, $L^{s}_{g}=P_{s}^{g}-R^{g}_{s}$, $P^{s}_{g}$ is a conformally invariant operator, $R^{s}_{g}$ is the $s$-scalar curvature on $(\mathbb{S}^{n},g)$;

$\tilde{f}:[0,+\infty)\rightarrow[0,+\infty)$ is a continuous function verifying:

1. $(H1)$

$\tilde{f}(t)t^{-\frac{n+2s}{n-2s}}$ is nonincreasing positive function in $(0,+\infty)$; 2. $(H2)$

there exists $1<p<\frac{n+2s}{n-2s}$ such that $\lim_{t\rightarrow+\infty}\tilde{f}(t)t^{-p}=c_{1}>0$; 3. $(H3)$

there exist $0<\mu\leq p-1$ and $C>0$ such that $\left|\frac{\tilde{f}(t)-\tilde{f}(r)}{t-r}\right|\leq C(t^{\mu}+t^{p-1})$ for all $0\leq r<t$;

$\tilde{f}_{1},\tilde{f}_{2}:[0,+\infty)\times[0,+\infty)\rightarrow[0,+\infty)$ are continuous functions verifying for $j=1,2$:

1. $(F1)$

$\tilde{f}_{j}(r,t)$ is nondecreasing in $r$ and $t$; 2. $(F2)$

${\tilde{f}_{j}(r,t)}{(r+t)^{-\frac{n+2}{n-2}}}$ is nonincreasing in one variable fixing the other variable; 3. $(F3)$

there exist $0<\mu<p-1$, $1<p<\frac{n+2}{n-2}$ such that

[TABLE]

for all $r_{1}>r_{2}\geq 0$, $t_{1}>t_{2}\geq 0$ and some $C>0$; 4. $(F4)$

there exist continuous functions $\psi_{1},\psi_{2}:[0,+\infty)\times[0,+\infty)\rightarrow[0,+\infty)$ such that

[TABLE]

$\psi_{1}(a_{1},a_{2})+\psi_{2}(a_{1},a_{2})>0$ provided that $a_{1}+a_{2}>0$.

Recall that Brezis and Li [7], in the spirit of Lin-Ni’s conjecture, showed that there exists a $\overline{\lambda}>0$ such that (1.15) has only a constant solution if $(M,g)=(\mathbb{S}^{n},g)$, $s=1$, $0<\lambda<\overline{\lambda}$, $\tilde{f}(t)=t^{p}$ and $p\leq\frac{n+2}{n-2}$. Later, Hebey [24] showed the same result for $n\geq 4$ and $p=\frac{n+2}{n-2}$ with positive scalar curvature. Such kind of uniqueness results of (1.15) on a compact manifold $M$ without boundary, with $\tilde{f}(t)=t^{p}$, $s=1$ and $p<\frac{n+2}{n-2}$, it was also shown for by Licois and Véron [29], and by Gidas and Spruck [17].

Denoting

[TABLE]

Since all matrix norms are equivalent we will use the above notation indiscriminately. Our results about uniqueness in this case are:

Theorem 1.5**.**

Let $(M,g)=(\mathbb{S}^{n}_{+},g)$ and $g\in[g_{\mathbb{S}^{n}_{+}}]$. Assume that $\tilde{f}_{1}$ satisfies (H1)-(H3) for $s=1$. Then there exists some $\lambda^{*}=\lambda^{*}(n,g,\mathbb{S}^{n}_{+})>0$ such that:

1. $(i)$

for all $0<\lambda<\lambda^{*}$, any positive continuous solution of (1.15) is constant. In particular, $u\equiv\lambda^{\frac{1}{p-1}}$ is the only positive solution of (1.5) for $\lambda$ small and $1<p<\frac{n+2}{n-2}$; 2. $(ii)$

for all $0<\|A\|<\lambda^{*}$, any pair $(u_{1},u_{2})$ of positive continuous solutions of (1.16) is given constant solutions.

Note that, compared to Corollary 1.2, the values $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ given by Theorem 1.5 (ii) does not necessarily depend on the sign in the case where $\tilde{f}_{1}(t,\cdot)=t^{p_{1}}$, $\tilde{f}_{2}(\cdot,t)=t^{p_{2}}$ and $p_{1}$, $p_{2}$ are subcritical values.

To establish some uniqueness results of solutions for problems involving conformal operators on conformal spheres, we are going to assume throughout this paper that:

[TABLE]

where $s\in(1,n/2)$, $H^{s}(\mathbb{S}^{n},g)$ is the completion of the space of smooth functions $C^{\infty}(\mathbb{S}^{n})$ under the norm $\|\cdot\|_{s,g}$ defined by

[TABLE]

and \overline{u}:=\mathop{\ooalign{\int_{\mathbb{S}^{n}}$\cr$-}}u~{}d\upsilon_{g}. When $s=1$, the value $\lambda_{1,1,g}$ is the first positive eigenvalue of the operator $P^{g}_{1}-R^{g}_{1}=-\Delta_{g}$ on $(\mathbb{S}^{n},g)$. If $s\in(0,1)$, it is not difficult to check (Lemma 5.1) that (1.17) holds. In the case $g=g_{\mathbb{S}^{n}}$ and $0<2s<n$, the value of $\lambda_{1,s,g}$ is determined by spherical harmonics (see [35]). Motivated by the results on the uniqueness of solutions mentioned above, we have:

Theorem 1.6**.**

Let $(M,g)=(\mathbb{S}^{n},g)$, $g\in[g_{\mathbb{S}^{n}}]$ and $0<2s<n$. Assume that $\tilde{f}_{1}$ satisfies (H1)-(H3) and $\tilde{f}_{1},\tilde{f}_{2}$ satisfy (F1)-(F4). If $R_{s}^{g}$ is positive for $s>1$, then there exists some $\lambda^{*}=\lambda^{*}(n,s,g,\mathbb{S}^{n})>0$ such that:

1. $(i)$

for all $0<\lambda<\lambda^{*}$, any positive continuous solution of (1.15) is constant. 2. $(ii)$

for all $0<\|A\|<\lambda^{*}$, and nonnegative values $\lambda_{2}$ and $\lambda_{3}$, any pair $(u_{1},u_{2})$ of positive continuous solutions of (1.16) is given by constant solutions.

Some examples related to the Ni-Li’s conjecture and Brezis-Nirenberg type problems for the fractional case are

[TABLE]

and

[TABLE]

with similar assumptions as in the Corollaries 1.1 and 1.2 we have:

Corollary 1.3**.**

Let $g\in[g_{\mathbb{S}^{n}}]$ be a conformal metric on $\mathbb{S}^{n}$. Assume that $R_{s}^{g}$ is positive for $s>1$ and $\lambda_{1},-\lambda_{2},-\lambda_{3},\lambda_{4}$ are nonnegative values.

1. $(i)$

If $p<\frac{n+2s}{n-2s}$, then the only positive solution of (1.19) is the constant $u\equiv\lambda^{\frac{1}{p-1}}$ for $\lambda$ enough small. 2. $(ii)$

If $p_{j}<\frac{n+2s}{n-2s}$ for $j=1,2$, then any pair $(u_{,}u_{2})$ of positive solutions of (1.20) is given by constant solutions.

Recently in [1], it was shown that Corollary 1.3 (i) holds in the critical case $p=\frac{n+2s}{n-2s}$ with $s$-curvature $R^{g}_{s}$ being positive.

In the proof of Theorem 1.6 for $1<s<n/2$, we used some kind of Schauder type estimates for the solutions of (1.15) and (1.16). To determine these estimates, we will show in Section 3 that the positive solutions of these problems are also positive solutions of the problems

[TABLE]

and

[TABLE]

provided that $R_{s}^{g}$ is positive on $\mathbb{S}^{n}$, where $g=\varphi^{\frac{4}{n-2s}}g_{\mathbb{S}^{n}}$, $\varphi\in C^{\infty}(\mathbb{S}^{n})$ is a positive function, $\mathcal{K}(\zeta,\omega)=\varphi(\zeta)\varphi(\omega)|\zeta-\omega|^{n-2s}$ and $|\cdot|$ is the Euclidean distance on $\mathbb{R}^{n+1}$. On the other hand, the converse holds regardless of the sign of $R^{g}_{s}$, i.e., positive solutions of (1.21) and (1.22) are also positive solutions of (1.15) and (1.16). This leads to the following result:

Theorem 1.7**.**

Let $(M,g)=(\mathbb{S}^{n},g)$, $g\in[g_{\mathbb{S}^{n}}]$ and $0<2s<n$. Then there exists some $\lambda^{*}=\lambda^{*}(n,s,g,\mathbb{S}^{n})>0$ such that:

1. $(i)$

for all $0<\lambda<\lambda^{*}$, any positive continuous solution of (1.21) is constant; 2. $(ii)$

for all $0<\|A\|<\lambda^{*}$, any pair $(u_{1},u_{2})$ of positive continuous solutions of (1.22) is given by constant solutions.

In the next section we prove Theorems 1.1 and 1.2, whereas, in Section 3 we prove Theorems 1.3 and 1.4. To prove these theorems we use the method of moving planes along with some integral inequalities, which were originally based on the ideas of S. Terracini [43]. This type of argument has been widely used to obtain some Liouville-type theorems in fractional differential or integral equations and systems, see, e.g. [3, 12, 22, 23, 44, 45] and references therein. However, to obtain some Liouville-type theorems it is necessary to use the Kelvin transformation, but in our case, the use of this method is not useful. So, the symmetry of $\mathbb{S}^{n}$ will be essential. Finally, in Section 4 we prove Theorems 1.5, 1.6 and 1.7, which are based on the arguments developed by Licois and Véron [29].

2. Proof of Theorems 1.1 and 1.2

Let $\zeta$ be an arbitrary point on $\partial\mathbb{S}^{n}_{+}$, which we will rename the north pole $N$. Let $\mathcal{F}^{-1}:\overline{\mathbb{S}^{n}_{+}}\backslash\{N\}\rightarrow\overline{\mathbb{R}^{n}_{+}}$ be the stereographic projection

[TABLE]

For each $0<2s<n$, we set

[TABLE]

Let $u$ be a solution of (1.1). Considering the new unknown $v$ defined on ${\mathbb{R}}^{n}_{+}$ by

[TABLE]

we have

[TABLE]

From (1.1) and standard computations we get

[TABLE]

In order to show Theorem 1.1 we used the moving plane method to prove symmetry with respect to the axis $y^{n}$ of any positive solution of problem (2.3). The following results are based on [3, 45].

Lemma 2.1**.**

Let $u$ be a positive solution on $\overline{\mathbb{S}^{n}_{+}}$ of (1.1). Under the assumptions of Theorem 1.1, $v=\xi_{1}(u\circ\mathcal{F})$ is symmetric with respect to the axis $y^{n}$.

Proof.

Given $t\in\mathbb{R}$, we set

[TABLE]

where $y_{t}:=I_{t}(y):=(2t-y^{1},y^{\prime})$ is the image of point $y=(y^{1},y^{\prime})$ by the reflection through the hyperplane $U_{t}$. We define the reflected function by $v^{t}(y):=v(y_{t})$. The proof is carried out in three steps. In the first step we show that

[TABLE]

is well-defined, i.e. $\Lambda<+\infty$. The second step consists in proving that if $\Lambda>0$ then $v\equiv v^{\Lambda}$ in $Q_{\Lambda}$. In the third step we conclude that $v$ is a symmetric function in the first $(n-1)$-variables.

Step 2.1.1**.**

$\Lambda<+\infty$.

