Semilinear elliptic equations with Hardy potential and gradient nonlinearity

TL;DR

This paper investigates positive solutions to a class of semilinear elliptic equations with Hardy potential and gradient nonlinearities, establishing existence, uniqueness, and singularity behavior depending on the nonlinearity exponent.

Contribution

It introduces a singular integral condition for existence and characterizes solution behavior across subcritical and supercritical regimes based on a critical exponent.

Findings

Existence and uniqueness of solutions with prescribed boundary data.

Identification of a critical exponent $q_$ for the nonlinearity.

Description of solutions with isolated boundary singularities.

Abstract

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta + \frac{\mu}{\delta^2} $, $\mu \in (0,\frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $\nu$. When $g(t)=t^q$ with $q \in (1,2)$, we show that equation (E) admits a critical exponent $q_\mu$ (depending only on $N$ and $\mu$). In the subcritical case, namely $1<q<q_\mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $\partial \Omega$. In the supercritical case, i.e. $q_\mu\leq q<2$, we demonstrate a removability result in terms of Bessel capacities.

Equations

$$-\Delta u-\frac{\mu }{{\delta }^2}u+g(|\nabla u|)=0$$

$$L_{\mu }=L_{\mu }^{\Omega }:=\Delta +\frac{\mu }{{\delta }^2}.$$

$$-L_{\mu }u=0\text{in }\Omega .$$

$$-\Delta u+g(|\nabla u|)=0\text{in }\Omega .$$

$$-L_{\mu }u+u^p=0\text{in }\Omega$$

$$\alpha :=\frac{1}{2}+\sqrt{\frac{1}{4}-\mu }$$

$$C_H(\Omega ):=\underset{H_0^1(\Omega )}{\inf }\frac{{\int }_{\Omega }|\nabla u{|}^{2}dx}{{\int }_{\Omega }(u/\delta {)}^{2}dx}.$$

$$\{\begin{array}{rlrl}-L_{\mu }u& =0& & \text{in }D,\\ u& =h& & \text{on }\partial D,\end{array}$$

$$u(x_0)={\int }_{\partial D}h(y)d{\omega }_D^{x_0}(y).$$

$$\underset{n\to \infty }{\lim }{\int }_{\partial {\Omega }_n}\varphi ud{\omega }_{{\Omega }_n}^{x_0}={\int }_{\partial \Omega }\varphi d\nu \forall \varphi \in C(\overline{\Omega }).$$

$${\lambda }_{\mu }:=\underset{\phi \in H_0^1(\Omega )\setminus \{0\}}{\inf }\frac{{\int }_{\Omega }(|\nabla \phi {|}^2-\frac{\mu }{{\delta }^2}{\phi }^2)dx}{{\int }_{\Omega }{\phi }^{2}dx}>0.$$

$${\mathbb{G}}_{\mu }^{\Omega }[\tau ](x):={\int }_{\Omega }{G}_{\mu }^{\Omega }(x,y)d\tau (y)\forall \tau \in \mathfrak{M}(\Omega ,{\delta }^{\alpha }),$$

$${\mathbb{K}}_{\mu }^{\Omega }[\nu ](x):={\int }_{\partial \Omega }{K}_{\mu }^{\Omega }(x,z)d\nu (z)\forall \nu \in \mathfrak{M}(\partial \Omega ).$$

$$\{\begin{array}{rl}-L_{\mu }u& =\tau \text{in }\Omega ,\\ \mathrm{tr}(u)& =\nu .\end{array}$$

$$-{\int }_{\Omega }u{L}_{\mu }\zeta dx={\int }_{\Omega }\zeta d\tau -{\int }_{\Omega }{\mathbb{K}}_{\mu }^{\Omega }[\nu ]L_{\mu }\zeta dx\forall \zeta \in {\mathbf{X}}_{\mu }(\Omega ),$$

$${\mathbf{X}}_{\mu }(\Omega ):=\{\zeta \in H_{loc}^1(\Omega ):{\delta }^{-\alpha }\zeta \in H^1(\Omega ,{\delta }^{2\alpha }),{\delta }^{-\alpha }{L}_{\mu }\zeta \in L^{\infty }(\Omega )\}.$$

$$\{\begin{array}{rl}-L_{\mu }u+g(|\nabla u|)& =0\text{in }\Omega ,\\ \mathrm{tr}(u)& =\nu .\end{array}$$

$$-{\int }_{\Omega }u{L}_{\mu }\zeta dx+{\int }_{\Omega }g(|\nabla u|)\zeta dx=-{\int }_{\Omega }{\mathbb{K}}_{\mu }[\nu ]L_{\mu }\zeta dx,\forall \zeta \in {\mathbf{X}}_{\mu }(\Omega ).$$

$$\Vert \nabla {\mathbb{G}}_{\mu }^{\Omega }[|\tau |]{\Vert }_{L_w^{\frac{N+\gamma }{N+\theta -1}}(\Omega ,{\delta }^{\gamma })}\le c\Vert \tau {\Vert }_{\mathfrak{M}(\Omega ,{\delta }^{\theta })}\tau \in \mathfrak{M}(\Omega ,{\delta }^{\theta }),$$

$$\nabla {\mathbb{G}}_{\mu }^{\Omega }[\tau ](x):={\int }_{\Omega }{\nabla }_x{G}_{\mu }^{\Omega }(x,y)d\tau (y).$$

$$\Vert \nabla {\mathbb{K}}_{\mu }^{\Omega }[|\nu |]{\Vert }_{L_w^{\frac{N+\gamma }{N+\alpha -1}}(\Omega ,{\delta }^{\gamma })}\le c\Vert \nu {\Vert }_{\mathfrak{M}(\partial \Omega )}\nu \in \mathfrak{M}(\partial \Omega ),$$

$$\nabla {\mathbb{K}}_{\mu }^{\Omega }[\nu ](x):={\int }_{\partial \Omega }{\nabla }_x{K}_{\mu }^{\Omega }(x,z)d\nu (z).$$

$$q_{\mu }:=\frac{N+\alpha }{N+\alpha -1}.$$

$${\int }_1^{\infty }g(s)s^{-1-q_{\mu }}ds<\infty .$$

$$u+{\mathbb{G}}_{\mu }^{\Omega }[g(|\nabla u|)]={\mathbb{K}}_{\mu }^{\Omega }[\nu ].$$

$$\underset{x\to y}{\lim }\frac{u(x)}{{\mathbb{K}}_{\mu }^{\Omega }[\nu ](x)}=1\text{non tangentially},\text{for }\nu \text{-a.e. }y\in \partial \Omega .$$

$$|g(t)-g(t^{\prime })|\le C(|t{|}^{q-1}+|t^{\prime }{|}^{q-1})|t-t^{\prime }|\forall t,t^{\prime }\ge 0,$$

$$g(\epsilon t)\le \epsilon g(t)\forall t>0\text{and }\forall \epsilon \in (0,1].$$

$$W(x):=\{\begin{array}{rlrl}& \delta (x{)}^{1-\alpha }& & \text{if }\mu <\frac{1}{4},\\ & \delta (x{)}^{\frac{1}{2}}|\ln \delta (x)|& & \text{if }\mu =\frac{1}{4}.\end{array}$$

$$\underset{x\in \Omega ,x\to \xi }{\lim }\frac{u(x)}{W(x)}=0\forall \xi \in \partial \Omega \setminus \{0\},$$

Full text

Key words: Hardy potential, Martin kernel, boundary trace, critical exponent, gradient term, isolated singularities, removable singularities.

2000 Mathematics Subject Classification: 35J60, 35J75, 35J10, 35J66.

1. 1 Introduction

2. 2 The linear problem

3. 2.1 Eigenvalue and eigenfunction

4. 2.2 Green kernel and Martin kernel

5. 2.3 Boundary trace

6. 3 Estimates of the gradient of Green kernel and Martin kernels

7. 4 Subcritical absorption

8. 4.1 Existence

9. 4.2 Regularity

10. 4.3 Comparison principle

11. 4.4 Some estimates

12. 5 Isolated boundary singularities

13. 5.1 A priori estimates

14. 5.2 Weak singularities

15. 5.3 Strong singularities

16. 6 Supercritical Case

17. A Barrier

1. Introduction

In this paper, we are concerned with the boundary value problem with measure data for the following equation

[TABLE]

in a $C^{2}$ bounded domain $\Omega$ in $\mathbb{R}^{N}$ ($N\geq 3$), where $\mu\in(0,\frac{1}{4}]$, $\delta(x)=\delta_{\Omega}(x):=\mathrm{dist}\,(x,\partial\Omega)$ and $g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a nondecreasing, continuous function with $g(0)=0$. Put

[TABLE]

We say that $u$ is $L_{\mu}$ harmonic (resp. subharmonic, superharmonic) if $u$ is a distributional solution (resp. subsolution, supersolution) of

[TABLE]

When $\mu=0$, $L_{\mu}\equiv\Delta$ and equation (1.1) becomes

[TABLE]

The boundary value problem with measure data for (1.4) was first studied by Nguyen and Véron in [21] where the existence of a positive solution with a prescribed measure boundary datum was obtained under a so-called subcriticality integral condition on $g$. In case that $g$ is a purely power function, i.e. $g(t)=t^{q}$, it was shown that equation (1.4) admits the critical exponent $q^{*}=\frac{N+1}{N}$ and the structure of the class of solutions with a boundary isolated singularity was fully depicted in the subcritical case $q\in(1,q^{*})$. These results were then extended to a much more intricate equations where the nonlinearity depends on both solutions and their gradient (see [17, 19]). An attempt to extend the mentioned results was carried out in [5] (see also the references therein) to the case of $p$-laplacian where the analysis is complicated and requires heavy computations due to the nonlinearity of the operator.

The case $\mu<0$ (and the case of more general potentials) was investigated by Ancona in [2].

When $\mu>0$, the semilinear equation with absorption power term

[TABLE]

has been recently studied in different directions. If $\mu\leq\frac{1}{4}$, Bandle et al. [3] gave a classification of large solutions, i.e. positive solutions of (1.5) which blow up on $\partial\Omega$, according to their boundary behavior, in connection to the exponent

[TABLE]

and the Hardy constant

[TABLE]

Afterwards, Marcus and Nguyen dealt with moderate solutions of (1.3) and (1.5) by introducing a concept of normalized boundary trace (see [16, Definition 1.2]). An advantage of this notion is that it allows to overcome the difficulty originating from the presence of the Hardy potential $\frac{\mu}{\delta^{2}}$ and hence enables to characterize $L_{\mu}$ harmonic functions in terms of their boundary behavior.

This notion of boundary trace was then extended by Marcus and Moroz in [15] to the case $\mu<\frac{1}{4}$ due to the fact that in this case there exists a local $L_{\mu}$ superharmonic function in a neighborhood of $\partial\Omega$ and then it was used to study the nonlinear problem (1.5). See also [4] and references therein.

In parallel, Gkikas and Véron [11] treated the boundary value problem for (1.3) and (1.5) in a slightly different setting, giving a complete description of singular solutions of (1.5) by introducing a notion of boundary trace in a dynamic way which is recalled below.

Let $D\Subset\Omega$ and $x_{0}\in D.$ If $h\in C(\partial D)$ then the following problem

[TABLE]

admits a unique solution which allows to define the $L_{\mu}$ harmonic measure $\omega_{D}^{x_{0}}$ on $\partial D$ by

[TABLE]

A sequence of domains $\{\Omega_{n}\}$ is called a smooth exhaustion of $\Omega$ if $\partial\Omega_{n}\in C^{2}$, $\overline{\Omega_{n}}\subset\Omega_{n+1}$, $\cup_{n}\Omega_{n}=\Omega$ and $\mathcal{H}^{N-1}(\partial\Omega_{n})\to\mathcal{H}^{N-1}(\partial\Omega)$. For each $n$, let $\omega_{\Omega_{n}}^{x_{0}}$ be the $L_{\mu}^{\Omega_{n}}$ harmonic measure on $\partial\Omega_{n}.$

Definition 1.1**.**

Let $\mu\in(0,\frac{1}{4}]$. A function $u$ possesses a boundary trace if there exists a measure $\nu\in\mathfrak{M}(\partial\Omega)$ such that for any smooth exhaustion $\{\Omega_{n}\}$ of $\Omega$, there holds

[TABLE]

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

It was showed in [10] that when $\mu\in(0,C_{H}(\Omega))$ the notion of boundary trace in Definition 1.1 coincides with the notion of normalized boundary trace introduced in [16]. Since we would like to deal with the whole range $(0,\frac{1}{4}]$, we will employ Definition 1.1. However, we need an additional condition as follows:

[TABLE]

Throughout the present paper, we assume that $\mu\in(0,\frac{1}{4}]$ and (1.11) holds. Under this condition, the Representation Theorem (see [16, 11]) is valid, which allows to develop a theory for linear equations (see [10]).

