Boundary singularities of solutions to semilinear fractional equations

TL;DR

This paper investigates boundary singularities and solutions to semilinear fractional equations, establishing existence results for solutions with prescribed boundary measures and analyzing critical exponents for specific nonlinearities.

Contribution

It introduces new existence results for solutions with boundary measure data and characterizes critical exponents for power nonlinearities in fractional equations.

Findings

Existence of solutions with prescribed boundary Radon measures.

Identification of critical exponents for power nonlinearities.

Analysis of boundary trace for positive moderate solutions.

Abstract

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $\mu$ is a Dirac mass, we prove the existence of several critical exponents p.

Equations

$$(-\Delta {)}^{s}u(x):=\underset{\epsilon \to 0}{\lim }(-\Delta {)}_{\epsilon }^{s}u(x)$$

$$(-\Delta {)}_{\epsilon }^{s}u(x):=a_{N,s}{\int }_{{\mathbb{R}}^N\setminus B_{\epsilon }(x)}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}dy,a_{N,s}:=\frac{\Gamma (N/2+s)}{{\pi }^{N/2}\Gamma (2-s)}s(1-s).$$

$${\Omega }_{\beta }:=\{x\in \Omega :\rho (x)<\beta \},D_{\beta }:=\{x\in \Omega :\rho (x)>\beta \},{\Sigma }_{\beta }:=\{x\in \Omega :\rho (x)=\beta \}.$$

$$\underset{\beta \to 0}{\lim }{\beta }^{1-s}{\int }_{{\Sigma }_{\beta }}|u-{\mathbb{M}}_s^{\Omega }[\mu ]|dS=0.$$

$$\{\begin{array}{rl}(-\Delta {)}^{s}u+f(u)& =\tau \text{in }\Omega \\ {\mathrm{tr}}_s(u)& =\mu \\ u& =0\text{in }{\Omega }^c.\end{array}$$

$${\int }_{\Omega }(u(-\Delta {)}^s\xi +f(u)\xi )dx={\int }_{\Omega }\xi d\tau +{\int }_{\Omega }{\mathbb{M}}_s^{\Omega }[\mu ](-\Delta {)}^s\xi dx,\forall \xi \in {\mathbb{X}}_s(\Omega ).$$

$$\{\begin{array}{rlrl}-\Delta u+f(u)& =0& & \text{in }\Omega \\ u& =\mu & & \text{on }\partial \Omega ,\end{array}$$

$${\int }_1^{\infty }[f(t)-f(-t)]s^{-1-p_c}dt<\infty ,$$

$$\{\begin{array}{rlrl}(-\Delta {)}^{s}u& =\tau & & \text{in }\Omega \\ {\mathrm{tr}}_s(u)& =\mu & & \\ u& =0& & \text{in }{\Omega }^c.\end{array}$$

$$u={\mathbb{G}}_s^{\Omega }[\tau ]+{\mathbb{M}}_s^{\Omega }[\mu ].$$

$${\Vert u\Vert }_{L^1(\Omega )}\le c({\Vert \tau \Vert }_{\mathfrak{M}(\Omega ,{\rho }^s)}+{\Vert \mu \Vert }_{\mathfrak{M}(\partial \Omega )}).$$

$$u={\mathbb{G}}_s^{\Omega }[\tau -f(u)]+{\mathbb{M}}_s^{\Omega }[\mu ]\text{in }\Omega ,$$

$$-{\mathbb{G}}_s^{\Omega }[{\tau }^{-}]-{\mathbb{M}}_s^{\Omega }[{\mu }^{-}]\le u\le {\mathbb{G}}_s^{\Omega }[{\tau }^{+}]+{\mathbb{M}}_s^{\Omega }[{\mu }^{+}]\text{in }\Omega .$$

$$p_1^{\ast }:=\frac{N+2s}{N},p_2^{\ast }:=\frac{N+s}{N-s},p_3^{\ast }:=\frac{N}{N-2s}.$$

$${\int }_1^{\infty }[f(s)-f(-s)]s^{-1-p_2^{\ast }}ds<\infty .$$

$$\{\begin{array}{rl}(-\Delta {)}^{s}u+f(u)& =0\text{in }\Omega \\ {\mathrm{tr}}_s(u)& =k{\delta }_z\\ u& =0\text{in }{\Omega }^c.\end{array}$$

$$\underset{\Omega \ni x\to z}{\lim }\frac{u_{z,k}^{\Omega }(x)}{M_s^{\Omega }(x,z)}=k.$$

$$\left\{\begin{array}[]{rll}{(-\Delta)^{s}}u+u^{p}&=0&\text{in }\Omega\\ \mathrm{tr\,}_{s}(u)&=k\delta_{0}\\ u&=0&\text{in }\Omega^{c}.\end{array}\right.$$

$$\left\{\begin{array}[]{rll}{(-\Delta)^{s}}u+u^{p}&=0&\text{in }\mathbb{R}^{N}_{+}\\ u&=0&\text{in }\overline{\mathbb{R}^{N}_{-}}\end{array}\right.$$

$$S^{N-1}:=\left\{\sigma =(\cos \varphi {\sigma }^{\prime },\sin \varphi ):{\sigma }^{\prime }\in S^{N-2},-\frac{\pi }{2}\le \varphi \le \frac{\pi }{2}\right\}$$

$$\left\{\begin{array}[]{lll}{\mathcal{A}}_{s}\omega-{\mathcal{L}}_{s,\frac{2s}{p-1}}\omega+\omega^{p}=0&\text{in }\;S_{+}^{N-1}\\[5.69054pt] \phantom{{\mathcal{A}}_{s}\omega-{\mathcal{L}}_{s,\frac{2s}{p-1}}+\omega^{p}}\omega=0&\text{in }\;\overline{S_{-}^{N-1}},\end{array}\right.$$

$$\left\{\begin{array}[]{rll}{(-\Delta)^{s}}u+u^{p}&=0&\text{in }\Omega\\ u&=0&\text{in }\Omega^{c}.\end{array}\right.$$

$$u_{\infty }^{{\mathbb{R}}_{+}^N}(x)=|x{|}^{-\frac{2s}{p-1}}{\omega }^{\ast }(\sigma ),\text{with }\sigma =\frac{x}{|x|}\forall x\in {\mathbb{R}}_{+}^N.$$

$$\lim_{\tiny\begin{array}[]{c}\Omega\ni x\to 0\\ \frac{x}{|x|}=\sigma\in S^{N-1}_{+}\end{array}}|x|^{\frac{2s}{p-1}}u_{\infty}^{\Omega}(x)=\omega^{*}(\sigma),$$

$$c^{-1}\rho (x{)}^s|x{|}^{-\frac{(p+1)s}{p-1}}\le u_{\infty }^{\Omega }(x)\le c\rho (x{)}^s|x{|}^{-\frac{(p+1)s}{p-1}}\forall x\in \Omega .$$

$$E^0{e}^{i\xi X_t}=e^{-t|\xi {|}^{2s}},\xi \in {\mathbb{R}}^N,t\ge 0.$$

$$P_t(x,A)=P^x(X_t\in A)={\mu }_t(A-x),$$

$$E^{x}u(X_{t_D})=E^x\{u(X_{t_D}):t_D<\infty \}.$$

$$u(x)=E^{x}u(X_{t_D}),x\in D.$$

$${\mathcal{D}}_s:=\left\{u:{\mathbb{R}}^N\mapsto \mathbb{R}:\text{Borel measurable such that}{\int }_{{\mathbb{R}}^N}\frac{|u(x)|}{(1+|x|{)}^{N+2s}}\right\}.$$

Full text

2010 Mathematics Subject Classification. 35J66, 35J67, 35R06, 35R11.

Key words: $s$-harmonic functions, semilinear fractional equations, boundary trace.

1. 1 Introduction

2. 2 Linear problems

3. 2.1 $s$-harmonic functions

4. 2.2 Green kernel, Poisson kernel and Martin kernel

5. 2.3 Boundary trace

6. 2.4 Weak solutions of linear problems

7. 3 Nonlinear problems

8. 3.1 Subcritical absorption

9. 3.2 Power absorption

10. A Appendix - Separable solutions

11. A.1 Separable $s$-harmonic functions

12. A.2 The nonlinear problem

13. A.2.1 Separable solutions in $\mathbb{R}^{N}$

14. A.2.2 Separable solutions in $\mathbb{R}_{+}^{N}$

1. Introduction

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with $C^{2}$ boundary and $s\in(0,1)$. Define the $s$-fractional Laplacian as

[TABLE]

where

[TABLE]

We denote by $G_{s}^{\Omega}$ and $M_{s}^{\Omega}$ the Green kernel and the Martin kernel of ${(-\Delta)^{s}}$ in $\Omega$ respectively. Denote by $\mathbb{G}_{s}^{\Omega}$ and $\mathbb{M}_{s}^{\Omega}$ the Green operator and the Martin operator (see section 2 for more details). Further, for $\phi\geq 0$, denote by $\mathfrak{M}(\Omega,\phi)$ the space of Radon measures $\tau$ on $\Omega$ satisfying $\int_{\Omega}\phi d|\tau|<\infty$ and by $\mathfrak{M}(\partial\Omega)$ the space of bounded Radon measures on $\partial\Omega$. Let $\rho(x)$ be the distance from $x$ to $\partial\Omega$. For $\beta>0$, set

[TABLE]

Definition 1.1**.**

We say that a function $u\in L^{1}_{loc}(\Omega)$ possesses a $s$-boundary trace on $\partial\Omega$ if there exists a measure $\mu\in\mathfrak{M}(\partial\Omega)$ such that

[TABLE]

The $s$-boundary trace of $u$ is denoted by $\mathrm{tr\,}_{s}(u)$.

Let $\tau\in\mathfrak{M}(\Omega,\rho^{s})$, $\mu\in\mathfrak{M}(\partial\Omega)$ and $f\in C(\mathbb{R})$ be a nondecreasing function with $f(0)=0$. In this paper, we study boundary singularity problem for semilinear fractional equation of the form

[TABLE]

We denote by $\mathbb{X}_{s}(\Omega)\subset C(\mathbb{R}^{N})$ the space of test functions $\xi$ satisfying

(i) $\mbox{\rm supp}\,(\xi)\subset\bar{\Omega}$,

(ii) ${(-\Delta)^{s}}\xi(x)$ exists for all $x\in\Omega$ and $|{(-\Delta)^{s}}\xi(x)|\leq C$ for some $C>0$,

(iii) there exists $\varphi\in L^{1}(\Omega,\rho^{s})$ and $\epsilon_{0}>0$ such that $|(-\Delta)_{\epsilon}^{s}\xi|\leq\varphi$ a.e. in $\Omega$, for all $\epsilon\in(0,\epsilon_{0}]$.

Definition 1.2**.**

Let $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\mu\in\mathfrak{M}(\partial\Omega)$. A function $u$ is called a weak solution of (1.2) if $u\in L^{1}(\Omega)$, $f(u)\in L^{1}(\Omega,\rho^{s})$ and

[TABLE]

The boundary value problem with measure data for semilinear elliptic equations

[TABLE]

was first studied by A. Gmira and L. Véron in [18] and then the typical model, i.e. problem (1.4) with $f(u)=u^{p}$ ($p>1$), has been intensively investigated by numerous authours (see [22, 23, 24, 25, 26] and references therein). They proved that if $f$ is a continuous, nondecreasing function satisfying

[TABLE]

where $p_{c}:=\frac{N+1}{N-1}$, then problem (1.4) admits a unique weak solution. In particular, when $f(u)=u^{p}$ with $1<p<p_{c}$ and $\mu=k\delta_{0}$ with $0\in\partial\Omega$ and $k>0$, there exists a unique solution $u_{k}$ of (1.4). It was showed [22, 26] that the sequence $\{u_{k}\}$ is increasing and converges to a function $u_{\infty}$ which is a solution of the equation in (1.4).

