Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

TL;DR

This paper studies the regularity of semi-stable, radially symmetric solutions to quasilinear elliptic equations on Riemannian manifolds, providing uniform bounds and estimates independent of boundary conditions and non-linearities.

Contribution

It establishes regularity results for solutions of p-Laplace type equations on Riemannian models, including extremal solutions with zero Dirichlet conditions, regardless of boundary or non-linearity specifics.

Findings

Proves uniform boundedness of solutions.

Provides Lebesgue and Sobolev estimates.

Shows regularity of extremal solutions.

Abstract

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions.

Equations

$$\mathrm{d}{s}^2=\mathrm{d}{r}^2+\psi (r{)}^2\mathrm{d}{\theta }^2\text{for }r\in (0,R)\text{ and }\theta \in {\mathbb{S}}^{N-1},$$

$$\displaystyle\begin{array}[]{lccllll}(i)&\psi(r)&=&\sinh r,&\quad K_{\psi}=-1&\quad\left(\text{Hyperbolic space}\right)\\ (ii)&\psi(r)&=&r,&\quad K_{\psi}=0&\quad\left(\text{Euclidean space}\right)\\ (iii)&\psi(r)&=&\sin r,&\quad K_{\psi}=1&\quad\left(\text{Elliptic space}\right)\end{array}$$

$$J_{\delta }(u)=\frac{1}{p}{\int }_{{\mathcal{B}}_1\setminus \overline{{\mathcal{B}}_{\delta }}}|{\nabla }_{g}u{|}^p\mathrm{d}{v}_g-{\int }_{{\mathcal{B}}_1\setminus \overline{{\mathcal{B}}_{\delta }}}F(u)\mathrm{d}{v}_g,\text{ where }F(x,t)={\int }_0^{t}f(x,s)\mathrm{d}s.$$

$$J_{\delta }(u)\le J_{\delta }(u+\varphi ).$$

$${\int }_{{\mathcal{B}}_1}\left[(p-1)|\nabla u{|}^{p-2}|\nabla \xi {|}^2-f^{\prime }(u){\xi }^2\right]\mathrm{d}{v}_g\ge 0.$$

$$-\text{div}(|{\nabla }_{g}u{|}^{p-2}{\nabla }_{g}u)=f(u)\text{ in }{\mathcal{B}}_1\setminus \left\{\mathcal{O}\right\}.$$

$$\Vert u{\Vert }_{L^{\infty }({\mathcal{B}}_1)}\le C_{N,p,\alpha ,\psi }\Vert u{\Vert }_{L^p({\mathcal{B}}_1)}.$$

$$\Vert u{\Vert }_{L^q({\mathcal{B}}_1)}\le C_{N,\psi ,p,q}\Vert u{\Vert }_{L^p({\mathcal{B}}_1)}\text{mbox}foranyq<q_0:=\frac{Np}{N-p-2-2\sqrt{\frac{N-1}{p-1}}}.$$

$$\Vert u{\Vert }_{W^{1,q}({\mathcal{B}}_1)}\le C_{N,\psi ,p,q}\Vert u{\Vert }_{L^p({\mathcal{B}}_1)}\text{mbox}foranyq<q_1:=\frac{Np}{N-2-2\sqrt{\frac{N-1}{p-1}}}.$$

$${\int }_0^{\delta }|u_r{|}^p{\psi }^{N-1-2\alpha }\mathrm{d}r\le C_{N,p,\alpha ,\psi }\Vert u{\Vert }_{L^p({\mathcal{B}}_1)}^p,$$

$$\{\begin{array}{rlrl}-\text{div}(|{\nabla }_{g}u{|}^{p-2}{\nabla }_{g}u)=& \lambda h(u)& \text{in}& {\mathcal{B}}_1,\\ u>& 0& \text{in}& {\mathcal{B}}_1,\\ u=& 0& \text{on}& \partial {\mathcal{B}}_1,\end{array}$$

$$\underset{t\to +\infty }{\lim }\frac{h(t)}{t^{p-1}}=+\infty .$$

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

$$\{\begin{array}{rlrl}-{\Delta }_{g}u=& \lambda h(u)& \text{in}& \Omega ,\\ u=& 0& \text{on}& \partial \Omega ,\end{array}$$

$$\{\begin{array}{rlrl}{\Delta }_{g}u+\lambda u+u^p=& 0& \text{in}& \mathcal{A},\\ u=& 0& \text{on}& \mathcal{A},\end{array}$$

$$\{\begin{array}{rlrl}-{\Delta }_{p}u=& \lambda h(u)& \text{in}& \Omega ,\\ u>& 0& \text{in}& \Omega ,\\ u=& 0& \text{on}& \partial \Omega ,\end{array}$$

$$I_{p,q}(v;\Omega )={\left({\int }_{\Omega }\left[{\left(\frac{1}{p^{\prime }}|{\nabla }_{T,v}|\nabla v{|}^{p/q}|\right)}^q+|H_v{|}^q|\nabla v{|}^p\right]dx\right)}^{1/p},p,q\ge 1,$$

$${\int }_{{\mathcal{B}}_1}|\nabla u{|}^{p-2}\nabla u\cdot \nabla \varphi \mathrm{d}{v}_g={\int }_{{\mathcal{B}}_1}h(u)\varphi \mathrm{d}{v}_g,$$