Related results for semilinear elliptic equations with Hardy potential and source term can be found in [6, 20].

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 $\mathfrak{M}^{+}(\Omega,\phi)$ the positive cone of $\mathfrak{M}(\Omega,\phi)$. Denote by $\mathfrak{M}(\partial\Omega)$ the space of bounded Radon measures on $\partial\Omega$ and by $\mathfrak{M}^{+}(\partial\Omega)$ the positive cone of $\mathfrak{M}(\partial\Omega)$. Denote $L^{p}_{w}(\Omega,\tau)$, $1\leq p<\infty$, $\tau\in\mathfrak{M}^{+}(\Omega)$, the weak $L^{p}$ space (or Marcinkiewicz space) with weight $d\tau$; see [18] for more details.

Let $G_{\mu}^{\Omega}$ and $K_{\mu}^{\Omega}$ be respectively the Green kernel and Martin kernel of $-L_{\mu}$ in $\Omega$ (see Section 2.2 for more details). The Green operator and Martin operator are defined as follows:

[TABLE]

[TABLE]

These operators play an important role in the study of the boundary value problem for the linear equation

[TABLE]

Definition 1.2**.**

Let $(\tau,\nu)\in\mathfrak{M}(\Omega,\delta^{\alpha})\times\mathfrak{M}(\partial\Omega)$. We say that $u$ is a weak solution of (1.14) if $u\in L^{1}(\Omega,\delta^{\alpha})$ and

[TABLE]

where the space of test function ${\bf X}_{\mu}(\Omega)$ is defined by

[TABLE]

The existence and uniqueness result for (1.14), which was established in [10], is an important ingredient in the investigation of the boundary value problem for (1.1)

[TABLE]

We reveal that the presence of the Hardy potential $\frac{\mu}{\delta^{2}}$ in the linear part of the equation means that the problem cannot be handled via classical elliptic PDEs methods as the singularity of the potential at the boundary is too strong. Moreover, the presence of the gradient term, which leads to the lack of monotonicity property of the nonlinearity, makes the analysis much intricate. The interplay between the Hardy potential $\frac{\mu}{\delta^{2}}$ and the gradient term yields substantial new difficulties and requires new methods.

Before stating main results of the paper, let us give the definition of weak solutions of (1.17).

Definition 1.3**.**

Let $\nu\in\mathfrak{M}(\partial\Omega)$. A function $u$ is called a weak solution of (1.17) if $u\in L^{1}(\Omega,\delta^{\alpha})$, $g(|\nabla u|)\in L^{1}(\Omega,\delta^{\alpha})$ and

[TABLE]

Crucial ingredients in the study of (1.17) are estimates of $\nabla\mathbb{G}_{\mu}^{\Omega}$ and $\nabla\mathbb{K}_{\mu}^{\Omega}$, which are established in the next proposition.

Proposition A. (i) Let $\theta\in[0,\alpha]$ and $\gamma\in[0,\frac{\theta N}{N-1})$. Then there exists a positive constant $c=c(N,\mu,\theta,\gamma,\Omega)$ such that

[TABLE]

where

[TABLE]

(ii) Let $\gamma\geq 0$. Then there exists a positive constant $c=c(N,\mu,\gamma,\Omega)$ such that

[TABLE]

where

[TABLE]

A main feature of problem (1.17) is that, in general, it is not solvable for any measure $\nu\in\mathfrak{M}(\partial\Omega)$. This occurs only when $q$ is smaller than the critical exponent given by

[TABLE]

Theorem B. (Existence)  Assume that $g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is continuous, nondecreasing and satisfies

[TABLE]

Then for any $\nu\in\mathfrak{M}^{+}(\partial\Omega)$ problem (1.17) admits a nonnegative weak solution $u=u_{\nu}$. Moreover,

[TABLE]

Let us briefly discuss the idea of the proof. Because of the presence of the Hardy potential, we first construct a solution of (1.1) in a subdomain $D\subset\subset\Omega$ due to a combination of the idea in [13] and the Schauder fixed point theorem. This result is used to obtain the existence of an approximate solution of the equation with truncated nonlinearity in the whole domain $\Omega$. Finally, we employ Proposition A and Vitali convergence theorem in the limit process to derive the existence of a solution of (1.17).

A combination of (1.23) and Schrödinger theory (see [11, 16] and references therein) asserts that any weak solution of (1.17) behaves like $\mathbb{K}_{\mu}[\nu]$ on $\partial\Omega$.

Proposition C. (Boundary behavior) Let $\nu\in\mathfrak{M}^{+}(\partial\Omega)$. If $u$ is a nonnegative weak solution of (1.17) then

[TABLE]

Following is the monotonicity result which clearly implies the uniqueness the solution of (1.17).

Theorem D. (Monotonicity) Assume that $g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is continuous, nondecreasing and satisfies

[TABLE]

for some $q\in(1,q_{\mu})$ and

[TABLE]

*Assume $\nu_{i}\in\mathfrak{M}^{+}(\partial\Omega)$, $\nu_{1}\leq\nu_{2}$ and $u_{i}$ be a nonnegative solution of (1.17) with $\nu=\nu_{i}$, $i=1,2$. If $\nu_{1}\leq\nu_{2}$ then $u_{1}\leq u_{2}$ in $\Omega$. *

Note that the classical method can not be applied to our setting because of the lack of monotonicity stemming from the presence of the gradient term and the fact that a constant is not a solution of (1.17). To overcome the difficulties, we develop a new method which is based on an estimates on the gradient of subsolutions, Kato’s inequality and a comparison principle in a subdomain of $\Omega$.

When $g(t)=t^{q}$ with $q\in(1,q_{\mu})$ then $g$ satisfies (g1)–(g3). In this case, the class of solutions with isolated singularity has an interesting structure which is exploited below.

Put

[TABLE]

Assume that $0\in\partial\Omega$. The following proposition provides universal pointwise estimates on solutions with isolated singularity at [math], as well as their gradient. The proof is obtained thanks to the barrier constructed in the Appendix and the scaling argument in [18].

Proposition E. (A priori estimates) Assume $0\in\partial\Omega$ and let $u$ be a positive solution of (1.1) in $\Omega,$ with $g(t)=t^{q},$ such that

[TABLE]

locally uniformly in $\partial\Omega\setminus\{0\}$. Then there exists a constant $C=C(N,\mu,q,\Omega)$ such that,

[TABLE]

[TABLE]

In case that the boundary trace is a Dirac measure concentrated at [math], a shaper estimates can be achieved, which is the content of the following theorem.

Theorem F. (Weak singularity) Assume $1<q<q_{\mu}$. Let $0\in\partial\Omega$ and $k>0$. Let $u_{0,k}^{\Omega}$ be the solution of

[TABLE]

where $\delta_{y}$ is the Dirac measure concentrated at [math]. Then

[TABLE]

*Furthermore the mapping $k\mapsto u_{0,k}^{\Omega}$ is increasing. *

From Theorem F, it is natural to analyze the behavior of $\lim_{k\to\infty}u_{0,k}^{\Omega}$. This task consists of some intermediate steps. The first one is to consider a separable solution of (1.1) in the case $\Omega=\mathbb{R}_{+}^{N}$ and then to translate equation (1.1) to an equation on the upper hemisphere

[TABLE]

The second one is to investigate the existence and uniqueness of the corresponding problem on $S_{+}^{N-1}$; at this step the exact behavior of $\lim_{k\to\infty}u_{0,k}^{\mathbb{R}_{+}^{N}}$ can be derived. In the last step, the scaling argument is employed to obtain the behavior of $\lim_{k\to\infty}u_{0,k}^{\Omega}$. These steps are described in more details below.

We denote by $x=(r,\sigma)\in\mathbb{R}_{+}\times S^{N-1}$, with $r=|x|$ and $\sigma=r^{-1}x$, the spherical coordinates in $\mathbb{R}^{N}$ and we recall the following representation

[TABLE]

where $\nabla^{\prime}$ denotes the covariant derivative on $S^{N-1}$ identified with the tangential derivative and $\Delta^{\prime}$ is the Laplace-Beltrami operator on $S^{N-1}$.

We look for a particular solution of

[TABLE]

under the separable form

[TABLE]

It follows from a straightforward computation that $\omega$ satisfies

[TABLE]

where

[TABLE]

where ${\bf e}_{N}$ is the unit vector pointing toward the North pole.

Let $\kappa_{\mu}=\alpha(N+\alpha-2)$ be the first eigenvalue of $-{\mathcal{L}}_{\mu}$ in $S_{+}^{N-1}$ and $\phi_{\mu}$ be the corresponding eigenfunction $\phi_{\mu}(\sigma)=({\bf e}_{N}\cdot\sigma)^{\alpha}$ for $\sigma\in S_{+}^{N-1}$. Denote

[TABLE]

It is asserted below that $q_{\mu}$ is a critical exponent for the existence of a positive solution of (1.32).

Theorem G. (i) If $q\geq q_{\mu}$ then there exists no nontrivial solution of (1.32).

(ii) If $1<q<q_{\mu}$ then problem (1.32) admits a unique positive solution $\omega\in{\bf Y}_{\mu}(S^{N-1}_{+})$. Moreover,

[TABLE]

*where $C=C(N,q,\mu)$. *

Denote $u_{0,\infty}^{\Omega}:=\lim_{k\to\infty}u_{0,k}^{\Omega}$. A combination of Proposition E, Theorem F and Theorem G ensures that $u_{0,\infty}^{\Omega}$ is a solution of (1.1). Moreover, $u_{0,\infty}^{\Omega}$ possesses richer properties as stated in the following theorem.

Theorem H. (Strong singularity) Assume $0\in\partial\Omega$ and $1<q<q_{\mu}$. Let $u_{0,\infty}^{\Omega}$ be defined as above. Then $u_{0,\infty}^{\Omega}$ is a solution of

[TABLE]

There exists a constant $c=c(N,\mu,q,\Omega)>0$ such that

[TABLE]

[TABLE]

Moreover

[TABLE]

*locally uniformly on $S^{N-1}_{+}$, where $\omega$ is the unique solution of (1.32). *

We next consider the supercritical case, i.e. $q\geq q_{\mu}$. For any Borel set $E\subset\mathbb{R}^{N-1}$, we denote by $C_{\alpha,p}^{\mathbb{R}^{N-1}}(E)$ the Bessel capacity of $E$ associated to the Bessel space $L_{\alpha,p}(\mathbb{R}^{N})$ (see Section 6 for more details).