To our knowledge, few papers concerning boundary singularity problem for nonlinear fractional elliptic equation have been published in the literature. The earliest works in this direction are the papers [17, 10] by P. Felmer et al. which deal with the existence, nonexistence and asymptotic behavior of large solutions for equations involving fractional Laplacian. Afterwards, N. Abatangelo [1] presented a suitable setting for the study of fractional Laplacian equations in a measure framework and provided a fairly comprehensive description of large solutions which improve the results in [17, 10]. Recently, H. Chen et al. [9] investigated semilinear elliptic equations involving measures concentrated on the boundary by employing approximate method.

In the present paper, we aim to establish the existence and uniqueness of weak solutions of (1.2). To this end, we develop a theory for linear equations associated to (1.2)

[TABLE]

Existence and uniqueness result for (1.6) is stated in the following proposition.

Proposition A. Assume $s\in(\frac{1}{2},1)$. Let $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\mu\in\mathfrak{M}(\partial\Omega)$. Then problem (1.6) admits a unique weak solution. The solution is given by

[TABLE]

Moreover, there exists a positive constant $c=c(N,s,\Omega)$ such that

[TABLE]

This proposition allows to study semilinear equation (1.2). We first deal with the case of $L^{1}$ data.

Theorem B. Assume $s\in(\frac{1}{2},1)$. Let $f\in C(\mathbb{R})$ be a nondecreasing function satisfying $tf(t)\geq 0$ for every $t\in\mathbb{R}$.

I. Existence and uniqueness. For every $\tau\in L^{1}(\Omega,\rho^{s})$ and $\mu\in L^{1}(\partial\Omega)$, problem (1.2) admits a unique weak solution $u$. Moreover,

[TABLE]

[TABLE]

II. Monotonicity. The mapping $(\tau,\mu)\mapsto u$ is nondecreasing.

Remark. The restriction $s\in(\frac{1}{2},1)$ is due to the fact that in this range of $s$, $\mathrm{tr\,}_{s}(\mathbb{G}[\tau])=0$ for every $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ (see Proposition 2.11). We conjecture that this still holds if $s\in(0,\frac{1}{2}]$.

We reveal that, in measures framework, because of the interplay between the nonlocal operator ${(-\Delta)^{s}}$ and the nonlinearity term $f(u)$, the analysis is much more intricate and there are 3 critical exponents

[TABLE]

This yields substantial new difficulties and leads to disclose new types of results. The new aspects are both on the technical side and on the one of the new phenomena observed.

Theorem C. Assume $s\in(\frac{1}{2},1)$. Let $f\in C(\mathbb{R})$ be a nondecreasing function, $tf(t)\geq 0$ for every $t\in\mathbb{R}$ and

[TABLE]

I. Existence and Uniqueness. For every $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\mu\in\mathfrak{M}(\partial\Omega)$ there exists a unique weak solution of (1.2). This solution satisfies (1.9) and (1.10). Moreover, the mapping $(\tau,\mu)\mapsto u$ is nondecreasing.

II. Stability. *Assume $\{\tau_{n}\}\subset\mathfrak{M}(\Omega,\rho^{s})$ converges weakly to $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\{\mu_{n}\}\subset\mathfrak{M}(\partial\Omega)$ converges weakly to $\mu\in\mathfrak{M}(\partial\Omega)$. Let $u$ and $u_{n}$ be the unique weak solutions of (1.2) with data $(\tau,\mu)$ and $(\tau_{n},\mu_{n})$ respectively. Then $u_{n}\to u$ in $L^{1}(\Omega)$ and $f(u_{n})\to f(u)$ in $L^{p}(\Omega,\rho^{s})$. *

If $\mu$ is a Dirac mass concentrated at a point on $\partial\Omega$, we obtain the behavior of the solution near that boundary point.

Theorem D. Under the assumption of Theorem C, let $z\in\partial\Omega$, $k>0$ and $u_{z,k}^{\Omega}$ be the unique weak solution of

[TABLE]

Then

[TABLE]

We next assume that $0\in\partial\Omega$. Let $0<p<p_{2}^{*}$ and denote by $u_{k}^{\Omega}$ the unique weak solution of

[TABLE]

By Theorem C, $u_{k}^{\Omega}\leq kM_{s}^{\Omega}(\cdot,0)$ and $k\mapsto u_{k}^{\Omega}$ is increasing. Therefore, it is natural to investigate $\lim_{k\to\infty}u_{k}^{\Omega}$. This is accomplishable thanks to the study of separable solutions of

[TABLE]

with $p>1$. Denote by

[TABLE]

the unit sphere in $\mathbb{R}^{N}$ and by $S_{+}^{N-1}:=S^{N-1}\cap\mathbb{R}_{+}^{N}$ the upper hemisphere. Writing separable solution under the form $u(x)=u(r,\sigma)=r^{-\frac{2s}{p-1}}\omega(\sigma)$, with $r>0$ and $\sigma\in S^{N-1}_{+}$, we obtain that $\omega$ satisfies

[TABLE]

where ${\mathcal{A}}_{s}$ is a nonlocal operator naturally associated to the $s$-fractional Laplace-Beltrami operator and ${\mathcal{L}}_{s,\frac{2s}{p-1}}$ is a linear integral operator with kernel. In analyzing the spectral properties of ${\mathcal{A}}_{s}$ we prove

Theorem E. Let $N\geq 2$, $s\in(0,1)$ and $p>p_{1}^{*}$.

I- If $p_{2}^{*}\leq p<p_{3}^{*}$ there exists no positive solution of (1.16) belonging to $W^{s,2}_{0}(S^{N-1}_{+})$.

II- If $p_{1}^{*}<p<p_{2}^{*}$ there exists a unique positive solution $\omega^{*}\in W^{s,2}_{0}(S^{N-1}_{+})$ of (1.16).

As a consequence of this result we obtain the behavior of $u_{k}^{\Omega}$ when $k\to\infty$.

Theorem F Assume $s\in(\frac{1}{2},1)$. Let $\Omega=\mathbb{R}^{N}_{+}$ or $\Omega$ be a bounded domain with $C^{2}$ boundary containing [math].

I- If $p\in(p_{1}^{*},p_{2}^{*})$ then $u_{\infty}^{\Omega}:=\lim_{k\to 0}u_{k}^{\Omega}$ is a positive solution of

[TABLE]

(i) If $\Omega=\mathbb{R}_{+}^{N}$ then

[TABLE]

(ii) If $\Omega$ is a bounded $C^{2}$ domain with $\partial\Omega$ containing [math] then

[TABLE]

locally uniformly on $S^{N-1}_{+}$. In particular, there exists a positive constant $c$ depending on $N$, $s$, $p$ and the $C^{2}$ norm of $\partial\Omega$ such that

[TABLE]

*II- Assume $p\in(0,p_{1}^{*}]$. Then $\lim_{k\to\infty}u_{k}^{\Omega}=\infty$ in $\Omega$. *

The main ingredients of the present study: estimates on Green kernel and Martin kernel, theory for linear fractional equations in connection with the notion $s-$boundary trace as mentioned above, similarity transformation and the study of equation (1.16).

The paper is organized as follows. In Section 2, we present important properties of $s$-boundary trace and prove Proposition A. Theorems B,C,D and F are obtained in Section 3. Finally, in Appendix, we discuss separable solutions of (1.15) and demonstrate Theorem E.

2. Linear problems

Throughout the present paper, we denote by $c,c^{\prime},c_{1},c_{2},C,...$ positive constants that may vary from line to line. If necessary, the dependence of these constants will be made precise.

2.1. $s$-harmonic functions

We first recall the definition of $s$-harmonic functions (see [3, page 46], [4, page 230], [6, page 20]). Denote by $(X_{t},P^{x})$ the standard rotation invariant $2s$-stable Lévy process in $\mathbb{R}^{N}$ (i.e. homogeneous with independent increments) with characteristic function

[TABLE]

Denote by $E^{x}$ the expectation with respect to the distribution $P^{x}$ of the process starting from $x\in\mathbb{R}^{N}$. We assume that sample paths of $X_{t}$ are right-continuous and have left-hand limits a.s. The process $(X_{t})$ is Markov with transition probabilities given by

[TABLE]

where $\mu_{t}$ is the one-dimensional distribution of $X_{t}$ with respect to $P^{0}$. It is well known that ${(-\Delta)^{s}}$ is the generator of the process $(X_{t},P^{x})$.

For each Borel set $D\subset\mathbb{R}^{N}$, set $t_{D}:=\inf\{t\geq 0:X_{t}\not\in D\}$, i.e. $t_{D}$ is the first exit time from $D$. If $D$ is bounded then $t_{D}<\infty$ a.s. Denote

[TABLE]

Definition 2.1**.**

Let $u$ be a Borel measurable function in $\mathbb{R}^{N}$. We say that $u$ is $s$-harmonic in $\Omega$ if for every bounded open set $D\Subset\Omega$,

[TABLE]

We say that $u$ is singular $s$-harmonic in $\Omega$ if $u$ is $s$-harmonic and $u=0$ in $\Omega^{c}$.

Put

[TABLE]

The following result follows from [5, Corollary 3.10 and Theorem 3.12] and [6, page 20] (see also [20]).

Proposition 2.2**.**

Let $u\in{\mathcal{D}}_{s}$.

(i)* $u$ is $s$-harmonic in $\Omega$ if and only if ${(-\Delta)^{s}}u=0$ in $\Omega$ in the sense of distributions.*

(ii)* $u$ is singular $s$-harmonic in $\Omega$ if and only if $u$ is $s$-harmonic in $\Omega$ and $u=0$ in $\Omega^{c}$.*

2.2. Green kernel, Poisson kernel and Martin kernel

In what follows the notation $f\sim g$ means: there exists a positive constant $c$ such that $c^{-1}f<g<cf$ in the domain of the two functions or in a specified subset of this domain.

Denote by $G_{s}^{\Omega}$ the Green kernel of ${(-\Delta)^{s}}$ in $\Omega$. Namely, for every $y\in\Omega$,

[TABLE]

where $\delta_{y}$ is the Dirac mass at $y$. By combining [1, Lemma 3.2] and [14, Corollary 1.3]), we get

Proposition 2.3**.**

(i) $G_{s}^{\Omega}$ is in continuous, symmetric, positive in $\{(x,y)\in\Omega\times\Omega:x\neq y\}$ and $G_{s}^{\Omega}(x,y)=0$ if $x$ or $y$ belongs to $\Omega^{c}$.

(ii) ${(-\Delta)^{s}}G_{s}^{\Omega}(x,\cdot)\in L^{1}(\Omega^{c})$ for every $x\in\Omega$ and ${(-\Delta)^{s}}G_{s}^{\Omega}(x,y)\leq 0$ for every $x\in\Omega$ and $y\in\Omega^{c}$.

(iii) There holds

[TABLE]

The similarity constant in the above estimate depends only on $\Omega$ and $s$.

Denote by $\mathbb{G}_{s}^{\Omega}$ the associated Green operator

[TABLE]

Put

[TABLE]

H. Chen and L. Véron obtained the following estimate for Green operator [12, Proposition 2.3 and Proposition 2.6].

Lemma 2.4**.**

Assume $\gamma\in[0,s]$ and $k_{s,\gamma}$ be as in (2.2).