Assume that there is $t>0$ such that $v^{t}(y)\geq v(y)$ for some $y\in Q_{t}\cup\partial\tilde{Q}_{t}$. Since $|y|<|y_{t}|$, we have $\frac{v(y)}{\xi_{1}(y)}<\frac{v^{t}(y)}{\xi_{1}^{t}(y)}$, and by (1.3),

[TABLE]

and

[TABLE]

where the last inequalities of (2.4) and (2.5) are consequence of $h_{1}(\frac{v}{\xi}),k_{1}(\frac{v}{\xi})\in L^{\infty}(\mathbb{R}^{n}_{+})$. Since $v(y)\rightarrow 0$ as $|y|\rightarrow\infty$, for $\varepsilon>0$ fixed, we can take $(v^{t}-v-\varepsilon)^{+}$ as test function with compact support in ${Q_{t}}\cup\partial\tilde{Q}_{t}$. Then, from (2.4) and (2.5) we obtain

[TABLE]

Using (2.2), we obtain that the right hand side of the above inequality is limited by the integral of a function that independent of $\varepsilon$. In fact, if $(v^{t}(y)-v(y)-\varepsilon)^{+}>0$ for some $y\in Q_{t}$, then $v^{t}(y)>v(y)$,

[TABLE]

and

[TABLE]

We denote

[TABLE]

By Fatou’s lemma, Sobolev inequality, (2.6), dominate convergence theorem and Hölder inequality, we obtain

[TABLE]

where

[TABLE]

Arguing as in the above computations and using Sobolev trace inequality,

[TABLE]

Since $v^{\frac{2n}{n-2}}\in L^{1}(\mathbb{R}^{n}_{+})$ and $v^{\frac{2(n-1)}{n-2}}\in L^{1}(\partial\mathbb{R}^{n}_{+})$, we readily deduce that $\lim_{t\rightarrow+\infty}\phi_{i}(t)=0$ for $i=1,2$. We can choose $t_{1}>0$ large enough such that $C\phi_{i}(t_{1})<1/4$, which from (2.7) and (2.8) implies

[TABLE]

Whence $(v^{t}-v)^{+}\equiv 0$ in $Q_{t}\cup\partial\tilde{Q}_{t}$ for $t>t_{1}$. Therefore $\Lambda$ is well defined, i.e. $\Lambda<+\infty$.

Step 2.1.2**.**

If $\Lambda>0$ then $v\equiv v_{\Lambda}$ in $Q_{\Lambda}$.

By definition of $\Lambda$ and the continuity of the solution, we get $v\geq v^{\Lambda}$ and $\xi_{1}>\xi_{1}^{\Lambda}$ in $Q_{\Lambda}$.

Suppose there is a point $y_{0}\in Q_{\Lambda}\cup\partial\tilde{Q}_{\Lambda}$ such that $v(y_{0})=v^{\Lambda}(y_{0})$. As a consequence there is $r>0$ sufficiently small so that $\frac{v}{\xi_{1}}<\frac{v^{\Lambda}}{{\xi_{1}^{\Lambda}}}$ in $B(y_{0},r)\cap(Q_{\Lambda}\cup\partial\tilde{Q}_{\Lambda})$. Hence, for $y\in B(y_{0},r)\cap Q_{\Lambda}$,

[TABLE]

where $C$ is a non-negative constant. The last inequality is consequence of $v$, $v^{\Lambda}$, $h_{1}(\frac{v^{\Lambda}}{{\xi^{\Lambda}}})\in L^{\infty}(\mathbb{R}^{n})$. If $y_{0}\in Q_{\Lambda}$, then it follows from maximum principle (see [23, Proposition 3.7]) that $v\equiv v^{\Lambda}$ in $B(y_{0},r)$ for $r$ small, and since the set $\{y\in Q_{\Lambda};~{}v(y)=v^{\Lambda}(y)\}$ is open and closed in $Q_{\Lambda}$, it is concluded that $v\equiv v^{\Lambda}$ in ${Q}_{\Lambda}$. On the other hand, if $y_{0}\in\partial\tilde{Q}_{\Lambda}$, then

[TABLE]

and from (2.9) along with the proof of [23, Proposition 3.7], one can show that

[TABLE]

which contradicts (2.10).

Now, we assume that $v>v^{\Lambda}$ in $Q_{\Lambda}\cup\partial\tilde{Q}_{\Lambda}$. We can choose two compacts $K\subset Q_{\Lambda}$ and $W\subset\partial\tilde{Q}_{\Lambda}$, and a number $\delta>0$ such that for all $t\in(\Lambda-\delta,\Lambda)$, we have $K\subset Q_{t}$, $W\subset\partial\tilde{Q}_{t}$ and

[TABLE]

Moreover, there exists $0<\delta_{1}<\delta$, so that

[TABLE]

Using (2.11) and following as in Step 2.1.1, by considering the integrals over $Q_{t}\backslash K\cup\partial\tilde{Q}_{t}\backslash W$, we see that $(v^{t}-v)^{+}\equiv 0$ in $Q_{t}\backslash(K\cup W)$. By (2.12) we get $v>v^{t}$ in ${Q}_{t}$ for all $t\in(\Lambda-\delta_{1},\Lambda)$, which contradicts the definition of $\Lambda$.

Step 2.1.3**.**

Symmetry.

If $\Lambda>0$, then

[TABLE]

This is a contradiction. Thus $\Lambda=0$. By continuity of $v$, we have $v^{0}(y)\leq v(y)$ for all $y\in Q_{0}$. We can also perform the moving plane argument to the left side to get a corresponding $\Lambda^{\prime}$. An analogue to Step 2.1.1 and Step 2.1.2 give us $\Lambda^{\prime}=0$. Then we get $v^{0}(y)\geq v(y)$ for $y\in Q_{0}$. This fact and the opposite inequality implies that $v$ is symmetric with respect to $U_{0}$. Therefore, we obtain $\Lambda=\Lambda^{\prime}=0$ for all directions that are vertical to the $y^{n}$ direction. Thus $v$ is symmetric with respect to the axis $y^{n}$, which completes the proof. ∎

Before continuing with the proof of Theorem 1.1, we want to comment on the proof of Lemma 2.1. In [45], the functions $f$ and $b$ was required to be nondecreasing continuous to get the symmetry of the positive solutions on $\overline{\mathbb{R}^{n}_{+}}$ of (2.3). Thanks to maximum principle [23, Proposition 3.7] and its proof, we don’t need this assumption here.

Lemma 2.2**.**

Let $u$ be a positive solution on $\overline{\mathbb{S}^{n}_{+}}$ of (1.1). Assume as in Lemma 2.1. Then for each $r\in[0,\pi/2]$, $u$ is constant in

[TABLE]

where $|\cdot|_{{\mathbb{S}^{n}}}$ is the distance in ${\mathbb{S}^{n}}$.

Proof.

Let $\zeta_{1}$, $\zeta_{2}\in A_{r}$, $r>0$. There is $\zeta\in\partial\mathbb{S}^{n}_{+}$ so that $|\zeta_{1}-\zeta|_{{\mathbb{S}^{n}}}=|\zeta_{2}-\zeta|_{{\mathbb{S}^{n}}}$. Let $\mathcal{F}^{-1}:\overline{\mathbb{S}^{n}_{+}}\backslash\{\zeta\}\rightarrow\overline{\mathbb{R}^{n}_{+}}$ be the stereographic projection. Then $\mathcal{F}(\zeta_{1})$ and $\mathcal{F}(\zeta_{2})$ are symmetrical points with respect to the axis $y^{n}$. We define $v=\xi_{1}(u\circ\mathcal{F})$ in $\mathbb{R}^{n}_{+}$. From Lemma 2.1 we obtain that $v$ is symmetric with respect to the axis $y^{n}$, and by the definition of $v$, and by the symmetry of $\xi_{1}$, we have $u(\zeta_{1})=u(\zeta_{2})$.

Therefore $u$ is constant in $A_{r}$ for each $r\in(0,\pi/2]$. By continuity of $u$, we have $u$ is constant in $A_{0}$. ∎

Proof of Theorem 1.1. Case (i): let $u$ be a positive solution on $\overline{\mathbb{S}^{n}_{+}}$ of (1.1). For an arbitrary point $\zeta\in\partial\mathbb{S}^{n}_{+}$, let $\mathcal{F}^{-1}:\overline{\mathbb{S}^{n}_{+}}\backslash\{\zeta\}\rightarrow\overline{\mathbb{R}^{n}_{+}}$ be the stereographic projection. We consider the equation (2.3), where $v=\xi_{1}(u\circ\mathcal{F})$ in $\mathbb{R}^{n}_{+}$. By Lemma 2.1 we have that $u$ is constant in each circle parallel to the equator. Assume now that there exists $\zeta_{0}\in\partial\mathbb{S}^{n}_{+}$ such that $b(u(\zeta_{0}))=0$. By Lemma 2.1, $\frac{\partial u}{\partial\nu}\equiv 0$ on $\partial\mathbb{S}^{n}_{+}$, and, therefore, $\frac{\partial v}{\partial\nu}\equiv 0$ on $\partial\mathbb{R}^{n}_{+}$. Define

[TABLE]

where $y=(y^{\prime},y^{n})\in\mathbb{R}^{n}_{+}$. We easily can check that $v^{*}$ satisfies in the weak sense

[TABLE]

Adapting the Lemma 2.1 for $v^{*}$ in the whole space $\mathbb{R}^{n}$, we conclude that $v^{*}$ is radially symmetric. Hence

[TABLE]

where $C$ is a constant.

On the other hand, the set $\mathcal{F}(\{y\in\mathbb{R}^{n}_{+};~{}|y|=1\})$ intersects perpendicularly to $A_{r}$ for any $r\in(0,\pi/2]$. From (2.14) we obtain that $u$ is constant in $\mathcal{F}(\{y\in\mathbb{R}^{n}_{+};|y|=1\})$, which jointly with Lemma 2.2 implies $u$ is constant in $\mathbb{S}^{n}_{+}$.

Case (ii): let $u$ be a nonnegative solution of (1.1). Then $v=\xi_{1}(u\circ\mathcal{F})$, where $\mathcal{F}^{-1}$ is the stereographic projection with $\zeta\partial\mathbb{S}^{n}_{+}$ being the north pole, resolves (2.3). From maximum principle, $u$ is zero or positive in $\overline{\mathbb{S}^{n}_{+}}$. Now, consider $\tilde{v}=\xi_{1}(u\circ\tilde{\mathcal{F}})$ where $\tilde{\mathcal{F}}^{-1}$ is the stereographic projection with $e^{n}=(0,...,1)\in\mathbb{R}^{n+1}$ being the south pole. By standard computations we have

[TABLE]

where $B^{n}_{1}(0)$ is the unit ball with center at [math] in the Euclidean space $\mathbb{R}^{n}$. By Lemma 2.2, $\tilde{v}$ is radially symmetrical. Therefore, the case (ii) follows from the assumptions on $f$ and $b$, and from the maximum principle to $\tilde{v}$. $\square$

Now, we consider the functions $v_{1}$, $v_{2}$ defined on $\mathbb{R}^{n}_{+}$ by

[TABLE]

where $\xi_{1}$ is defined in (2.1) and $(u_{1},u_{2})$ is a pair of solutions of (1.2). Hence

[TABLE]

From (1.2) and standard computations gives

[TABLE]

To prove the Theorem 1.2, we will use the same arguments that were used in the proof of the Theorem 1.1. The following lemma shows the symmetry of solutions of the above system.