Definition 1.4**.**

Let $\nu\in\mathfrak{M}^{+}(\partial\Omega).$ We will say that $\nu$ is absolutely continuous with respect to the Bessel capacity $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{q}-\alpha,q^{\prime}}$ where $q^{\prime}=\frac{q}{q-1}$, if

[TABLE]

Theorem I. (Absolute continuity) Assume $q_{\mu}\leq q<2$ and $\nu\in\mathfrak{M}^{+}(\partial\Omega)$ such that the problem

[TABLE]

has a solution. Then

(i) If $q\neq\alpha+1$ then $\nu$ is absolutely continuous with respect to $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{q}-\alpha,q^{\prime}}.$

*(ii) If $q=\alpha+1$ then for any $\varepsilon\in(0,\min\{\alpha+1,\frac{(N-1)\alpha}{\alpha+1}-(1-\alpha)\}),$ $\nu$ is absolutely continuous with respect to $C^{\mathbb{R}^{N-1}}_{\varepsilon+1-\alpha,\frac{\alpha+1}{\alpha}}.$ *

Theorem J. (Removability) Assume $q_{\mu}\leq q<2.$ Let $K\subset\partial\Omega$ be compact such that

(i) $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{q}-\alpha,q^{\prime}}(K)=0$ if $q\neq\alpha+1$

or

(ii) $C^{\mathbb{R}^{N-1}}_{\varepsilon+1-\alpha,q^{\prime}}(K)=0,$ for some $\varepsilon\in(0,\min\{\alpha+1,\frac{(N-1)\alpha}{\alpha+1}-(1-\alpha)\}),$ if $q=\alpha+1.$

Then any nonnegative solution $u\in C^{2}(\Omega)\cap C(\overline{\Omega}\setminus K)$ of

[TABLE]

such that

[TABLE]

*is identically zero. *

The paper is organized as follows. In Section 2, we recall main properties of the boundary trace and some facts about linear equations. In Section 3, we establish estimates of the gradient of solutions in weak $L^{p}$ spaces (see Proposition A). Section 4 is devoted to the proof of Theorem B, Proposition C and Theorem D. Moreover, in this section, we also provide some estimates of solutions of (1.1). In Section 5, we demonstrate Proposition E and Theorems F, G and H. In Section 6 we deal with the supercritical case and provide the proof of Theorems I and J. Finally, in Appendix we construct a barrier for solutions of (1.1) which serves to obtain Proposition E.

Notation. In what follows the notation $f\approx g$ 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. Of course, in the later case, the constant depends on the subset.

For $\beta>0$, put

[TABLE]

Acknowledgements. The authors are grateful to Professor L. Véron for his useful comments.

2. The linear problem

2.1. Eigenvalue and eigenfunction

Throughout the paper we assume that $\mu\in(0,\frac{1}{4}]$ and (1.11) holds.

We recall important facts of the eigenvalue $\lambda_{\mu}$ of $-L_{\mu}$ and the associated eigenfunction $\varphi_{\mu}$ which can be found in [9].

If $0<\mu<\frac{1}{4}$ then the minimizer $\varphi_{\mu}\in H_{0}^{1}(\Omega)$ of (1.11) exists and satisfies

[TABLE]

where $\alpha$ is defined by (1.6).

If $\mu=\frac{1}{4}$, there is no minimizer of (1.11) in $H_{0}^{1}(\Omega)$, but there exists a nonnegative function $\varphi_{\frac{1}{4}}\in H_{loc}^{1}(\Omega)$ such that

[TABLE]

and satisfies

[TABLE]

in the sense of distributions. In addition, $\delta^{-\frac{1}{2}}\varphi_{\frac{1}{4}}\in H^{1}_{0}(\Omega,\delta)$.

2.2. Green kernel and Martin kernel

Let $G_{\mu}^{\Omega}$ and $K_{\mu}^{\Omega}$ be respectively the Green kernel and Martin kernel of $-L_{\mu}$ in $\Omega$ (see [16, 11]) for more details). We recall that

[TABLE]

[TABLE]

Denote $L^{p}_{w}(\Omega,\tau)$, $1\leq p<\infty$, $\tau\in\mathfrak{M}^{+}(\Omega)$, the weak $L^{p}$ space (or Marcinkiewicz space) with weight $\tau$; see [18] for more details. Notice that, for every $s>-1$,

[TABLE]

Moreover for any $u\in L_{w}^{p}(\Omega,\delta^{s})$ ($s>-1$),

[TABLE]

Let $\mathbb{G}_{\mu}^{\Omega}$ and $\mathbb{K}_{\mu}^{\Omega}$ be the Green operator and Martin operator of $-L_{\mu}$ in $\Omega$ which are given in (1.12), (1.13).

We recall estimate of Green kernel and Martin kernel in weak $L^{p}$ spaces (see [10]).

Proposition 2.1**.**

(i)* Let $\gamma\in(-\frac{\alpha N}{N+2\alpha-2},\frac{\alpha N}{N-2})$. There exists a constant $c=c(N,\mu,\gamma,\Omega)$ such that*

[TABLE]

(ii)* Let $\gamma>-1$. Then there exists a constant $c=c(N,\mu,\gamma,\Omega)$ such that*

[TABLE]

2.3. Boundary trace

In this subsection we recall main properties of the boundary trace in connection with of $L_{\mu}$ harmonic functions. It is worth emphasizing that the below results are valid for $\mu\in(0,\frac{1}{4}]$ (under the condition that the first eigenvalue $\lambda_{\mu}$ of $-L_{\mu}$ is positive).

Proposition 2.2**.**

([10])* (i) For any $\tau\in\mathfrak{M}(\Omega,\delta^{\alpha}),\,\mathrm{tr\,}(\mathbb{G}_{\mu}^{\Omega}[\tau])=0$ and for any $\nu\in\mathfrak{M}(\partial\Omega)$, $\mathrm{tr\,}(\mathbb{K}_{\mu}^{\Omega}[\nu])=\nu$.*

(ii) Let $w$ be a nonnegative $L_{\mu}$ subharmonic function in $\Omega$. Then $w$ is dominated by an $L_{\mu}$ superharmonic function if and only if $w$ has a boundary trace $\nu\in\mathfrak{M}(\partial\Omega)$. Moreover, if $w$ has a boundary trace then $L_{\mu}w\in\mathfrak{M}^{+}(\Omega,\delta^{\alpha})$. In addition, if $\mathrm{tr\,}(w)=0$ then $w=0$.

(iii) Let $u$ be a nonnegative $L_{\mu}$ superharmonic function. Then there exist $\nu\in\mathfrak{M}^{+}(\partial\Omega)$ and $\tau\in\mathfrak{M}^{+}(\Omega,\delta^{\alpha})$ such that

[TABLE]

(iv) Let $(\tau,\nu)\in\mathfrak{M}(\Omega,\delta^{\alpha})\times\mathfrak{M}(\partial\Omega)$. Then there exists a unique weak solution $u$ of (1.14). The solution is given by (2.9). Moreover, there exists $c=c(N,\mu,\Omega)$ such that

[TABLE]

In addition, for any $\zeta\in\mathbf{X}_{\mu}(\Omega)$, $\zeta\geq 0$,

[TABLE]

[TABLE]

3. Estimates of the gradient of Green kernel and Martin kernels

We begin this section by recalling well-known geometric properties of a $C^{2}$ bounded domain $\Omega$.

Proposition 3.1**.**

There exists a positive constant $\beta_{0}$ such that $\delta\in C^{2}(\overline{\Omega}_{4\beta_{0}})$. Moreover, for any $x\in\Omega_{4\beta_{0}}$ there exists a unique $\xi_{x}\in\partial\Omega$ such that

a) $\delta(x)=|x-\xi_{x}|$ and ${\bf n}_{\xi_{x}}=-\nabla\delta(x)=-\frac{x-\xi_{x}}{|x-\xi_{x}|}.$

b) $x(s):=x+s\nabla\delta(x)\in\Omega_{\beta_{0}}$ and $\delta(x(s))=|x(s)-\xi_{x}|=\delta(x)+s,$ for any $0<s<4\beta_{0}-\delta(x).$

Lemma 3.2**.**

Let $D$ be a $C^{2}$ bounded domain in $\mathbb{R}^{N}$. If $u$ is a nonnegative $L_{\mu}$ harmonic function in $D$ then

[TABLE]

Proof.

Take an arbitrary point $x_{*}\in D$ and put

[TABLE]

Note that if $x\in B_{d}(x_{*})$ then $y=\frac{1}{d}x\in B_{1}(y_{*})$ and $1\leq\mathrm{dist}\,(y,\partial(\frac{1}{d}D))\leq 3$. In $B_{1}(y_{*})$,

[TABLE]

By local estimate for elliptic equations [12, Theorem 8.32] and the Harnack inequality [12, Theorem 8.20], there exist positive constants $c_{i}=c_{i}(N,\mu)$, $i=1,2$ such that

[TABLE]

In particular,

[TABLE]

which implies (3.1). ∎

Let us recall a result from [7] which will be useful in the sequel.

Proposition 3.3**.**

([7, Lemma 2.4])* Let $\omega$ be a nonnegative bounded Radon measure in $D=\Omega$ or $\partial\Omega$ and $\eta\in C(\Omega)$ be a positive weight function. Let $H$ be a continuous nonnegative function on $\{(x,y)\in\Omega\times D:\;x\neq y\}.$ For any $\lambda>0$ we set*

[TABLE]

Suppose that there exist $C>0$ and $k>1$ such that $m_{\lambda}(y)\leq C\lambda^{-k}$ for every $\lambda>0$. Then the operator

[TABLE]

belongs to $L^{k}_{w}(\Omega,\eta)$ and

[TABLE]

Lemma 3.4**.**

Let $\theta\in[0,\alpha]$ and $\gamma\in[0,\frac{\theta N}{N-1})$. Then there exists a positive constant $c=c(N,\mu,\theta,\gamma,\Omega)$ such that (1.19) holds.

Proof.

By Lemma 3.2, there exists $C=C(\mu,N,\Omega)>0$ such that

[TABLE]

Set

[TABLE]

Then it is easy to see that

[TABLE]

which implies

[TABLE]

Since $\Omega$ is $C^{2}$ there exists $\beta_{0}>0$ such that for any $x\in\Omega_{3\beta_{0}}$ there exists a unique $\xi\in\partial\Omega$ satisfies $|x-\xi|=\delta(x).$ Furthermore there exists a $C^{2}$ function $\Gamma:\mathbb{R}^{N-1}\rightarrow\mathbb{R}$ such that (upon relabeling and reorienting the coordinate axes if necessary) we have

[TABLE]

Step 1. We will show that there exists $C=C(N,\mu,\theta,\gamma,\Omega)$ such that

[TABLE]

To prove that, we note, for any $x,\;y\in\Omega$, $x\neq y$, that

[TABLE]

By (2.3) and the above inequality we obtain

[TABLE]

where $C=C(N,\mu,\Omega)$, which implies $A_{\lambda,1}(y)\subset\widetilde{A}_{\lambda,1}(y)$ for every $y\in\Omega$ where

[TABLE]

Set

[TABLE]

Without loss of the generality we assume that $0\in\partial\Omega,$ $\delta(y)=|y|$ and there exists a $C^{2}$ function $\Gamma:\mathbb{R}^{N-1}\rightarrow\mathbb{R}$ such that $\Gamma(0)=0,\;\nabla\Gamma(0)=0$ and

[TABLE]

Let $\lambda>C$ where $C$ is the constant in (3.6).

Set $Q_{\lambda}:={\lambda}^{\frac{1}{N+\theta-1}}\Omega$ and let $\delta_{Q_{\lambda}}$ be the distance function to $\partial Q_{\lambda}$. For any $\psi\in Q_{\lambda}$, there exists $x\in\Omega$ such that $\psi={\lambda}^{\frac{1}{N+\theta-1}}x.$ In addition we have $\delta_{Q_{\lambda}}(\psi)={\lambda}^{\frac{1}{N+\theta-1}}\delta(x)$ and

[TABLE]

Set $\psi={\lambda}^{\frac{1}{N+\theta-1}}x$ and $y_{\lambda}={\lambda}^{\frac{1}{N+\theta-1}}y,$ then by change of variables we have

[TABLE]

where

[TABLE]

Set $d_{\lambda}(\psi)=\psi_{N}-{\lambda}^{\frac{1}{N+\theta-1}}\Gamma({\lambda}^{-\frac{1}{N+\theta-1}}\psi^{\prime}),$ it can be checked that

[TABLE]

where $c_{0}=\sup_{|x^{\prime}|\leq M}|\nabla\Gamma(x^{\prime})|$ with $M=\sup_{(x^{\prime},x_{N})\in\partial\Omega}|x^{\prime}|.$

Set $z=(z^{\prime},z_{N})$ with $z_{N}=d_{\lambda}(\psi)$ and $z^{\prime}=\psi^{\prime}$. By change of variables and by the above arguments we obtain

[TABLE]

where $c_{i}=c_{i}(N,\mu,\theta,\Omega)$, $i=1,2$ and $c_{3}(N,\mu,\theta,\gamma,\Omega)$.