(i)* There exists a constant $c=c(N,s,\gamma,\Omega)>0$ such that*

[TABLE]

(ii)* Assume $\{\tau_{n}\}\subset\mathfrak{M}(\Omega,\rho^{\gamma})$ converges weakly to $\tau\in\mathfrak{M}(\Omega,\rho^{\gamma})$. Then $\mathbb{G}_{s}^{\Omega}[\tau_{n}]\to\mathbb{G}_{s}^{\Omega}[\tau]$ in $L^{p}(\Omega,\rho^{s})$ for any $p\in[1,k_{s,\gamma})$.*

Let $P_{s}^{\Omega}$ be the Poisson kernel of ${(-\Delta)^{s}}$ defined by (see [7])

[TABLE]

The relation between $P_{s}^{\Omega}$ and $G_{s}^{\Omega}$ is expressed in [1, Proposition 2] (see also [14, Theorem 1.4], [4, Lemma 2], [14, Theorem 1.5]).

Proposition 2.5**.**

(i) $P_{s}^{\Omega}(x,y)=-{(-\Delta)^{s}}G_{s}^{\Omega}(x,y)$ for every $x\in\Omega$ and $y\in{\overline{\Omega}}^{c}$. Moreover, $P_{s}^{\Omega}$ is continuous in $\Omega\times{\overline{\Omega}}^{c}$.

(ii) There holds

[TABLE]

The similarity constant in the above estimate depends only on $\Omega$ and $s$.

Denote by $\mathbb{P}_{s}^{\Omega}$ the corresponding operator defined by

[TABLE]

Fix a reference point $x_{0}\in\Omega$ and denote by $M_{s}^{\Omega}$ the Martin kernel of ${(-\Delta)^{s}}$ in $\Omega$, i.e.

[TABLE]

By [15, Theorem 3.6], the Martin boundary of $\Omega$ can be identified with the Euclidean boundary $\partial\Omega$. Denote by $\mathbb{M}_{s}^{\Omega}$ the associated Martin operator

[TABLE]

The next result [4, 15] is important in the study of $s$-harmonic functions, which give a unique presentation of $s-$harmonic functions in terms of Martin kernel.

Proposition 2.6**.**

(i)* The mapping $(x,z)\mapsto M_{s}^{\Omega}(x,z)$ is continuous on $\Omega\times\partial\Omega$. For any $z\in\partial\Omega$, the function $M_{s}^{\Omega}(.,z)$ is singular $s$-harmonic in $\Omega$ with $M_{s}^{\Omega}(x_{0},z)=1$. Moreover, if $z,z^{\prime}\in\partial\Omega$, $z\neq z^{\prime}$ then $\lim_{x\to z^{\prime}}M_{s}^{\Omega}(x,z)=0$.*

(ii)* There holds*

[TABLE]

The similarity constant in the above estimate depends only on $\Omega$ and $s$.

(iii)* For every $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ the function $\mathbb{M}_{s}^{\Omega}[\mu]$ is singular $s$-harmonic in $\Omega$ with $u(x_{0})=\mu(\mathbb{R}^{N})$. Conversely, if $u$ is a nonnegative singular $s$-harmonic function in $\Omega$ then there exists a unique $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ such that $u=\mathbb{M}_{s}^{\Omega}[\mu]$ in $\mathbb{R}^{N}$.*

(iv)* If $u$ is a nonnegative $s$-harmonic function in $\Omega$ then there exists a unique $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ such that*

[TABLE]

Lemma 2.7**.**

(i)* There exists a constant $c=c(N,s,\gamma,\Omega)$ such that*

[TABLE]

(ii)* If $\{\mu_{n}\}\subset\mathfrak{M}(\partial\Omega)$ converges weakly to $\mu\in\mathfrak{M}(\partial\Omega)$ then $\mathbb{M}_{s}^{\Omega}[\mu_{n}]\to\mathbb{M}_{s}^{\Omega}[\mu]$ in $L^{p}(\Omega,\rho^{\gamma})$ for every $1\leq p<\frac{N+\gamma}{N-s}$.*

Proof.

(i) By using (2.5) and a similar argument as in the proof of [2, Theorem 2.5], we obtain (2.6).

(ii) By combining the fact that $M_{s}^{\Omega}(x,z)=0$ for every $x\in\Omega^{c}$, $z\in\partial\Omega$ and Proposition 2.6 (i) we deduce that for every $x\in\mathbb{R}^{N}$, $M_{s}^{\Omega}(x,\cdot)\in C(\partial\Omega)$. It follows that $\mathbb{M}_{s}^{\Omega}[\mu_{n}]\to\mathbb{M}_{s}^{\Omega}[\mu]$ everywhere in $\Omega$. Due to (i) and the Holder inequality, we deduce that, for any $1\leq p\leq\frac{N+\gamma}{N-s}$, $\{\mathbb{M}_{s}^{\Omega}[\mu_{n}]\}$ is uniformly integrable with respect to $\rho^{\gamma}dx$. By invoking Vitali’s theorem, we obtain the convergence in $L^{p}(\Omega,\rho^{\gamma})$.           $\square$

2.3. Boundary trace

We recall that, for $\beta>0$,

[TABLE]

The following geometric property of $C^{2}$ domains can be found in [26].

Proposition 2.8**.**

There exists $\beta_{0}>0$ such that

(i) For every point $x\in\overline{\Omega}_{\beta_{0}}$, there exists a unique point $z_{x}\in\partial\Omega$ such that $|x-z_{x}|=\rho(x)$. This implies $x=z_{x}-\rho(x){\bf n}_{z_{x}}$.

(ii) The mappings $x\mapsto\rho(x)$ and $x\mapsto z_{x}$ belong to $C^{2}(\overline{\Omega}_{\beta_{0}})$ and $C^{1}(\overline{\Omega}_{\beta_{0}})$ respectively. Furthermore, $\lim_{x\to z_{x}}\nabla\rho(x)=-{\bf n}_{z_{x}}$.

Proposition 2.9**.**

Assume $s\in(0,1)$. Then there exist positive constants $c=c(N,\Omega,s)$ such that, for every $\beta\in(0,\beta_{0})$,

[TABLE]

Proof.

For $r_{0}>0$ fixed, by (2.5),

[TABLE]

which implies

[TABLE]

Note that for $r_{0}$ fixed, the rate of convergence is independent of $y$.

In order to prove (2.7) we may assume that the coordinates are placed so that $y=0$ and the tangent hyperplane to $\partial\Omega$ at [math] is $x_{N}=0$ with the $x_{N}$ axis pointing into the domain. For $x\in\mathbb{R}^{N}$ put $x^{\prime}=(x_{1},\cdots,x_{N-1})$. Pick $r_{0}\in(0,\beta_{0})$ sufficiently small (depending only on the $C^{2}$ characteristic of $\Omega$) so that

[TABLE]

Hence if $x\in\Sigma_{\beta}\cap B_{r_{0}}(0)$ then $\frac{1}{4}(|x^{\prime}|+\beta)\leq|x|$. Combining this inequality and (2.5) leads to

[TABLE]

Therefore, for $\beta<r_{0}$,

[TABLE]

By combining estimates (2.8) and (2.10), we obtain the second estimate in (2.7). The first estimate in (2.7) follows from (2.5).           $\square$

As a consequence, we get the following estimates.

Corollary 2.10**.**

Assume $s\in(0,1)$. For every $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ and $\beta\in(0,\beta_{0})$, there holds

[TABLE]

with $c$ is as in (2.7).

Proposition 2.11**.**

Assume $s\in(\frac{1}{2},1)$. Then there exists a constant $c=c(s,N,\Omega)$ such that for any $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and any $0<\beta<\beta_{0}$,

[TABLE]

Moreover,

[TABLE]

Proof.

Without loss of generality, we may assume that $\tau>0$. Denote $v:=\mathbb{G}_{s}^{\Omega}[\tau]$. We first prove (2.12). By Fubini’s theorem and (2.5),

[TABLE]

Note that, if $x\in\Sigma_{\beta}\cap B_{\frac{\beta}{2}}(y)$ then $\beta/2\leq\rho(y)\leq 3\beta/2$. Therefore

[TABLE]

where the last inequality holds since $s>\frac{1}{2}$. On the other hands, we have

[TABLE]

Combining the above estimates, we obtain (2.12).

Next we demonstrate (2.13). Given $\epsilon\in(0,\left\|\tau\right\|_{\mathfrak{M}(\Omega,\rho^{s})})$ and $\beta_{1}\in(0,\beta_{0})$ put $\tau_{1}=\tau\chi_{{}_{\bar{D}_{\beta_{1}}}}$ and $\tau_{2}=\tau\chi_{{}_{\Omega_{\beta_{1}}}}$. We can choose $\beta_{1}=\beta_{1}(\epsilon)$ such that

[TABLE]

Thus the choice of $\beta_{1}$ depends on the rate at which $\int_{\Omega_{\beta}}\rho^{s}\,d\tau$ tends to zero as $\beta\to 0$.

Put $v_{i}:=\mathbb{G}_{s}^{\Omega}[\tau_{i}]$. Then, for $0<\beta<\beta_{1}/2$,

[TABLE]

which yields

[TABLE]

On the other hand, due to (2.12),

[TABLE]

From (2.15) and (2.16), we obtain (2.13).           $\square$

Lemma 2.12**.**

Assume $s\in(\frac{1}{2},1)$. Let $u,w\in{\mathcal{D}}_{s}$ be two nonnegative functions satisfying

[TABLE]

If $u\leq w$ in $\mathbb{R}^{N}$ then ${(-\Delta)^{s}}u\in\mathfrak{M}(\Omega,\rho^{s})$ and there exists a measure $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ such that

[TABLE]

Moreover, if $\mu=0$ then $u=0$.

Proof.

By the assumption, there exists a nonnegative Radon measure $\tau$ on $\Omega$ such that ${(-\Delta)^{s}}u=-\tau$.

We first prove that $\tau\in\mathfrak{M}^{+}(\Omega,\rho^{s})$. Define

[TABLE]

By [1, page 5547], there holds is a positive constant $c=c(\Omega,s)$ such that

[TABLE]

where the similarity constant depends only on $\Omega$ and $s$. This follows

[TABLE]

We define

[TABLE]

For any $\beta\in(0,\beta_{0})$, denote by $\tau_{\beta}$ the restriction of $\tau$ to $D_{\beta}$ and by $v_{\beta}$ the restriction of $u$ on $\Sigma_{\beta}$. By [1, Theorem 1.4], there exists a unique solution $v_{\beta}$ of

[TABLE]

Moreover, the solution can be written as

[TABLE]

By the maximum principle [1, Lemma 3.9], $v_{\beta}=u$ and $\mathbb{P}_{s}^{D_{\beta}}[u|_{D_{\beta}^{c}}]\leq w$ a.e. in $\mathbb{R}^{N}$. This, together with (2.22), implies that $\mathbb{G}_{s}^{D_{\beta}}[\tau_{\beta}]\leq w$ in $D_{\beta}$. Letting $\beta\to 0$ yields $\mathbb{G}_{s}^{\Omega}[\tau]<\infty$. For fixed $x_{0}\in\Omega$, by (2.1), $G_{s}^{\Omega}(x_{0},y)>c\rho(y)^{s}$ for every $y\in\Omega$. Hence the finiteness of $\mathbb{G}_{s}^{\Omega}[\tau]$ implies that $\tau\in\mathfrak{M}^{+}(\Omega,\rho^{s})$.