Lemma 2.3**.**

Let $(u_{1},u_{2})$ be a pair of solutions of (1.2). Then $v_{1}=\xi_{1}(u_{1}\circ\mathcal{F})$ and $v_{2}=\xi_{1}(u_{2}\circ\mathcal{F})$ are symmetric with respect to the axis $y^{n}$.

Proof.

As in Lemma 2.1, for $t\in\mathbb{R}$ we define the same tools we need; let $Q_{t}$, $\partial\tilde{Q}_{t}$, $U_{t}$, $y_{t}:=I_{t}(y):=(2t-y^{1},y^{\prime})$, and the reflected functions by $v_{1}^{t}(y):=v_{1}(y_{t})$, $v_{2}^{t}:=v_{2}(y_{t})$. The proof will be carried out in three steps. In the first step we show that

[TABLE]

is well-defined, i.e. $\Lambda<+\infty$. The second step consists in proving that if $\Lambda>0$ then $v_{1}\equiv v_{1}^{\Lambda}$ or $v_{2}\equiv v_{2}^{\Lambda}$ in $Q_{\Lambda}$. The third step we conclude the symmetries of $v_{1}$ and $v_{2}$.

Step 2.3.1**.**

$\Lambda<+\infty$.

Assume that there is $t>0$ such that $v_{1}^{t}(y)\geq v_{1}(y)$ for some $y\in Q_{t}\cup\partial\tilde{Q}_{t}$. Since $|y|<|y_{t}|$, we have, $\frac{v_{1}}{\xi_{1}}<\frac{v_{1}^{t}}{\xi_{1}^{t}}$.

If $v_{2}>v_{2}^{t}$, then from (1.4) we have

[TABLE]

where $C$ is a non-negative constant.

If $v_{2}^{t}>v_{2}$, then $\frac{v_{1}^{t}v_{2}^{t}}{\xi_{1}^{t}}>\frac{v_{1}v_{2}}{\xi_{1}}$ and

[TABLE]

where the last inequality is consequence of $(u_{1}\circ\mathcal{F})^{t}((u_{2}\circ\mathcal{F})^{t})^{-1}\in L^{\infty}(\mathbb{R}^{n}_{+})$ and $C$ is a non-negative constant. Making similar computations, we have

[TABLE]

where $C$ is a nonnegative constant. Since $v_{1}(y)\rightarrow 0$ as $|y|\rightarrow\infty$, for fixed $\varepsilon>0$, we can take $(v_{1}^{t}-v_{1}-\varepsilon)^{+}$ as test function with compact support in $Q_{t}\cup\partial\tilde{Q}_{t}$ for (2.17)-(2.19). Then we obtain

[TABLE]

By (2.15), we obtain that the right hand side of the above inequality is limited by the integral of a function independent of $\varepsilon$, namely,

[TABLE]

We denote

[TABLE]

Using Fatou’s lemma, Sobolev inequality, (2.20), dominate convergence theorem and Hölder inequality, we obtain

[TABLE]

where

[TABLE]

Arguing as in the above computations and using Sobolev trace inequality,

[TABLE]

Making a similar computation, we have

[TABLE]

[TABLE]

where

[TABLE]

By (2.15) we deduce that $\lim_{t\rightarrow\infty}\phi_{i,j}(t)=0$ for all $i,j=1,2$. Then we can choose a $t_{1}\in\mathbb{R}$ such that $C\phi_{i,j}(t)<\frac{1}{8}$ for all $t>t_{1}$ and $i,j=1,2$. From (2.21) - (2.24), we deduce

[TABLE]

It follows that $(v_{1}^{t}-v_{1})^{+}\equiv 0\equiv(v_{2}^{t}-v_{2})^{+}$ on $Q_{t}\cup\partial\tilde{Q}_{t}$ for all $t>t_{1}$. Thus $\Lambda$ is well defined, i.e. $\Lambda<+\infty$.

Step 2.3.2**.**

If $\Lambda>0$ then $v_{1}\equiv v_{1}^{\Lambda}$ or $v_{2}\equiv v_{2}^{\Lambda}$ in $Q_{\Lambda}$.

By definition of $\Lambda$ and continuity of solutions, we get $v_{1}\geq v_{1}^{\Lambda}$ and $v_{2}\geq v_{2}^{\Lambda}$ in $Q_{\Lambda}$. Then, for $i=1,2$,

[TABLE]

From maximum principle, we obtain either $v_{i}\equiv v_{i}^{\Lambda}$ in $Q_{\Lambda}$ for some $i=1,2$, or $v_{i}>v_{i}^{\Lambda}$ in $Q_{\Lambda}$ for all $i=1,2$. Suppose $v_{1}>v_{1}^{\Lambda}$ and $v_{2}>v_{2}^{\Lambda}$ in $Q_{\Lambda}$. We can choose compacts $K\subset Q_{\Lambda}$, $W\subset\partial\tilde{Q}_{\lambda}$ and a number $\delta>0$ such that for all $t\in(\Lambda-\delta,\Lambda)$ we have $K\subset Q_{t}$, $W\subset\partial\tilde{Q}_{t}$ and

[TABLE]

On the other hand, there exists $0<\delta_{1}<\delta$, such that

[TABLE]

Using (2.25) and following as in Step 2.3.1, since the integrals are over $Q_{t}\backslash K$, we see that $(v_{i}^{t}-v_{i})^{+}\equiv 0$ in $Q_{t}\backslash K$ for $i=1,2$. By (2.26) we get $v_{1}>v_{1}^{t}$ and $v_{2}>v_{2}^{t}$ in $Q_{t}$ for all $t\in(\Lambda-\delta_{1},\Lambda)$, contradicting the definition of $\Lambda$.

Step 2.3.3**.**

Symmetry.

Suppose $\Lambda>0$. From Step 2.3.2 we can assume $v_{1}=v_{1}^{\Lambda}$. Then

[TABLE]

This is a contradiction. Thus $\Lambda=0$. By continuity of $v_{1}$ and $v_{2}$, we have $v_{1}^{0}(y)\leq v_{1}(y)$ and $v_{2}^{0}(y)\leq v_{2}(y)$ for all $y\in Q_{0}$ . We can also perform the moving plane method to the left side and find a corresponding $\Lambda^{\prime}$. By a similar argument as in Step 2.3.1 and Step 2.3.2, we can show that $\Lambda^{\prime}=0$. Then we get $v_{1}^{0}(y)\geq v_{1}(y)$ and $v_{2}^{0}(y)\geq v_{1}(y)$ for $y\in Q_{0}$. This fact and the above inequality imply that $v_{1}$ and $v_{2}$ are symmetric with respect to $U_{0}$. Therefore, if $\Lambda=\Lambda^{\prime}=0$ for all directions that are vertical to the $y^{n}$ direction, then $v_{1}$ and $v_{2}$ are symmetric with respect to the axis $y^{n}$. ∎

Lemma 2.4**.**

Let $(u_{1},u_{2})$ be a solution of (1.2). Then for each $r\in[0,\pi/2]$, we have that $u_{1}=u_{1}(r)$ and $u_{2}=u_{2}(r)$ in $A_{r}$, where $A_{r}$ is defined by (2.13).

Proof.

The arguments for the proof are the same as the Lemma 2.2. ∎

Proof of Theorem 1.2.

Since $h_{i1}(at,bt)$ and $h_{i1}(at,bt)t^{\frac{n+2}{n-2}}$ are decreasing and nondecreasing in $t$, respectively, it follows that

[TABLE]

Fixing $t$ and taking $r\rightarrow 0$, we have $h_{i1}(at,bt)\geq 0$, and therefore, $h_{i1}$ is nonnegative in $(0,+\infty)\times(0,+\infty)$ for $i=1,2$. Similarly $k_{i1}$ is nonnegative in $(0,+\infty)\times(0,+\infty)$ for $i=1,2$. Therefore, using Lemma 2.4 and maximum principle, the prove of Theorem 1.2 is entirely similar to the proof of Theorem 1.1, so we omit the details here. $\square$

3. Proof of Theorems 1.3 and 1.4

3.1. Problems involving the fractional Laplace operator

We start with a brief review of the operator $(-\Delta)^{s}$ and the relationship between differential equations and integral equations.

Let $\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of the $C^{\infty}$ rapidly decreasing functions in $\mathbb{R}^{n}$. Given $\psi\in\mathcal{S}(\mathbb{R}^{n})$ and $0<2s<n$, the fractional Laplacian operator $(-\Delta)^{s}$ is defined by

[TABLE]

where $~{}\widehat{}~{}$ denotes the Fourier transform. In particular, for $0<s<1$, the above definition is equivalent up to constants to (see [13])

[TABLE]

where $P.V.$ denotes $\lim_{\varepsilon\rightarrow 0^{+}}\int_{|x-y|>\varepsilon}$. Henceforth we will write for $s\notin\mathbb{N}$ as $s=m+\gamma$, with $m\in\mathbb{N}$ and $0<\gamma<1$. By definition of the fractional Laplacian, $(-\Delta)^{s}=(-\Delta)^{m}(-\Delta)^{\gamma}=(-\Delta)^{\gamma}(-\Delta)^{m}$. Moreover, it is very simple to check that

[TABLE]

where $C$ is a positive constant that depends on $n$, $s$ and $\psi$. One can extend this operator to a wider space of distributions as the following. Let

[TABLE]

For $v\in\mathcal{L}_{s}$, we define $(-\Delta)^{s}v$ as a distribution:

[TABLE]

Consider the following equation

[TABLE]

where $f:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function. We say that $v\in\mathcal{L}_{s}$ is a solution of (3.1) in the distributional sense if

[TABLE]

and if $v$ belong to the $H^{s}(\mathbb{R}^{n})$ classical fractional Sobolev space in $\mathbb{R}^{n}$, then we say that $v$ is a weak solution of (3.1). We also consider a class of solutions where the fractional Laplacian is defined in a little bit more stronger sense, that is, $v\in\mathcal{L}_{s}$ is a strong (classical) solution of (3.1) if $v\in C_{loc}^{2m+\lfloor 2\gamma\rfloor,2\gamma-\lfloor 2\gamma\rfloor+\varepsilon}(\mathbb{R}^{n})$, for some $\varepsilon>0$, and $(-\Delta)^{s}v=(-\Delta)^{m}(-\Delta)^{\gamma}v$ is pointwise well-defined and continuous in the whole $\mathbb{R}^{n}$ satisfying (3.1). Moreover, due to the nonlocal feature of $(-\Delta)^{\gamma}$, we have that $(-\Delta)^{\gamma}v\in C^{2m}(\mathbb{R}^{n})$ (see [41]).

The equation (3.1) is closely related to the following integral equation

[TABLE]

where $c_{n,s}=\frac{\Gamma(\frac{n}{2}-s)}{\pi^{\frac{n}{2}}2^{2s}\Gamma(s)}$. In [12], it showed that $v\in H^{s}(\mathbb{R}^{n})$ is a solution of (3.1) if and only if $v$ is a solution of (3.2) provided that $f(\cdot,v)\in L^{\frac{2n}{n+2s}}(\mathbb{R}^{n})$. We will show the same result for $v\in\mathcal{L}_{s}$, a much weaker restriction. The Riesz potential is defined by

[TABLE]

where $0<\alpha<n$. The following properties can be found in [39, Chapter 5]:

1. (P1)

$\|\mathcal{I}_{\alpha}v\|_{L^{q}(\mathbb{R}^{n})}\leq C\|v\|_{L^{p}(\mathbb{R}^{n})}$ if $1<p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$; 2. (P2)

$\mathcal{I}_{\alpha+\beta}v=\mathcal{I}_{\alpha}\mathcal{I}_{\beta}v$ if $v\in L^{p}(\mathbb{R}^{n})$, $1<p<\frac{n}{\alpha+\beta}$, $\alpha,\beta>0$ and $\alpha+\beta<n$; 3. (P3)

$(-\Delta)^{\frac{\alpha}{2}}\mathcal{I}_{\alpha}v=v$ if $v\in L^{p}(\mathbb{R}^{n})$ and $1<p<\frac{n}{\alpha}$.

