Anisotropic singularities to semilienar elliptic equations in a measure framework

TL;DR

This paper investigates very weak solutions to a semilinear elliptic equation with anisotropic singularities in a measure framework, establishing existence, uniqueness, and asymptotic behavior under subcritical growth conditions.

Contribution

It introduces a novel approach to analyze very weak solutions with measure data involving anisotropic singularities, extending the scope to broader nonlinearities.

Findings

Existence and uniqueness of solutions under subcritical growth

Solutions exhibit anisotropic singularity at the origin

Limit behavior of solutions as parameters tend to infinity

Abstract

The purpose of this article is to study very weak solutions of elliptic equation $$ -\Delta u+g(u)=2k\frac{\partial \delta_0}{\partial x_N }+j\delta_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $\delta_0$ is the Dirac mass concentrated at the origin and $\frac{\partial\delta_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial \delta_0}{\partial x_N},\zeta\rangle=\frac{\partial\zeta(0)}{\partial x_N} , \qquad \forall \zeta\in C^1_0(B_1(0)). $$ We obtain that the above problem admits a unique very weak solution $u_{k,j}$ under the integral subcritical assumption $$\int_1^{\infty}g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ Furthermore, we prove that $u_{k,j}$ has anisotropic singularity at the origin and we consider the odd property $u_{k,0}$ and limit of $\{u_{k,0}\}_k$ as $k\to\infty$. We pose the constraint on nonlienarity $g(u)$ that we only require integrability in the principle value sense, due to the singularities only at the origin. This makes us able to search the very weak solutions in a larger scope of the nonlinearity.

Equations

$$\left\{\begin{array}[]{lll}-\Delta u+g(u)=2k\frac{\partial\delta_{0}}{\partial x_{N}}+j\delta_{0}&{\rm in}\quad\ \ B_{1}(0),\\[5.69054pt] \phantom{-\Delta+g(u)}u=0&{\rm on}\quad\ \ \partial B_{1}(0),\end{array}\right.$$

$$⟨\frac{\partial {\delta }_0}{\partial x_N},\zeta ⟩=\frac{\partial \zeta (0)}{\partial x_N},\forall \zeta \in C_0^1(B_1(0)).$$

$${\int }_1^{\infty }g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty .$$

$$-\Delta u+(u-\lambda {)}_{+}^{\frac{3}{2}}=\sum _{i=1}^n{m}_i{\delta }_{a_i}\mathrm{in}{\mathbb{R}}^3$$

$$-\Delta u+(u-\lambda {)}_{+}^{\frac{3}{2}}=0\mathrm{in}{\mathbb{R}}^3\setminus \{a_1,\cdots ,a_n\}.$$

$$-\Delta u+|u{|}^{p-1}u=0\mathrm{in}\Omega \setminus \{0\}$$

$$-\Delta u+u^p=k{\delta }_0\mathrm{in}({\mathrm{C}}_{\mathrm{c}}^{\infty }(\mathrm{\Omega }){)}^{\prime },$$

$$\Gamma_{N}(x)=\left\{\begin{array}[]{lll}c_{N}|x|^{2-N}&{\rm if}\quad N\geq 3,\\[5.69054pt] \phantom{}c_{N}\log|x|&{\rm if}\quad N=2.\end{array}\right.$$

$$-{\Delta }_{S^{N-1}}\omega +|\omega {|}^{p-1}\omega =\lambda \omega ,$$

$$\left\{\begin{array}[]{lll}\displaystyle-\Delta u=u^{p}\quad{\rm in}\quad\Omega\setminus\{0\},\\[5.69054pt] \displaystyle u>0\quad{\rm in}\quad\Omega\setminus\{0\},\qquad u=0\quad{\rm on}\quad\partial\Omega\end{array}\right.$$

$$-\Delta u-u^p=\sum _{|a|=0}^{\infty }{k}_a{D}^a{\delta }_0\mathrm{in}(C_c^{\infty }(\Omega ){)}^{\prime }$$

$$\left\{\begin{array}[]{lll}\displaystyle-\Delta u=u^{p}+k\delta_{0}&{\rm in}\quad\Omega,\\[5.69054pt] \phantom{-\Delta}\displaystyle u=0&{\rm on}\quad\partial\Omega.\end{array}\right.$$

$$P_N(x)={\tilde{c}}_N\frac{x_N}{|x{|}^N},\forall x\in {\mathbb{R}}^N\setminus \{0\},$$

$$⟨\frac{\partial {\delta }_0}{\partial x_N},\zeta ⟩=\frac{\partial \zeta (0)}{\partial x_N},\forall \zeta \in C_0^1(B_1(0)).$$