We next show that there exists a measure $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ such that (2.18) holds. Put $v=u+\mathbb{G}_{s}^{\Omega}[\tau]$ then $v$ is a nonnegative singular $s$-harmonic in $\Omega$ due to the fact that $\mathbb{G}_{s}^{\Omega}[\tau]=0$ in $\Omega^{c}$. By Proposition 2.2 and Proposition 2.6 (iii), there exists $\mu\in\mathfrak{M}^{+}(\partial\Omega)$ such that $v=\mathbb{M}_{s}^{\Omega}[\mu]$ in $\mathbb{R}^{N}$. By Proposition 2.11, we obtain (2.18). If $\mu=0$ then $v=0$ and thus $u=0$.           $\square$

Definition 2.13**.**

A function $u$ possesses a $s$-boundary trace on $\partial\Omega$ if there exists a measure $\mu\in\mathfrak{M}(\partial\Omega)$ such that

[TABLE]

The $s$-boundary trace of $u$ is denoted noted by $\mathrm{tr\,}_{s}(u)$.

Remark. (i) The notation of $s$-boundary trace is well defined. Indeed, suppose that $\mu$ and $\mu^{\prime}$ satisfy (2.23). Put $v=(\mathbb{M}_{s}^{\Omega}[\mu-\mu^{\prime}])^{+}$. Clearly $v\leq\mathbb{M}_{s}^{\Omega}[|\mu|+|\mu^{\prime}|]$, $v=0$ in $\Omega^{c}$ and $\lim_{\beta\to 0}\beta^{1-s}\int_{\Sigma_{\beta}}|v|dS=0$. By Kato’s inequality [8, Theorem 1.2], ${(-\Delta)^{s}}v\leq 0$ in $\Omega$. Therefore, we deduce $v\equiv 0$ from Lemma 2.12. This implies $\mathbb{M}_{s}^{\Omega}[\mu-\mu^{\prime}]\leq 0$. By permuting the role of $\mu$ and $\mu^{\prime}$, we obtain $\mathbb{M}_{s}^{\Omega}[\mu-\mu^{\prime}]\geq 0$. Thus $\mu=\mu^{\prime}$.

(ii) It is clear that for every $\mu\in\mathfrak{M}(\partial\Omega)$, $\mathrm{tr\,}_{s}(\mathbb{M}_{s}^{\Omega}[\mu])=\mu$. Moreover, if $s>\frac{1}{2}$, by Proposition 2.11, for every $\tau\in\mathfrak{M}(\Omega,\rho^{s})$, $\mathrm{tr\,}_{s}(\mathbb{G}_{s}^{\Omega}[\tau])=0$.

(iii) This kind of boundary trace was first introduced by P.-T. Nguyen and M. Marcus [21] in order to investigate semilinear elliptic equations with Hardy potential. In the present paper we prove that it is still an effective tools in the study of nonlocal fractional elliptic equations.

2.4. Weak solutions of linear problems

Definition 2.14**.**

Let $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\mu\in\mathfrak{M}(\partial\Omega)$. A function $u$ is called a weak solution of (1.6) if $u\in L^{1}(\Omega)$ and

[TABLE]

Proof of Proposition A. The uniqueness follows from [12, Proposition 2.4]. Let $u$ be as in (1.7). By [12],

[TABLE]

This implies (2.24) and therefore $u$ is the unique solution of (1.6). Since $s\in(\frac{1}{2},1)$, by Proposition 2.11, $\mathrm{tr\,}_{s}(u)=\mathrm{tr\,}_{s}(\mathbb{M}_{s}^{\Omega}[\mu])=\mu$. Finally, estimate (1.8) follows from Lemma 2.4 and Lemma 2.7.           $\square$

3. Nonlinear problems

In this section, we study the nonlinear problem (1.2). The definition of weak solutions of (1.2) is given in Definition 1.2.

3.1. Subcritical absorption

Proof of Theorem B.

Monotonicity. Let $\tau,\tau^{\prime}\in L^{1}(\Omega,\rho^{s})$, $\mu,\mu^{\prime}\in L^{1}(\partial\Omega)$ and $u$ and $u^{\prime}$ be the weak solutions of (1.2) with data $(\tau,\mu)$ and $(\tau^{\prime},\mu^{\prime})$ respectively. We will show that if $\tau\leq\tau^{\prime}$ and $\mu\leq\mu^{\prime}$ then $u\leq u^{\prime}$ in $\Omega$. Indeed, put $v:=(u-u^{\prime})^{+}$, it is sufficient to prove that $v\equiv 0$. Since (1.9) holds, it follows

[TABLE]

Similarly

[TABLE]

Therefore

[TABLE]

By Kato inequality, the assumption $\tau\leq\tau^{\prime}$ and the monotonicity of $f$, we obtain

[TABLE]

Therefore

[TABLE]

Since $\mu\leq\mu^{\prime}$, it follows that $\mathrm{tr\,}_{s}(v)=0$. By Lemma 2.12, $v=0$ and thus $u\leq u^{\prime}$.

Existence.

Step 1: Assume that $\tau\in L^{\infty}(\Omega)$ and $\mu\in L^{\infty}(\partial\Omega)$.

Put $\hat{f}(t):=f(t+\mathbb{M}_{s}^{\Omega}[\mu])-f(\mathbb{M}_{s}^{\Omega}[\mu])$ and $\hat{\tau}:=\tau-f(\mathbb{M}_{s}^{\Omega}[\mu])$. Then $\hat{f}$ is nondecreasing and $t\hat{f}(t)\geq 0$ for every $t\in\mathbb{R}$ and $\hat{\tau}\in L^{1}(\Omega,\rho^{s})$. Consider the problem

[TABLE]

By [11, Proposition 3.1] there exists a unique weak solution $v$ of (3.1). It means that $v\in L^{1}(\Omega)$, $\hat{f}(v)\in L^{1}(\Omega,\rho^{s})$ and

[TABLE]

Put $u:=v+\mathbb{M}_{s}^{\Omega}[\mu]$ then $u\in L^{1}(\Omega)$ and $f(u)\in L^{1}(\Omega,\rho^{s})$. By (3.2) $u$ satisfies (1.3).

Step 2: Assume that $0\leq\tau\in L^{1}(\Omega,\rho^{s})$ and $0\leq\mu\in L^{1}(\partial\Omega)$.

Let $\{\tau_{n}\}\subset C^{1}(\overline{\Omega})$ be a nondecreasing sequence converging to $\tau$ in $L^{1}(\Omega,\rho^{s})$ and $\{\mu_{n}\}\subset C^{1}(\partial\Omega)$ be a nondecreasing sequence converging to $\mu$ in $L^{1}(\partial\Omega)$. Then $\{\mathbb{M}_{s}^{\Omega}[\mu_{n}]\}$ is nondecreasing and by Lemma 2.7 (ii) it converges to $\mathbb{M}_{s}^{\Omega}[\mu]$ a.e. in $\Omega$ and in $L^{p}(\Omega,\rho^{s})$ for every $1\leq p<p_{2}^{*}$. Let $u_{n}$ be the unique solution of (1.2) with $\tau$ and $\mu$ replaced by $\tau_{n}$ and $\mu_{n}$ respectively. By step 1 and the monotonicity of $f$, we derive that $\{u_{n}\}$ and $\{f(u_{n})\}$ are nondecreasing. Moreover

[TABLE]

Let $\eta\in C(\overline{\Omega})$ be the solution of

[TABLE]

then $c^{-1}\rho^{s}<\eta<c\rho^{s}$ in $\Omega$ for some $c>1$. By choosing $\xi=\eta$ in (3.3), we get

[TABLE]

Hence $\{u_{n}\}$ and $\{f(u_{n})\}$ are uniformly bounded in $L^{1}(\Omega)$ and $L^{1}(\Omega,\rho^{s})$ respectively. By the monotone convergence theorem, there exists $u\in L^{1}(\Omega)$ such that $u_{n}\to u$ in $L^{1}(\Omega)$ and $f(u_{n})\to f(u)$ in $L^{1}(\Omega,\rho^{s})$. By letting $n\to\infty$ in (3.3), we deduce that $u$ satisfies (1.3), namely $u$ is a weak solution of (1.2).

The uniqueness follows from the monotonicity.

Step 3: Assume that $\tau\in L^{1}(\Omega,\rho^{s})$ and $\mu\in L^{1}(\partial\Omega)$.

Let $\{\tau_{n}\}\subset C^{1}(\overline{\Omega})$ be a sequence such that $\{\tau_{n}^{+}\}$ and $\{\tau_{n}^{-}\}$ are nondecreasing and $\tau_{n}^{\pm}\to\tau^{\pm}$ in $L^{1}(\Omega,\rho^{s})$. Let $\{\mu_{n}\}\subset C^{1}(\partial\Omega)$ be a sequence such that $\{\mu_{n}^{+}\}$ and $\{\mu_{n}^{-}\}$ are nondecreasing and $\mu_{n}^{\pm}\to\mu^{\pm}$ in $L^{1}(\partial\Omega)$. Let $u_{n}$ be the unique weak solution of (1.2) with data $(\tau_{n},\mu_{n})$, then

[TABLE]

Let $w_{1,n}$ and $w_{2,n}$ be the unique weak solutions of (1.2) with data $(\tau_{n}^{+},\mu_{n}^{+})$ and $(-\tau_{n}^{-},-\mu_{n}^{-})$ respectively. Then

[TABLE]

Moreover, for any $n\in\mathbb{N}$, $w_{2,n}\leq 0\leq w_{1,n}$ and

[TABLE]

It follows that

[TABLE]

This, together with (3.7), implies

[TABLE]

Put $v_{n}:=\mathbb{G}_{s}^{\Omega}[\tau_{n}-f(u_{n})]$. By (3.10), the sequence $\{\tau_{n}-f(u_{n}\}$ is uniformly bounded in $L^{1}(\Omega,\rho^{s})$. Hence by [12, Proposition 2.6], the sequence $\{v_{n}\}$ is relatively compact in $L^{q}(\Omega)$ for $1\leq q<\frac{N}{N-s}$. Consequently, up to a subsequence, $\{v_{n}\}$ converges in $L^{q}(\Omega)$ and a.e. in $\Omega$ to a function $v$. On the other hand, by Lemma 2.7 ii), up to a subsequence, $\{\mathbb{M}_{s}^{\Omega}[\mu_{n}]\}$ converges in $L^{q}(\Omega,\rho^{s})$ for $1\leq q<p_{2}^{*}$ and a.e. in $\Omega$ to $\mathbb{M}_{s}^{\Omega}[\mu]$. Due to (3.6), we deduce that $\{u_{n}\}$ converges a.e. in $\Omega$ to $u=v+\mathbb{M}_{s}^{\Omega}[\mu]$. Since $f$ is continuous, $\{f(u_{n})\}$ converges a.e. in $\Omega$ to $f(u)$.

By step 2, the sequences $\{w_{1,n}\}$, $\{f(w_{1,n})\}$, $\{-w_{2,n}\}$ and $\{-f(w_{2,n})\}$ are increasing and converge to $w_{1}$ in $L^{1}(\Omega)$, $f(w_{1})$ in $L^{1}(\Omega,\rho^{s})$, $-w_{2}$ in $L^{1}(\Omega)$ and $-f(w_{2})$ in $L^{1}(\Omega,\rho^{s})$ respectively. In light of (3.9) and the generalized dominated convergence theorem, we obtain that $\{u_{n}\}$ and $\{f(u_{n})\}$ converge to $u$ and $f(u)$ in $L^{1}(\Omega)$ and $L^{1}(\Omega,\rho^{s})$ respectively. By passing to the limit in (3.3), we derive that $u$ satisfies (1.3).