By the above properties we can see that if $v\in\mathcal{L}_{s}$ is a smooth solution of (3.2) such that $f(\cdot,v(\cdot))\in L^{p}(\mathbb{R}^{n})$ with $1<p<\frac{n}{2s}$, then $v$ is a strong solution of (3.1). We need some additional conditions on $f$ or $v$ to get the converse.

Theorem 3.1**.**

Let $f:\mathbb{R}^{n}\times[0,+\infty)\rightarrow[0,+\infty)$ be a continuous function and let $v\in\mathcal{L}_{s}$ be a nonnegative strong solution of (3.1). If either $f(\cdot,t)$ is nondecreasing in $t>0$ and $\inf_{x\in\mathbb{R}^{n}}f(x,t)>0$ for any $t>0$ or $\lim_{x\rightarrow\infty}v(x)=0$, then $v$ is a nonnegative solution of (3.2) provided $f(\cdot,v(\cdot))\in L^{p}(\mathbb{R}^{n})$ for some $1<p<\frac{n}{2s}$.

Proof.

In our proof, we follows close the arguments of [47]. Let $v$ be a nonnegative solution of (3.1). The case $0<s<1$ is similar to case $s\geq 1$. Assume $s=1+\gamma$, $0\leq\gamma<1$. Denote $v_{0}=:v$, $v_{1}=:(-\Delta)^{\gamma}v_{0}$. Then

[TABLE]

From [10, 22], $v_{0}$ and $v_{1}$ are nonnegatives. Let

[TABLE]

where $G^{1}_{R}(x,y)$ is Green’s function on the ball $B_{R}$:

[TABLE]

It is easy to verify that

[TABLE]

By (3.4) and (3.5), we have

[TABLE]

From the Maximum Principle we have that

[TABLE]

Since $f(\cdot,v_{0})\in L^{p}(\mathbb{R}^{n})$ for $1<p<\frac{n}{2s}<\frac{n}{2}$, it follows from (P1), dominate convergence theorem and (P3) that $\mathcal{I}_{2}(f(\cdot,v_{0}))(x)<+\infty$ for all $x\in\mathbb{R}^{n}$,

[TABLE]

This implies that $v_{1}=w_{1}+C$ on whole space $\mathbb{R}^{n}$, where $C$ is constant nonnegative. Assume $\gamma>0$ and let

[TABLE]

where $G^{\gamma}_{R}(x,y)$ is Green’s function on the ball $B_{R}$ (see e.g. [47]):

[TABLE]

Moreover,

[TABLE]

From the Maximum Principle [41] we have that

[TABLE]

Since $G^{\gamma}_{R}(x,y)\geq 0$ and $\lim_{R\rightarrow+\infty}G^{\gamma}_{R}(x,y)=c_{n,\gamma}|x-y|^{2\gamma-n}$, we have from Fatou’s lemma that

[TABLE]

which implies $v_{1}\equiv w_{1}$ in $\mathbb{R}^{n}$. By (P1) and a simple computation shows that

[TABLE]

By (P3), it follows that

[TABLE]

From [47] we have $v_{0}=w_{0}+C$, where $C$ is a nonnegative constant (This Liouville Theorem type has been first proved by Bogdan, Kulczycki, and Nowak [4]). Since either $\lim_{x\rightarrow\infty}v=0$ or $f(\cdot,v)\geq f(\cdot,C)>0$, we have that $C=0$, and by (P2),

[TABLE]

For the case $s=m+\gamma$, with $m\in\mathbb{N}$, one can prove the theorem using iteratively the above same arguments. ∎

The previous theorem holds if we substitute the condition $f(\cdot,v)\in L^{p}(\mathbb{R}^{n})$ by $0\leq f\in C_{loc}^{0,\varepsilon}$, where $0<\varepsilon<1$. This last condition guarantees

[TABLE]

in the proof of Theorem 3.1 without using (P3). Note that if $s=1$ and

[TABLE]

we can and argue as in [18, Lemmas 4.1 and 4.2] to obtain

[TABLE]

Moreover, making an inductive argument and considering (see [42])

[TABLE]

one can show for $s=m\in\mathbb{N}$ that if $w_{m}(x):=\mathcal{I}_{2m}(f(\cdot,v))(x)<+\infty$ for all $x\in\mathbb{R}^{n}$, then

[TABLE]

In the case $0<s=\gamma<1$, it is not difficult to show that if $w_{\gamma}:=\mathcal{I}_{2\gamma}(f\cdot,v)<+\infty$, then $w_{\gamma}:=\mathcal{I}_{2\gamma}(f\cdot,v)\in C_{loc}^{\lfloor 2\gamma\rfloor,2\gamma-\lfloor 2\gamma\rfloor+\varepsilon}(\mathbb{R}^{n})$. In fact, let be a smooth cutoff function such that $\eta(x)\in[0,1]$ for each $x\in\mathbb{R}^{n}$, $\text{supp}(v)\subset B_{1}$, and $\eta=1$ in $B_{7/8}$. We can write $w_{\gamma}$ as

[TABLE]

By the standard Riesz potential theory it follows that

[TABLE]

On the other hand, we can argue as in [18] to obtain $w_{2,\gamma}\in C^{\lfloor 2\gamma\rfloor}_{loc}(\mathbb{R}^{n})$. Then, for $x_{1},x_{2}\in\overline{B_{1/2}}$ and $2\gamma\leq 1$, we have

[TABLE]

where $x_{3}$ is on the line from $x_{1}$ to $x_{2}$; and for $2\gamma>1$,

[TABLE]

So, $\|w_{\gamma}\|_{C^{\lfloor 2\gamma\rfloor,2\gamma-\lfloor 2\gamma\rfloor+\varepsilon}(\overline{B_{1/2}})}<+\infty$.

Finally, if $w_{\gamma}\in\mathcal{L}_{\gamma}$ then by the Fubini theorem and (P3),

[TABLE]

and therefore,

[TABLE]

Therefore, from the above discussed and following the proof of Theorem 3.1 we have the following result:

Theorem 3.2**.**

Let $f:\mathbb{R}^{n}\times[0,+\infty)\rightarrow[0,+\infty)$ be a $C_{loc}^{0,\varepsilon}$ function and $0<s=m+\gamma<n/2$.

1. (i)

Any $\mathcal{L}_{s}\cap C_{loc}^{2m+\lfloor 2\gamma\rfloor,2\gamma-\lfloor 2\gamma\rfloor+\varepsilon}$ solution of (3.2) is a strong solution of (3.2). 2. (ii)

Let $v\in\mathcal{L}_{s}$ be a nonnegative strong solution of (3.1). If either $f(\cdot,t)$ is nondecreasing in $t\geq 0$ and $\inf_{x\in\mathbb{R}^{n}}f(x,t)>0$ for any $t>0$ or $\lim_{x\rightarrow\infty}v(x)=0$, then $v$ is a nonnegative solution of (3.2).

The above theorem can be adapted for systems

[TABLE]

such that its pair $(v_{1},v_{2})$ of solutions is also a pair of solutions of

[TABLE]

Theorem 3.3**.**

Let $f_{1},f_{2}:\mathbb{R}^{n}\times[0,\infty)\times[0,+\infty)\rightarrow[0,+\infty)$ be $C_{loc}^{0,\varepsilon}$ functions and $0<s=m+\gamma<n/2$.

1. $(i)$

Any pair of $\mathcal{L}_{s}\cap C_{loc}^{2m+\lfloor 2\gamma\rfloor,2\gamma-\lfloor 2\gamma\rfloor+\varepsilon}$ solutions of (3.8) is a pair of strong solutions of (3.7). 2. $(ii)$

Let $(v_{1},v_{2})\in\mathcal{L}_{s}\times\mathcal{L}_{s}$ be a pair of nonnegative strong solutions of (3.7). If either $f_{j}(\cdot,r,t)$ is nondecreasing in $r\geq 0$ and $t\geq 0$ for $j=1,2$, and $\inf_{x\in\mathbb{R}^{n}}(f_{1}+f_{2})(x,r,t)>0$ for any $r+t>0$ or $\lim_{x\rightarrow\infty}v_{j}(x)=0$ for $j=1,2$, then $(v_{1},v_{2})$ is a pair of nonnegative solutions of (3.8).

3.2. Problems involving the conformally operator $P^{g}_{s}$

The operator $P_{s}:=P_{s}^{g_{\mathbb{S}^{n}}}$ can be seen more concretely on $\mathbb{R}^{n}$ using stereographic projection (see for more details [5, 35]). Let $\zeta$ be an arbitrary point on ${\S}^{n}$, which we will rename the north pole $N$ and let $\mathcal{F}^{-1}:{\S}^{n}\backslash\{N\}\rightarrow\mathbb{R}^{n}$ be the stereographic projection. Then

[TABLE]

where $s\in(0,n/2)$, $\xi_{s}$ is defined in (2.1) and $(-\Delta)^{s}$ is the fractional Laplacian operator. Moreover, since the set $C^{\infty}(\mathbb{S}^{n})$ is dense in $H^{s}(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$, it follows that the above identity holds in $H^{s}(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$.

Consider the problem

[TABLE]

where $0<2s<n$, $n>2$, $g\in[g_{\mathbb{S}^{n}}]$, $R_{s}^{g}$ is the $s$-curvature and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function.

From here on we will write the metric $g\in[g_{\mathbb{S}^{n}}]$ as $g=\varphi^{\frac{4}{n-2s}}g_{\mathbb{S}^{n}}$ where $\varphi\in C^{\infty}(\mathbb{S}^{n})$ is a positive function. By (1.7), the above problem is equivalent to

[TABLE]

Now, let $u$ be a smooth solution of (3.10). We define

[TABLE]

Then

[TABLE]

By (3.9), (3.10) and (3.11), $v$ resolves

[TABLE]

Moreover, if $f(t)+R^{g}_{s}t$ is a nonnegative function in $t\geq 0$, then by Theorem 3.1, $v$ is a solution of

[TABLE]

where$c_{n,s}=\frac{\Gamma(\frac{n}{2}-s)}{\pi^{\frac{n}{2}}2^{2s}\Gamma(s)}$. Thus, by the stereographic projection, $u$ solves

[TABLE]

where $\mathcal{K}(\zeta,\omega)=\varphi(\zeta,\omega)|\zeta-\omega|^{n-2s}$ and $|\cdot|$ is the Euclidean distance in $\mathbb{R}^{n+1}$. On the other hand, by the stereographic projection, (3.14), the property (P3) and (3.9), any continuous positive solution of (3.15) is also a continuous positive solution of (3.10). In the case $s>1$, we will use the expression (3.14) to determine the symmetry of the solutions of (3.13).

Therefore, we can write the above mentioned as:

Theorem 3.4**.**

Let $g$ be a conformal metric in $[g_{\mathbb{S}^{n}}]$ and $0<2s<n$.