$$\left\{\begin{array}[]{lll}-\Delta u+g(u)=2k\frac{\partial\delta_{0}}{\partial x_{N}}+j\delta_{0}&{\rm in}\quad\ \ B_{1}(0),\\[7.11317pt] \phantom{-\Delta+g(u)}u=0&{\rm on}\quad\ \ \partial B_{1}(0),\end{array}\right.$$

$${\int }_{B_1(0)}[u(-\Delta )\xi +g(u)\xi ]dx=2k\frac{\partial \xi (0)}{\partial x_N}+j\xi (0),\forall \xi \in C_0^{1,1}(B_1(0)).$$

$${\int }_1^{\infty }g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty$$

$$g(s+t)-g(s)\le c_1\left[\frac{g(s)}{1+|s|}t+g(t)\right],s\in \mathbb{R},t\ge 0$$

$$\underset{t\to 0^{+}}{\lim }\frac{u_{k,j}(te)}{P_N(te)}=2k\mathrm{for}e=(e_1,\cdots ,e_N)\in \partial B_1(0),e_N\ne 0$$

$$\underset{t\to 0^{+}}{\lim }\frac{u_{k,j}(te)}{{\Gamma }_N(te)}=j\mathrm{for}e\in \partial B_1(0),e_N=0;$$

$$\left\{\begin{array}[]{lll}-\Delta u+g(u)=0&{\rm in}\quad\ \ B_{1}(0)\setminus\{0\},\\[5.69054pt] \phantom{-\Delta+g(u)}u=0&{\rm on}\quad\ \ \partial B_{1}(0).\end{array}\right.$$

$$u_k(x^{\prime },x_N)=-u_k(x^{\prime },-x_N),\forall (x^{\prime },x_N)\in B_1(0)\setminus \{0\}.$$

$$\left\{\begin{array}[]{lll}-\Delta u+g(u)=0&{\rm in}\quad\ \ B_{1}^{+}(0),\\[5.69054pt] \phantom{-\Delta+g(u)}u=k\delta_{0}&{\rm on}\quad\ \ \partial B_{1}^{+}(0),\end{array}\right.$$

$${\int }_{B_1^{+}(0)}u(-\Delta )\xi dx+{\int }_{B_1^{+}(0)}g(u)\xi dx=2k\frac{\partial \xi (x_0)}{\partial x_N},\forall \xi \in C_0^{1,1}(B_1^{+}(0)).$$

$$u_{\infty }(x)=\underset{k\to \infty }{\lim }{u}_k(x),\forall x\in B_1(0)\setminus \{0\}.$$

$$\left\{\begin{array}[]{lll}-\Delta u+|u|^{p-1}u=0&{\rm in}\quad B_{1}(0)\setminus\{0\},\\[5.69054pt] \phantom{-\Delta+|u|^{p-1}u}u=0&{\rm on}\quad\partial B_{1}(0)\end{array}\right.$$

$$\underset{t\to 0^{+}}{\lim }{u}_{\infty }(te)t^{\frac{2}{p-1}}=\phi (e),\forall e\in \partial B_1(0),$$

$$\phi (e)>0\mathrm{if}{e}_N>0.$$

$$w(x)=0,\forall x\in \{(x^{\prime },0)\in B^{\ast }\}.$$

$$\left\{\begin{array}[]{lll}-\Delta u+g(u)=f&{\rm in}\quad\ \ B_{1}(0),\\[5.69054pt] \phantom{-\Delta+g(u)}u=0&{\rm on}\quad\ \ \partial B_{1}(0)\end{array}\right.$$

Full text

Anisotropic singularities to semi-linear elliptic equations in a measure framework

Huyuan Chen111chenhuyuan@yeah.net

Department of Mathematics, Jiangxi Normal University,

Nanchang, Jiangxi 330022, PR China

Abstract. The purpose of this article is to study very weak solutions of elliptic equation

[TABLE]

where $k>0$, $j\geq 0$, $B_{1}(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^{N}$ with $N\geq 2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^{1}$ function, $\delta_{0}$ is the Dirac mass concentrated at the origin and $\frac{\partial\delta_{0}}{\partial x_{N}}$ is defined in the distribution sense that

[TABLE]

We obtain that problem (1) admits a unique very weak solution $u_{k,j}$ under the integral subcritical assumption

[TABLE]

Furthermore, we prove that $u_{k,j}$ has anisotropic singularity at the origin and we consider the odd property $u_{k,0}$ and limit of $\{u_{k,0}\}_{k}$ as $k\to\infty$.

We pose the constraint on nonlienarity $g(u)$ that we only require integrability in the principle value sense, due to the singularities only at the origin. This makes us able to search the very weak solutions in a larger scope of the nonlinearity.

††AMS Subject Classifications: 35R06, 35B40, 35Q60. ††Key words: Anisotropic singularity; Very weak solution; Uniqueness.