$$u^{\ast }=\underset{\lambda \nearrow {\lambda }^{\ast }}{\lim }{u}_{\lambda }$$

$${\int }_{{\mathcal{B}}_1}|{\nabla }_{g}w{|}^{p-2}{\nabla }_{g}w\cdot {\nabla }_g\varphi \mathrm{d}\sigma ={\int }_{{\mathcal{B}}_1}\varphi \mathrm{d}\sigma ,\forall \varphi \in C_0^{\infty }({\mathcal{B}}_1).$$

$${\int }_{{\mathcal{B}}_1}|{\nabla }_{g}w{|}^{p-2}{\nabla }_{g}w\cdot {\nabla }_g\varphi \mathrm{d}\sigma ={\int }_{{\mathcal{B}}_1}\varphi \mathrm{d}\sigma \ge {\int }_{{\mathcal{B}}_1}\lambda h(w)\varphi \mathrm{d}\sigma$$

$$\{\begin{array}{rlrl}-\text{div}(|{\nabla }_g{v}_1{|}^{p-2}{\nabla }_g{v}_1)=& {\lambda }_1|v_1{|}^{p-2}{v}_1& \text{in}& {\mathcal{B}}_1,\\ v_1=& 0& \text{in}& \partial {\mathcal{B}}_1,\end{array}$$

$$-\text{div}(|{\nabla }_g{v}_1{|}^{p-2}{\nabla }_g{v}_1)={\lambda }_1{v}_1^{p-1}\le {\lambda }_{1}h(0)<\lambda h(u)=-\text{div}(|{\nabla }_{g}u{|}^{p-2}{\nabla }_{g}u).$$

$$\{\begin{array}{rlrl}-\text{div}(|{\nabla }_g{v}_2{|}^{p-2}{\nabla }_g{v}_2)=& ({\lambda }_1+ϵ)v_1^{p-1}& \text{in}& {\mathcal{B}}_1,\\ v_2=& 0& \text{in}& \partial {\mathcal{B}}_1,\end{array}$$

$$-\text{div}(|{\nabla }_g{v}_2{|}^{p-2}{\nabla }_g{v}_2)=({\lambda }_1+ϵ)v_1^{p-1}\le ({\lambda }_1+ϵ)u^{p-1}\le \lambda h(u)=-\text{div}(|{\nabla }_{g}u{|}^{p-2}{\nabla }_{g}u).$$

$$\{\begin{array}{rlrl}-\text{div}(|{\nabla }_g{v}_n{|}^{p-2}{\nabla }_g{v}_n)=& ({\lambda }_1+ϵ)v_{n-1}^{p-1}& \text{in}& {\mathcal{B}}_1,\\ v_n=& 0& \text{in}& \partial {\mathcal{B}}_1.\end{array}$$

$$\{\begin{array}{rlrl}-\text{div}(|{\nabla }_g{u}_{\lambda }{|}^{p-2}{\nabla }_g{u}_{\lambda })=& ({\lambda }_1+ϵ)u_{\lambda }^{p-1}& \text{in}& {\mathcal{B}}_1,\\ u_{\lambda }=& 0& \text{in}& \partial {\mathcal{B}}_1,\end{array}$$

$$-(p-1)|u_r{|}^{p-2}{u}_{rr}-\frac{(N-1){\psi }^{\prime }}{\psi }|u_r{|}^{p-2}{u}_r=f(u)\text{ with }r\in (0,1).$$

$${\int }_{{\mathcal{B}}_1}|u_r{|}^p\left[(p-1)|{\eta }_r{|}^2+\frac{\partial }{\partial r}\left(\frac{(N-1){\psi }^{\prime }}{\psi }\right){\eta }^2\right]\mathrm{d}{v}_g\ge 0.$$

$$0\le {\int }_{{\mathcal{B}}_1}$$

Full text

Research supported in part by INCTmat/MCT/Brazil, CNPq and CAPES

Corresponding author: J. M. do Ó

1. Introduction

Let $(\mathcal{M},g)$ be a Riemannian model of dimension $N\geq 2$, that is, a manifold $\mathcal{M}$ admitting a pole $\mathcal{O}$ and whose metric $g$ is given, in polar coordinates around $\mathcal{O}$, by

[TABLE]

where $r$ is by construction the Riemannian distance between the point $P=(r,\theta)$ to the pole $\mathcal{O}$, $\mathrm{d}\theta^{2}$ is the canonical metric on the unit sphere $\mathbb{S}^{N-1}$ and $\psi$ is a smooth function in $[0,R)$ and positive in $(0,R)$ for some $R\in(0,+\infty]$, and $\psi(0)=\psi^{\prime\prime}(0)=0\text{ and }\psi^{\prime}(0)=1$. As examples we have the important cases of space forms, i.e., the unique complete and simply connected Riemannian manifold of constant sectional curvature $K_{\psi}$ corresponding to the choice of $\psi$ namely,

[TABLE]

Let us denote the geodesic ball of radius r with center at the pole $\mathcal{O}$ by $\mathcal{B}_{r}$ and $W_{r}^{1,p}(\mathcal{B}_{1})$ the elements of the Sobolev space which are radially symmetric with respect to the pole $\mathcal{O}$. For $u\in W_{r}^{1,p}(\mathcal{B}_{1})$, let us consider the energy functional