1. (i)

Any continuous nonnegative solution of (3.15) is also solution of (3.10). 2. (ii)

If $f(t)+R^{g}_{s}t$ is continuous nonnegative in $\mathbb{S}^{n}$ and $t\geq 0$, then any nonnegative smooth solution of (3.10) is also solution of (3.15).

3.3. Proof of Theorem 1.3 for the case $0<s<1$

Let $g=g_{\mathbb{S}^{n}}$ be the standard metric on $\mathbb{S}^{n}$ and $0<s<1$ such that $2s<n$. Let $u$ be a positive solution of (3.10). Taking an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, (3.13) becomes

[TABLE]

where $v=\xi_{s}(u\circ\mathcal{F})$, $\mathcal{F}^{-1}$ is the stereographic projection with $\zeta_{0}$ being the north pole and

[TABLE]

First we will prove Theorem 1.3 for case $s\in(0,1)$. Given $t\in\mathbb{R}$ we set

[TABLE]

where $y_{t}:=(2t-y^{1},y^{\prime})$ is the image of point $y=(y^{1},y^{\prime})$ under of reflection through the hyperplane $U_{t}$. We define the reflected function by $v^{t}(y):=v(y_{t})$. We define the following functions for $t\geq 0$ and $\varepsilon>0$:

[TABLE]

where $(v^{t}-v)^{+}=\max\{v^{t}-v,0\}$ and $(v^{t}-v)^{-}=\min\{v^{t}-v,0\}$. The following result is a Sobolev inequality type for the function $w_{\varepsilon}^{t}$, which will be used to demonstrate the symmetry of $v$.

Lemma 3.1**.**

Under the assumptions of Theorem 1.3 for $s\in(0,1)$, there exists a positive constant $C=C(n,s)>0$ such that

[TABLE]

for each $t>0$.

Proof.

See [15, Lemma 4.2]. ∎

Proof of Theorem 1.3. Let $u$ be the solution of problem (1.10). We take an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, and let $\mathcal{F}^{-1}:{\mathbb{S}^{n}}\backslash\{\zeta\}\rightarrow\mathbb{R}^{n}$ be the stereographic projection. We define $v=\xi_{s}(u\circ\mathcal{F})$ in $\mathbb{R}^{n}$. The proof is carried out in three steps. In the first step we show that

[TABLE]

is well-defined, i.e. $\Lambda<+\infty$. The second step consists in proving that if $\Lambda=0$. In the third step we conclude the proof.

Step 3.1.1**.**

$\Lambda<+\infty$.

For $\varepsilon>0$ and $t>0$, we consider the functions $w_{\varepsilon}^{t}$ and $w^{t}$ defined by (3.18). Using the same arguments as in (2.4), we have

[TABLE]

Following the computations in (2.7), we use Fatou’s lemma, Lemma 3.1, (3.20), Hölder’s and Sobolev’s inequalities, and dominate convergence theorem to obtain

[TABLE]

where $\phi(t)=C(\int_{Q_{t}}(v^{t})^{\frac{2n}{n-2s}}dy)^{\frac{2s}{n}}$. Since $v^{\frac{2n}{n-2s}}\in L^{1}(\mathbb{R}^{n})$, then $\lim_{t\rightarrow+\infty}\phi(t)=0$. Thus, choosing $t_{1}>0$ large sufficiently such that $\varphi(t_{1})<1$, we have from (3.21)

[TABLE]

This implies $(v^{t}-v)^{+}\equiv 0$ in $Q_{t}$ for $t>t_{1}$. Therefore $\Lambda$ is well defined, i.e. $\Lambda<+\infty$.

Step 3.1.2**.**

$\Lambda=0$.

Assume $\Lambda>0$. By definition of $\Lambda$ and continuity of the solution, we get $v\geq v^{\Lambda}$ and $\xi_{s}>\xi_{s}^{\Lambda}$ in $Q_{\Lambda}$.

Suppose there is a point $y_{0}\in Q_{\Lambda}$ such that $v(y_{0})=v^{\Lambda}(y_{0})$. Using the fact of $h$ is decreasing, we have

[TABLE]

On the other hand,

[TABLE]

which contradicts (3.22). As a sequence, $v>v^{\Lambda}$ in $Q_{\Lambda}$.

We can choose a compact $K\subset Q_{\Lambda}$ and a number $\delta>0$ such that $\forall t\in(\Lambda-\delta,\Lambda)$ we have $K\subset Q_{t}$ and

[TABLE]

On the other hand, there exists $0<\delta_{1}<\delta$, such that

[TABLE]

Using (3.23), (3.24) and following as in Step 3.1.1, in (3.21), since the integrals are over $Q_{t}\backslash K$, we see that $(v^{t}-v)^{+}\equiv 0$ in $Q_{t}\backslash K$. By (3.24) we get $v>v^{t}$ in $Q_{t}$ for all $t\in(\Lambda-\delta_{1},\Lambda)$. This is a contradiction with the definition of $\Lambda$.

Step 3.1.3**.**

Symmetry.

By Step 3.1.2 we have $\Lambda=0$ for all directions. This implies that $v$ is radially symmetrical in $\mathbb{R}^{n}$. By definition of $v$, we obtain that $u$ is constant on every $(n-1)$-sphere whose elements $\omega\in{\mathbb{S}^{n}}$ satisfy $|\omega-N|=\text{constant}$. Since $\zeta\in{\mathbb{S}^{n}}$ is arbitrary on ${\mathbb{S}^{n}}$, $u$ is constant. $\square$

Before continuing with the proof of Theorem 1.3 for the case $s>1$, we want to comment on the above proof. In [44], the function $f$ was required to be a nondecreasing continuous to get the symmetry of the positive solutions on $\mathbb{R}^{n}$ of (3.16). Thanks to Lemma 3.1 and the proof of Step 3.1.2, we don’t need this assumption here.

3.4. Proof of Theorem 1.3 for the case $1<s<n/2$

Let $g=g_{\mathbb{S}^{n}}$ be the standard metric on $\mathbb{S}^{n}$. Let $u$ be a positive solution of (3.10). Taking an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, $v=\xi_{s}(u\circ\mathcal{F})$ resolves (3.16), where $\mathcal{F}^{-1}$ is the stereographic projection with $\zeta_{0}$ being the north pole. Assume that $h_{s}(t)$ and $h_{s}(t)t^{\frac{n+2s}{n-2s}}$ are nonincreasing and nondecreasing in $t>0$, respectively. Then, for $0<r<t$,

[TABLE]

Taking $t\rightarrow+\infty$ in the above inequality, we have $h_{s}(r)\geq 0$ for all $r>0$. It follows from (3.16) and Theorem 3.1 that $v$ resolves (up to constants)

[TABLE]

To prove Theorem 1.3 in the case $s\in(1,n/2)$ we need the following result, which is essential to show the symmetry of the solutions of (3.25).

Lemma 3.2**.**

[TABLE]

Proof.

See the proof of Lemma 1 in [44]. ∎

Proof of Theorem 1.3. Let $u$ be the solution of problem (1.10). We take an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, and let $\mathcal{F}^{-1}:{\mathbb{S}^{n}}\backslash\{\zeta_{0}\}\rightarrow\mathbb{R}^{n}$ be the stereographic projection. We define $v=\xi_{s}(u\circ\mathcal{F})$ in $\mathbb{R}^{n}$. It follows that $v$ solves (3.25). Similarly as in the proof of this theorem for case $s\in(0,1)$, this one consists of three steps. Let

[TABLE]

where $Q_{t}$ is defined by (3.17).

Step 3.2.1**.**

$\Lambda<+\infty.$**

As in Step 2.1.1, we assume that there is a $t>0$ such that: if $v^{t}(y)\geq v(y)$ for some $y\in Q_{t}$, then

[TABLE]

or if $v^{t}(y)\leq v(y)$, then

[TABLE]

where the last inequality in (3.26) is consequence of $h_{s}(\frac{v}{\xi})\in L^{\infty}(\mathbb{R}^{n})$. From Lemma 3.2 and the above inequalities, we have

[TABLE]

where $|y-z|<|y_{t}-z|$ for all $z\in Q_{t}$ and $\chi_{Q_{t}}$ is the indicator function.

By the Hardy-Littlewood-Sobolev [30] and Hölder inequalities, it follows that

[TABLE]

where $1<r<q$ and $(q-r)/(rq)=2s/n$. Since $v\in L^{\frac{2n}{n-2s}}(\mathbb{R}^{n})$, we can choose a $t_{1}\in\mathbb{R}$ such that

[TABLE]

for all $t>t_{1}$. Then we conclude $(v^{t}-v)^{+}\equiv 0$ in $Q_{t}$ for all $t>t_{1}$. Therefore $\Lambda$ is well defined.

Step 3.2.2**.**

$\Lambda=0$.

Assume $\Lambda>0$. By definition of $\Lambda$, we have $v\geq v^{\Lambda}$ in $Q_{t}$. We can infer from (3.27) and Lemma 3.2 that $v>v^{\Lambda}$ in $Q_{\Lambda}$. The conclusion follows from the arguments used in the last part of the proof of Step 3.1.2.

Step 3.2.3**.**

Symmetry.

This step is similar to Step 3.1.3. $\square$

3.5. Uniqueness results for systems

Based on the arguments presented in the above subsections we can extend Theorem 1.3 for systems. Let $(u_{1},u_{2})$ be a pair of positive solutions of problem (1.11). We take an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, and let $\mathcal{F}^{-1}:{\mathbb{S}^{n}}\backslash\{\zeta_{0}\}\rightarrow\mathbb{R}^{n}$ be the stereographic projection. We define

[TABLE]

Then we have that

[TABLE]

By (1.11) and (3.9), $(v_{1},v_{2})$ solves

[TABLE]

where

[TABLE]

Following the proof of Theorem 1.2, we have that $h_{1s}$, $h_{2s}$ are nonnegatives functions. Then by Theorem 3.3, we have that $(v_{1},v_{2})$ is a par of positive solutions of (up to constants)

[TABLE]

We define the reflected functions $v_{1}^{t}(y):=v_{1}(y_{t})$ and $v_{2}^{t}(y):=v_{2}(y_{t})$ with respect to the hyperplane $U_{t}$. The following result is a version of Lemma 3.2 for systems.

Lemma 3.3**.**

[TABLE]

Proof of Theorem 1.4. Let $(u_{1},u_{2})$ be a pair of solutions of problem (1.11). We take an arbitrary point $\zeta_{0}\in\mathbb{S}^{n}$, and let $\mathcal{F}^{-1}:{\mathbb{S}^{n}}\backslash\{\zeta_{0}\}\rightarrow\mathbb{R}^{n}$ be the stereographic projection. We define $v_{i}=\xi_{s}(u_{j}\circ\mathcal{F})$ in $\mathbb{R}^{n}$, $j=1,2$. It follows that $(v_{1},v_{2})$ solves (3.31). Similarly as in the proofs of Theorem 1.3 and Lemma 2.3, this one consists of three steps. Let

[TABLE]

where $Q_{t}$ is defined by (3.17).

Step 3.3.1**.**

$\Lambda<+\infty$.

Following the computations in (2.17) and (2.18), we have for $j=1,2$,

[TABLE]

By Lemma 3.3 and arguing as Step 3.2.1, we obtain

[TABLE]

where $q>1$. Since $v_{j}\in L^{\frac{2n}{n-2s}}(\mathbb{R}^{n})$, we can choose a $t_{1}\in\mathbb{R}$ such that

[TABLE]

for all $t>t_{1}$. Then we conclude $(v_{j}^{t}-v_{j})^{+}\equiv 0$ in $Q_{t}$ for all $t>t_{1}$ and $j=1,2$. Therefore $\Lambda$ is well defined.