1. Introduction

As early as in 1977, Lieb-Simon in [10] studied the very weak solutions to equation

[TABLE]

in the description of the Thomas-Fermi theory of electric field potential determined by the nuclear charge and distribution of electrons in an atom, where $\lambda\geq 0$, $t_{+}=\max\{t,0\}$, $m_{i}>0$, $a_{i}\in\mathbb{R}^{3}$ and $\delta_{a_{i}}$ is the Dirac mass at $a_{i}$ for $i=1,2,\cdots,n$. In fact, the solution of (1.1) turns out to be a classical singular solution of

[TABLE]

As a fundamental PDE’s model, the isolated singular problem

[TABLE]

has been studied extensively, where $\Omega$ is a domain in $\mathbb{R}^{N}$ with $N\geq 3$. Brezis-Véron in [3] showed that problem (1.2) admits no isolated singular solution when $p\geq\frac{N}{N-2}$. A complete classification of the isolated singularities at the origin for (1.2) was given by Véron in [18] when $\frac{N+1}{N-1}\leq p<\frac{N}{N-2}$ as follows:

(i) either $|x|^{\frac{2}{p-1}}u(x)$ converges to a constant which can take only two values $\pm[\frac{2}{p-1}(\frac{2p}{p-1}-N)]^{1/(p-1)}$ as $x\to 0$,

(ii) or $|x|^{N-2}u(x)$ converges to a constant $c_{N}k$, and the couple $(u,\,k)$ is related to the weak solution of

[TABLE]

where $c_{N}$ is the normalized constant of the fundamental solution $\Gamma_{N}$ of $-\Delta u=\delta_{0}$ in $\mathbb{R}^{N}$, that is,

[TABLE]

For $1<p<\frac{N+1}{N-1}$, the above classification holds under the restriction of nonnegative solutions of (1.2) and all above singular solutions are isotropic. A conjucture states that there is a rich structure of the singularities for (1.2) without the restriction of nonnegativity for $1<p<\frac{N+1}{N-1}$. Véron in [18] partially answered this conjucture and showed that the anisotropic singular solutions could be constructed by considering the following nonlinear eigenvalue problem on $S^{N-1}$

[TABLE]

where $S^{N-1}$ is the sphere of unit ball in $\mathbb{R}^{N}$ and $\Delta_{S^{N-1}}$ is the Laplace-Beltrami operator. Later on, Chen-Matano-Véron in [5] provideds the anisotropic singular solutions of (1.2) by analyzing the corresponding Laplace-Beltrami equations in the sphere. More singularities analysis see the references [12, 16, 17, 20].

In contrast with the absorption nonlinearity, the isolated singular solutions of elliptic problem with source nonlinearity

[TABLE]

was classified by Lions in [11], by using Schwartz’s Theorem to build that

[TABLE]

and then by choosing suitable test functions in $C^{\infty}_{c}(\Omega)$ to kill all $D^{a}\delta_{0}$ with $a$ multiple index and $|a|\geq 1$, finally building the connections with the weak solutions of

[TABLE]

Lions in [11] proved that when $p\in(1,\frac{N}{N-2})$ with $N\geq 3$, any solution of (1.4) is a weak solution of (1.6) for some $k\geq 0$, and when $p\geq\frac{N}{N-2}$, the parameter $k=0$. Essentially, $D^{a}\delta_{0}$ with $|a|\geq 1$ is killed in (1.5) because it is anisotropic singular source, that is, this source makes the solutions anisotropic singular. For instance, the fundamental solution of $-\Delta u=D^{a}\delta_{0}$ with $a=(0,\cdots,0,1)\in\mathbb{R}^{N}$ is

[TABLE]

where $\tilde{c}_{N}$ are the normalized constants, see [4]. Obviously, $P_{N}$ has anisotropic singularities.

Inspired by (1.5), we observe that $D^{a}\delta_{0}$ with $|a|\geq 1$ could provide anisotropic source and our motivation in this article is to make use of this kind of sources to construct anisotropic singular solutions for (1.2) and to find the criteria for more general nonlinearity. Let $B_{1}(0)$ be the unit ball centered at the origin in $\mathbb{R}^{N}$ with $N\geq 2$, denote by $\delta_{0}$ the Dirac mass concentrated at the origin, to be convenient, $\frac{\partial\delta_{0}}{\partial x_{N}}=D^{a}\delta_{0}$ with $a=(0,\cdots,0,1)\in\mathbb{R}^{N}$ and then

[TABLE]

So our concern is to study the isolated singular solutions of semilinear elliptic problem

[TABLE]

where parameters $k>0$ and $j\geq 0$.

Before we state our main results, we introduce the definition of weak solution to (1.8) as follows.