[TABLE]

Definition 1.1**.**

We say that a decreasing function $u\in W_{r}^{1,p}(\mathcal{B}_{1})$ is a radial local minimizer of (1.6) if for any $0<\delta<1$ there exists $\epsilon=\epsilon(\delta)>0$ such that for all radial functions $\phi\in C_{c}^{1}\left(\mathcal{B}_{1}\setminus\overline{\mathcal{B}_{\delta}}\right)$ satisfying $\|\phi\|_{C^{1}}\leq\epsilon$, we have

[TABLE]

Definition 1.2**.**

Let $u\in W_{r}^{1,p}(\mathcal{B}_{1})$ be a critical point of (1.6). We say that $u$ is semi-stable if $u_{r}(r)<0$ for all $r\in(0,1)$ and for all radially symmetric function $\xi\in C_{c}^{1}(\mathcal{B}_{1}\setminus\left\{\mathcal{O}\right\})$ it holds

[TABLE]

We note that critical points of the functional $J_{\delta}$ correspond to weak solutions of the singular problem

[TABLE]

In particular, if $u$ is a radial local minimizer of $J_{\delta}$, then $u$ is a semi-stable solution of ($\mathcal{S}$). We are going to focus our analysis on the important case $1<p\leq N$, since for $p>N$, it holds $W^{1,p}(\mathcal{B}_{1})\hookrightarrow L^{\infty}(\mathcal{B}_{1})$. Precisely, if $N<p\leq+\infty$, from Morrey’s inequality we have $\|u\|_{C^{0,\gamma}(\mathcal{B}_{1})}\leq C\|u\|_{W^{1,p}(\mathcal{B}_{1})}$, where $C$ is a constant which depends on $p$ and $N$.

1.1. Main Results and Comments

The aim of the paper is twofold. Firstly, to establish a priori estimates for radial semi-stable classical solutions of ($\mathcal{S}$). Precisely, we establish $L^{\infty}$, $L^{q}$ and $W^{1,q}$ estimates for semi-stable, radially symmetric, and decreasing solutions of ($\mathcal{S}$) without assuming Dirichlet boundary condition or any other kind of boundary conditions. It should be an interesting question to study similar results for non-radial solutions. We stress that our results hold for any locally Lipschitz non-linearity $f(s)$ and metric $g$ satisfying (1.1).

Since the celebrated paper by B. Gidas, W. M. Ni and L. Nirenberg [24] the question of symmetry in non-linear partial differential equations has been the subject of intensive investigations. In $[16]$, by using a variant of moving planes method, it was established radial symmetry for non-negative solutions of quasilinear elliptic equations defined in geodesic balls of the hyperbolic space $\mathbb{H}^{n}$ with homogeneous Dirichlet boundary condition.

Theorem 1.1**.**

For any $f(s)$ locally Lipschitz function and $u$ semi-stable solution of ($\mathcal{S}$), it holds that:

1. $(a)$

If $N<p+4p/(p-1)$, then $u\in L^{\infty}(\mathcal{B}_{1})$ and

[TABLE] 2. $(b)$

If $N\geq p+4p/(p-1)$, then $u\in L^{q}(\mathcal{B}_{1})$ and

[TABLE]

Moreover, $u\in W^{1,q}(\mathcal{B}_{1})$ and

[TABLE]

Remark 1.1**.**

For our argument in the proof of Theorem 1.1 it was crucial the following key estimate for semi-stable solutions of ($\mathcal{S}$)

[TABLE]

where $\psi$ is the polar decomposition of $\mathrm{d}s^{2}$ given in (1.1) (see Proposition 3.1 below). We have proved this estimate by using the radial symmetry of the solution $u$ and by choosing an appropriated test function in the semi-stability inequality (1.7),

Remark 1.2**.**

From Theorem 1.1 $(a)$, one can see that Problem ($\mathcal{S}$) does not have any singular solution.

Remark 1.3**.**

Note that $q_{0}>p^{*}=(Np)/(N-p)$ (the critical Sobolev exponent) and $q_{1}>p$. Under the hypotheses of Theorem 1.1 if $N\geq p+4p/(p-1)$ then $u$ belongs to $L^{q}\left(\mathcal{B}_{1}\right)$ for all $q<q_{0}$. Since $q_{0}$ is greater than the critical Sobolev exponent, from Theorem 1.1 $(ii)$ we conclude that semi-stable radially symmetric and decreasing weak solutions of ($\mathcal{S}$) have a better regularity than the one expected by using the classical Sobolev embedding. Moreover, we established better regularity than $W^{1,p}$ for semi-stable solutions to Problem ($\mathcal{S}$), since our estimates shows an improvement in the Sobolev space $W^{1,q}$ for $q<q_{1}$.