The other two steps ($\Lambda=0$ and symmetry) are similar to Steps 3.2.2 and 3.2.3.

$\square$

4. Proof of Theorems 1.5, 1.6 and 1.7

We start with the following lemma that shows some estimates for the solutions of (1.15) and (1.16). Recall that (1.17) holds for all $0<s<n/2$ and $g=\varphi^{\frac{4}{n-2s}}g_{M}$ denotes the conformal metric on $M=\mathbb{S}^{n}_{+}$ or $M=\mathbb{S}^{n}$, where $\varphi\in C^{\infty}(M)$ is a positive function on $M$. In the case when $M=\mathbb{S}^{n}_{+}$, $\lambda_{1,1,g}=\lambda_{1,g}$ is the first nonzero Neumann eigenvalue of $-L^{1}_{g}=-\Delta_{g}$ on $(\mathbb{S}^{n}_{+},g)$.

Lemma 4.1**.**

Let $(M,g)=(\mathbb{S}^{n}_{+},g)$ or$(\mathbb{S}^{n},g)$, $g\in[g_{M}]$, $n>2$ and $0<2s<n$.

1. (i)

Suppose that $\tilde{f}$ satisfies (H1)-(H3). Then there exists a positive constant $C=C(n,s,\tilde{f})$ such that

[TABLE]

for any nonnegative solution $u$ of (1.15), where \overline{u}=\mathop{\ooalign{\int_{\mathbb{S}^{n}}$\cr$-}}u~{}d\upsilon_{g}. 2. (ii)

Suppose that $\tilde{f}_{j}$ satisfies (F1)-(F4) for each $j=1,2$. Then there exists a positive constant $C=C(n,s,\tilde{f}_{1},\tilde{f}_{2})$ such that

[TABLE]

for any pair $(u_{1},u_{2})$ of positive solutions of (1.16).

Proof.

Case (i). Let $u$ be a nonnegative solution of (1.15). By the assumptions on $\tilde{f}$, we have that

[TABLE]

Multiplying (1.15) by $(u-\overline{u})$ and integrating, we have

[TABLE]

Clearly we have

[TABLE]

By (4.2) and (4.3) we have

[TABLE]

The first part of the theorem follows from the above equality, (4.1), (1.17) for $M=\mathbb{S}^{n}$ or $\lambda_{1,g}$ for $M=\mathbb{S}^{n}_{+}$.

Case (ii). Let $(u_{1},u_{2})$ a pair of nonnegative solutions of (1.16). By the assumptions on $\tilde{f}_{1},\tilde{f}_{2}$, we have

[TABLE]

for $j=1,2$. Following as in the proof of (i), we have

[TABLE]

and

[TABLE]

It follows from the above equalities that

[TABLE]

The second of the theorem follows from the above equality, (4.4), (1.17) if $M=\mathbb{S}^{n}$ or $\lambda_{1,g}$ if $M=\mathbb{S}^{n}_{+}$. ∎

The following result, whose proof is easily demonstrated, will be used in the proof of Theorems 1.5, 1.6 and 1.7 for the system case.

Lemma 4.2**.**

Assume that $f_{1}$, $f_{2}$ satisfy (F1)-(F4). Then $\psi_{1},\psi_{2}:[0,+\infty)\times[0,+\infty)\rightarrow[0,+\infty)$ satisfy the following:

1. (i)

$\psi_{1}(r,t)$, $\psi_{2}(r,t)$ are nondecreasing in $r\geq 0$ and $t\geq 0$; 2. (ii)

${\psi_{1}(at,bt)}{t^{-\frac{n+2s}{n-2s}}}$, ${\psi_{2}(at,bt)}{t^{-\frac{n+2s}{n-2s}}}$ are nonincreasing in $t>0$ for any $a\geq 0$, $b\geq 0$; 3. (iii)

there exist $C>0$ such that

[TABLE] 4. (iv)

$(\psi_{1}+\psi_{2})(r,t)>0$* if $r+t>0$;* 5. (v)

if $\lim_{k\rightarrow+\infty}a_{k}=a$ and $\lim_{k\rightarrow+\infty}b_{k}=b$,

[TABLE]

Remark 4.1**.**

If $(v_{1},v_{2})$ is a pair of nonnegative strong solutions of

[TABLE]

where $0<2s<n$ and $\psi_{1},\psi_{2}$ are given by Lemma 4.2, then by [10, 22] and maximum principle we have that $v_{j}=0$ or $v_{j}>0$ in $\mathbb{R}^{n}$ for $j=1,2$. Moreover, if $v_{1}+v_{2}\neq 0$ then by Theorems 3.2 and 3.3, either $v_{j}$ is positive solution of

[TABLE]

or $(v_{1},v_{2})$ is a pair of positive solutions of

[TABLE]

4.1. Constants solutions for some elliptic problems on $\mathbb{S}^{n}_{+}$

In this subsection we will prove Theorem 1.5. The following result shows a type of a priori estimate for the solutions of (1.15).

Lemma 4.3**.**

Let $(M,g)=(\mathbb{S}^{n}_{+},g)$ and $g\in[g_{\mathbb{S}^{n}_{+}}]$. Suppose that $\tilde{f}$ satisfies (H1)-(H3) for $s=1$. Then there exists a positive constant $C=C(n,g,\mathbb{S}^{n}_{+},\tilde{f})$ such that for any $\lambda\geq 0$, any nonnegative continuous solution $u$ of (1.15) satisfies

[TABLE]

Proof.

Let us suppose that (4.6) does not hold. Then there exist four sequences $C_{k}$, $\lambda_{k}$, $\zeta_{k}$ and $u_{k}$ such that $\lambda_{k}\geq 0$ and $u_{k}\neq 0$ is a nonnegative solution of

[TABLE]

with the following properties:

[TABLE]

There are three possibilities:

Case 4.3.1**.**

$\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$* tends to zero when $k$ tends to infinity.*

Then there exists some $k_{0}$ such that

[TABLE]

By Lemma 4.1, $u_{k}$ is constant for $k$ large. From (4.7) and (4.9) it follows that

[TABLE]

and, since $u_{k}<1$ for $k$ large and $\tilde{f}(t)t^{-\frac{n+2}{n-2}}$ is nonincreasing positive in $t>0$, we have that $1\geq C^{\frac{n-2}{4}}_{k}\tilde{f}(1)$, which contradicts (4.8).

Case 4.3.2**.**

$\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$* tends to some nonzero limit $c$ when $k$ tends to infinity.*

From (4.8) - (4.9), $\lambda_{k}$ tends to 0. Setting $\tilde{u}_{k}=u_{k}/\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$ then

[TABLE]

Setting the inverse of stereographic projection $\mathcal{F}$ for some $\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole and denoting $H=\xi_{1}(\varphi\circ\mathcal{F})$, we can see that $v_{k}=H(\tilde{u}_{k}\circ\mathcal{F})$ resolves

[TABLE]

One can see that the right side of the above problem is in $L^{1}(\mathbb{R}^{n}_{+})$ and is uniformly bounded in $k$. Since $\tilde{f}\in C^{0,\gamma}_{loc}(\mathbb{R})$, it follows from [34, Theorems 8.1, 8.2 and 8.3] and after passing to a subsequence that

[TABLE]

where $v$ and $\tilde{u}$ are non-negative smooth functions defined, respectively, in $\overline{\mathbb{R}^{n}_{+}}$ and $\overline{\mathbb{S}^{n}_{+}}$, with $0<\beta<1$. If we consider $-\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole, and following the above arguments, we have after passing to a subsequence that $\tilde{u}$ solves

[TABLE]

and $\tilde{u}(\zeta_{0})=1$, which is impossible because the integral on $\mathbb{S}^{n}_{+}$ on the right side of (4.10) is zero while the integral of the left side is not zero.

Case 4.3.3**.**

$\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$* tends to infinity when $k$ tends to infinity.*

There exists a sequence $\{\zeta_{k}^{\prime}\}_{k\in\mathbb{N}}$ on $\overline{\mathbb{S}^{n}_{+}}$ such that

[TABLE]

By (4.7), $v_{k}=H(u_{k}\circ\mathcal{F})$ resolves

[TABLE]

where $H=\xi_{1}(\varphi\circ\mathcal{F})$ and $\mathcal{F}$ is the inverse of stereographic projection for some $\zeta\in\partial\mathbb{S}^{n}_{+}$, $\zeta\neq\zeta_{0}^{\prime}$, being the north pole. Then there exist $R>0$ and $C>0$ such that $x_{k}:=\mathcal{F}^{-1}(\zeta_{k}^{\prime})\in D_{R}:=B_{R}\cap\overline{\mathbb{R}^{n}_{+}}$ for all $k$,

[TABLE]

Let us define the following scaling

[TABLE]

where $\alpha_{k}$ is defined by

[TABLE]

By (4.11), $\tilde{v}_{k}$ satisfies $\|\tilde{v}_{k}\|_{L^{\infty}(D_{R})}>\tilde{C}>0$ and

[TABLE]

where

[TABLE]

By the conditions on $\tilde{f}$, the right side of (4.12) is in $L^{1}(\mathbb{R}^{n}_{+})$ and is uniformly bounded in $k$. Since $\tilde{f}\in C^{0,\gamma}_{loc}([0,+\infty))$, it follows from the regularity theory in [34] that

[TABLE]

for some nonnegative function $\tilde{v}$ and $\beta\in(0,1)$. Moreover, since

[TABLE]

it gets that

[TABLE]

Therefore, since $\lim_{t\rightarrow+\infty}\tilde{f}(t)t^{-p}=c_{1}$, it follows that

[TABLE]

and $\tilde{v}$ solves

[TABLE]

and $\|\tilde{v}\|_{L^{\infty}(D_{R})}>0$, which is impossible from [45] since $p<(n+2)/(n-2)$. ∎

We also have a type of a priori estimate for the solutions of (1.16).

Lemma 4.4**.**

Let $(M,g)=(\mathbb{S}^{n}_{+},g)$ and $g\in[g_{\mathbb{S}^{n}_{+}}]$. Suppose that $\tilde{f}_{j}$ satisfies (F1)-(F4) for $s=1$ and $j=1,2$. Then there exists a positive constant $C=C(n,g,\mathbb{S}^{n}_{+},\tilde{f}_{1},\tilde{f}_{2})$ such that for any $\|A\|$, and any nonnegative continuous solution pair $(u_{1},u_{2})$ of (1.16) with $M=\mathbb{S}^{n}_{+}$ satisfies

[TABLE]

Proof.

We will follow the arguments used in the proof of Lemma 4.3. Let us suppose that (4.13) does not hold. Then there exist sequences $C_{k}$, $\zeta_{k}$, $\omega_{k}$, $u_{1,k}$, $u_{2,k}$ and

[TABLE]

such that $U_{k}:=(u_{1,k},u_{2,k})\neq(0,0)$ is a pair of nonnegative solutions of

[TABLE]

with the following properties:

[TABLE]

for some $\zeta_{0}$, $\omega_{0}\in\overline{\mathbb{S}^{n}_{+}}$. There are three possibilities.