The uniqueness follows from the monotonicity.           $\square$

Define

[TABLE]

This space is endowed with the norm

[TABLE]

We say that a sequence $\{\tau_{n}\}\subset\mathfrak{M}(\Omega,\rho^{s})$ converges weakly to a measure $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ if

[TABLE]

Proof of Theorem C.

Monotonicity. The monotonicity can be proved by using a similar argument as in the proof of Theorem B.

Existence. Let $\{\tau_{n}\}\subset C^{1}(\Omega)$ and $\{\mu_{n}\}\subset C^{1}(\partial\Omega)$ such that $\tau_{n}^{\pm}\to\tau^{\pm}$ weakly and $\mu_{n}^{\pm}\to\mu^{\pm}$ weakly. Then there is a positive constant $c$ independent of $n$ such that

[TABLE]

Let $u_{n}$, $w_{1,n}$ and $w_{2,n}$ as in the proof of Theorem B. Then

[TABLE]

This, together with (2.3), (2.6) and (3.11), implies that

[TABLE]

We have

[TABLE]

From this it follows

[TABLE]

We infer from (3.9) and the estimate $c^{-1}\rho^{s}\leq\eta\leq c\rho^{s}$ that

[TABLE]

This implies that $\{u_{n}\}$ and $\{f(u_{n})\}$ are uniformly bounded in $L^{1}(\Omega)$ and $L^{1}(\Omega,\rho^{s})$ respectively. By a similar argument as in step 3 of the proof of Theorem B, we deduce that, up to a subsequence, $\{u_{n}\}$ converges a.e. in $\Omega$ to a function $u$ and $\{f(u_{n})\}$ converges a.e. in $\Omega$ to $f(u)$. By Hölder inequality, we infer that $\{u_{n}\}$ is uniformly integrable in $L^{1}(\Omega)$.

Next we prove that $\{f\circ u_{n}\}$ is uniformly integrable in $L^{1}(\Omega,\rho^{s})$. Define $\tilde{f}(s):=f(|s|)-f(-|s|)$, $s\in\mathbb{R}$. Then $\tilde{f}$ is nondecreasing in $\mathbb{R}$ and $|f(s)|\leq\tilde{f}(s)$ for every $s\in\mathbb{R}$. For $\ell>0$ and $n\in\mathbb{N}$, set

[TABLE]

We take an arbitrary Borel set $D\subset\Omega$ and estimate

[TABLE]

On one hand, we have

[TABLE]

From (3.13), we infer $a_{n}(s)\leq\tilde{c}\,s^{-p_{2}^{*}}$ where $\tilde{c}$ is a positive constant independent of $n$. Hence, for any $l>\ell$,

[TABLE]

By assumption (1.11), there exists a sequence $\{l_{k}\}$ such that $l_{k}\to\infty$ and $l_{k}^{-p_{2}^{*}}\tilde{f}(l_{k})\to 0$ as $k\to\infty$. Taking $l=l_{k}$ in (3.18) and then letting $k\to\infty$, we obtain

[TABLE]

From assumption (1.11), we see that the right hand-side of (3.19) tends to [math] as $\ell\to\infty$. Therefore, for any $\epsilon>0$, one can choose $\ell>0$ such that the right hand-side of (3.19) is smaller than $\epsilon/2$. Fix such $\ell$, one then can choose $\delta>0$ small such that if $\int_{D}\rho^{s}dx<\delta$ then $\tilde{f}(\ell)\int_{D}\rho^{s}dx<\epsilon/2$. Therefore, from (3.17), we derive

[TABLE]

This means $\{f\circ u_{n}\}$ is uniformly integrable in $L^{1}(\Omega,\rho^{s})$.

By Vitali convergence theorem, we deduce that, up to a subsequence, $u_{n}\to u$ in $L^{1}(\Omega)$ and $f(u_{n})\to f(u)$ in $L^{1}(\Omega,\rho^{s})$. Since $u_{n}$ satisfies (3.3), by passing to the limit, we deduce that $u$ is a weak solution of (1.2).

Stability. Assume $\{\tau_{n}\}\subset\mathfrak{M}(\Omega,\rho^{s})$ converges weakly to $\tau\in\mathfrak{M}(\Omega,\rho^{s})$ and $\{\mu_{n}\}\subset\mathfrak{M}(\partial\Omega)$ converges weakly to $\mu\in\mathfrak{M}(\partial\Omega)$. Let $u$ and $u_{n}$ be the unique weak solution of (1.2) with data $(\tau,\mu)$ and $(\tau_{n},\mu_{n})$ respectively. Then by a similar argument as in Existence part, we deduce that $u_{n}\to u$ in $L^{1}(\Omega)$ and $f(u_{n})\to f(u)$ in $L^{p}(\Omega,\rho^{s})$.           $\square$.

Proposition 3.1**.**

Assume $f$ is a continuous nondecreasing function on $\mathbb{R}$ satisfying $f(0)=0$ and (1.11). Then for every $z\in\partial\Omega$,

[TABLE]

Proof.

By (2.1),

[TABLE]

Hence

[TABLE]

Put

[TABLE]

[TABLE]

For every $y\in{\mathcal{D}}_{1}$, $|x-z|\leq 2|y-z|$, therefore

[TABLE]

Hence

[TABLE]

We next estimate $I_{2}$. For every $y\in{\mathcal{D}}_{2}$, $|x-z|\leq 2|x-y|$, hence

[TABLE]

Therefore, by (1.11),

[TABLE]

Finally, we estimate $I_{3}$. For every $y\in{\mathcal{D}}_{3}$, $|y-z|\leq 3|x-y|$, therefore

[TABLE]

Put

[TABLE]

If $\lim_{r\to 0}g_{1}(r)<\infty$, then $\lim_{x\to z}I_{3}=0$ by (3.25). Otherwise, $\lim_{r\to 0}g_{1}(r)=\infty=\lim_{r\to 0}g_{2}(r)$. Therefore, by L’ Hôpital’s rule,

[TABLE]

By combining (3.25) and (3.26) we obtain

[TABLE]

We deduce (3.21) by gathering (3.23), (3.24) and (3.27).           $\square$

Proof of Theorem D. From Theorem C we get

[TABLE]

which implies

[TABLE]

We derive (1.13) due to Proposition 3.1.           $\square$

3.2. Power absorption

In this subsection we assume that $0\in\partial\Omega$. Let $0<p<p_{2}^{*}$ and denote by $u_{k}^{\Omega}$ the unique solution of (1.14). By Theorem C, $u_{k}^{\Omega}\leq kM_{s}^{\Omega}(\cdot,0)$ and $k\mapsto u_{k}^{\Omega}$ is increasing. Therefore, it is natural to investigate $\lim_{k\to\infty}u_{k}^{\Omega}$.

For any $\ell>0$, put

[TABLE]

If $u$ is a solution of (1.17) in $\Omega$ then $T_{\ell}[u]$ is a solution of (1.17) in $\Omega_{\ell}$.

By Corollary A.9, the function

[TABLE]

where $\ell_{s,p}$ is a positive constant, is a radial singular solution of

[TABLE]

Lemma 3.2**.**

Assume $p\in(p_{1}^{*},p_{2}^{*})$. Then there exists a positive constant $C$ depending on $N$, $s$, $p$ and the $C^{2}$ characteristic of $\Omega$ such that the following holds. If $u$ is a positive solution of (1.17) satisfying $u\leq U$ in $\Omega$ then there holds

[TABLE]

Proof.

Let $P\in(\partial\Omega\setminus\{0\})\cap B_{1}(0)$ and put

[TABLE]

Put

[TABLE]

Then $u_{d}$ is a solution of

[TABLE]

Moreover

[TABLE]

Put $P_{d}=d^{-1}P$ and let $\beta_{0}$ be the constant in Proposition 2.8. We may assume $\beta_{0}\leq\frac{1}{4}$. Let $\zeta_{P}\in C^{\infty}(\mathbb{R}^{N})$ such that $0\leq\zeta\leq 1$ in $\mathbb{R}^{N}$, $\zeta=0$ in $B_{\beta_{0}}(P_{d})$ and $\zeta=1$ in $\mathbb{R}^{N}\setminus B_{2\beta_{0}}(P_{d})$. Let $\eta_{d}\in C(\overline{\Omega}_{d})$ be the solution of (3.4) with $\Omega$ replaced by $\Omega_{d}$. For $l>0$, denote

[TABLE]

We will compare $u_{d}$ with $V_{d,l}$.

Step 1: We show that $V_{d,l}$ is a super solution of (3.32) for $l$ large enough.

For $y\in\Omega_{d}\setminus B_{4\beta_{0}}(P_{d})$, $\zeta_{P}(y)=1$ and hence

[TABLE]

where $c_{26}=c_{26}(N,s,p,\beta_{0})$. Since $(\Omega_{d}\cap B_{2\beta_{0}}(0))\subset(\Omega_{d}\setminus B_{4\beta_{0}}(P_{d}))$, it follows that, for any $y\in\Omega_{d}\cap B_{2\beta_{0}}(0)\setminus\{0\}$,

[TABLE]

Therefore if we choose $l\geq c_{26}$ then

[TABLE]

Next we see that there exists $c_{27}>0$ such that

[TABLE]

Consequently,

[TABLE]

Therefore if we choose $l\geq c_{27}$ then

[TABLE]

By combining (3.33) and (3.34), for $l\geq\max\{c_{26},c_{27}\}$, we deduce that $V_{d,l}$ is a super solution of (3.32).

Step 2: We show that $u_{d}\leq V_{d,l}$ in $\Omega_{d}$. By contradiction, we assume that there exists $x_{0}\in\Omega_{d}$ such that

[TABLE]

Then ${(-\Delta)^{s}}(u_{d}-V_{d,l})(x_{0})\geq 0$. It follows that

[TABLE]

This contradiction implies that $u_{d}\leq V_{d,l}$ in $\Omega_{d}$.

Step 3: End of proof. From step 2, we deduce that

[TABLE]

We note that $\eta_{d}(y)\leq c\mbox{\rm dist}\,(y,\partial\Omega_{d})^{s}$ for every $y\in\Omega_{d}$. Here the constant $c$ depends on $N$, $s$ and the $C^{2}$ characteristic of $\Omega_{d}$. Since $d<\frac{1}{2}$, a $C^{2}$ characteristic of $\Omega_{d}$ can be taken as a $C^{2}$ characteristic of $\Omega$. Therefore the constant $c$ can be taken independently of $P$. Consequently,

[TABLE]

This implies

[TABLE]

Put

[TABLE]

If $x\in{\mathcal{F}}_{1}$ then let $P\in\partial\Omega\setminus\{0\}$ such that $\rho(x)=|x-P|$. It follows that

[TABLE]

By combining (3.35) and (3.36), we get

[TABLE]

If $x\in{\mathcal{F}}_{2}$ then (3.31) follows from the assumption $u\leq U$. Thus (3.31) holds for every $x\in\Omega_{\beta_{0}}\cap B_{\frac{1}{1+\beta_{0}}}(0)$. If $x\in\Omega\setminus B_{\frac{1}{1+\beta_{0}}}(0)$ then by a similar argument as in Step 1 and Step 2 without similarity transformation, we deduce that there exist constants $c$ and $\tilde{\beta}\in(0,\frac{1}{2(1+\beta_{0})})$ depending on $N$, $s$, $p$ and the $C^{2}$ characteristic of $\Omega$ such that (3.31) holds in $B_{\tilde{\beta}}(P)\cap\Omega$ for every $P\in\partial\Omega\setminus B_{\frac{1}{1+\beta_{0}}}(0)$. Finally, since $u\leq U$, (3.31) holds in $D_{\frac{\tilde{\beta}}{2}}=\{x\in\Omega:\rho(x)>\frac{\tilde{\beta}}{2}\}$. Thus (3.31) holds in $\Omega$.           $\square$

Lemma 3.3**.**

Let $p\in(0,p_{2}^{*})$. There exists a constant $c=c(N,s,p,\Omega)>0$ such that for any $x\in\Omega$ and $z\in\partial\Omega$, there holds

[TABLE]

Proof.