Our second purpose of this work is to apply the elliptic estimates obtained in Theorem 1.1 to prove regularity results for the following class of quasilinear elliptic problems

[TABLE]

where $\lambda$ is a positive parameter and $h(s)$ is an increasing $C^{1}-$function such that $h(0)>0$ and

[TABLE]

The study of this class of problems with various boundaries conditions has received considerable attention in recent years under the influence of the pioneering works of I. Gelfand [22], D. Joseph and T. Lundgren [29], J. Keener and H. Keller [31], M. Crandall and P. Rabinowitz [12], F. Mignot and J.-P. Puel [37]. First, we would like to mention the progress involving Laplacian

[TABLE]

where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$. Non-linear elliptic problems like (1.9) appear naturally in several physical phenomena, just to mention some applications, it arises in the theory of non-linear diffusion generated by non-linear sources [26, 27, 28], thermal ignition of a chemically active mixture of gases [22], membrane buckling [7] and gravitation equilibrium [11]. We refer the reader to [13, 17, 18] for a recent survey on this subject.

In recent years, regularity issues about this class of singular elliptic problems have been the focus of an active research area. The parameter $\lambda$ measure the non-dimensional strength of the non-linearity. It is well known that if $h$ is super-linear, there exists $\lambda^{*}\in(0,+\infty)$ such that if $\lambda\in(0,\lambda^{*})$, then problem (1.9) admits a semi-stable solution $u_{\lambda}$ and if $\lambda>\lambda^{*}$, then problem (1.9) admits no regular solution. This allows one to define the extremal solution $u^{*}:=\lim_{\lambda\nearrow\lambda^{*}}u_{\lambda}$, which is a weak solution of (1.9). In [40], G. Nedev proved regularity results for extremal solutions of (1.9) in dimensions $2$ and $3$ and $L^{q}$ estimates for every $q<N/(N-4)$ when $N\geq 4$ just assuming that $h(s)$ is a positive convex function with $h(0)>0$ and $h^{\prime}(0)\geq 0$. For a related problem still in the Euclidean case see [5], where X. Cabré assuming $h(s)$ to be a $C^{1}$ non-decreasing super-linear non-linearity with $h(0)>0$, proved boundedness of the extremal solution for Problem (1.9) in dimension $N\leq 4$. In dimension $2$ the domain $\Omega$ can be general but, in contrast with Nedev’s result, in dimensions $3$ and $4$ the domain is assumed to be convex. After that, X. Cabré and M. Sanchón [6] completed the analysis in [5] when they proved that if $N\geq 5$ and $\Omega$ is a convex bounded domain of Euclidean space $\mathbb{R}^{N}$ then the extremal solution belongs to $L^{\frac{2N}{N-4}}$.

Recently, there has been growing interest on singular elliptic partial differential equations on Riemannian manifolds. The problem involving the Laplace-Beltrami operator

[TABLE]

where $\Omega$ is a bounded domain was studied recently by D. Castorina and M. Sanchón in [8] for the inhomogeneous context. They proved qualitative properties for semi-stable solutions and they established $L^{\infty}$, $L^{q}$ and $W^{1,q}$ estimates which do not depends on the non-linearity $h(s)$. Furthermore, the authors obtained regularity results for the extremal solution for exponential and power non-linearities. A similar setting has been considered by E. Berchio, A. Ferrero and G. Grillo [2] in order to study uniqueness and qualitative properties of radial entire solutions of the Lane–Emden–Fowler equation $-\Delta u=|u|^{m-1}u$ with $m>1$ on certain classes of Cartan–Hadamard manifolds where the so-called Joseph-Lundgren exponent is involved in the stability of solutions. The existence of a stable solution to the semi-linear equation $-\Delta_{g}u=f(u)$ on a complete, non-compact, boundaryless Riemannian manifold with non-negative Ricci curvature and $f\in C^{1}$ was studied by A. Farina, L. Mari and E. Valdinoci [19]. They classify both the solution and the manifold and also discuss the classification of monotone solutions with respect to the direction of some Killing vector field, in the spirit of a conjecture of De Giorgi. In [39], F. Morabito investigated the existence and uniqueness of positive radial solutions of the problem

[TABLE]

when $\lambda<0$, $\mathcal{A}$ is an annular domain in a Riemannian manifold of dimension $N$ endowed with the metric $\mathrm{d}s^{2}=\mathrm{d}r^{2}+S^{2}(r)\mathrm{d}\theta^{2}$ under suitable assumptions on the function $S^{2}(r)$. He also show that there exist positive non-radial solutions arising by bifurcation from the radial solution, where $\lambda$ and $p$ are the bifurcation parameters.

Many non-linear problems in physics and mechanics are formulated in equations that contain the $p-$Laplacian, for example on Non-Newtonian Fluids, Glaceology and Non-linear Elasticity (see [14]). For some problems of non-linear partial differential equations on Riemannian manifold we refer to [25, 30, 23]. Gelfand type problems involving the $p-$Laplacian in the homogeneous case of the form

[TABLE]

was studied by J. García-Azorero, I. Peral and J. Puel [20, 21] where $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$. They proved that for every $p>1$ and $h(s)=e^{s}$, the extremal solution $u^{*}$ is an energy solution for every dimension and that it is bounded in some range of dimensions. For a more general non-linearity, X. Cabré and M. Sanchón [3] proved that every semi-stable solution is bounded for a explicit exponent which is optimal for the boundedness of semi-stable solutions and, in particular, it is bigger than the critical Sobolev exponent $p^{*}-1$. For general $h(s)$ and $p>1$ the interested reader can see [4, 10, 41, 43] for more regularity results about the extremal solution. In [4], X. Cabré, A. Capella and M. Sanchón treated the delicate issue about regularity of extremal solutions $u^{*}$ of (1.10) at $\lambda=\lambda^{*}$ when $\Omega$ is the unit ball of $\mathbb{R}^{N}$. Among other results, they established pointwise, $L^{q}$ and $W^{1,q}$ estimates which are optimal and do not depend on the non-linearity $h(s)$.