Definition 1.1**.**

A function $u\in L^{1}(B_{1}(0))$ is called a very weak solution of (1.8), if $g(u)$ be integrable in the principle value sense near the origin, $g(u)\in L^{1}(B_{1}(0),\,|x|dx)$ and

[TABLE]

We note that $k\frac{\partial\delta_{0}}{\partial x_{N}}$ and $j\delta_{0}$ are both visible in the distributional identity (1.9). When $k=0$, the definition of very weak solution of (1.8) requires $g(u)\in L^{1}(B_{1}(0),\rho dx)$, see the references [19]. Since both sources have the support at the origin, so in this article we pose the constraint for the very weak solution that $g(u)$ is integrable in the principle value sense near the origin means, i.e. $\lim_{\epsilon\to 0^{+}}\int_{B_{1}(0)\setminus B_{\epsilon}(0)}g(u)\,dx$ exists, that provides higher possibility for searching the sign-changing singular solutions of (1.8).

Now we are ready to state our first theorem on the existence and asymptotic behavior of very weak solutions to problem (1.8).

Theorem 1.1**.**

Assume that $k>0$, $j\geq 0$, $\Gamma_{N},$ $P_{N}$ is given in (1.3), (1.7) respectively, and the nonlinearity $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and Lipschitz function satisfying

[TABLE]

and

[TABLE]

for some $c_{1}>0$.

Then (1.8) admits a very weak solution $u_{k,j}$ such that

$(i)$*    $u_{k,j}\geq 0$ in $B_{1}^{+}(0)=\{x=(x^{\prime},x_{N})\in B_{1}(0):\,x_{N}>0\}$;*

$(ii)$*   $u_{k,j}$ has the following singularity at the origin*

[TABLE]

and

[TABLE]

$(iii)$*   $u_{k,j}$ is a classical solution of*

[TABLE]

In the particular case that $j=0$, denoting $u_{k}$ the solution $u_{k,0}$ of (1.8) with $j=0$, $u_{k}$ is $x_{N}$-odd, that is,

[TABLE]

Furthermore, the $x_{N}$-odd very weak solution is unique.

We note that $g(s)=|s|^{p-1}s$ with $p\in(0,\frac{N}{N-1})$ verifies (1.10) and (1.11). It follows by (1.12) that $|u_{k,j}|^{p-1}u_{k,j}\in L^{1}(B_{1}(0))$, but for $p\in[\frac{N}{N-1},\frac{N+1}{N-1})$ and $k>0$, $|u_{k,j}|^{p-1}u_{k,j}\not\in L^{1}(B_{1}(0))$. We can’t able to obtain the uniqueness of the very weak solution to (1.8), due to the failure of application the Kato’s inequality, which requires that the nonlinearity term $g(u)\in L^{1}(B_{1}(0))$.

For the existence of very weak solutions, the normal method is to approximate the Radon measure by $C^{1}_{0}$ functions and consider the limit of the corresponding classical solutions. When $k>0$, we use a sequence of Dirac measures $\frac{\delta_{te_{N}}-\delta_{-te_{N}}}{t}$ to approach the source $\frac{\partial\delta_{0}}{\partial x_{N}}$ and in this approximation, the biggest challenge is to find a uniform estimate. To overcome this difficulty, our strategy is to to consider $x_{N}$-odd property of solutions when $j=0$ to derive the uniform bound in this approaching process.

We next state the nonexistence of very weak solution of (1.8).

Theorem 1.2**.**

Assume that $k>0$, $j=0$, $g(s)=|s|^{p-1}s$ with $p\geq\frac{N+1}{N-1}$, then there is no $x_{N}$-odd weak solution for problem (1.8).

Our strategy here is to make use of $x_{N}-$odd property to deduce (1.8) into boundary data problem

[TABLE]

in the distributional sense that

[TABLE]

It is interesting but still open to derive the nonexistence when $j\not=0$.

Finally, we analyze the limit of the weak solutions $\{u_{k}\}_{k}$ as $k\to\infty$. From the monotonicity of $u_{k}$ in $B_{1}^{+}(0)$ and $B_{1}^{-}(0)$ respectively, the limit of $\{u_{k}\}_{k}$ as $k\to\infty$ exists in $B_{1}(0)\setminus\{0\}$, denote

[TABLE]

Theorem 1.3**.**

Assume that $k>0$, $j=0$, $g(s)=|s|^{p-1}s$ with $p>1$, $u_{k}$ is the unique $x_{N}$-odd very weak solution of (1.8) and $u_{\infty}$ is given by (1.16). Then $u_{\infty}$ is a classical solution of

[TABLE]

and satisfies that

[TABLE]

where $\varphi:\partial B_{1}(0)\to\mathbb{R}$ is a continuous $x_{N}$-odd function such that for unit vector $e=(e_{1},\cdots,e_{N})$

[TABLE]

The rest of this paper is organized as follows. In Section 2, we analyze the $x_{N}$-odd property. Section 3 is devoted to study the $x_{N}$-odd very weak solution in subcritical case when $j=0$ and the nonexistence the $x_{N}$-odd very weak solution in the subcritical case. In Section 4, we consider the limit of the unique $x_{N}$-odd weak solutions $u_{k}$ of (1.8) with $j=0$ as $k\to\infty$. Finally, we prove the existence of non $x_{N}$-odd very weak solution when $j>0$ in Section 5.