Combining the above estimates leads to

[TABLE]

where $c=c(\Omega,\mu,N,\theta,\gamma)$ and $C$ is the constant in (3.6).

Next, we estimate

[TABLE]

where $c^{\prime}$ and $c^{\prime\prime}$ depend on $N,\mu,\theta,\gamma,\Omega$.

Thus (3.4) follows by (3.11) and (3.12).

Step 2.We will show that there exists a constant $C=C(N,\mu,\theta,\gamma,\Omega)$ such that

[TABLE]

where $D_{\beta_{0}}$ is defined in (1.44).

Indeed, from (3.5) it is easy to see that

[TABLE]

From (2.3), there exists $C=C(N,\mu,\Omega)$ such that, for every $(x,y)\in\Omega\times\Omega,\;x\neq y$,

[TABLE]

which implies

[TABLE]

This leads to

[TABLE]

where $C$ is the constant in (3.16).

Thus by (3.14) and (3.17), for every $\lambda\geq 1$, we have

[TABLE]

which yields (3.13).

Step 3. We will show that there exists a constant $C=C(N,\mu,\theta,\gamma,\Omega)$ such that

[TABLE]

From (2.3), there exists $C=C(N,\mu,\Omega)$ such that, for every $(x,y)\in\Omega\times\Omega,\;x\neq y$,

[TABLE]

For any $x\in A_{\lambda,2}(y)$, by (3.15) and (3.19),

[TABLE]

This ensures

[TABLE]

where in the above inequality we have used the fact that $\gamma<\frac{\theta N}{N-1}$.

Step 4. End of proof.

We infer from (3.4), (3.13), (3.18) and (3.3) that

[TABLE]

From that we can deduce that (3.21) also holds for every $\lambda>0$. Therefore by applying Proposition 3.3 with $D=\Omega,$ $\eta=\delta^{\gamma}$ with $\gamma\geq 0$, $d\omega=\delta^{\theta}d\tau$ and

[TABLE]

we obtain

[TABLE]

Thus the result follows by (3.2). ∎

Lemma 3.5**.**

Let $\gamma\geq 0$. Then there exists a positive constant $c=c(N,\mu,\gamma,\Omega)$ such that (1.21) holds.

Proof.

We use a similar argument as in the proof of Lemma 3.4. By Lemma 3.2, there exists $C=C(N,\mu,\Omega)>0$ such that

[TABLE]

A similar, and simpler, argument as in the proof of Lemma 3.4 justifies (1.21) and hence we omit the proof. ∎

Proof of Proposition A. The proposition follows from Lemma 3.4 and Lemma 3.5. ∎

4. Subcritical absorption

4.1. Existence

Let $g:\mathbb{R}\to\mathbb{R}$ be a locally Lipschitz continuous nonnegative and nondecreasing function vanishing at [math]. In this subsection, we deal with the existence of a solution of (1.17).

Proof of Theorem B.

Step 1: First we assume that $\sup_{t\in\mathbb{R}}|g(t)|=M<\infty.$ Let $D\subset\subset\Omega$ be a smooth open domain and consider the equation

[TABLE]

First we note that $u_{1}=0$ is supersolution of (4.1) and $u_{2}=-\mathbb{K}_{\mu}^{\Omega}[\nu]$ is a solution of (4.1). Let

[TABLE]

In this step we use the idea in [13] in order to construct a solution $v\in W^{1,\infty}(D)$ of the following problem

[TABLE]

which satisfies

[TABLE]

Let $d_{D,\Omega}:=\mathrm{dist}\,(\partial D,\partial\Omega)$ and $u\in W^{1,1}(D)$. By the standard elliptic theory, there exists a unique solution of the problem

[TABLE]

Recall that $\delta=\mathrm{dist}\,(\cdot,\partial\Omega)$.

We define an operator $\mathbb{A}$ as follows: to each $u\in W^{1,1}(D)$, we associate the unique solution $\mathbb{A}[u]$ of (4.5). Furthermore, since

[TABLE]

by the standard elliptic estimates we can obtain the existence of a positive constant $C=C(N,\mu,d_{D,\Omega},D)$ such that

[TABLE]

Also, by (4.6) and standard elliptic estimates, we have that there exists a positive constant $C=C(N,\mu,d_{D,\Omega},D)$ such that

[TABLE]

We will use the fixed point theorem to prove the existence of a fixed point of $\mathbb{A}$ by examining the following criteria.

We claim that $\mathbb{A}$ is continuous. Indeed, if $u_{n}\rightarrow u$ in $W^{1,1}(D)$ then since $g\in C(\mathbb{R}_{+})\cap L^{\infty}(\mathbb{R}_{+})$, it follows that $g(|\nabla u_{n}+\nabla\mathbb{K}_{\mu}^{\Omega}[\nu]|)\to g(|\nabla u+\nabla\mathbb{K}_{\mu}^{\Omega}[\nu]|)$ and $T[u_{n}]\to T[u]$ in $W^{1,1}(D)$. Hence $\mathbb{A}[u_{n}]\to\mathbb{A}[u]$ in $W^{1,1}(D)$.

Next we claim that $\mathbb{A}$ is compact. Indeed, let $\{u_{n}\}$ be a sequence in $W^{1,1}(D)$ then by (4.6) and (4.7), $\{\mathbb{A}[u_{n}]\}$ is uniformly bounded in $W^{1,\infty}(D)$. Therefore there exists $\psi\in W^{1,p}_{loc}(D)$ and a subsequence still denoted by $\{\mathbb{A}[u_{n}]\}$ such that $\mathbb{A}[u_{n}]\rightarrow\psi$ in $L^{p}_{loc}(D)$ and $\nabla\mathbb{A}[u_{n}]\to\nabla\psi$ weakly in $L^{p}_{loc}(D)$ and a.e. in $D$. By dominated convergence theorem we deduce that $\mathbb{A}[u_{n}]\rightarrow\psi$ in $W^{1,1}(D)$.

Now set

[TABLE]

Then ${\mathcal{K}}$ is a closed, convex subset of $W^{1,1}(D)$ and $\mathbb{A}({\mathcal{K}})\subset{\mathcal{K}}$. Thus we can apply Schauder fixed point theorem to obtain the existence of a function $v\in{\mathcal{K}}$ such that $\mathbb{A}[v]=v$. This means $v$ is a weak solution of (4.5).

By the standard elliptic theory, we can easily deduce that $v,u_{2}\in C^{2}(D)\cap C(\overline{D})$. Moreover, it can be seen that $v\leq 0$.

Now we allege that $v\geq u_{2}$ by employing an argument of contradiction. Suppose $x_{0}\in D$ is such that

[TABLE]

Then $\nabla v(x_{0})=\nabla u_{2}(x_{0})$, $-\Delta(v-u_{2})(x_{0})\leq 0$ and $T[v](x_{0})=T[u_{2}](x_{0})=u_{2}(x_{0})$. But

[TABLE]

which is clearly a contradiction.

As a consequence, $T(v)=v$ and therefore $v$ is a solution of of (4.3).

Step 2: Let $\{\Omega_{n}\}$ be a smooth exhaustion of $\Omega$ and let $v_{n}$ be the solution of (4.3) in $D=\Omega_{n}$ satisfying (4.4). Then

[TABLE]

for some positive constant $C=C(N,\mu,\Omega)$, where $M=\sup_{t\in\mathbb{R}}|g(t)|$. This implies that there exists a subsequence, still denoted by $\{v_{n}\},$ such that $v_{n}\rightarrow v$ in $W^{1,p}_{loc}(\Omega)$ and $v$ satisfies

[TABLE]

Furthermore,

[TABLE]

Setting $u=v+\mathbb{K}_{\mu}^{\Omega}[\nu],$ then $u$ is a solution of (1.17) satisfying $0\leq u\leq\mathbb{K}_{\mu}^{\Omega}[\nu]$ in $\Omega$.

Step 3: Set $g_{n}:=\min(g,n)$ and let $u_{n}$ be a nonnegative solution of

[TABLE]

satisfying

[TABLE]

Then $u_{n}$ satisfies

[TABLE]

[TABLE]

Choosing $\zeta=\varphi_{\mu}$, we have by (2.11)

[TABLE]

Now by (4.13), Lemma 3.4 and Lemma 3.5 we obtain

[TABLE]

Thus by (4.14) we have

[TABLE]

Similarly we can show that

[TABLE]

By 2.5, for any $1<p<q_{\mu}$, $\{u_{n}\}$ is uniformly bounded in $W^{1,p}(\Omega,\delta^{\alpha})$. Thus there exist $u\in W^{1,p}_{loc}(\Omega)$ and a subsequence still denoted by $\{u_{n}\}$ such that $u_{n}\to u$ a.e. in $\Omega$ and $\nabla u_{n}\to\nabla u$ a.e. in $\Omega$. Then (4.11) and the dominated convergence theorem guarantees that $u_{n}\rightarrow u$ in $L^{1}(\Omega,\delta^{\alpha})$.

For $s>0$, set $E_{n}(s)=\{x\in\Omega:|\nabla u_{n}(x)|>s\}$. Then by (2.6) and (4.17),

[TABLE]

Let $G\subset\Omega$ be a Borel subset. Then for any $s_{0}>0$

[TABLE]

But

[TABLE]

Thus we have proved

[TABLE]

We obtain easily, using (g1) and fixing $s_{0}$ first, that for any $\epsilon>0$, there exists $\kappa>0$ such that

[TABLE]

Thus we invoke Vitali convergence theorem to derive that $g_{n}(|\nabla u_{n}|)\rightarrow g(|\nabla u|)$ in $L^{1}(\Omega,\delta^{\alpha})$. From (4.11) we deduce that $0\leq u_{n}\rightarrow u$ in $L^{1}(\Omega,\delta^{\alpha})$. Letting $n\to\infty$ in identity (4.12), we deduce that $u$ is a weak solution of (1.17). ∎

The next results asserts that any weak solution $u$ of (1.17) behaves like $\mathbb{K}_{\mu}^{\Omega}[\nu]$ on the boundary.

Proof of Proposition C.

Since $u$ is a nonnegative solution of (1.17), formulation (1.23) holds. By a similar argument as in [16, Proposition I], $\mathbb{G}_{\mu}^{\Omega}[g(|\nabla u|)]$ is an $L_{\mu}$ potential (i.e. $\mathbb{G}_{\mu}^{\Omega}[g(|\nabla u|)]$ does not dominate any positive $L_{\mu}$ harmonic function). Consequently, in view of [16, Theorem 2.6],

[TABLE]

This and (1.23) imply (1.24). ∎

4.2. Regularity

This subsection is devoted to the regularity property of distributional solutions.

Definition 4.1**.**

Let $D$ be a subdomain of $\Omega$. A function $u$ is called a (distributional) subsolution (resp. supersolution) of

[TABLE]

if $u,g(|\nabla u|)\in L_{loc}^{1}(D)$ and

[TABLE]

A (distributional) solution in $D$ is a distributional subsolution and supersolution in $D$.

Lemma 4.2**.**

Let $1\leq q<\frac{N}{N-1}$ and assume that $g$ is a locally Lipschitz function satisfying

[TABLE]

for some $C>0$. If $u$ is a distributional solution of (4.31) then $u\in C^{2}(\Omega)$.

Proof.