We use a similar argument as in the proof of Proposition 3.1. It is easy to see that for every $x\in\Omega$ and $z\in\partial\Omega$,

[TABLE]

Let ${\mathcal{D}}_{i}$, $i=1,2,3$ be as in (3.22) and put

[TABLE]

By proceeding as in the proof of Proposition 3.1 we deduce easily that there is positive constants $c_{24}=c_{24}(N,s,p,\Omega)$ such that

[TABLE]

and

[TABLE]

Combining (3.39) and (3.40) implies (3.37).           $\square$

Proposition 3.4**.**

Assume $p\in(p_{1}^{*},p_{2}^{*})$. Then $u_{\infty}^{\Omega}:=\lim_{k\to 0}u_{k}^{\Omega}$ is a positive solution of (1.17). Moreover, there exists $c=c(N,s,p,\Omega)>0$ such that

[TABLE]

Proof.

We first claim that for any $k>0$,

[TABLE]

Indeed, by (2.5),

[TABLE]

Since $p<p_{2}^{*}$, it follows that

[TABLE]

By proceeding as in Step 2 of the proof of Lemma 3.2, we deduce that $u_{k}^{\Omega}\leq U$ in $\Omega$.

Consequently, $u_{\infty}^{\Omega}:=\lim_{k\to\infty}u_{k}^{\Omega}$ is a solution of (1.17) vanishing on $\partial\Omega\setminus\{0\}$ and satisfying $u_{\infty}^{\Omega}\leq U$ in $\Omega$. In light of Lemma 3.2, we obtain the upper bound in (3.41).

Next we prove the lower bound in (3.41). By (2.5) and Lemma 3.3, for any $k>0$ and $x\in\Omega$, we have

[TABLE]

For $x\in\Omega$, one can choose $r>0$ such that $x\in\Omega\cap(B_{2r}(0)\setminus B_{r}(0))$. Choose $k=ar^{-\frac{N+s-(N-s)p}{p-1}}$, where $a>0$ will be made precise later on, then

[TABLE]

By choosing $a=(2c_{29}c_{30})^{-\frac{1}{p-1}}$, we deduce for any $x\in\Omega$ there exists $k>0$ depending on $|x|$ such that

[TABLE]

Since $u_{\infty}^{\Omega}\geq u_{k}^{\Omega}$ in $\Omega$ we obtain the first inequality in (3.41).           $\square$

Proposition 3.5**.**

Assume $0<p\leq p_{1}^{*}$. There exist $k_{0}=k_{0}(N,s,p)$ and $c=c(N,s,p,\Omega)$ such that the following holds. There exists a decreasing sequence of positive numbers $\{r_{k}\}$ such that $\lim_{k\to\infty}r_{k}=0$ and for any $k>k_{0}$,

[TABLE]

Proof.

For any $\ell>0$, we have

[TABLE]

Case 1: $p\in(\frac{s}{N-s},p_{1}^{*})$. Put $k_{1}:=(2c_{29}c_{30})^{\frac{s}{N+2s-Np}}$ and take $k>k_{1}$. For $\ell>0$, put $r_{\ell}=\ell^{-\frac{1}{s}}$, then $\ell=r_{\ell}^{-s}$. Take arbitrarily $x\in\Omega\setminus B_{r_{k}}(0)$ then one can choose $\ell\in(\max(2^{-s}k,k_{1}),k)$ such that $x\in\Omega\cap(B_{r_{\ell}}(0)\setminus B_{\frac{r_{\ell}}{2}}(0))$. From (3.44), (2.5) and (3.37), we get

[TABLE]

Here the first estimate holds since $\frac{N}{N-s}<p<p_{2}^{*}$ and the third estimate holds since $p<p_{1}^{*}$ and $\ell>k_{1}$. Since $k>\ell$, we deduce that

[TABLE]

Case 2: $p=\frac{s}{N-s}$. Put $k_{2}=(\frac{2c_{29}c_{30}(1+s)}{s})^{\frac{s}{N-sp}}$ and take $k>k_{2}$. For $\ell>0$, put $r_{\ell}=\ell^{-\frac{1}{s}}$, then $\ell=r_{\ell}^{-s}$. Take arbitrarily $x\in\Omega\setminus B_{r_{k}}(0)$ then one can choose $\ell\in(\max(2^{-s}k,k_{2}),k)$ such that $x\in\Omega\cap(B_{r_{\ell}}(0)\setminus B_{\frac{r_{\ell}}{2}}(0))$. From (3.44), (2.5) and Lemma 3.3, we get

[TABLE]

Here the third estimate holds since $\ell>k_{2}$ and $N-sp>0$. Therefore (3.45) holds.

Case 3: $p\in(0,\frac{s}{N-s})$. Put $k_{3}=(2c_{29}c_{30})^{\frac{s}{N+s-sp}}$ and take $k>k_{3}$. For $\ell>0$, put $r_{\ell}=\ell^{-\frac{1}{s}}$, then $\ell=r_{\ell}^{-s}$. Take arbitrarily $x\in\Omega\setminus B_{r_{k}}(0)$ then one can choose $\ell\in(\max(2^{-s}k,k_{3}),k)$ such that $x\in\Omega\cap(B_{r_{\ell}}(0)\setminus B_{\frac{r_{\ell}}{2}}(0))$. From (3.44), (2.5) and (3.37), we get

[TABLE]

Here the third estimate holds since $\ell>k_{3}$ and $N+s-sp>0$. Therefore (3.45) holds.

Case 4: $p=p_{1}^{*}$. Put $k_{4}=\exp((2c_{29}c_{30})^{\frac{s}{N+s-(N-s)p}})$ and take $k>k_{4}$. For $\ell>0$, put $r_{\ell}=(\ell\ln(\ell))^{-\frac{1}{s}}$, then $\ell\ln(\ell)=r_{\ell}^{-s}$ and $\ell<r_{\ell}^{-s}$ when $\ell>3$. Take arbitrarily $x\in\Omega\setminus B_{r_{k}}(0)$ then one can choose $\ell\in(\max(2^{-s}k,k_{4}),k)$ such that $x\in\Omega\cap(B_{r_{\ell}}(0)\setminus B_{\frac{r_{\ell}}{2}}(0))$. From (3.44), (2.5) and (3.37), we get

[TABLE]

Here the inequality holds since $p=p_{1}^{*}$ and the last estimate follows from the following estimate

[TABLE]

Since $u_{k}^{\Omega}(x)\geq u_{\ell}^{\Omega}(x)$, we derive

[TABLE]

By putting $k_{0}:=\max(k_{1},k_{2},k_{3},k_{4})$, we obtain (3.43).           $\square$

Proposition 3.6**.**

Assume $0<p\leq p_{1}^{*}$. Then $\lim_{k\to\infty}u_{k}^{\Omega}(x)=\infty$ for every $x\in\Omega$.

Proof.

The proposition can be obtained by adapting the argument in the proof of [9, Theorem 1.2]. Let $r_{0}>0$ and put

[TABLE]

Then

[TABLE]

which implies

[TABLE]

Fix $y_{0}\in\Omega\setminus\overline{B}_{r_{0}}(0)$ and set $\delta:=\frac{1}{2}\min\{\rho(y_{0}),|y_{0}|-r_{0}\}$. By [13, Lemma 2.4] there exists a unique classical solution $w_{k}$ of the following problem

[TABLE]

By [13, Lemma 2.2],

[TABLE]

Next put $\tilde{w}_{k}:=w_{k}-\chi_{B_{r_{0}}(0)}u_{k}$ then $\tilde{w}_{k}=w_{k}$ in $B_{\delta}(y_{0})$. Moreover, for $x\in B_{\delta}(y_{0})$

[TABLE]

where $A=(|y_{0}|+r_{0})^{-N-2s}$. It follows that, for $x\in B_{\delta}(y_{0})$,

[TABLE]

Therefore $\tilde{w}_{k}\in C(\overline{B_{\delta}(y_{0})})$ is a supersolution of

[TABLE]

Let $\eta_{0}\in C(\overline{B_{\delta}(y_{0})})$ be the unique solution of

[TABLE]

We can choose $k$ large enough so that the function

[TABLE]

is a subsolution of (3.51). By [13, Lemma 2.2] we obtain

[TABLE]

Put

[TABLE]

then we derive from (3.53) that

[TABLE]

By combining (3.46), (3.48) and (3.54), we deduce that

[TABLE]

This implies

[TABLE]

$\square$

Theorem 3.7**.**

Assume $p\in(1,p_{2}^{*})$ and either $\Omega=\mathbb{R}_{+}^{N}:=\{x=(x^{\prime},x_{N}):x_{N}>0\}$ or $\partial\Omega$ is compact with $0\in\partial\Omega$. Then, for any $k>0$, there exists a unique solution solution $u_{k}^{\Omega}$ of problem (1.14) satisfying $u_{k}^{\Omega}\leq kM_{s}^{\Omega}(\cdot,0)$ in $\Omega$ and

[TABLE]

Moreover, the map $k\mapsto u_{k}^{\Omega}$ is increasing.

Proof.

Step 1: Existence. For $R>0$ we set $\Omega_{R}=\Omega\cap B_{R}$ and let $u:=u^{\Omega_{R}}_{k}$ be the unique solution of

[TABLE]

Then

[TABLE]

Since $R\mapsto M_{s}^{\Omega_{R}}(.,0)$ is increasing, it follows from (1.13) that $R\mapsto u^{\Omega_{R}}_{k}$ is increasing too with the limit $u^{*}$ and there holds

[TABLE]

From (3.57), we deduce that

[TABLE]

where $c$ depends only on $N$, $s$ and the $C^{2}$ characteristic of $\Omega$. Hence by the regularity up to the boundary [27], $\{u^{\Omega_{R}}_{k}\}$ is uniformly bounded in $C_{loc}^{s}(\overline{\Omega}\setminus B_{\epsilon})$ and in $C_{loc}^{2s+\alpha}(\Omega\setminus B_{\epsilon})$ for any $\epsilon>0$. Therefore, $\{u_{k}^{\Omega_{R}}\}$ converges locally uniformly, as $R\to\infty$, to $u^{*}\in C(\overline{\Omega}\setminus\{0\})\cap C^{2s+\alpha}(\Omega)$. Thus $u^{*}$ is a positive solution of (1.17). Moreover by combining (1.13), (3.57), the fact that $M_{s}^{\Omega_{R}}\uparrow M_{s}^{\Omega}$ and $u_{k}^{\Omega_{R}}\uparrow u_{k}^{\Omega}$, we deduce that $\mathrm{tr\,}_{s}(u^{*})=k\delta_{0}$ and

[TABLE]

Step 2: Uniqueness. Suppose $u$ and $u^{\prime}$ are two weak solutions of (1.17) satisfying $\max\{u,u^{\prime}\}\leq kM_{s}^{\Omega}(\cdot,0)$ in $\Omega$ and

[TABLE]

Take $\epsilon>0$ and put $u_{\epsilon}:=(1+\epsilon)u^{\prime}+\epsilon$, $v:=(u-u_{\epsilon})_{+}$. Then by (3.59) there exists a smooth bounded domain $G\subset\Omega$ such that $v=0$ in $G^{c}$ and $\mathrm{tr\,}_{s}^{G}(v)=0$. In light of Kato’s inequality, we derive ${(-\Delta)^{s}}v\leq 0$ in $G$. Moreover, $v\leq kM_{s}^{\Omega}(\cdot,0)$ in $G$. By Lemma 2.12 we obtain $v=0$ in $G$ and therefore $u\leq(1+\epsilon)u^{\prime}+\epsilon$ in $\Omega$. Letting $\epsilon\to 0$ yields $u\leq u^{\prime}$ in $\Omega$. By permuting the role of $u$ and $u^{\prime}$, we derive $u=u^{\prime}$ in $\Omega$.