2. Preliminary

We start this section from the $x_{N}$-odd property. Notice that an $x_{N}$-odd function $w$ defined in a $x_{N}$-symmetric domain $B^{*}$ satisfies

[TABLE]

In what follows, we denote by $c_{i}$ a generic positive constant.

Lemma 2.1**.**

Assume that $f\in C^{1}(\overline{B_{1}(0)})$ is an $x_{N}$-odd function, $g\in C^{1}(\mathbb{R})$ is an odd and nondecreasing function.

Then

[TABLE]

admits a unique classical solution $w_{f}$. Moreover,

$(i)$* $w_{f}$ is $x_{N}$-odd in $B_{1}(0)$;*

$(ii)$* assume more that $f\geq 0$ in $B_{1}^{+}(0)=\{x\in B_{1}(0):\ x_{N}>0\}$ and $f\not\equiv 0$ in $B_{1}^{+}(0)$, then $w_{f}>0$ in $B_{1}^{+}(0)$.*

Proof. Since $g$ is an odd and nondecreasing function, then it is standard to obtain the existence of solution by the method of super and sub solutions.

Uniqueness. Let $w_{f},\tilde{w}_{f}$ be two solutions of (2.1), $w=w_{f}-\tilde{w}_{f}$ in $B_{1}(0)$ and $A_{+}=\{x\in B_{1}(0):\,w(x)>0\}$. We claim that $A_{+}=\emptyset$. In fact, if $A_{+}\not=\emptyset$, we observe that $w$ is a solution of

[TABLE]

By applying Maximum Principle, we have that

[TABLE]

which contradicts the definition of $A_{+}$. Then $A_{+}=\emptyset$. Similarly, $\{x\in B_{1}(0):w(x)<0\}$ is empty. Therefore, $w_{f}=\tilde{w}_{f}$ in $B_{1}(0)$ and the uniqueness holds.

$(i)$* * Let $v(x^{\prime},x_{N})=-w_{f}(x^{\prime},-x_{N})$, and by direct computation, we derive that

[TABLE]

then $v$ is a solution of (2.1). It follows from the uniqueness of solution of (2.1) that

[TABLE]

$(ii)$* * We observe that $w_{f}=0$ on $\partial B_{1}^{+}(0)$ and then $w_{f}$ is a classical solution of

[TABLE]

We now claim that $w_{f}\geq 0$ in $B_{1}^{+}(0)$. Indeed, if not, we have that $\min_{B_{1}^{+}(0)}w_{f}<0$. Let $A_{-}=\{x\in B_{1}^{+}(0):\ w_{f}(x)<\frac{1}{2}\min_{B_{1}^{+}(0)}w_{f}\}$, then $\psi:=w_{f}+\frac{1}{2}\min_{B_{1}^{+}(0)}w_{f}$ satisfies

[TABLE]

By Maximum Principle, we have that

[TABLE]

which contradicts the definition of $A_{-}$.

We next prove that $w_{f}>0$ in $B_{1}^{+}(0)$. Problem (2.2) could be seen as

[TABLE]

where $\phi(x)=\frac{g(w_{f}(x))}{w_{f}(x)}$ if $w_{f}(x)\not=0$ and $\phi(x)=g^{\prime}(0)$ if $w_{f}(x)=0$. It follows by $g\in C^{1}(\mathbb{R})$ that $\phi$ is continuous and $\phi\geq 0$ in $B_{1}^{+}(0)$. Since $f\geq 0$ in $B_{1}^{+}(0)$, it follows by strong maximum principle that $w_{f}>0$ or $w_{f}\equiv 0$ in $B_{1}^{+}(0)$, then we exclude $w_{f}\equiv 0$ in $B_{1}^{+}(0)$ by the fact that $f\not\equiv 0$ in $B_{1}^{+}(0)$.   $\Box$

Corollary 2.1**.**

Assume that $f\in C(\overline{B_{1}(0)})$ is an $x_{N}$-odd function such that $f\geq 0$ in $B_{1}^{+}(0)$ and $g\in C^{1}(\mathbb{R})$ is an odd and nondecreasing function. Let $w_{f}$ be the solution of (2.1) and $\mathbb{G}_{B_{1}(0)}[f]$ be the unique solution of

[TABLE]

Then $\mathbb{G}_{B_{1}(0)}[f]$ is $x_{N}$-odd and

[TABLE]

Proof. By applying Lemma 2.1 with $g\equiv 0$, we have that $\mathbb{G}_{B_{1}(0)}[f]$ is $x_{N}$-odd and