Furthermore, D. Castorina and M. Sanchón [9] obtain a priori estimates for semi-stable solutions of the reaction-diffusion problem $-\Delta_{p}u=h(u)$ in $\Omega$ while the reaction term is driven by any positive $C^{1}$ non-linearity $h$ and, as a main tool, they develop Morrey-type and Sobolev-type inequalities that involve the functional

[TABLE]

where $v\in C^{\infty}_{0}(\overline{\Omega})$. In $\eqref{11}$, $H_{v}(x)$ denotes the mean curvature at $x$ of the hyper-surface $\{y\in\Omega\>|v(y)|=|v(x)|\}$ and $\nabla_{T,v}$ is the tangential gradient along a level set of $|v|$. In addition to being of independent interest, these geometric inequalities are used, together with judicious choice of test functions in the semi-stability condition, to obtain their a priori estimates for semi-stable solutions.

In this paper we investigate similar results in the inhomogeneous context of a Riemannian manifold. We use some ideas of [3], comparison principle for $-\Delta_{p}$ (because it is uniformly elliptic) and the positivity of the first eigenvalue (as well the corresponding eigenfunction) of $-\Delta_{p}$ on $\Omega$ (cf. [1, 36, 38]). We point out that the regularity results achieved in this paper represent a geometrical extension of the ones obtained for the Euclidean case in [4].

Before we state our main result on the regularity of semi-stable solutions for ($\mathcal{P}_{\lambda}$), let us introduce some basic definitions. We say that $u\in W_{0}^{1,p}(\mathcal{B}_{1})$ is a weak solution of ($\mathcal{P}_{\lambda}$) if $h(u)\in L^{1}(\mathcal{B}_{1})$ and

[TABLE]

for all $\phi\in C_{0}^{\infty}(\mathcal{B}_{1})$. Furthermore, by minimal solution we mean smaller than any other super-solution of the Problem ($\mathcal{P}_{\lambda}$) and regular solution means that a weak solution $u$ of ($\mathcal{P}_{\lambda}$) is $C^{1,\beta}(\mathcal{B}_{1})$.

Let us state the existence and basic properties of touchdown parameter.

Theorem 1.2**.**

There exist $\lambda^{*}\in(0,\infty)$ such that

1. (i)

For $0<\lambda<\lambda^{*}$, the problem ($\mathcal{P}_{\lambda}$) has a regular minimal solution $u_{\lambda}$. 2. (ii)

For $\lambda>\lambda^{*}$, ($\mathcal{P}_{\lambda}$) admits no weak solution. 3. (iii)

The map $\lambda\rightarrow u_{\lambda}$ is increasing.

As a consequence of Theorem 1.2, the increasing limit

[TABLE]

is well defined by the point-wise increasing property. If $u^{*}$ is a weak solution of $(\mathcal{P}_{\lambda^{*}})$, then $u^{*}$ is called the extremal solution. Since the extremal solutions can be obtained as the limit of classical minimal solutions, our next result is useful in order to prove that $u^{*}$ has the same regularity properties as the ones stated in Theorem 1.1. For this, we need to bound $u^{p-1}$ and $h(u)$ in $L^{1}(\mathcal{B}_{1})$ uniformly in $\lambda$. This is possible because we have the growth condition ($H_{1}$) on $h(s)$ and the radially decreasing property of the minimal solutions $u_{\lambda}$. Let us now state precisely our results for ($\mathcal{P}_{\lambda}$).

Theorem 1.3**.**

Suppose that $N<p+4p/(p-1)$ and let $h(s)$ be a positive and increasing $C^{1}-$function satisfying ($H_{1}$). Then $u^{*}$ is a semi-stable solution of $(P_{\lambda^{*}})$ and $u^{*}\in L^{\infty}(\mathcal{B}_{1})$.

1.2. Outline

In the next section we bring basic facts about the p-Laplace Beltrami operator which will be used through the paper. In Section 2 we prove existence of extremal parameter $\lambda^{*}$ and minimal solutions of ($\mathcal{P}_{\lambda}$) for $0<\lambda<\lambda^{*}$. In Section 3 we prove that key-estimate (1.8) by using a suitable choice of test functions under the semi-stability property. In Section 4 we use (1.8) to prove our main theorem about regularity for radially symmetric and decreasing semi-stable solutions of Problem ($\mathcal{S}$) and apply this results for the study of the regularity of extremal solutions of ($\mathcal{P}_{\lambda}$).

2. Proof of Theorem 1.2

Our first proposition establishes the analog of the classical results for ($\mathcal{P}_{\lambda}$) in the Euclidean case. Using some ideas coming from X. Cabré and M. Sanchón [3] and X. Luo, D. Ye and F. Zhou [35] we prove the existence of a critical parameter $\lambda^{*}$ which is related with the resolvability of ($\mathcal{P}_{\lambda}$).