Case 4.4.1**.**

$\|u_{1,k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}+\|u_{2,k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$* tends to zero when $k$ tends to infinity.*

From (4.15)-(4.17), we have $\lim_{k\rightarrow+\infty}\|A_{k}\|=0$. Then there exists $k_{0}$ such that for $k\geq k_{0}$, it has that

[TABLE]

By Lemma 4.1, $u_{1,k}$ and $u_{2,k}$ are constants. From (4.14),

[TABLE]

Since $\tilde{f}_{j}(r,t)(r+t)^{-\frac{n+2}{n-2}}$ is nonincreasing in $(r,t)$ for $j=1,2$, and $u_{1,k}<1$, $u_{2,k}<1$ for k large, it follows from (4.16) that

[TABLE]

which is a contradiction since $\tilde{f}_{1}(1,1)+\tilde{f}_{2}(1,1)>0$.

Case 4.4.2**.**

$\|u_{1,k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$* and $\|u_{2,k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$ tend to some limit $c_{1}\neq 0$ and $c_{2}$, respectively, when $k$ tends to infinity.*

From (4.15)-(4.17), $\|A_{k}\|$ tend to [math]. Following the proof of Lemma 4.3 (Case 4.3.2), we set $\tilde{u}_{1,k}=u_{1,k}/\|u_{1,k}\|_{L^{\infty}(\mathbb{S}^{n}_{+})}$ and then $v_{1,k}=H(\tilde{u}_{1,k}\circ\mathcal{F})$, $v_{2,k}=H(u_{2,k}\circ\mathcal{F})$ resolve

[TABLE]

where $\mathcal{F}$ is the inverse of stereographic projection for some $\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole, $H=\xi_{1}(\varphi\circ\mathcal{F})$ in $\mathbb{R}^{n}$,

[TABLE]

One can see that the right hand side of the above system is in $L^{1}(\mathbb{R}^{n})$ and is uniformly bounded in $k$. Then, since $\tilde{f}_{1},\tilde{f}_{2}\in C^{0,\gamma}_{loc}([0,+\infty)\times[0,+\infty))$, it follows from [34, Theorems 8.1, 8.2 and 8.3] and after passing to a subsequence that

[TABLE]

where $v_{1}$, $\tilde{u}_{1}$, $v_{2}$ and $\tilde{u}_{2}$ are non-negative functions and $0<\beta<1$. If we consider $-\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole and following the same previous arguments, we have that $(\tilde{u}_{1},u_{2})$ solves

[TABLE]

and $\tilde{u}_{1}(\zeta_{0})=1$, which is impossible because either $\tilde{u}_{1}\equiv 0$ or the integral of $L^{1}_{g}\tilde{u}_{1}+c_{1}^{-1}L^{1}_{g}{u}_{2}$ on $\mathbb{S}^{n}_{+}$ is zero while the integral of $\tilde{f}_{1}(c_{1}\tilde{u}_{1},u_{2})+\tilde{f}_{2}(c_{1}\tilde{u}_{1},u_{2})$ is not zero.

Case 4.4.3**.**

$\|u_{1,k}\|_{L^{\infty}(\mathbb{S}^{n})}$* or $\|u_{2,k}\|_{L^{\infty}(\mathbb{S}^{n})}$ tend to infinity when $k$ tends to infinity.*

Assume after passing to a subsequence that

[TABLE]

There exist sequences $\{\zeta_{k}^{\prime}\}_{k\in\mathbb{N}},\{\omega_{k}^{\prime}\}_{k\in\mathbb{N}}$ on $\overline{\mathbb{S}^{n}_{+}}$ such that

[TABLE]

By (4.14), $v_{1,k}=H(u_{1,k}\circ\mathcal{F}]$ and $v_{2,k}=H(u_{2,k}\circ\mathcal{F})$ resolve

[TABLE]

where $\mathcal{F}^{-1}$ is the stereographic projection for some $\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole such that $\zeta\neq\zeta_{0}^{\prime}$, $\zeta\neq\omega_{0}^{\prime}$, and $\tilde{a}_{k}$, $\tilde{b}_{k}$, $\tilde{c}_{k}$, $\tilde{d}_{k}$ are given by (4.18). Following the proof of Lemma 4.3, there exist $R>0$ and $C>0$ such that $x_{k}:=\mathcal{F}^{-1}(\zeta_{k}^{\prime}),~{}y_{k}:=\mathcal{F}^{-1}(\omega_{k}^{\prime})\in D_{R}:=B_{R}\cap\overline{\mathbb{R}^{n}_{+}}$ for all $k$,

[TABLE]

Let us define the following scaling

[TABLE]

where $\alpha_{k}$ is defined by $\alpha_{k}^{\frac{2}{p_{1}-1}}\|v_{1,k}\|_{L^{\infty}(\mathbb{R}^{n})}=1$. By (4.19),

[TABLE]

From (4.20), $(\tilde{v}_{1,k},\tilde{v}_{2,k})$ satisfies $\|\tilde{v}_{1,k}\|_{L^{\infty}(D_{R})}\geq\tilde{C}>0$ for all $k$ and

[TABLE]

By the conditions on $\tilde{f}_{1}$ and $\tilde{f_{2}}$, the right side of (4.22) and (4.23) are in $L^{1}(\mathbb{R}^{n}_{+})$ and are uniformly bounded in $k$. Since $\tilde{f}_{1},\tilde{f}_{2}\in C^{0,\gamma}_{loc}([0,+\infty)\times[0,\infty))$, it follows from the regularity theory in [34] that

[TABLE]

for some nonnegative functions $\tilde{v}_{1}$, $\tilde{v}_{2}$ and $\beta\in(0,1)$. Moreover,

[TABLE]

for $\tilde{v}_{1}(x)\tilde{v}_{2,k}(x)\neq 0$. By Lemma 4.2, it follows that

[TABLE]

for $j=1,2$, and $(\tilde{v}_{1},\tilde{v}_{2})$ solves

[TABLE]

for some $\tilde{c}_{1},\tilde{c}_{2}$ positive constants and $\|\tilde{v}_{1}\|_{L^{\infty}(D_{R})}>0$, which is impossible from maximum principle and [45] since $\psi_{j}$ satisfies Lemma 4.2 (iii) with $p<\frac{n+2}{n-2}$. ∎

Proof of Theorem 1.6. It follows from Lemma 4.3, Lemma 4.4 and Lemma 4.1 for $\lambda$ and $\|A\|$ small values. $\square$

4.2. Constant solutions for problems involving the operator $P^{g}_{s}$ on $\mathbb{S}^{n}$

In this subsection we will prove Theorems 1.6 and 1.7. We begin by showing the following lemma that is used to determine Schauder type estimates of the solutions of (1.15), (1.16), (1.21) and (1.22).

Lemma 4.5**.**

Let $H:\mathbb{R}^{n}\rightarrow(0,+\infty)$ be a $C^{\infty}(\mathbb{R}^{n})$ function such that $\|H\|_{C^{0,\gamma}(\mathbb{R}^{n})}<C_{1}$, $|\nabla H(x)|\leq C_{1}H(x)$ for all $x\in\mathbb{R}^{n}$ and

[TABLE]

where $C_{1}$ depends on $n$ and $s$. Assume that $\tilde{f}$ and $\tilde{f}_{1},\tilde{f}_{2}$ satisfy (H1)-(H3) and (F1)-(F4), respectively. Then:

1. (i)

for any $v\in C^{0,\gamma}(\mathbb{R}^{n})$ nonnegative function,

[TABLE]

where $C$ depends on $C_{1}$, $\tilde{f}$ and $\|v\|_{C^{0,\gamma}(\mathbb{R}^{n})}$; 2. (ii)

for any $v_{1},v_{2}\in C^{0,\gamma}(\mathbb{R}^{n})$ nonnegative functions,

[TABLE]

where $C$ depends on $C_{1}$, $\tilde{f}_{1}$, $\tilde{f}_{2}$, $\|v_{1}\|_{C^{0,\gamma}(\mathbb{R}^{n})}$ and $\|v_{2}\|_{C^{0,\gamma}(\mathbb{R}^{n})}$.

Proof.

We will prove only (i), since the proof of (ii) is similar to (i). By the assumptions on $\tilde{f}$, we have

[TABLE]

Let $x,y\in\mathbb{R}^{n}$. Assume first that $|x-y|\geq 1$. Then

[TABLE]

Now assume that $|x-y|<1$. Then

[TABLE]

where $0\leq\theta\leq 1$. Therefore, the lemma follows from (4.24) and (4.25). ∎

The following lemma shows a type of a priori estimation of the solutions of (1.15). Recall that the condition $R^{g}_{s}$ is used to transform the problems (1.15), (1.16) into integral problems of the type (1.21), (1.22), and thus, to be able to determine some Schauder type estimates for the solutions of the problems addressed.

Lemma 4.6**.**

Let $(M,g)=(\mathbb{S}^{n},g)$ and $g\in[g_{\mathbb{S}^{n}}]$. Suppose that $\tilde{f}$ satisfies (H1)-(H3) for $0<2s<n$ and $R^{g}_{s}$ is positive for $s>1$. Then there exists a positive constant $C=C(n,s,g,\mathbb{S}^{n},\tilde{f})$ such that for any $\lambda\in[0,+\infty)$ if $0<s\leq 1$ or $\lambda\in[0,\min R^{g}_{s}]$ if $s>1$, any nonnegative smooth solution $u$ of (1.15) satisfies

[TABLE]

Proof.

Let us suppose that (4.26) does not hold. Then there exist four sequences $C_{k}$, $\lambda_{k}$, $\zeta_{k}$ and $u_{k}$ such that $\lambda_{k}\geq 0$ and $u_{k}$ is a nonnegative solution of

[TABLE]

with the following properties:

[TABLE]

for some $M\in\mathbb{R}^{n}$ and $\zeta_{0}\in\mathbb{S}^{n}$. Following the proof of Lemma 4.3, there are three possibilities:

Case 4.6.1**.**

$\|u_{k}\|_{L^{\infty}(M)}$* tends to zero when $k$ tends to infinity.*

This case is similar to Case 4.3.1.

Case 4.6.2**.**

$\|u_{k}\|_{L^{\infty}(M)}$* tends to some nonzero limit $c$ when $k$ tends to infinity.*

From (4.28) - (4.30), $\lambda_{k}$ tends to 0. Setting $\tilde{u}_{k}=u_{k}/\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n})}$ then

[TABLE]

Setting the inverse of stereographic projection $\mathcal{F}$ for some $\zeta\in\mathbb{S}^{n}$ being the north pole and denoting $H_{s}=\xi_{s}(\varphi\circ\mathcal{F})$, it can be deduced from (1.7) and (4.27) that $v_{k}=H_{s}(\tilde{u}_{k}\circ\mathcal{F})$ resolves

[TABLE]

One can see that the right side of the above problem is in $L^{1}(\mathbb{R}^{n})$ and is uniformly bounded in $k$. By (4.31), standard regularity theory if $s=1$, [41, Theorems 2.8 and 2.9] if $0<s<1$, Theorem 3.1 and Theorem 5.2 if $s>1$, we have that $\{\|v_{k}\|_{C^{1,\gamma_{0}}(\mathbb{R}^{n})}\}_{k\in\mathbb{N}}$ is uniformly bounded for some $\gamma_{0}\in(0,1)$. It follows from (4.31), Lemma 4.5, regularity theory if $0<s\leq 1$, Theorems 3.1 and 5.1 if $s>1$ and after passing to a subsequence that

[TABLE]

where $v$ and $\tilde{u}$ are non-negative functions defined, respectively, in $\mathbb{R}^{n}$ and $\mathbb{S}\backslash\{-\zeta_{0}\}$, $0<\beta<1$ with $2(s-\lfloor s\rfloor)<1+\beta$, and $C$ does not depend on $k$. Using for $s\notin\mathbb{N}$ the following estimate

[TABLE]

where $\chi$ is the indicator function on the variable $y$, and by the dominated convergence theorem, it follows (up to constants) that

[TABLE]

Tending $k$ to infinity, we have that $v=H_{s}(\tilde{u}\circ\mathcal{F})$ solves

[TABLE]

If we consider $-\zeta\in\partial\mathbb{S}^{n}_{+}$ being the north pole, and following the above argument, we have, after passing to a subsequence, that $\tilde{u}$ solves

[TABLE]

and $\tilde{u}(\zeta_{0})=1$. Since $P^{g}_{s}$ is self adjoint in $L^{2}(\mathbb{S}^{n},g)$, one can see that the integral on $\mathbb{S}^{n}$ of the left side of (4.32) is zero while the integral of the left side is not zero, which is impossible.