Let $x_{0}\in\Omega$ and $r_{0}=\delta(x_{0})$. Let $G_{i}(x,y)$ and $K_{i}(x,y)$ be the Green kernel and Poisson Kernel of $-\Delta$ in $B(x_{0},\frac{r_{0}}{2^{i}})$ respectively. Since $u,g(|\nabla u|)\in L^{1}(B(x_{0},\frac{r_{0}}{2}))$, it follows that for any $x\in B(x_{0},\frac{r_{0}}{2^{i}})$,

[TABLE]

From the above formula, for any $x\in B(x_{0},\frac{r_{0}}{2^{i}})$,

[TABLE]

Combining (4.22), (4.23), [7, Theorem 2.5 and Theorem 2.6] and the fact that

[TABLE]

we deduce that $u,\,|\nabla u|\in L^{p}(B(x_{0},\frac{r_{0}}{2^{2}}))$ for any $q<p<\frac{N}{N-1}$.

Let $q<p<\frac{N}{N-1}$ and put $p_{i}:=(\frac{p}{q})^{i-1}p,\;i\in\mathbb{N}$.

Claim: There hold

[TABLE]

We will prove the claim by induction. Indeed, (4.25) holds for $i=1$. Suppose that (4.25) true for some $i\in\mathbb{N}$. We will show that (4.25) holds for $i+1$. By [7, Proposition 2.1], for every $x,y\in B(x_{0},\frac{r_{0}}{2^{i}}),\;x\neq y$,

[TABLE]

From (4.22)–(4.26) and the assumption on $g$, we can easily obtain the following estimates for any $x\in B(x_{0},\frac{r_{0}}{2^{i+2}})$

[TABLE]

By (4.27) and Holder inequality, for every $x\in B(x_{0},\frac{r_{0}}{2^{i+2}})$,

[TABLE]

By integrating over $B(x_{0},\frac{r_{0}}{2^{i+2}})$ and keeping in mind that $p<\frac{N}{N-1}$, we obtain

[TABLE]

Similarly, since $p<\frac{N}{N-1}$, we deduce from (4.28) that

[TABLE]

Therefore (4.25) holds true for $i+1$. Thus we have proved the claim.

Now fix $i$ large enough such that $\frac{qp}{p-1}<p_{i-1}.$ Then $u,|\nabla u|\in L^{p_{i-1}}(B(x_{0},\frac{r_{0}}{2^{i}}))$. We will estimate the terms on the right hand-side of (4.22). By (4.26) and Holder inequality,

[TABLE]

Similarly, by (4.26), Holder inequality and the assumption (4.21),

[TABLE]

Combining (4.29), (4.30) and (4.24), we obtain that $u\in L^{\infty}(B(x_{0},\frac{r_{0}}{2^{i+1}}))$. By a similar argument, one can show that $|\nabla u|\in L^{\infty}(B(x_{0},\frac{r_{0}}{2^{i+1}}))$. Thus the desired regularity result follows by standard elliptic regularity theory. ∎

4.3. Comparison principle

Lemma 4.3**.**

Let $g$ be a locally Lipschitz function. If $u\in C^{2}(\Omega)$ is a nonnegative solution

[TABLE]

and there exists $x_{0}\in\Omega$ such that $u(x_{0})=0$ then $u\equiv 0$.

Proof.

Since $g$ is locally Lipschitz function and $u\in C^{2}(\Omega)$ we can write

[TABLE]

where ${\bf b}$ has the following property: for any $\beta\in(0,\beta_{0})$, there exists $C_{\beta}$ such that

[TABLE]

Let $\beta\in(0,\beta_{0})$ be small enough such that $x_{0}\in D_{\beta}$. Note that $u$ is a nonnegative solution of

[TABLE]

Thus, by the maximum principle, $u$ cannot achieve a nonpositive minimum in $D_{\beta}.$ Thus the result follows straight forward. ∎

Next we state the comparison principle for (4.19).

Lemma 4.4**.**

Let $g$ be a locally Lipschitz function and satisfy (g3). We assume that $D\subset\Omega$ and $u_{1},u_{2}\in C^{2}(D)$ are respectively nonnegative subsolution and positive supersolution of (4.19) in $D$ such that

[TABLE]

Then $u_{1}\leq u_{2}$ in $D$.

Proof.

Suppose by contradiction that

[TABLE]

By (4.32), we deduce that there exists $x_{0}\in D$ such that

[TABLE]

Let $r>0$ be such that $B(x_{0},r)\subset D$. Then we see that

[TABLE]

Since $g$ is locally Lipschitz, we can write

[TABLE]

where ${\bf b}$ satisfies $\sup_{x\in B(x_{0},\frac{r}{2})}|\textbf{b}(x)|\leq C$. Hence,

[TABLE]

and by maximum principle $m^{-1}u_{1}-u_{2}$ can not achieve a non-negative maximum in $B(x_{0},\frac{r}{2}).$ This is a contradiction. Thus $u_{1}\leq u_{2}$ in $D$. ∎

Next we will prove the comparison principle for (1.17).

In order to demonstrate Theorem D, we need the following auxiliary result.

Lemma 4.5**.**

Let $\tau\in\mathfrak{M}(\Omega,\delta^{\alpha})$ and $v\geq 0$ satisfies

[TABLE]

Then for any $1<q<q_{\mu}$, there exists a constant $c=c(N,\Omega,\mu)$ such that

[TABLE]

Proof.

We notice that $(v-\mathbb{G}_{\mu}^{\Omega}[\tau])_{+}$ is a nonnegative $L_{\mu}$ subharmonic function with $\mathrm{tr\,}((v-\mathbb{G}_{\mu}^{\Omega}[\tau])_{+})=0$. By Propositions 2.2, $(v-\mathbb{G}_{\mu}^{\Omega}[\tau])_{+}=0$, i.e. $v\leq\mathbb{G}_{\mu}^{\Omega}[\tau]$ a.e. in $\Omega$. Put $\tilde{v}=\mathbb{G}_{\mu}^{\Omega}[\tau]-v$ then $\tilde{v}$ is a nonnegative $L_{\mu}$ superharmonic function in $\Omega$ and $\mathrm{tr\,}(\tilde{v})=0$. Due to Proposition 2.2, there exists $\tilde{\tau}\in\mathfrak{M}^{+}(\Omega,\delta^{\alpha})$ such that

[TABLE]

This implies that $\tilde{v}=\mathbb{G}_{\mu}^{\Omega}[\tilde{\tau}]$ and hence $v=\mathbb{G}_{\mu}^{\Omega}[\tau-\tilde{\tau}]$. Therefore, by Lemma 3.4,

[TABLE]

By using $\mathbb{G}_{\mu}^{\Omega}[1]$ as a test function for (4.34), keeping in mind the estimate

[TABLE]

we obtain

[TABLE]

Combining (4.35) and (4.37), we deduce (4.33). ∎

We turn to the

Proof of Theorem D.

Since $u_{i}$ is a solution of (1.17), $g(|\nabla u_{i}|)\in L^{1}(\Omega,\delta^{\alpha})$, $i=1,2$. Moreover, from Lemma 3.4 and Lemma 3.5, we deduce that

[TABLE]

Without loss of generality we assume that $\nu_{2}\neq 0,$ thus by Lemma 4.3 $u_{2}(x)>0$ for any $x\in\Omega.$ In addition, by Lemma 4.2, $u_{i}\in C^{2}(\Omega).$ Finally by the representation formula we have

[TABLE]

Let $0<\varepsilon\leq 1,$ then

[TABLE]

which implies

[TABLE]

Due to (g3), $\varepsilon u_{1}$ is a subsolution of (1.1). Also since $u_{i}\in C^{2}(\Omega)$ and $u_{2}>0$ in $\Omega$, it follows that

[TABLE]

where $\beta>0$ is small enough. Without loss of generality we assume that $C_{\beta}>1.$ Set $\varepsilon_{\beta}=\frac{1}{C_{\beta}}<1$. Then $\varepsilon_{\beta}u_{1}-u_{2}\leq 0$ in $D_{\beta}$.

Claim: For any $\beta>0$ small enough, there holds

[TABLE]

Indeed, from (g3), we observe that

[TABLE]

Since $g$ is locally Lipschitz, there holds

[TABLE]

with the estimate $\sup_{x\in D_{\beta}}|\textbf{b}(x)|\leq C$. Hence

[TABLE]

By the maximum principle $\varepsilon_{\beta}u_{1}-u_{2}$ can not achieve a nonnegative maximum in $D_{\beta}$ and thus the claim follows.

Due to Kato’s inequality [18], we get

[TABLE]

where $E_{\beta}=\{x\in\Omega:\varepsilon_{\beta}u_{1}-u_{2}>0\}$. By (4.38) we derive that $E_{\beta}\subset\Omega_{\beta}$.

Applying Lemma 4.5 and Holder’s inequality, thanks to (g2), we get

[TABLE]

Since $E_{\beta}\subset\Omega_{\beta}$ and $|\nabla u_{i}|\in L^{q}(\Omega,\delta^{\alpha})$, we can choose $\beta_{*}$ small enough such that

[TABLE]

By the above inequality and (4.40) we obtain that

[TABLE]

for some constant $c_{*}\geq 0$ and since $(\varepsilon_{\beta_{*}}u_{1}-u_{2})_{+}=0$ on $\overline{D}_{\beta_{*}}$ we have that $c_{*}=0$, namely $\varepsilon_{\beta_{*}}u_{1}\leq u_{2}$ in $\Omega$. As a consequence,

[TABLE]

This implies the existence of $x_{*}\in\partial D_{\beta_{*}}$ such that

[TABLE]

Next we take $\beta<\beta_{*}$, then $\varepsilon_{\beta}\leq\varepsilon_{\beta_{*}}$. On the other hand, we infer from (4.42) that $\varepsilon_{\beta}\geq\varepsilon_{\beta_{*}}$ and hence $\varepsilon_{\beta}=\varepsilon_{\beta_{*}}$. Therefore (4.43) contradicts (4.38). ∎

4.4. Some estimates

Lemma 4.6**.**

Assume $g(t)=t^{q}$ with $1<q<\frac{N}{N-1}$. If $u$ is a nonnegative solution of

[TABLE]

then

[TABLE]

where $M_{\beta_{0}}:=\sup_{\overline{D}_{\beta_{0}}}u$, $C=C(N,\mu,q,\beta_{0},M_{\beta_{0}})$ and $C^{\prime}=C^{\prime}(N,\mu,q,\beta_{0},M_{\beta_{0}})$.

Proof.  For $\beta\in(0,\beta_{0})$, put

[TABLE]

By a simple computation, we deduce that for $C_{1}=C_{1}(N,\mu,\beta_{0},q)$ large enough

[TABLE]

By Lemma 4.4, $u\leq w_{\beta}$ in $D_{\beta}\setminus D_{\beta_{0}}$. Consequently, by letting $\beta\to 0$, we obtain (4.45).

Next we prove (4.46). Fix $x_{0}\in\Omega$ and set $d_{0}=\frac{1}{3}\delta(x_{0})$, $y_{0}=\frac{1}{d_{0}}x_{0}$ and

[TABLE]

Then $\max\{v(y):y\in B_{2}(y_{0})\}=1$, $m_{0}^{q-1}d_{0}^{2-q}<C(N,\mu,q,\beta_{0},M_{\beta_{0}})<\infty$, and

[TABLE]

By [14], there exists a positive constant $c=c(N,\mu,q,\beta_{0},M_{\beta_{0}})$ such that

[TABLE]

Consequently,

[TABLE]

which implies (4.46). ∎

5. Isolated boundary singularities

In this section, we assume that $0\in\partial\Omega$ and study the behavior near [math] of solutions of (4.44) which vanish on $\partial\Omega\setminus\{0\}$.

5.1. A priori estimates

We first establish pointwise a priori estimates for solutions with isolated singularity at [math], as well as their gradient.

Proof of Proposition E.

We first prove (1.26).