Proof of Theorem 1.2.

For $(i)$, let $w\in W_{0}^{1,p}(\mathcal{B}_{1})$ a weak solution of $-\mathrm{div}(|\nabla_{g}w|^{p-2}\nabla_{g}w)=1$ in $\mathcal{B}_{1}$, that is

[TABLE]

We can see that $w\in C^{1,\alpha}(\overline{\mathcal{B}_{1}})$ by using $C^{1,\alpha}$-regularity results (see [33, 15, 42]). By the Maximum principle [1, Theorem 3.3] $w$ is non-negative in $\mathcal{B}_{1}$. It is easy verify that [math] is sub-solution of ($\mathcal{P}_{\lambda}$) and if $\lambda\leq\lambda_{0}:=1/h(\max_{\overline{\mathcal{B}_{1}}}w),$

[TABLE]

that is, $w$ is a super-solution of ($\mathcal{P}_{\lambda}$). Thus, for any $\lambda\leq\lambda_{0}$, Problem ($\mathcal{P}_{\lambda}$) has a weak solution $u\in W_{0}^{1,p}(\mathcal{B}_{1})$ given by Sub and Super-solution Method (see [32]) with $0\leq u\leq w$ in $\overline{\mathcal{B}_{1}}$. This implies that $u\in C^{1,\alpha}(\overline{\mathcal{B}_{1}})$. As any regular solution $u$ of ($\mathcal{P}_{\lambda}$) is also a super-solution for $(P_{\mu})$ if $\mu\in(0,\lambda)$, the set of $\lambda$ for which ($\mathcal{P}_{\lambda}$) admits a regular solution is an interval. For $(ii)$ we will show that for $\lambda$ sufficient large, there is no regular solution for ($\mathcal{P}_{\lambda}$), so $\lambda^{*}<+\infty.$ It is well known that for the non-linear eigenvalue problem

[TABLE]

there exists a smaller positive and simple eigenvalue $\lambda_{1}$ with a positive eigenfunction $v_{1}$ in $\mathcal{B}_{1}$. Now, suppose that ($\mathcal{P}_{\lambda}$) admits a regular solution $u$ for $\lambda>\lambda_{1}$. The regularity result in [33] give that $v_{1}\in C^{1,\alpha}(\overline{\mathcal{B}_{1}})$. By homogeneity, we can assume that $\|v_{1}\|_{\infty}<h(0)^{\frac{1}{p-1}}$. Note that

[TABLE]

By the comparison principle [1] we have that $v_{1}\leq u$. Let us to take $v_{2}$ a solution of

[TABLE]

where $\epsilon$ is a positive constant. For $\lambda>\frac{\lambda_{1}+\epsilon}{h(0)}\max_{\overline{\mathcal{B}_{1}}}u^{p-1}$ we obtain

[TABLE]

Using the comparison principle again we obtain $v_{1}\leq v_{2}\leq u.$ Now, let us define recursively $u_{n}$ as the unique solution of

[TABLE]

By comparison principle we obtain $v_{1}\leq...\leq v_{n-1}\leq v_{n}\leq u\in C^{1,\alpha}(\overline{\mathcal{B}_{1}}).$ This implies that $v_{n}\rightharpoonup u_{\lambda}$ in $W_{0}^{1,p}(\mathcal{B}_{1})$ and consequently $u_{\lambda}$ satisfies

[TABLE]

which is impossible since the first eigenvalue of p-Laplace Beltrami operator is isolated (see [44, 34]). Define the critical threshold $\lambda^{*}$ as the supremum of $\lambda>0$ for which ($\mathcal{P}_{\lambda}$) admits a regular solution. Thus we have that $\lambda^{*}<+\infty.$ Note that, by the construction above, $u_{\lambda}$ is independent of the choice of the super-solution. Since any regular solution of ($\mathcal{P}_{\lambda}$) is also a super-solution of ($\mathcal{P}_{\lambda}$), we can conclude that $u_{\lambda}$ is a regular minimal solution of ($\mathcal{P}_{\lambda}$). In order to check (iii), let $\lambda\leq\mu$. Thus, $u_{\mu}$ is a super-solution of ($\mathcal{P}_{\lambda}$), which implies that $u_{\lambda}\leq u_{\mu}$, that is, the map $\lambda\rightarrow u_{\lambda}$ is increasing. ∎

3. A priori estimates

In this section we prove the principal estimate (1.8), which as we already mention it is the crucial in our argument to obtain the regularity of the semi-stable solutions in Theorem 1.1. The main idea is to apply an appropriate test function in the stability inequality (see Proposition 3.1). The radial form of ($\mathcal{S}$) can be written as follows

[TABLE]

Let us discuss a few preliminary estimates which will be used in our argument. In the next Lemma we prove that the second variation of energy associated to ($\mathcal{S}$) is independent of the non-linearity $f(s)$.

Lemma 3.1**.**

Let $u\in W_{r}^{1,p}(\mathcal{B}_{1})$ be a semi-stable solution of ($\mathcal{S}$) satisfying $u_{r}(r)<0$ for all $r\in(0,1)$. Then, for all radially symmetric function $\eta\in C_{c}^{1}(\mathcal{B}_{1}\setminus\left\{\mathcal{O}\right\})$ it holds

[TABLE]

Proof.