By a similar argument as in step 2, we can show that $k\mapsto u_{k}^{\Omega}$ is increasing.           $\square$

Proof of Theorem F. (i) Case 1: $p_{1}^{*}<p<p_{2}^{*}$.

Since $\partial\Omega\in C^{2}$, there exist two open balls $B$ and $B^{\prime}$ such that $B\subset\Omega\subset B^{\prime c}$ and $\partial B\cap\partial B^{\prime}=\{0\}$. Since $M_{s}^{B}(x,0)\leq M_{s}^{\Omega}(x,0)\leq M_{s}^{B^{\prime c}}(x,0)$ it follows from Theorem 3.7 that

[TABLE]

where the first inequality holds in $B$ and the second inequality holds in $\Omega$.

Let ${\mathcal{O}}$ be $B$, $\Omega$ or $B^{\prime c}$. Because of uniqueness, we have

[TABLE]

with ${\mathcal{O}}_{\ell}=\ell^{-1}{\mathcal{O}}$. By Theorem 3.7, the sequence $\{u^{\mathcal{O}}_{k}\}$ is increasing and by (3.42), $u^{\mathcal{O}}_{k}\leq U$. It follows that $\{u^{\mathcal{O}}_{k}\}$ converges to a function $u^{\mathcal{O}}_{\infty}$ which is a positive solution of (1.17) with $\Omega$ replaced by ${\mathcal{O}}$.

Step 1: ${\mathcal{O}}:=\mathbb{R}^{N}_{+}$. Then ${\mathcal{O}}_{\ell}=\mathbb{R}_{+}^{N}$. Letting $k\to\infty$ in (3.61) yields to

[TABLE]

Therefore $u_{\infty}^{\mathbb{R}^{N}_{+}}$ is self-similar and thus it can be written in the separable form

[TABLE]

where $r=|x|$, $\sigma=\frac{x}{|x|}\in S^{N-1}$ and $\omega$ satisfies (1.16). Since $p_{1}^{*}<p<p_{2}^{*}$, it follows from Theorem E that $\omega=\omega^{*}$, the unique positive solution of (1.16). This means

[TABLE]

This implies (3.41).

Step 2: ${\mathcal{O}}:=B$ or $B^{\prime c}$. In accordance with our previous notations, we set $B_{\ell}=\ell^{-1}B$ and $(B^{\prime c})_{\ell}=\ell^{-1}B^{\prime c}$ for $\ell>0$ and we have,

[TABLE]

and

[TABLE]

When $\ell\to 0$, $u_{\infty}^{B_{\ell}}\uparrow\underline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ and $u_{\infty}^{(B^{\prime c})_{\ell}}\downarrow\overline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ where $\underline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ and $\overline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ are positive solutions of (3.42) in $\mathbb{R}^{N}_{+}$ such that

[TABLE]

Furthermore there also holds for $\ell,\ell^{\prime}>0$,

[TABLE]

Letting $\ell\to 0$ and using (3.64) and the above convergence, we obtain

[TABLE]

Again this implies that $\underline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ and $\overline{u}_{\infty}^{\mathbb{R}^{N}_{+}}$ are separable solutions of (1.15). Since $p_{1}^{*}<p<p_{2}^{*}$, by Theorem E,

[TABLE]

Step 3: End of the proof. From (3.60) and (3.64) there holds

[TABLE]

Since the left-hand side and the right-hand side of (3.69) converge to the same function $u_{\infty}^{\mathbb{R}^{N}_{+}}$, we obtain

[TABLE]

and this convergence holds in any compact subset of $\Omega$. Take $|x|=1$, we derive (1.18). Estimate (3.41) follows from Proposition 3.4.

(ii) Case 2: $0<p\leq p_{1}^{*}$. Then by Proposition 3.6, $\lim_{k\to\infty}u_{k}^{\Omega}(x)=\infty$ for every $x\in\Omega$.           $\square$

Appendix A Appendix - Separable solutions

A.1. Separable $s$-harmonic functions

We denote by $(r,\sigma)\in\mathbb{R}_{+}\times S^{N-1}$ the spherical coordinates in $\mathbb{R}^{N}$, consider the following parametric representation of the unit sphere

[TABLE]

hence $x_{N}=r\sin\phi$. We define the spherical fractional Laplace-Beltrami operator ${\mathcal{A}}_{s}$ by

[TABLE]

with

[TABLE]

where $\overrightarrow{\sigma}=(1,\sigma)$. If $u:(r,\sigma)\mapsto u(r,\sigma)=r^{-\beta}\omega(\sigma)$ is $s$-harmonic in $\mathbb{R}^{N}\setminus\{0\}$, it satisfies, at least formally,

[TABLE]

where ${\mathcal{L}}_{s,\beta}$ is the integral operator

[TABLE]

whenever this integral is defined. We will see in the next two lemmas that the role of the exponent $\beta_{0}=N$ is fundamental for the definition of ${\mathcal{L}}_{s,\beta}\omega$ since we have

Lemma A.1**.**

If $N\geq 2$, $s\in(0,1)$, $\beta<N$ and $(\sigma,\eta)\in\mathbb{R}^{N-1}\times\mathbb{R}^{N-1}$ such that $\langle\sigma,\eta\rangle\neq 1$, we define

[TABLE]

Then

(i)ii $B_{s,\beta}(\sigma,\eta)<0\Longleftrightarrow\beta<N-2s$,

(ii)i $B_{s,\beta}(\sigma,\eta)=0\Longleftrightarrow\beta=N-2s$,

(iii) $B_{s,\beta}(\sigma,\eta)>0\Longleftrightarrow\beta>N-2s$.

Proof.

Since $\beta<N$, the integral in (A.7) is absolutely convergent. We write

[TABLE]

By the change of variable $\tau\mapsto\tau^{-1}$

[TABLE]

where $c_{s}=\beta+2s-N$. Since

[TABLE]

the claim follows.          $\square$

As a byproduct of (A.7) we have the following monotonicity formula

Lemma A.2**.**

If $N\geq 2$ and $s\in(0,1)$, then for any $(\sigma,\eta)\in S^{N-1}\times S^{N-1}$ the mapping $\beta\mapsto B_{s,\beta}(\sigma,\eta)$ is continuous and increasing from $(N-2s,N)$ onto $(0,\infty)$.

In the next result we analyze the behavior of $B_{s,\beta}(\sigma,\eta)$ when $\sigma-\eta\to 0$ on $S^{N-1}$.

Lemma A.3**.**

Assume $N\geq 2$, $s\in(0,1)$ and $\beta<N$ with $\beta\neq N-2s$, then

I- If $N\geq 3$, there exists $c=c(N,\beta,s)>0$ such that

[TABLE]

*II- If $N=2$, *

(i) either $s>\frac{1}{2}$ and (A.8) holds with $N=2$,

(ii) either $s=\frac{1}{2}$ and

[TABLE]

(iii) or $0<s<\frac{1}{2}$ and

[TABLE]

Proof.

First, notice that the quantity

[TABLE]

is uniformly bounded with respect to $(\sigma,\eta)$. The only possible singularity in the expression given in (A.7) occurs when $\langle\sigma,\eta\rangle=1$ and $\tau=1$. We write $\langle\sigma,\eta\rangle=1-\frac{1}{2}\kappa^{2}$ and $t=1-\tau$, hence

[TABLE]

as $t\to 0$. Moreover

[TABLE]

Hence

[TABLE]

If $N=2$ and $s<\frac{1}{2}$,

[TABLE]

for some $M>0$ independent of $\kappa$. If $N=2$ and $s=\frac{1}{2}$

[TABLE]

and if $N=3$ or $N=2$ and $s>\frac{1}{2}$,

[TABLE]

as $\kappa\to 0$. Since $\sigma,\eta\in S^{N-1}$ there holds $\kappa^{2}=2(1-\langle\sigma,\eta\rangle)=\left|\sigma-\eta\right|^{2}$. Thus the claim follows.          $\square$

Proposition A.4**.**

Assume $N\geq 2$, $s\in(0,1)$ and $\beta<N$ with $\beta\neq N-2s$. Then $\omega\mapsto{\mathcal{L}}_{s,\beta}\omega$ is a continuous linear operator from $L^{q}(S^{N-1})$ into $L^{r}(S^{N-1})$ for any $1\leq q,r\leq\infty$ such that

[TABLE]

Furthermore, ${\mathcal{L}}_{s,\beta}$ is positive (resp. negative) operator if $\beta<N-2s$ (resp. $N-2s<\beta<N$).

Proof.

By Lemma A.3, for any $\eta\in S^{N-1}$, $B_{s,\beta}(.,\eta)\in L^{a}(S^{N-1})$ for all $1<a<\frac{N-1}{N+2s-3}$ if $N\geq 3$ or $N=2$ and $s>\frac{1}{2}$; $B_{s,\beta}(.,\eta)\in\bigcap_{1\leq a<\infty}L^{a}(S^{1})$ if $N=2$ and $s=\frac{1}{2}$ and $B_{s,\beta}(.,\eta)$ is uniformly bounded on $S^{1}$ if $N=2$ and $0<s<\frac{1}{2}$. The continuity result follows from Young’s inequality and the sign assertion from Lemma A.1.           $\square$

The above calculations justifies the name of fractional Laplace-Beltrami operator given to ${\mathcal{A}}_{s}$ since we have the following relation.

Lemma A.5**.**

Assume $N\geq 2$ and $s\in(0,1)$, then

[TABLE]

where ${\mathcal{B}}_{s}$ is a bounded linear operator from $L^{q}(S^{N-1})$ into $L^{r}(S^{N-1})$ for $q$, $r$ satisfying (A.11) and

[TABLE]

Proof.

If $(\sigma,\eta)\in S^{N-1}\times S^{N-1}$, we set $\langle\sigma,\eta\rangle=1-\frac{1}{2}\kappa^{2}$. Then

[TABLE]

Then we put $t=1-\tau$, hence, when $t\to 0$, we have after some straightforward computation

[TABLE]

This implies

[TABLE]

Since $\kappa=\left|\sigma-\eta\right|$, the claim follows from Proposition A.4 and the kernel estimate in Lemma A.3.          $\square$

Lemma A.6**.**

Under the assumption of Lemma A.5 there holds

[TABLE]

where

[TABLE]

Proof.

There holds by Cauchy-Schwarz inequality

[TABLE]

Since, by invariance by rotation, we have

[TABLE]

we derive (A.15).          $\square$

We denote the upper hemisphere of the unit sphere in $\mathbb{R}^{N}$ by $S_{+}^{N-1}=S^{N-1}\cap\mathbb{R}_{+}^{N}$.

Proposition A.7**.**

Let $N\geq 2$, $s\in(0,1)$ and $N-2s<\beta<N$. Then there exist a unique $\lambda_{s,\beta}>0$ and a unique (up to an homothety) positive $\psi_{1}\in W^{s,2}_{0}(S^{N-1}_{+})$, such that

[TABLE]

Furthermore the mapping $\beta\mapsto\lambda_{s,\beta}$ is continuous and decreasing from $(N-2s,N)$ onto $(0,\infty)$. Finally $\lambda_{s,\beta}=1$ if and only if $\beta=N-s$ and $\psi_{1}(\sigma)=(\sin\phi)^{s}$.