Step 1. Let $\beta_{0}$ be the constant in Proposition A.1. Let $x_{i}\in\partial\Omega$ be such that $|x_{i}|\geq\frac{\beta_{0}}{16},$

[TABLE]

for some $n\in\mathbb{N}$. Notice that there exists a constant $\varepsilon_{0}=\varepsilon_{0}(\beta_{0})>0$ such that $\mathrm{dist}\,(\partial A,\partial\Omega)>\varepsilon_{0}.$

Let $w_{i}$ be the function constructed in Proposition A.1 in $B(x_{i},\frac{\beta_{0}}{16})$ for $R=\frac{\beta_{0}}{16},\;i=1,...,n.$ Then by the maximum principle (see [11, Proposition 2.13 and 2.14]), we have that

[TABLE]

As a consequence, there is a positive constant $C_{0}=C_{0}(N,\mu,q,\Omega,\beta_{0})$ such that

[TABLE]

Set

[TABLE]

We will show that $v(x)\geq u(x)$ for every $x\in\Omega\setminus A$. Indeed, by a direct computation,

[TABLE]

[TABLE]

and

[TABLE]

Gathering estimates (5.1)–(5.3) leads to, for $C_{1}=C_{1}(N,\mu,q,\beta_{0},\Omega)>0$ large enough,

[TABLE]

Moreover, we have

[TABLE]

By Lemma 4.4 we deduce that $u\leq v$ in $\Omega\setminus\overline{A},$ which implies that

[TABLE]

Thus by Lemma 4.6 there exists $C_{2}=C_{2}(\Omega,N,\mu,q,\beta_{0})>0$ such that

[TABLE]

Step 2. For $\ell>0$, put

[TABLE]

Let $\xi\in\partial\Omega\setminus\{0\}$ and put $d=d(\xi):=\frac{1}{2}|\xi|$. We assume that $d\leq 1$. Denote $u_{d}:=T_{d}[u]$ then $u_{d}$ is a solution of (4.44) in $\Omega^{d}=\frac{1}{d}\Omega$. Let $R_{0}=\frac{\beta_{0}}{16},$ where $\beta_{0}$ is the constant in Proposition A.1.

Then the solution $w_{\xi,\frac{3R_{0}}{4}}$ mentioned in Proposition A.1 satisfies

[TABLE]

Thus $u_{d}$ is bounded above in $B_{\frac{3R_{0}}{5}}(\xi)\cap\Omega^{d}$ by a constant $C>0$ depending only on $N,q,\mu$ and the $C^{2}$ characteristic of $\Omega^{d}$. As $d\leq 1$ a $C^{2}$ characteristic of $\Omega$ is also a $C^{2}$ characteristic of $\Omega^{d}$, therefore the constant $C$ can be taken to be independent of $\xi$. We note here that the constant $R_{0}\in(0,1)$ depends on $C^{2}$ characteristic of $\Omega$.

Now put

[TABLE]

Then we infer from (1.25) that

[TABLE]

Since $u_{d}$ is a positive $L_{\mu}$ subharmonic in $\Omega^{d}$, in view of the proof of [11, Propositions 2.11 and 2.12], we deduce that

[TABLE]

By proceeding as in the proof of [9, Theorem 2.12 ], we deduce that there exists $C=C(N,\mu,q)>0$ such that

[TABLE]

Hence

[TABLE]

Let $x\in\Omega_{\frac{R_{0}}{2}}$ and $\xi$ be the unique point in $\partial\Omega\setminus\{0\}$ such that $|x-\xi|=\delta(x)$. Put $d=\frac{1}{2}|\xi|$.

Case 1: $|x|<\frac{1}{1+R_{0}}$. If $\delta(x)\leq\frac{R_{0}|x|}{16}$ then

[TABLE]

As a consequence,

[TABLE]

In the last inequality we have used the fact that $R_{0}<1$. By combining (5.7)–(5.9), we obtain

[TABLE]

If $\delta(x)>\frac{R_{0}|x|}{16}$ then by (5.4) we have

[TABLE]

From (5.10) and (5.11) we deduce that (1.26) holds for every $x\in\Omega_{\frac{R_{0}}{2}}$ such that $|x|<\frac{1}{1+R_{0}}$.

Case 2: $|x|\geq\frac{1}{1+R_{0}}$. By an argument similar to the one used to obtain (5.6), we can prove that

[TABLE]

If $x\in\Omega\setminus\Omega_{\frac{R_{0}}{2}}$ then (1.26) follows directly from (5.4).

Next we prove (1.27). Let $x_{0}\in\Omega$ such that $\delta(x_{0})<\min(\frac{\beta_{0}}{36},1)$. By proceeding as in the proof of (4.46), we can deduce

[TABLE]

This and (1.26) imply (1.27).

If $\delta(x_{0})\geq\min(\frac{\beta_{0}}{36},1)=:r_{0},$ by [14] and (1.26), there exists a positive constant $c=c(N,\mu,q,\beta_{0})$ such that

[TABLE]

and the result follows. ∎

5.2. Weak singularities

Proof of Theorem F.

Let $u=u_{0,k}^{\Omega}$ be the solution of (1.28). By Theorem B and Lemma 4.3, $0<u\leq kK_{\mu}^{\Omega}(\cdot,0)$ in $\Omega$. Moreover,

[TABLE]

By proceeding as in the proof of (4.46), we obtain

[TABLE]

This follows that

[TABLE]

Case 1: $\alpha+(\alpha-1)q\geq 0.$

By the assumption and (2.3), we have

[TABLE]

Since $q<q_{\mu}$, it follows that

[TABLE]

Combining (5.15), (5.16) and (2.4) yields

[TABLE]

As a consequence,

[TABLE]

Case 2: $-1+\alpha<\alpha+(\alpha-1)q<0.$

Note that $q_{\mu}<\frac{1}{1-\alpha}.$ By (5.14) and (2.3) we have

[TABLE]

where

[TABLE]

Let $\beta\in(0,\beta_{0})$ such that $\delta\in C^{2}(\overline{\Omega_{\beta}}).$ We consider the cut-off function $\phi\in C^{\infty}(\overline{\Omega_{\frac{\beta}{2}}}),$ such that $0\leq\phi\leq 1,$ $\phi=1$ in $\Omega_{\frac{\beta}{4}}$ and $\phi=0$ in $\Omega\setminus\overline{\Omega_{\frac{\beta}{2}}}.$ Then

[TABLE]

By the definition of $\phi$ and using the inequality

[TABLE]

we obtain

[TABLE]

Let $\widetilde{\beta}\in(0,\frac{\beta}{4})$ be such that $|x-y|>r_{0}>0$ for any $y\in\Omega_{\widetilde{\beta}}$. Let $\varepsilon>0$ be such that

[TABLE]

and $0<\tilde{\varepsilon}<\varepsilon$ be such that $(\alpha-1)q+1-\tilde{\varepsilon}>0$. Then by (5.21), we have

[TABLE]

Note that by the choice of $\tilde{\varepsilon}$, $N-2-N+1+(\varepsilon-\tilde{\varepsilon})>-1$, which implies

[TABLE]

Combining the above estimates, we deduce

[TABLE]

Now note that

[TABLE]

On the other hand

[TABLE]

By collecting the above estimates, we obtain

[TABLE]

It follows from integration by parts, (5.23) and (5.24) that

[TABLE]

We will estimate $M(x)$ and $N(x)$ successively. By putting $e_{x}=|x|^{-1}x$ and $\eta=|x|^{-1}y$, we obtain

[TABLE]

The last integral is finite and independent of $x$ since $q<q_{\mu},$ $0<\alpha<1$ and $N\geq 3$. This and (2.4) imply that

[TABLE]

Similarly we have

[TABLE]

Combining (5.14), (5.20), (5.22), (5.25) – (5.27) implies that there exists a positive constant $C=C(\Omega,N,\mu,q)>0$ such that

[TABLE]

By the above inequality and (5.12) we can easily prove (1.29).

By combining (5.18) and (5.12), we obtain (1.29). The monotonicity comes from Theorem B. ∎

5.3. Strong singularities

We recall that ${\mathcal{L}}_{\mu}$ is defined in (1.33). Notice that the eigenvalue $\kappa_{\mu}$ is explicitly determined as follows

[TABLE]

and the corresponding eigenfunction $\phi_{\mu}(\sigma)=(\frac{x_{N}}{|x|}\lfloor_{S^{N-1}_{+}})^{\alpha}=({\bf e}_{{}_{N}}\cdot\sigma)^{\alpha}$ solves

[TABLE]

Notice that equation $(\ref{Eq7})$ admits a unique positive solution with supremum $1$ and if $\mu=0$ then $\alpha=1,$ which means that $\phi_{0}(\sigma)={\bf e}_{{}_{N}}\cdot\sigma$ is the first eigenfunction of $-\Delta^{\prime}$ in $H^{1}_{0}(S^{N-1}_{+})$. Moreover,

[TABLE]

By [8, Theorem 6.1] the infimum exists since $\phi_{0}(\sigma)={\bf e}_{{}_{N}}\cdot\sigma$ is the first eigenfunction of $-\Delta^{\prime}$ in $H^{1}_{0}(S^{N-1}_{+})$. The minimizer $\phi_{\mu}$ belongs to $H^{1}_{0}(S^{N-1}_{+})$ only if $1<\mu<\frac{1}{4}$. Furthermore $\phi_{\mu}\in{\bf Y}_{\mu}(S^{N-1}_{+})$ where ${\bf Y}_{\mu}(S^{N-1}_{+})$ is defined in (1.34).

Finally by (5.30) the following expression holds

[TABLE]

Indeed, since $\phi_{\frac{1}{4}}=\phi_{0}^{\frac{1}{2}}$ we have

[TABLE]

and

[TABLE]

Taking into account that $\kappa_{\frac{1}{4}}-\frac{N-1}{2}=-\frac{1}{4},$ by the above equalities we obtain (5.32).

Proof of Theorem G.

Step 1. Existence. Set

[TABLE]

then the function $\overline{\omega}:=\gamma_{1}\phi_{\mu}$ is a supersolution of (1.32). Indeed by (5.30) and (5.32),

[TABLE]

Let $\alpha_{0}\in(\alpha,1)$ be such that

[TABLE]

We note that $\phi_{\mu_{0}}=\phi_{0}^{\alpha_{0}},$ where

[TABLE]

We allege that there exists a positive constant $\gamma_{2}=\gamma_{2}(N,q,\mu,\mu_{0})\leq\gamma_{1}$ such that the function $\underline{\omega}=\gamma_{2}\phi_{\mu_{0}}$ is a subsolution of (1.32). Indeed by (5.30) and (5.32) we have

[TABLE]

provided $\gamma_{2}$ is small enough. Notice that we can choose $\gamma_{2}\leq\gamma_{1}$.

For $t\in(0,1)$, set $S_{t}:=\{\sigma\in S_{+}^{N-1}:\phi_{0}(\sigma)<t\}$ and $\widetilde{S}_{t}:=S_{+}^{N-1}\setminus S_{t}$. In view of the proof of [13, Theorem 6.5], there exists a solution $\omega_{t}\in W^{2,p}(\widetilde{S}_{t})$ to (1.32) such that

[TABLE]

Therefore, by the standard elliptic theory, there exist a function $\widetilde{w}$ and a sequence $t_{n}\searrow 0$ such that $\omega_{t_{n}}\rightarrow\widetilde{\omega}$ locally uniformly in $C^{1}(S^{N-1}_{+})$ and $\widetilde{\omega}$ satisfies

[TABLE]

Furthermore by (5.33) we have that

[TABLE]

Set $\tilde{u}(x)=|x|^{-\frac{2-q}{q-1}}\widetilde{\omega}(\sigma)$ then $\tilde{u}$ satisfies

[TABLE]

and

[TABLE]

Let $x_{0}=(x_{0}^{\prime},0)$ be such that $|x_{0}^{\prime}|=1$ then in view of the proof of (4.46), there exists a constant $C_{1}=C(N,\mu,q)$ such that

[TABLE]

This implies

[TABLE]

Step 2. Uniqueness.