[TABLE]

Denote $v=\mathbb{G}_{B_{1}(0)}[f]-w_{f}$, then $v=0$ on $\partial B_{1}^{+}(0)$ and $-\Delta v=g(w_{f})\geq 0$, by Maximum Principle, we have that $v\geq 0$ in $B_{1}^{+}(0)$, which ends the proof.   $\Box$

Corollary 2.2**.**

Assume that $f\in C(\overline{B_{1}(0)})$ is an $x_{N}$-odd function such that $f\geq 0$ in $B_{1}^{+}(0)$ and $g_{1},\,g_{2}\in C^{1}(\mathbb{R})$ are odd and nondecreasing functions satisfying

[TABLE]

Let $w_{f,i}$ be the solutions of (2.1) replaced by $g$ by $g_{i}$ with $i=1,\,2$ respectively.

Then

[TABLE]

Proof. By applying Lemma 2.1 and Corollary 2.1, we have that $w_{f,1},w_{f,2}$ are $x_{N}$-odd and are nonnegative in $B_{1}^{+}(0)$. We denote $w=w_{f,1}-w_{f,2}$, then $w$ satisfies that

[TABLE]

We first claim that $w\geq 0$ in $B_{1}^{+}(0)$. If not, we have that $\min_{B_{1}^{+}(0)}w<0$. Let us define

[TABLE]

then $\tilde{w}:=w+\frac{1}{2}\min_{B_{1}^{+}(0)}w$ satisfies that

[TABLE]

By Maximum Principle, we have that

[TABLE]

which contradicts the definition of $A_{-}$.   $\Box$

Proposition 2.1**.**

Let $f_{1}$ and $f_{2}$ be $x_{N}$-odd functions in $C^{1}_{loc}(\overline{B_{1}(0)}\setminus\{0\})\cap L^{1}(B_{1}(0),|x|dx)$ satisfying $f_{2}\geq f_{1}\geq 0$ in $B_{1}^{+}(0)$, then the problem

[TABLE]

admits a unique $x_{N}$-odd weak solution $u_{i}$ with $i=1,2$ in the sense that $u_{i}\in L^{1}(B_{1}(0))$,

[TABLE]

Moreover, $u_{i}$ is a classical solution of

[TABLE]

and

[TABLE]

Proof. Uniqueness. Let $w$ satisfy that

[TABLE]

for any $\xi\in C^{1,1}_{0}(B_{1}(0))$ such that $\xi(0)=0$. Since $0\in L^{1}(B_{1}(0))$, then the test function could be improved into $C^{1,1}_{0}(B_{1}(0))$ without the restriction that $\xi(0)=0$. Denote by $\eta_{1}$ the solution of

[TABLE]

Then $\eta_{1}\in C^{1,1}_{0}(B_{1}(0))$ and then

[TABLE]

This implies $w=0$ in $B_{1}(0)\setminus\{0\}$.

Existence. Let $f_{i,\epsilon}=f_{i}\chi_{B_{1}(0)\setminus B_{\epsilon}(0)}$, where $i=1,2$, $\chi_{B_{1}(0)\setminus B_{\epsilon}(0)}=1$ in $B_{1}(0)\setminus B_{\epsilon}(0)$ and $\chi_{B_{1}(0)\setminus B_{\epsilon}(0)}=0$ in $B_{\epsilon}(0)$, then $f_{i,\epsilon}$ is an $x_{N}$-odd function in $L^{\infty}(B_{1}(0))$ such that $f_{2,\epsilon}\geq f_{1,\epsilon}\geq 0$ in $B^{+}_{1}(0)$, then

[TABLE]

admits a unique solution $u_{i,\epsilon}$ satisfying

[TABLE]

Moreover, from Lemma 2.1 with $g\equiv 0$, we have that

[TABLE]

[TABLE]

and for any $x\in B_{1}^{+}(0)$,

[TABLE]

where $\tilde{y}=(y^{\prime},-y_{N})$. We observe that $G_{B_{1}(0)}(x,y)=\frac{c_{N}}{|x-y|^{N-2}}-\tilde{G}_{B_{1}(0)}(x,y)$, where $\tilde{G}_{B_{1}(0)}(x,y)$ is a harmonic function in $B_{1}(0)$ with the boundary value $\frac{c_{N}}{|x-y|^{N-2}}$ for $y\in\partial B_{1}(0)$. Therefore, for $x,y\in B_{1}^{+}$, we have that $\tilde{G}_{B_{1}(0)}(x,y)-\tilde{G}_{B_{1}(0)}(x,\tilde{y})\geq 0$ and then

[TABLE]

Moreover, we see that

[TABLE]

and

[TABLE]

then

[TABLE]

Therefore, we obtain a uniform bound for $u_{2,\epsilon}$, together with monotonicity, then passing to the limit as $\epsilon\to 0^{+}$ in (2.10), we deduces that $u_{i}:=\lim_{\epsilon\to 0^{+}}u_{i,\epsilon}$ is a weak solution of (2.3) and it follows from standard stability theorem that $u_{i}$ is a classical solution of (2.5) for $i=1,2$. $\Box$

By direct extension, we have the following corollary.