Proof.

We first notice that

[TABLE]

for any $\omega\in C^{1}_{0}(S^{N-1}_{+})$. By Lemma A.5 and (A.11) with $r=q=2$,

[TABLE]

where

[TABLE]

Since, by Poincaré inequality [16], there holds

[TABLE]

we obtain that the right-hand side of (A.17) is bounded from above by $\left(\frac{1}{2}c_{36}+\frac{c_{37}}{2c_{38}}\right)\left\|\omega\right\|^{2}_{W^{s,2}_{0}(S^{N-1}_{+})}$. Next we use the expansion estimates in Lemma A.5 to obtain that

[TABLE]

where $\kappa=\left|\sigma-\eta\right|\leq 2$. Hence

[TABLE]

Therefore,

[TABLE]

Finally we obtain

[TABLE]

We consider the bilinear form in $W^{s,2}_{0}(S^{N-1}_{+})$

[TABLE]

Then $\mathbb{A}$ is symmetric and there holds

[TABLE]

and

[TABLE]

By Riesz theorem, for any $L\in W^{-s,2}(S^{N-1}_{+})$ there exists $\omega_{L}\in W^{s,2}_{0}(S^{N-1}_{+})$ such that

[TABLE]

We denote $\omega_{L}={\mathcal{A}}_{s}^{-1}(L)$. It is clear that ${\mathcal{A}}_{s}^{-1}$ is positive and since the the embedding of $W^{s,2}_{0}(S^{N-1}_{+})$ into $L^{2}(S^{N-1}_{+})$ is compact by Rellich-Kondrachov theorem [16], ${\mathcal{A}}_{s}^{-1}$ is a compact operator. Hence the operator

[TABLE]

is a compact positive operator (here we use the fact that $\beta>N-2s$ which makes ${\mathcal{B}}_{s,\beta}$ positive). By the Krein-Rutman theorem there exists $\mu>0$ and $\psi_{1}\in W^{s,2}_{0}(S^{N-1}_{+})$, $\psi_{1}\geq 0$ such that

[TABLE]

The function $\psi_{1}$ is the unique positive eigenfunction and $\mu$ is the only positive eigenvalue with positive eigenfunctions. Furthermore $\mu$ is the spectral radius of ${\mathcal{A}}_{s}^{-1}\circ{\mathcal{B}}_{s,\beta}$. If we set $\lambda_{s,\beta}=\mu^{-1}$, we obtain (A.16). It is also classical that $\lambda_{s,\beta}$ can be defined by

[TABLE]

Using (A.7), Lemma A.2 and monotone convergence theorem, we derive that the mapping

[TABLE]

is increasing and continuous. This implies that $\beta\mapsto\lambda_{s,\beta}$ is decreasing and continuous. Since ${\displaystyle\int_{S^{N-1}_{+}}}\omega{\mathcal{L}}_{s,\beta}\omega dS\to\infty$ when $\beta\uparrow N$, the expression (A.19) implies that $\lambda_{s,\beta}\to 0$ when $\beta\uparrow N$. Next, if $\omega\geq 0$ is an element of $W^{s,2}_{0}(S^{N-1}_{+})$ such that ${\displaystyle\int_{S^{N-1}_{+}}}\omega{\mathcal{L}}_{s,\beta}\omega dS=1$, we derive from Poincaré inequality [16] and (A.15),

[TABLE]

Since $c_{35}\to 0$ when $\beta\downarrow N-2s$, we infer that $\displaystyle\lim_{\beta\to N-2s}\lambda_{s,\beta}=\infty$. Consequently the mapping $\beta\mapsto\lambda_{s,\beta}$ is a decreasing homeomorphism from $(N-2s,N)$ onto $(0,\infty)$ and there exists a unique $\beta_{s}\in(N-2s,N)$ such that $\lambda_{s,\beta_{s}}=1$. The following expression of the Martin kernel in $\mathbb{R}^{N}_{+}$ is classical,

[TABLE]

hence, if $y=0$, it is a separable singular $s$-harmonic function expressed in spherical coordinates with $x=(r,\sigma)$ by

[TABLE]

This means that the function $\sigma\mapsto\omega(\sigma)=(\sin\phi)^{s}$, which vanishes on $\overline{S^{N-1}_{-}}$ and belongs to $W^{s,2}_{0}(S^{N-1}_{+})\cap L^{\infty}(S^{N-1}_{+})$, satisfies

[TABLE]

The uniqueness of the positive eigenfunction implies that this function is $\psi_{1}$ and $\beta=N-s$.           $\square$

A.2. The nonlinear problem

A.2.1. Separable solutions in $\mathbb{R}^{N}$

If we look for separable positive solutions of

[TABLE]

under the form $u(x)=r^{-\frac{2s}{p-1}}\omega(\sigma)$ where $x=(r,\sigma)\in\mathbb{R}_{+}\times S^{N-1}$, then $\omega$ satisfies

[TABLE]

Proposition A.8**.**

Assume $N\geq 2$ and $s\in(0,1)$.

(i) If $p\geq p_{3}^{*}$ then there exists no positive solution of (A.22).

(ii) If $p_{1}^{*}<p<p_{3}^{*}$ then the unique positive solution of (A.22) is a constant function with value

[TABLE]

where $c_{35}$ is the constant defined in Lemma A.6.

Proof.

If $p\geq p_{3}^{*}$, we assume that there exists a solution $\omega\geq 0$ of (A.22). Then $\omega$ satisfies

[TABLE]

Since $p\geq p_{3}^{*}$, we have $c_{s}\leq 0$ which implies

[TABLE]

Then $\omega=0$ since the two other integrals are nonnegative.

Next, if $p_{1}^{*}<p<p_{3}^{*}$ it is clear that if $\omega$ is a constant nonnegative solution of (A.22) then we have

[TABLE]

Using invariance by rotation of the integral term on $S^{N-1}$, we derive the claim. Conversely, assume $\omega$ is any bounded nonconstant positive solution, then it belongs to $C^{2}(S^{N-1})$ by [27]. Let $\sigma_{0}\in S^{N-1}$ where $\omega$ is maximal, then ${\mathcal{A}}_{s}\omega(\sigma_{0})\geq 0$ thus

[TABLE]

Hence $\omega(\sigma_{0})<\ell_{s,p}$. Similarly $\displaystyle\min_{S^{N-1}}\omega>\ell_{s,p}$, which is a contradiction.           $\square$

Corollary A.9**.**

Assume $N\geq 2$, $s\in(0,1)$ and $p_{1}^{*}<p<p_{3}^{*}$. Then the only positive separable solution $u$ of (A.21) in $\mathbb{R}^{N}\setminus\{0\}$ is

[TABLE]

A.2.2. Separable solutions in $\mathbb{R}_{+}^{N}$

If we consider separable solutions $x\mapsto u(x)=r^{-\frac{2s}{p-1}}\omega(\sigma)$ of problem (1.15) then $\omega$ satisfies (1.16).

Proof of Theorem E.

Step 1: Non-existence. Assume that such a solution $\omega\geq 0$ exists, then

[TABLE]

Hence

[TABLE]

If $\lambda_{s,\frac{2s}{p-1}}\geq 1$, equivalently $p\geq p_{2}^{*}$, the only nonnegative solution is the trivial one.

Step 2: Existence. Consider the following functional with domain $W^{s,2}_{0}(S^{N-1}_{+})\cap L^{p+1}(S^{N-1}_{+})$,

[TABLE]

Because of Lemma A.6, ${\mathcal{J}}(\omega)\to\infty$ when $\left\|\omega\right\|_{W^{s,2}_{0}(S^{N-1}_{+})}+\left\|\omega\right\|_{L^{p+1}(S^{N-1}_{+})}\to\infty$. Furthermore, for $\epsilon>0$, we have

[TABLE]

This implies that $\inf{\mathcal{J}}(\omega)<0$ if $\lambda_{s,\frac{2s}{p-1}}<1$, and thus the infimum of ${\mathcal{J}}$ in $W^{s,2}_{0}(S^{N-1}_{+})\cap L_{+}^{p+1}(S^{N-1}_{+})$ is achieved by a nontrivial nonnegative solution of (1.16).

Step 3: Uniqueness.

(i) Existence of a maximal solution. By [27] any solution $\omega$ is smooth. Hence, at its maximum $\sigma_{0}$, it satisfies ${\mathcal{A}}_{s}\omega(\sigma_{0})\geq 0$, thus

[TABLE]

This implies that $\sup\omega\leq\ell_{s,p}$. From the equation the set ${\mathcal{E}}\subset W^{s,2}_{0}(S^{N-1}_{+})$ of positive solutions of (1.16) is bounded in $W^{s,2}_{0}(S^{N-1}_{+})\cap L^{\infty}(S^{N-1}_{+})$ and thus in $C^{s}(S^{N-1})\cap C^{2}(S^{N-1}_{+})$ by [27]. We put $\overline{\omega}(\sigma)=\sup\{\omega(\sigma):\omega\in{\mathcal{E}}\}$. There exists a countable dense set ${\mathcal{S}}:=\{\sigma_{n}\}\subset S^{N-1}_{+}$ and a sequence of function $\{\omega_{n}\}\subset{\mathcal{E}}$ such that

[TABLE]

Furthermore, this sequence $\{\omega_{n}\}$ can be constructed such that $\{\omega_{n}(\sigma_{k})\}$ is nondecreasing for any $k$. Finally by local compactness estimate, $\{\omega_{n}\}$ converges to $\overline{\omega}$ in $C^{s-\delta}(S^{N-1})\cap C^{2}(S^{N-1}_{+})$ for any $\delta\in(0,s)$ and weakly in $W^{s,2}_{0}(S^{N-1}_{+})$. This implies that $\overline{\omega}$ belongs to ${\mathcal{E}}$. It follows from [27, Th 1.2] that any $\omega\in{\mathcal{E}}$ satisfies

[TABLE]

(ii) Existence of a minimal solution. This follows from Theorem 3.7 that $u_{k}^{\mathbb{R}_{+}^{N}}\uparrow u_{\infty}^{\mathbb{R}_{+}^{N}}$ and $u_{\infty}^{\mathbb{R}_{+}^{N}}$ is self-similar and it is the minimal solution of (1.17) in $\mathbb{R}^{N}_{+}$ which satisfies

[TABLE]

Thus $u_{\infty}^{\mathbb{R}_{+}^{N}}(r,\sigma)=r^{-\frac{2s}{p-1}}\underline{\omega}(\sigma)$ and $\underline{\omega}$ is the minimal positive solution of (1.16). Furthermore it follows from (3.41) that

[TABLE]

if $\phi=\phi(\sigma)$ is the latitude of $\sigma$.

(iii) End of the uniqueness proof. By combining (A.27) and (A.29) we infer that there exists $K>1$ such that

[TABLE]

Assume $\overline{\omega}\neq\underline{\omega}$, then

[TABLE]

is a positive supersolution (by convexity) of (1.16). Moreover

[TABLE]

is a positive subsolution of (1.16) smaller than $\omega_{1}$ hence also than $\underline{\omega}$. It follows by classical construction that there exists a solution $\tilde{\omega}$ of (1.16) which satisfies $\omega_{2}\leq\tilde{\omega}\leq\omega_{1}$, which contradicts the minimality of $\underline{\omega}$.           $\square$

Acknowledgements. The first author is supported by Fondecyt Grant 3160207. The second author is supported by collaboration programs ECOS C14E08.