We start considering $\eta\in C_{c}^{1}(\mathcal{B}_{1}\setminus\mathcal{O})$ be a radial function with compact support in $\mathcal{B}_{1}\setminus\mathcal{O}$ and choosing $\xi=u_{r}\eta$ as test function in (1.7) there holds

[TABLE]

On the other hand, multiplying (3.1) by $(\eta^{2}u_{r}\psi^{N-1})_{r}$, integrating and using integration by parts we are able to compute

[TABLE]

which together with $\partial_{r}(|u_{r}|^{p-2}u_{r})=(p-1)|u_{r}|^{p-2}u_{rr}$ implies

[TABLE]

Thus

[TABLE]

which yields

[TABLE]

Using (3.2) and (3.3) we have

[TABLE]

which is the desired conclusion. ∎

We obtain $L^{p}-$estimates for the radial derivative of semi-stable solutions of ($\mathcal{S}$) with the help of Lemma 3.1. For that we consider a suitable class of test functions to analyze the inhomogeneous context of a Riemannian manifold assuming that $p\leq N$ and $1\leq\alpha<1+\sqrt{(N-1)/(p-1)}$. To be more precise,

Proposition 3.1**.**

Let $u\in W_{r}^{1,p}(\mathcal{B}_{1})$ be a semi-stable solution in $\mathcal{B}_{1}\setminus\mathcal{O}$ of ($\mathcal{S}$) satisfying $u_{r}(r)<0$ for $r\in(0,1)$ and $\delta=\delta(\psi)\in(0,1/2)$ such that $\psi^{\prime}>0$ in $[0,\delta]$. Then

[TABLE]

for every $1\leq\alpha<1+\sqrt{(N-1)/(p-1)}$, where $C_{N,p,\alpha,\psi}$ is a constant depending only on $N,$ $p,$ $\alpha$ and $\psi$.

Proof.

Using the semi-stability condition of $u$ and applying Lemma 3.1 with $\psi\eta$ as test function we obtain

[TABLE]

Now, take $\alpha$ satisfying $1\leq\alpha<1+\sqrt{(N-1)/(p-1)}$, $\epsilon\in(0,1)$ sufficiently small and

[TABLE]

a Lipschitz function which vanishes on $\partial\mathcal{B}_{1}$. Choosing $\eta=\eta_{\epsilon}$ in the inequality (3.4) we have

[TABLE]

which can be written as

[TABLE]

Since $(N-p)\eta_{\epsilon}^{2}|u_{r}|^{p}(\psi^{\prime})^{2}\psi^{N-1}\,\mathrm{d}r\geq 0$ we obtain

[TABLE]

Throughout the proof, $\tilde{C}_{n,p,\alpha}$ (respectively $\tilde{C}_{n,p,\alpha,\psi}$) denote different positive constants depending only on $n$, $p$ and $\alpha$ (respectively on $n$, $p$, $\alpha$, $\psi$). Rewritten the above equation follows

[TABLE]

Observe that, by assumption, $\displaystyle\inf_{(0,\delta)}\psi^{\prime}$ and $\displaystyle\sup_{(0,\delta)}\psi^{\prime}$ are positive. Now, we can rearrange the terms in the integrals to obtain

[TABLE]

Taking $\epsilon\rightarrow 0$, follows that

[TABLE]

If we define

[TABLE]

using that $1\leq p\leq N$ and $\alpha$ satisfying $1\leq\alpha<1+\sqrt{(N-1)/(p-1)}$, we can check that $\lim_{t\rightarrow+\infty}\zeta(t)=0$ and $\lim_{t\rightarrow 0^{+}}\zeta(t)=-\infty$. Thus, by a compactness argument, $\zeta(t)$ is bounded from above. This implies that there exists $C_{N,p,\alpha,\psi}>0$ such that

[TABLE]

and (3.5) leads to

[TABLE]

On the other hand, since $u$ is radially decreasing follows that

[TABLE]

and using Mean value theorem for some $\tilde{\delta}\in(\delta,2\delta)$ it holds

[TABLE]

Thus, integrating (3.1) from $r\in(0,\delta)$ to $\tilde{\delta}$ and using (3.8) we obtain

[TABLE]

which together (3.7) implies

[TABLE]

Integrating from [math] to $\delta$ to obtain

[TABLE]

and going back to (3.6) we conclude that

[TABLE]

which completes the proof. ∎

4. Proof of Main Theorems

Proof of Theorem 1.1.

Let $\delta\in(0,1/2)$ as in Proposition 3.1. Since $u$ is radially symmetric and positive we can check that

[TABLE]

Using Hölder inequality we can estimate

[TABLE]

(i) In order to prove $L^{\infty}$ estimate, observe that, by monotonicity, for all $\delta\leq t<1$ we have

[TABLE]

Taking $t=0$ in (4.2), we can analyze the integral and check that

[TABLE]

when $(2\alpha-N+1)/(p-1)>-1$, that is, $\alpha>(N-p)/2$. Thus, for all $0<t<\delta$ we have

[TABLE]

whenever $\max\left\{(N-p)/2,1\right\}<\alpha<1+\sqrt{(N-1)/(p-1)}$. This occurs if, and only if, $N<p+4p/(p-1)$. Therefore, the desired $L^{\infty}$ estimate ($(a)$) holds true.