Let $\omega_{i}\in{\bf Y}_{\mu}(S^{N-1}_{+})$, $i=1,2,$ be two positive solutions of (1.32). Let $x_{0}=(x_{0}^{\prime},0)$ be such that $|x_{0}^{\prime}|=1,$ Then $u_{i}(x)=|x|^{-\frac{2-q}{q-1}}\omega_{i}\in H^{1}(B(x_{0},\frac{2}{3}),x_{N}^{2\alpha})$ and satisfies

[TABLE]

which implies $-L_{\mu}u_{i}\leq 0$ in $\mathbb{R}^{N}_{+}$. Since $0<v_{i}:=x_{N}^{-\alpha}u_{i}\in H^{1}(B(x_{0},\frac{1}{2}),\;x_{N}^{2\alpha})$ and satisfies

[TABLE]

by [9, Theorem 2.12], there exists a positive constant $C_{i}>0$ such that

[TABLE]

Therefore in view of the proof of (5.36), we deduce that there exists a positive constant $C_{0}$ such that

[TABLE]

and

[TABLE]

Set

[TABLE]

Without loss of generality we may assume that $b_{t_{0}}>1$ for some $t_{0}\in(0,1)$; thus by (5.37) we have

[TABLE]

In the sequel we consider $t\in(0,t_{0})$. Put $\psi:=\phi_{0}^{\alpha}-\frac{1}{2}\phi_{0}^{\alpha+\varepsilon},$ where $\varepsilon\in(0,1-\alpha)$ is a parameter that will be determined later. Then we have $\frac{1}{2}\phi_{0}^{\alpha}\leq\psi\leq\phi_{0}^{\alpha}.$ We recall that $\phi_{0}^{\alpha}=\phi_{\mu}$ and $\phi_{0}^{\alpha+\varepsilon}=\phi_{\mu_{\varepsilon}}$ where

[TABLE]

From the definition of $\psi$, it is easy to check that

[TABLE]

Now, let $\omega_{t}=b_{t}^{-1}\omega_{2}.$ We remark that $\omega_{t}$ is a subsolution of (1.32) and $\omega_{t}-\omega_{1}\leq 0$ in $\widetilde{S}_{t}$. Also, we have

[TABLE]

Since $1<q<2$, the following inequality holds for any nonnegative number $h_{1},h_{2},k_{1},k_{2}$

[TABLE]

By applying (5.41) with h_{1}=\Big{(}\frac{2-q}{q-1}\Big{)}\omega_{\delta}, $h_{2}=\left|\nabla^{\prime}\omega_{\delta}\right|$, k_{1}=\Big{(}\frac{2-q}{q-1}\Big{)}\omega_{1} and $k_{2}=\left|\nabla^{\prime}\omega_{1}\right|$ and keeping in mind estimates (5.37) and (5.38), we obtain

[TABLE]

Now set $V_{t}:=\psi^{-1}(\omega_{t}-\omega_{1})$. By (5.40), (5.42) and the definition of $\psi,$ we can easily deduce the existence of a positive constant $C=C(N,\mu,q,C_{0})$ such that

[TABLE]

Now since $\psi V_{t}\in{\bf Y}_{\mu}(S^{N-1}_{+})$ and $V_{t}(\sigma)\leq 0$ for any $\sigma\in\widetilde{S}_{t},$ multiplying the above inequality by $(V_{t})_{+}$ and integrating over $S_{+}^{N-1}$, we get

[TABLE]

By the definition of $\psi$ and (5.39), we have

[TABLE]

Note here that if $\varepsilon<1-\alpha$ then $q<2<\frac{2-\alpha-\varepsilon}{1-\alpha}$. This leads to

[TABLE]

By Young’s inequality, we deduce that

[TABLE]

where $C$ is the constant in (5.44) and $\hat{C}=\hat{C}(q,N,\mu)$.

Gathering (5.44), (5.45) and (5.47) yields

[TABLE]

where $C_{1}=C(q,N,\mu)$. By (5.46) and the above inequality we can find a positive constant $t_{1}=t_{1}(N,q,\mu,\varepsilon,C_{0})$ such that

[TABLE]

which implies $(V_{t_{1}})_{+}=0$ in $S_{t_{1}}$ since $(V_{t_{1}})_{+}=0$ on $\{\sigma\in S_{+}^{N-1}:\phi_{0}(\sigma)=t_{1}\}.$ Hence

[TABLE]

Thus we have proved that

[TABLE]

This means that $\omega_{1}-\omega_{t_{1}}(\sigma)\geq 0$ for any $\sigma\in S^{N-1}_{+}$ and

[TABLE]

But

[TABLE]

which implies

[TABLE]

By the above inequality and mean value theorem there exists $\bar{\Lambda}>0$ such that

[TABLE]

where $\overline{s}$ and $\overline{\xi}$ are functions with respect to $\sigma\in\widetilde{S}_{\frac{t_{1}}{2}}$ such that $\frac{\partial H(\overline{s},\overline{\xi})}{\partial\xi}\in L^{\infty}(\widetilde{S}_{\frac{t_{1}}{2}}).$ By the maximum principle, $\omega_{1}-\omega_{t_{1}}$ cannot achieve a non-positive minimum in $\widetilde{S}_{\frac{t_{1}}{2}}\setminus\partial\widetilde{S}_{\frac{t_{1}}{2}}$ which clearly contradicts (5.49).

The result follows by exchanging the role of $\omega_{1},\;\omega_{2}.$ ∎

By Theorem F and Proposition E, the sequence $\{u_{0,k}^{\Omega}\}$ is increasing and bounded from above, hence there exists $u_{0,\infty}^{\Omega}=\lim_{k\to\infty}u_{0,k}^{\Omega}$.

Proof of Theorem H.

By Proposition E, for every $k>0$,

[TABLE]

and

[TABLE]

where $c=c(N,\mu,q,\Omega)$. Moreover $\{u_{0,k}^{\Omega}\}$ is increasing. Therefore, by the standard regularity result, $u_{0,k}^{\Omega}\to u_{0,\infty}^{\Omega}$ in $C_{loc}^{1}(\Omega)$ and $u_{0,\infty}^{\Omega}$ is a positive solution of (4.44). Letting $k\to\infty$ in (5.50) and (5.51) yields the second estimate in (1.37) and estimate (1.38).

Next we infer from (5.12), (5.28) and (2.4) that, for every $k>0$,

[TABLE]

For $x\in\Omega$, choose $k=a|x|^{-\frac{N+\alpha-(N+\alpha-1)q}{q-1}}$, where $a>0$ will be made precise later on, then

[TABLE]

By choosing $a=(2c_{3})^{-\frac{1}{q-1}}$, we deduce for any $x\in\Omega$ there exists $k>0$ depending on $|x|$ such that

[TABLE]

Since $u_{0,\infty}^{\Omega}\geq u_{0,k}^{\Omega}$ in $\Omega$ we obtain the first inequality in (1.37).

Next we prove (1.39). Since $u_{0,k}^{\Omega}$ is the solution of (1.28), it follows that, for any $\ell>0$, $T_{\ell}[u_{0,k}^{\Omega}]$ is a solution of

[TABLE]

where $m_{q}=\frac{2-q}{q-1}-N+1$ and $T_{\ell}$ is given in (5.5). By the uniqueness,

[TABLE]

Sending $k\to\infty$ implies $T_{\ell}[u_{0,\infty}^{\Omega}]=u_{0,\infty}^{\Omega^{\ell}}$ in $\Omega^{\ell}$.

Since estimates (1.37) and (5.51) are invariant under the transformation $T_{\ell}$, it follows that

[TABLE]

[TABLE]

By local regularity results, we deduce that there exist a function $U$ and a subsequence $\{\ell_{n}\}$ such that $u_{0,\infty}^{\Omega^{\ell_{n}}}\to U$, as $\ell_{n}\to 0$, in $C_{loc}^{1}(\mathbb{R}_{+}^{N})$. Furthermore, $U$ is positive solution of (1.30).

From (5.53), for any $\ell^{\prime}>0$, we have

[TABLE]

By letting $k\to\infty$ and $\ell_{n}\to 0$, we obtain that $T_{\ell^{\prime}}[U]=U$ in $\mathbb{R}_{+}^{N}$ for every $\ell^{\prime}$. By putting $\omega(\sigma)=r^{\frac{2-q}{q-1}}U(x)$ with $r=|x|$, $\sigma=r^{-1}x$, we deduce that $\omega$ is a positive solution of (1.32). By Theorem G, $\omega$ is the unique solution of (1.32). Thus (1.39) follows. ∎

6. Supercritical Case

We start the section with an observation that when $g(t)=t^{q}$ the condition (g1) is fulfilled if and only if $q<q_{\mu}$; in which case the solvability of (1.41) holds for every $\mu\in\mathfrak{M}^{+}(\partial\Omega)$. On the contrary, in the supercritical case i.e. if $q\geq q_{\mu}$, a continuity condition with respect to some Besov capacity is needed to derive an existence result.

We recall below some notations concerning Besov space (see, e.g., [1, 22]). For $\sigma>0$, $1\leq p<\infty$, we denote by $W^{\sigma,p}(\mathbb{R}^{d})$ the Sobolev space over $\mathbb{R}^{d}$. If $\sigma$ is not an integer the Besov space $B^{\sigma,p}(\mathbb{R}^{d})$ coincides with $W^{\sigma,p}(\mathbb{R}^{d})$. When $\sigma$ is an integer we denote $\Delta_{x,y}f:=f(x+y)+f(x-y)-2f(x)$ and

[TABLE]

with norm

[TABLE]

Then

[TABLE]

with norm

[TABLE]

These spaces are fundamental because they are stable under the real interpolation method developed by Lions and Petree. For $\alpha\in\mathbb{R}$ we defined the Bessel kernel of order $\alpha$ by $G_{\alpha}(\xi)={\mathcal{F}}^{-1}(1+|.|^{2})^{-\frac{\alpha}{2}}{\mathcal{F}}(\xi)$, where ${\mathcal{F}}$ is the Fourier transform of moderate distributions in $\mathbb{R}^{d}$. The Bessel space $L_{\alpha,p}(\mathbb{R}^{d})$ is defined by

[TABLE]

with norm

[TABLE]

It is known that if $1<p<\infty$ and $\alpha>0$, $L_{\alpha,p}(\mathbb{R}^{d})=W^{\alpha,p}(\mathbb{R}^{d})$ if $\alpha\in\mathbb{N}$ and $L_{\alpha,p}(\mathbb{R}^{d})=B^{\alpha,p}(\mathbb{R}^{d})$ if $\alpha\notin\mathbb{N}$, always with equivalent norms. The Bessel capacity is defined for compact subsets $K\subset\mathbb{R}^{d}$ by

[TABLE]

It is extended to open sets and then Borel sets by the fact that it is an outer measure.

Let us recall the following result in [16].

Proposition 6.1**.**

Let $\nu\in\mathfrak{M}^{+}(\partial\Omega)$ and $\beta_{0}$ be the constant in Proposition (3.1). Then the following inequalities hold

[TABLE]

Proof.

Estimates (6.1) follows from [16, Corollary 2.10]. Estimate (6.2) can be obtained by a similar argument with some modifications and we omit it. ∎

Lemma 6.2**.**

Let $\nu\in\mathfrak{M}^{+}(\partial\Omega),$ $q\in(1,2)$ and $u\in C^{2}(\Omega)$ be a nonnegative solution of (1.41).

(i) If $q\neq\alpha+1$ then there exists a constant $\beta_{1}=\beta_{1}(N,\mu,q,\Omega)>0$ such that the following inequality holds

[TABLE]

where $C$ depends only on $N$, $\mu$, $q$, $\Omega$ and $\sup_{\Sigma_{\beta_{1}}}(\mathbb{K}_{\mu}^{\Omega}[\nu])^{q}.$

(ii) If $q=\alpha+1$ then for any $\varepsilon>0$ small enough there exists a constant $\beta_{1}=\beta_{1}(N,\mu,\Omega,\varepsilon)>0$ such that the following inequality holds

[TABLE]

where $C$ depends only on $N$, $\mu$, $\Omega$, $\varepsilon$ and $\sup_{\Sigma_{\beta_{1}}}(\mathbb{K}_{\mu}^{\Omega}[\nu])^{\alpha+1}.$

Proof.

Since $u$ is a nonnegative solution of (1.41) we have that $|\nabla u|\in L^{q}(\Omega,\delta^{\alpha})$. Let $\beta_{1}\in(0,\beta_{0})$ where $\beta_{0}$ is the constant in Proposition 3.1.