Case 4.6.3**.**

$\|u_{k}\|_{L^{\infty}(\mathbb{S}^{n})}$* tends to infinity when $k$ tends to infinity.*

There exists a sequence $\{\zeta_{k}^{\prime}\}_{k\in\mathbb{N}}$ in $\mathbb{S}^{n}$ such that

[TABLE]

for some $\zeta_{0}^{\prime}\in\mathbb{S}^{n}$. By (4.7) and (3.9), $v_{k}=H_{s}(u\circ\mathcal{F})$ resolves

[TABLE]

where $H_{s}=\xi_{s}(\varphi\circ\mathcal{F})$, $\mathcal{F}$ is the inverse of stereographic projection with $\zeta_{0}^{\prime}$ being the south pole. Then there exist $R>0$ such that for all $k$,

[TABLE]

Let us define the following scaling

[TABLE]

where $\alpha_{k}$ is defined by

[TABLE]

For each $k$, $\tilde{v}_{k}$ satisfies $\|\tilde{v}_{k}\|_{L^{\infty}(B_{R})}>\tilde{C}_{1}$ and

[TABLE]

where

[TABLE]

One can see that the right side of (4.34) is in $L^{1}(\mathbb{R}^{n})$ and is uniformly bounded in $k$. By Lemma 4.5,standard regularity theory if $0<s\leq 1$, Theorems 5.1 and 5.2 if $s>1$, and following the arguments of the above case, we have that

[TABLE]

Moreover,

[TABLE]

Therefore, since $\lim_{k\rightarrow+\infty}\alpha_{k}=0$ and $\lim_{t\rightarrow+\infty}\tilde{f}(t)t^{-p}=c_{1}$, it follows that

[TABLE]

and $\tilde{v}\in\mathcal{L}_{s}$ is a nonnegative strong solution of

[TABLE]

which is impossible from Theorem 3.2 and [44, Corollary 1] since $p<\frac{n+2s}{n-2s}$. ∎

Following the arguments used in the proofs of Lemma 4.6 and Lemma 4.4 along with Lemma 4.5, Lemma 4.2 and Remark 4.1, we can obtain the following result that shows a type of a priori estimate for the solutions of (1.16) on $M=\mathbb{S}^{n}$.

Lemma 4.7**.**

Let $(M,g)=(\mathbb{S}^{n},g)$ and $g\in[g_{\mathbb{S}^{n}}]$. Suppose that $\tilde{f}_{1},\tilde{f}_{2}$ satisfy (F1)-(F4) for $0<2s<n$ and $R^{g}_{s}$ is positive for $s>1$. Then there exists a positive constant $C=C(n,s,g,\mathbb{S}^{n},\tilde{f}_{1},\tilde{f}_{2})$ such that

[TABLE]

any nonnegative smooth solution pair $(u_{1},u_{2})$ of (1.16) satisfies

[TABLE]

Proof of Theorem 1.6. It follows from Lemma 4.6, Lemma 4.7 and Lemma 4.1 for $\lambda$ and $\|A\|$ smalls. $\square$

Remark 4.2**.**

By the projection stereographic, Theorems 5.1 and 5.2, Theorem 3.4 (i), Theorem 3.2 and the proofs of Lemma 4.6 and Lemma 4.7, we have that the conclusions of Lemmas 4.6 and 4.7 hold for the solutions of (1.21) and (1.22) without the hypothesis on the sign of $R^{g}_{s}$, $\lambda_{2}$ or $\lambda_{3}$.

Proof of Theorem 1.7. It follows from Remark 4.2, Theorem 3.4 and Lemma 4.1 for $\lambda$ and $\|A\|$ smalls. $\square$

5. Appendix

5.1. Conformally invariant operators on spheres

In this subsection, we recall some results for $P_{s}$ with $s\in(0,1)$. Pavlov and Samko [38] showed that

[TABLE]

where $c_{n,-s}=\frac{2^{2s}s\Gamma(\frac{n+2s}{2})}{\pi^{\frac{n}{2}}\Gamma(1-s)}$, $|\cdot|$ is the Euclidean distance in $\mathbb{R}^{n+1}$ and $P.V.\int_{\mathbb{S}^{n}}$ is understood as $\lim_{\varepsilon\rightarrow 0}\int_{|\zeta-\omega|>\varepsilon}$ and $P_{s}(1)=\frac{\Gamma(\frac{n}{2}+s)}{\Gamma(\frac{n}{2}-s)}$. Then, using arguments based on [13], we may write $P_{s}$ as a weighted second order differential quotient.

Proposition 5.1**.**

If $s\in(0,1)$ , then

[TABLE]

where $\theta(\zeta,\omega)$ denotes the symmetric value of $\omega$ on $\mathbb{S}^{n}$ with respect to $\zeta$.

The above proposition allows us to give an analogous expression for $\|\cdot\|_{s,g_{\mathbb{S}^{n}}}$ in terms of a singular integral operator:

Proposition 5.2**.**

If $s\in(0,1)$, then

[TABLE]

for all $u\in H^{s}(\mathbb{S},g_{\mathbb{S}^{n}})$.

Proof.

Denote by $\theta(\zeta,\omega)$ the symmetric value of $\omega$ on $\mathbb{S}^{n}$ with respect to $\zeta\in\mathbb{S}^{n}$. Given $u\in C^{2}(\mathbb{S}^{n})$, from Proposition 5.1 and the Fubini theorem, we have

[TABLE]

Therefore, the proposition follows immediately from the above equality and (5.1). ∎

The Proposition 5.2 is needed to show the existence of eigenvalues for the operator $P^{g}_{s}-R^{g}_{s}$, which is described as follows:

Lemma 5.1**.**

If $s\in(0,1)$, then the operator $P^{g}_{s}-R^{g}_{s}$ has a positive eigenvalue $\lambda_{1,s,g}$ such that (1.17) holds.

Proof.

Assume that $g=\varphi^{4/(n-2s)}g_{\mathbb{S}^{n}}$ with $0<\varphi\in C^{\infty}(\mathbb{S}^{n})$. Following the proof of Proposition 5.2, we define

[TABLE]

where \overline{u}=\mathop{\ooalign{\int_{\mathbb{S}^{n}}$\cr$-}}u. From the proof of [40, Proposition A.1] and the Sobolev embeddings (see [27, p.1587]), there exists a $u^{*}\in H^{s}(\mathbb{S}^{n},g)$ such that

[TABLE]

and

[TABLE]

Therefore, $u^{*}-\overline{u^{*}}$ is an eigenvector corresponding to the eigenvalue $\lambda_{1,s,g}=2I(u^{*})$. ∎

One can follow the arguments from [40] and obtain a sequence $\{\lambda_{i,s,g}\}_{i\in\mathbb{N}}$ in $\mathbb{R}$ such that each $\lambda_{i,s,g}$ is an eigenvalue of $P^{g}_{s}-R^{g}_{s}$,

[TABLE]

and

[TABLE]

5.2. Schauder type estimates

In this subsection we shall prove some local Schauder estimates for solutions of nonlocal equations on $\mathbb{R}^{n}$. Other Schauder type estimates can be found in [25, 26, 41].

Theorem 5.1**.**

Let $v=\mathcal{I}_{2s}(w)$ be the Riesz potential of $w\in L^{\infty}(\mathbb{R}^{n})$ and $s\in(0,n/2)$. Assume $w\in L^{1}(\mathbb{R}^{n})\cup C^{0,\alpha}(\mathbb{R}^{n})$, $v\in L^{\infty}(\mathbb{R}^{n})$, $\alpha\in(0,1)$ and $s=m+\gamma$ with $m\in\mathbb{N}$ and $\gamma\in[0,1)$.

1. (i)

If $2\gamma+\alpha\notin\mathbb{N}$ and $2\gamma+\alpha<1$, then $v\in C^{2m,2\gamma+\alpha}(\mathbb{R}^{n})$ and

[TABLE] 2. (ii)

If $2\gamma+\alpha\notin\mathbb{N}$ and $2\gamma+\alpha>1$, then $v\in C^{2m+1,2\gamma+\alpha-1}(\mathbb{R}^{n})$ and

[TABLE]

Proof.

We will show that $v$ has the corresponding regularity in a neighborhood of the origin. The same argument works for a neighborhood of every point; so we get, respectively, that $v\in C^{2m,2\gamma+\alpha}(\mathbb{R}^{n})$ or $v\in C^{2m+1,2\gamma+\alpha-1}(\mathbb{R}^{n})$. Let be a smooth cutoff function such that $\eta(x)\in[0,1]$ for each $x\in\mathbb{R}^{n}$, $\text{supp}(v)\subset B_{1}$, and $\eta=1$ in $B_{7/8}$. We can write $v$ as

[TABLE]

It is clear that $v_{1}\in C^{\infty}(\overline{B_{1/2}})$. By interpolation inequalities we have

[TABLE]

where $k=2m$ if $0<\beta=2\gamma+\alpha<1$, or $k=2m+1$ if $0<\beta=2\gamma+\alpha-1<1$. Then we see that

[TABLE]

On the other hand, it follows from the conditions on $2\gamma+\alpha$ and the standard Riesz potential theory that

[TABLE]

Therefore, the theorem follows from (5.2) and (5.3). ∎

Theorem 5.2**.**

Let $v:=\mathcal{I}_{2s}(w)$ be the Riesz potential of $w\in L^{\infty}(\mathbb{R}^{n})$ and $s\in(0,n/2)$. Assume $w\in L^{1}(\mathbb{R}^{n})$, $v\in L^{\infty}(\mathbb{R}^{n})$ and $s=m+\gamma$ with $m\in\mathbb{N}$ and $\gamma\in[0,1)$.

1. (i)

If $\gamma=0$, then $v\in C^{2m-1,\alpha}(\mathbb{R}^{n})$ for any $\alpha\in(0,1)$, and

[TABLE] 2. (ii)

If $0<2\gamma\leq 1$, then $v\in C^{2m,\alpha}(\mathbb{R}^{n})$ for any $\alpha\in(0,2\gamma)$, and

[TABLE] 3. (iii)

If $2\gamma>1$, then $v\in C^{2m+1,\alpha}(\mathbb{R}^{n})$ for any $\alpha\in(0,2\gamma-1)$, and

[TABLE]

Proof.

Following the proof of Theorem 5.1, it is sufficient to estimate $v_{2}$. Denote $K(x)=|x|^{2s-n}$ for all $x\in\mathbb{R}^{n}\backslash\{0\}$. By the intermediate value theorem, on the line from $x$ to $y$ there exists some $x_{0}$ with

[TABLE]

We put $\delta:=2|x-y|$. Then we have

[TABLE]

Taking

[TABLE]

we have

[TABLE]

This proves the theorem, because obviously we also have

[TABLE]

∎