Corollary 2.3**.**

Let $f$ be an $x_{N}$-odd function in $C^{1}_{loc}(\overline{B_{1}(0)}\setminus\{0\})\cap L^{1}(B_{1}(0),|x|^{i}dx)$ with $i\in\mathbb{Z}$ satisfying $f\geq 0$ in $B_{1}^{+}(0)$, then the problem

[TABLE]

admits a unique $x_{N}$-odd weak solution $u_{f}$ with $i\in\mathbb{N}$, $i\geq 2$ in the sense that $u_{f}\in L^{1}(B_{1}(0),|x|^{i-1}dx)$,

[TABLE]

for any $\xi\in C^{1,1}_{0}(B_{1}(0))$ s. t. $|\xi(x)|\leq c|x|^{i}$ for any $x\in B_{1}(0)$ and some $c>0$. Moreover,

[TABLE]

and $u_{f}$ is a classical solution of

[TABLE]

Remark 2.1**.**

The arguments in Proposition 2.1 and Corollary 2.3 hold when $B_{1}(0)$ is replaced by $\mathbb{R}^{N}$ and the boundary condition is done by

[TABLE]

In order to study the convergence of weak solutions, we recall the definition and basic properties of the Marcinkiewicz spaces.

Definition 2.1**.**

Let $\Omega\subset\mathbb{R}^{N}$ be a domain and $\mu$ be a positive Borel measure in $\Omega$. For $\kappa>1$, $\kappa^{\prime}=\kappa/(\kappa-1)$ and $u\in L^{1}_{loc}(\Omega,d\mu)$, we set

[TABLE]

and

[TABLE]

$M^{\kappa}(\Omega,d\mu)$ is called the Marcinkiewicz space with exponent $\kappa$ or weak $L^{\kappa}$ space and $\|.\|_{M^{\kappa}(\Omega,d\mu)}$ is a quasi-norm. The following property holds.

Proposition 2.2**.**

[1]** Assume that $1\leq q<\kappa<\infty$ and $u\in L^{1}_{loc}(\Omega,d\mu)$. Then there exists $C(q,\kappa)>0$ such that

[TABLE]

for any Borel set $E$ of $\Omega$.

3. $x_{N}$-odd very weak solution with $j=0$

3.1. Existence of very weak solution

In this subsection, we prove the existence and uniqueness of very weak solution to problem (1.8) when $j=0$.

Theorem 3.1**.**

Assume that $k>0$ $j=0$, the nonlinearity $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and Lipchitz continuous function satisfying (1.10).

Then (1.8) admits a unique $x_{N}$-odd very weak solution $w_{k}$ such that

$(i)$*    $w_{k^{\prime}}\geq w_{k}\geq 0$ in $B_{1}^{+}(0)$ for $k^{\prime}\geq k>0$;*

$(ii)$*   $w_{k}$ satisfies (1.12);*

$(iii)$*   $w_{k}$ is a classical solution of (1.14).*

Before proving Theorem 3.1, we need following preliminaries.

Lemma 3.1**.**

Assume that $k>0$, the nonlinearity $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and Lipchitz continuous function and $u$ is a very weak solution of (1.8), locally bounded in $B_{1}(0)\setminus\{0\}$. Then $u$ is a classical solution of (1.14).

Proof. Since $\frac{\partial\delta_{0}}{\partial x_{N}},\,\delta_{0}$ have the support in $\{0\}$, so for any open sets $O_{1},O_{2}$ in $B_{1}(0)$ such that $\bar{O}_{1}\subset O_{2}\subset\bar{O}_{2}\subset B_{1}(0)\setminus\{0\},$ $u$ is a very weak solution of

[TABLE]

where $u\in L^{\infty}(O_{2})$ and $g(u)\in L^{\infty}(O_{2})$. By standard regularity results, we have that $u$ satisfies (3.1) in $O_{1}$ in the classical sense.   $\Box$

For $n\in\mathbb{N}$, we consider $\{g_{n}\}_{n}$ of $C^{1}$, odd, nondecreasing functions defined in $\mathbb{R}$ satisfying

[TABLE]

Proposition 3.1**.**

Let $g_{n}$ be defined by (3.2) and

[TABLE]

Then for any $n\in\mathbb{N}$, problem

[TABLE]

admits a unique very weak solution $w_{k,n,t}$, which is a classical solution of

[TABLE]

Moreover, $w_{k,n,t}$ is $x_{N}$-odd in $B_{1}(0)\setminus\{te_{N},-te_{N}\}$,

[TABLE]

and

[TABLE]