(ii) On the other hand, since $u$ is decreasing, using (4.1) we have

[TABLE]

Taking $t\in(0,\delta)$ and using (4.2) we have

[TABLE]

Therefore, if $q<Np/(N-p-2-2\sqrt{(N-1)/(p-1)})$, choosing suitable $\alpha$ such that

[TABLE]

Taking this last inequality combined with (4.3) we obtain the desired $L^{q}$ estimate.

Now, we are looking for $W^{1,q}$ estimate. We use similar idea as can be found in the proof of Theorem 1.2 in [8]. For this, observe that every function $u\in W_{r}^{1,p}(\mathcal{B}_{1})$ also belongs to the Sobolev space $W^{1,p}(\delta,1)$. Thus, we have

[TABLE]

Now, using equation (3.1) we have

[TABLE]

Now, let $\tilde{\delta}\in(\delta,2\delta)$ such that (3.8) holds. Integrating the last inequality,

[TABLE]

and using (4.1),

[TABLE]

Using Hölder inequality and observing that we can use Proposition 3.1 because we can take our $\delta$ sufficiently small, follows

[TABLE]

for all $\alpha\in[1,1+\sqrt{(N-1)/(p-1)}]$. Thus, for $s\in(0,\delta)$,

[TABLE]

Now, observe that

[TABLE]

since $q<Np/(N-2-2\sqrt{(N-1)/(p-1)})$. Thus

[TABLE]

Using equations (4.4) and (4.5) we finish the proof. ∎

Proof of Theorem 1.3.

Let $\lambda\in(0,\lambda^{*})$. There exists $\rho_{\lambda}\in(1/2,1)$ such that mean value property holds, that is,

[TABLE]

Since $u_{\lambda}$ is decreasing, (see proof of Theorem 1.2), we have

[TABLE]

Thus

[TABLE]

follows by monotonicity. By using $\phi(r)=\min\left\{1,(2-4r)^{+}\right\}$ as test function and (4.6) we obtain

[TABLE]

Using the assumption ($H_{1}$), given $\delta>0$ we have for any $\lambda\in(\lambda^{*}/2,\lambda^{*})$ and for all $t>0$,

[TABLE]

where $C_{\delta}$ does not depends on $\lambda$. With this

[TABLE]

Since $u_{\lambda}$ is decreasing follows that

[TABLE]

Now, take $\delta$ sufficiently small and combine (4.8) with (4.9) to obtain

[TABLE]

where $C$ is a constant independent of $\lambda$. Repeating the argument in (4.9) we are able to obtain an estimate uniform in $\lambda$ for $\|u_{\lambda}^{p-1}\|_{L^{1}(B_{1})}$. Using this in (4.7) we obtain a estimate for $\|h(u_{\lambda})\|_{L^{1}(B_{1/4})}$. Again by monotonicity we can apply the same argument used above to control $\|h(u_{\lambda})\|_{L^{1}(B_{1})}$ uniformly in $\lambda$. Thus

[TABLE]

where $C$ is a constant independent of $\lambda$. Observe that every radial function $u\in W^{1,p}(\mathcal{B}_{1})$ also belongs to the Sobolev space $W^{1,p}(\delta,1)$ in one dimension for a given $\delta\in(0,1).$ Using the Sobolev embedding in one dimension, $u$ becomes a continuous function of $r=\mathrm{dist}(x,\mathcal{O})\in[\delta,1]$ and

[TABLE]

In view of this estimate, we can assume that $u>0=u(1)\text{ in }\mathcal{B}_{1}$. Take $\alpha$ satisfying $1\leq\alpha<1+\sqrt{(N-1)/(p-1)}$ and using Proposition 3.1,

[TABLE]

Now, choose $r_{0}$ such that $2C_{N,p,\psi}\leq\psi^{-2\alpha}$ in $r\in(0,r_{0})$ to obtain

[TABLE]

which implies

[TABLE]

Since $u$ is decreasing we have that

[TABLE]

Thus,

[TABLE]

We can conclude that

[TABLE]

By (4.10) we deduce a bound for $\|u_{\lambda}\|_{W^{1,p}(B_{1})}.$ By using the compactness and since $u_{\lambda}\rightarrow u^{*}$ as $\lambda\rightarrow\lambda^{*}$ follows that $u^{*}\in W_{0}^{1,p}(B_{1})$. We can pass to the limit and conclude that $u^{*}$ is a weak solution of ($\mathcal{P}_{\lambda}$). It is clear that $u^{*}$ is radially symmetric and decreasing. By Fatou’s Lemma we obtain that $u^{*}$ is semi-stable. Finally, we can pass to the limit and the regularity statement follows as a consequence of Theorem 1.1. ∎

Corollary 4.1**.**

The extremal solution $u^{*}$ has the same regularity stated in Theorem 1.1.

Proof.

The proof follows straightforward by using above estimates and passing to the limit as $\lambda\rightarrow\lambda^{*}$. ∎