(i) First we assume that $q\neq\alpha+1$ and let $\gamma\neq-1$. Then for $\beta\in(0,\beta_{1})$,

[TABLE]

Observe that for any $\gamma\in(\alpha-q,\max\{\frac{\alpha-1-q}{2},2(\alpha-q)+1\})$, we have

[TABLE]

Therefore, for such $\gamma$, we can choose $\beta_{1}=\beta_{1}(N,q,\mu,\Omega)$ such that

[TABLE]

Consequently, by Hölder inequality we can find a constant $C_{1}=C_{1}(N,q,\mu,\Omega)$ such that

[TABLE]

By the above estimates, there is a positive constant $C_{2}=C_{2}(N,\mu,q,\Omega)$ such that

[TABLE]

By (4.45), Proposition 6.1 and taking into account that $\gamma+q-1>\alpha-1$, we obtain

[TABLE]

Therefore, by letting $\beta\to 0$ in (6.6), we obtain

[TABLE]

By dominated convergence theorem, we can send $\gamma\to\alpha-q$ in (6.7) to obtain

[TABLE]

This implies (6.3).

The proof of (6.4) follows by similar arguments as the proof of (6.3) (with $\gamma=\varepsilon-1$) with the some modifications and we omit it. ∎

Next put $\Sigma=\partial\Omega$ and denote by $\Delta_{\Sigma}$ the Laplace-Beltrami operator on $\Sigma$.

Proof of Theorem I.

Let $\varepsilon\geq 0$ and $u$ be the solution of (1.41). If $\eta\in L^{\infty}(\Sigma)\cap B^{\varepsilon+\frac{\alpha+1}{q}-\alpha,q^{\prime}}(\Sigma)$, we denote by $H:=H[\eta]$ the solution of

[TABLE]

Let $h\in C^{\infty}(\mathbb{R}_{+})$ such that $0\leq h\leq 1$, $h^{\prime}\leq 0$, $h\equiv 1$ on $[0,\frac{\beta_{0}}{2}]$, $h\equiv 0$ on $[\beta_{0},\infty]$. The lifting we consider is expressed by

[TABLE]

with $x\approx(\delta,\sigma)=(\delta(x),\sigma(x))$.

Case 1: $q\neq\alpha+1.$ Set $\varepsilon=0$ and $\zeta=\varphi_{\mu}R[\eta]^{q^{\prime}}$ where $\varphi_{\mu}$ is the eigenfunction associated to the first eigenvalue $\lambda_{\mu}$ of $-L_{\mu}$ in $\Omega$ (see Section2.1). By proceeding as the proof of (3.46) in [11, Lemma 3.8], we obtain

[TABLE]

where

[TABLE]

Following the arguments of the proof of (3.48) in [11, Lemma 3.9] we can obtain

[TABLE]

By Lemma 6.3 we have

[TABLE]

where the constant $C$ depends on $\Omega,\mu$ and $N.$

Combining (6.11), (6.13) and (6.14) we obtain

[TABLE]

Let $K\subset\partial\Omega$ be a compact set. If $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{q}-\alpha,q^{\prime}}(K)=0$ then there exists a sequence $\{\eta_{n}\}$ in $C^{2}_{0}(\partial\Omega)$ with the following properties:

[TABLE]

This implies that $0\leq R[\eta_{n}]\leq 1$ and $\lim_{n\to\infty}R[\eta_{n}]=0$ a.e. in $\Omega$. Put $\zeta_{n}=\varphi_{\mu}R[\eta_{n}]^{q^{\prime}}$. Then

[TABLE]

From (6.15)–(6.17), we obtain

[TABLE]

This implies that $\nu$ is absolutely continuous with respect to $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{q}-\alpha,q^{\prime}}$.

Case 2: $q=\alpha+1.$ Let $0<\varepsilon<\alpha+1$ and $\zeta=\varphi_{\mu}R[\eta]^{q^{\prime}}.$ Proceeding as the proof of (6.11), we can prove

[TABLE]

where

[TABLE]

Using (6.4) and the ideas of the proof of (6.15) we can obtain the following inequality

[TABLE]

where the constant $C$ depends on $N,\mu,\Omega$ and $\varepsilon$.

The rest of the proof follows by using a similar argument as in the first case. ∎

Proposition 6.3**.**

Let $u\in C^{2}(\Omega)$ be a positive solution of (1.42). If $|\nabla u|\in L^{q}(\Omega,\delta^{\alpha})$ it possesses a boundary trace $\nu\in\mathfrak{M}(\partial\Omega),$ i.e., $u$ is the solution of the boundary value problem (1.41) with this measure $\nu.$

Proof.

If $v:=\mathbb{G}_{\mu}^{\Omega}[|\nabla u|^{q}]$ then $v\in L^{1}(\Omega,\delta^{\alpha})$ and $u+v$ is a positive $L_{\mu}$ harmonic function. Hence $u+v\in L^{1}(\Omega,\delta^{\alpha})$ and there exists a non-negative measure $\nu\in\mathfrak{M}(\partial\Omega)$ such that $u+v=\mathbb{K}_{\mu}^{\Omega}[\nu].$ By Proposition 2.2 we obtain the result. ∎

Proof of Theorem J.

In view of the proof of [11, Proposition A.2] we can obtain the following estimates

[TABLE]

[TABLE]

where $C$ depends on $N,\mu,q,\Omega$ and $\sup_{\Sigma_{\beta_{0}}}u$.

Case 1: Assume that $q\neq\alpha+1$ and $C^{\mathbb{R}^{N-1}}_{\frac{\alpha+1}{\alpha}-\alpha,q^{\prime}}(K)=0$. Then there exists a sequence $\{\eta_{n}\}$ in $C^{2}_{0}(\partial\Omega)$ satisfying (6.16). In particular, there exists a decreasing sequence $\{{\mathcal{O}}_{n}\}$ of relatively open subsets of $\partial\Omega$, containing $K$ such that $\eta_{n}=1$ on ${\mathcal{O}}_{n}$ and thus $\eta_{n}=1$ on $K_{n}:=\overline{\mathcal{O}}_{n}$. We set $\tilde{\eta}_{n}=1-\eta_{n}$ and $\tilde{\zeta}_{n}=\varphi_{\mu}R[\tilde{\eta}_{n}]^{2q^{\prime}}$ where $R$ is defined by (6.10). Then $0\leq\tilde{\eta}_{n}\leq 1$ and $\tilde{\eta}_{n}=0$ on $K_{n}$. Therefore

[TABLE]

Furthermore

[TABLE]

Proceeding as the proof of (3.65) in [11, Theorem 3.10] we have

[TABLE]

Using the expression of $L_{\mu}\tilde{\zeta}_{n}$, we derive from (6.25) that

[TABLE]

where

[TABLE]

By proceeding as in the proof of (3.75) in [11, Theorem 3.10] we can prove

[TABLE]

Next we recall that $\beta_{0}$ is the constant in Proposition 3.1.

**Claim: ** There exists a positive constant $\beta_{1}=\beta_{1}(q,\mu,N,\Omega)\in(0,\beta_{0})$ such that

[TABLE]

where $C$ depends only on $N,\mu,\Omega$ and $\max(\sup_{\Sigma_{\beta_{1}}}u,\sup_{\Sigma_{\beta_{0}}}u)$.

Indeed, let $\beta_{1}\in(0,\beta_{0})$. By integration by parts, we have

[TABLE]

where $C=C(N,\mu,q,\Omega)$. Now we choose $\beta_{1}$ small enough such that

[TABLE]

By Hölder inequality we have

[TABLE]

where $C_{1}=C_{1}(N,\mu,q,\Omega)$. In view of the proof of (3.53) in [11, Lemma 3.9], by (6.21) and Hölder inequality, we obtain

[TABLE]

where $C_{i}=C_{i}(N,\mu,q,\Omega,C_{0})$, $i=2,3$. Combining all above we can easily deduce

[TABLE]

where $C$ depends only on $N,\mu,q,\Omega$ and $\max(\sup_{\Sigma_{\beta_{1}}}u,\sup_{\Sigma_{\beta_{0}}}u)$.

Now by (6.21) and (6.23), we have

[TABLE]

hence letting $\beta\to 0$ in (6.30), we obtain the claim.

Combining (6.21), (6.28) and (6.29) leads to

[TABLE]

which implies

[TABLE]

Letting $n\to\infty$ and using the fact that $\eta_{n}\to 0$, we obtain by Fatou’s lemma that

[TABLE]

Combining this with the fact that $|\nabla u|$ is bounded in $D_{\frac{\beta_{1}}{2}}$ due to (6.22), we assert that $|\nabla u|\in L^{q}(\Omega,\delta^{\alpha}).$ Thus by Proposition 6.3 there exists a nonnegative Radon measure $\nu$ with support in $K$ such that

[TABLE]

In light of Theorem I, $\nu\equiv 0$ which implies $u=0$ and the result follows in this case.

Case 2: Assume that $q=\alpha+1$ and $C^{\mathbb{R}^{N-1}}_{\varepsilon+1-\alpha,q^{\prime}}(K)=0$ for $\varepsilon$ as in statement (ii). Then we can obtain the desired result by combining the ideas in Case 1 of this theorem and in Case 2 of Theorem I. ∎

Appendix A Barrier

In this section we will construct a barrier which plays an important role. This barrier will have the same properties as the barrier in [11, Proposition 6.1]. Let $\beta_{0}$ be the constant in Proposition 3.1.

Proposition A.1**.**

Let $\Omega\subset\mathbb{R}^{N}$ be a $C^{2}$ domain, $0<\mu\leq\frac{1}{4}$ and $q>1$. Then for any $z\in\partial\Omega$ and $0<R\leq\frac{\beta_{0}}{16}$, there exists a super solution $w:=w_{z,R}$ of $(\ref{Epower})$ in $\Omega\cap B_{R}(z)$ such that $w\in C(\overline{\Omega}\cap B_{R}(z))$, $w(x)\to\infty$ when $\mathrm{dist}\,(x,K)\to 0$, for any compact subset $K\subset\Omega\cap\partial B_{R}(z)$ and $w$ vanishes on $\partial\Omega\cap B_{R}(z)$. More precisely

[TABLE]

where $b\geq\max\{\frac{4-q}{q-1}+\gamma,\frac{N-2}{2},1\}$ and $c=c(N,\mu,q,b,\gamma)$.

Proof.

Take $z\in\partial\Omega$ and $R\in(0,\frac{\beta_{0}}{16}]$. Without loss of generality, we may assume $z=0$.

Case 1: $\mu<\frac{1}{4}$. Set

[TABLE]

where $b,\gamma>0$ to be chosen later on. By straightforward calculation we have

[TABLE]

Now let $x_{0}=x+(\frac{\beta_{0}}{4}-\delta(x))\nabla\delta(x)$ and $\xi_{x}$ be the unique point in $\partial\Omega$ such that $\delta(x)=|x-\xi_{x}|.$ Then $\delta(x_{0})=|x_{0}-\xi_{x}|=\frac{\beta_{0}}{4},$

[TABLE]

Furthermore

[TABLE]

We have

[TABLE]

Since $|x_{0}|^{2}\geq\frac{\beta_{0}^{2}}{16}$, it follows from the last inequality that $x\cdot x_{0}\geq 0$. Therefore

[TABLE]

where in the above inequality we have used the fact that $|x|\leq\frac{\beta_{0}}{16}$ and $|x-x_{0}|\geq\frac{3\beta_{0}}{16}$. By (A.3) and Hölder inequality we have

[TABLE]

Combining (A.4) and (A.2) we have

[TABLE]

The rest proof is similar to the proof of [11, Proposition 6.1] and we omit it.

Case 2: $\mu=\frac{1}{4}$. Set

[TABLE]

for some $\Lambda,b$ to be made precise later on. It is easy to calculate

[TABLE]

By (A.3) and Hölder inequality, we have

[TABLE]

Hence,

[TABLE]

Proceeding as the proof of [11, Proposition 6.1], we obtain the desired result. ∎