Proof. We observe that $\mu_{t}$ is a bounded Radon measure and $g_{n}$ is bounded, Lipschitz continuous and nondecreasing, then it follows from [19, Theorem 3.7] under the integral subcritical assumption (1.10) replaced $\frac{N}{N-1}$ by $\frac{N}{N-2}$ for $N\geq 3$ and the Kato’s inequality, problem (3.3) admits a unique weak solution $w_{k,n,t}$. Moreover, $w_{k,n,t}$ could be approximated by the classical solutions $\{w_{k,n,t,m}\}$ to problem

[TABLE]

where

[TABLE]

and $\{\sigma_{m}\}_{m}$ is a sequence of radially symmetric, nondecreasing smooth functions converging to $\delta_{0}$ in the distribution sense. Furthermore,

[TABLE]

Since $\mu_{t,m}$ is $x_{N}$-odd and nonnegative in $B_{1}^{+}(0)$, so is $w_{k,n,t,m}$ by Lemma 2.1. We observe that $(k+1)\sigma_{m}\geq k\sigma_{m}$, it follows from Lemma 2.1 that

[TABLE]

Since

[TABLE]

then it follows from Corollary 2.2 that

[TABLE]

From the proof of Theorem 2.9 in [19], we know that

[TABLE]

where $q\in[1,\frac{N}{N-2})$. By regularity results, any compact set $K$ and open set $O$ in $B_{1}(0)$ such that $K\subset O$, $\bar{O}\cap\{te_{N},-te_{N}\}=\emptyset$, there exist $c_{6},c_{7}>0$ independent of $m$ such that

[TABLE]

Therefore, up to some subsequence, there exists a measurable function $\tilde{w}$ such that

[TABLE]

where $q\in[1,\frac{N}{N-2})$. Then $\tilde{w}$ is $x_{N}$-odd, nonnegative in $B_{1}^{+}(0)$ and

[TABLE]

Passing to the limit in (3.6) as $m\to\infty$, we deduce that $\tilde{w}$ is a weak solution of (3.3). By the uniqueness of weak solution of (3.3), we obtain that $w_{n,t}=\tilde{w}$. Therefore, $w_{n,t}$ is $x_{N}$-odd, nonnegative in $B_{1}^{+}(0)$ and it follows from (3.7) and (3.4) that

[TABLE]

and

[TABLE]

This ends the proof.  $\Box$

We next passing to the limit of weak solutions as $t\to 0^{+}$.

Proposition 3.2**.**

Let $g_{n}$ be defined by (3.2). Then for any $n\in\mathbb{N}$, problem

[TABLE]

*admits a unique very weak solution $w_{k,n}$. Moreover,

$(i)$ $w_{k,n}$ is $x_{N}$-odd for any $n\in\mathbb{N}$ in $B_{1}(0)\setminus\{0\}$ and*

[TABLE]

and

[TABLE]

$(ii)$* $w_{k,n}$ is a classical solution of*

[TABLE]

Proof. It follows from Proposition 3.1 that problem (3.3) admits a unique very weak solution $w_{k,n,t}$, that is,

[TABLE]

On the one hand, we have that

[TABLE]

On the other hand, by Proposition 3.1, we have that

[TABLE]

By regularity results, for $\sigma\in(0,1)$ and any compact set $K$ and open set $O$ in $B_{1}(0)$ such that $K\subset O$, $\bar{O}\cap\{te_{N}:\ t\in(-\frac{1}{2},\frac{1}{2})\}=\emptyset$, there exist $c_{8},c_{9}>0$ independent of $t$ such that

[TABLE]

Moreover, by [4, Proposition 3.3], $\{\mathbb{G}_{B_{1}(0)}[\mu_{t}]\}$ is uniformly bounded in $M^{\frac{N}{N-1}}(B_{1}(0),dx)$ if $N\geq 3$ and is uniformly bounded in $M^{\frac{2}{1+\sigma}}(B_{1}(0),dx)$ for any $\sigma\in(0,\frac{1}{2})$ if $N=2$. Therefore, $\{w_{k,n,t}\}_{t}$ is relatively compact in $L^{p}({B_{1}(0)})$ for any $p\in[1,\frac{N}{N-1})$. There exists $w_{k,n}\in L^{1}(B_{1}(0))$ such that

[TABLE]

which implies that

[TABLE]

Therefore, up to some subsequence, passing to the limit as $t\to 0^{+}$ in the identity (3.9), it follows that $w_{k,n}$ is a very weak solution of (3.8). Moreover, $w_{k,n}$ is $x_{N}$-odd and nonnegative in $B_{1}^{+}(0)$.

Uniqueness. Let $v_{n}$ be a weak solution of (3.8) and then $\varphi_{n}:=w_{k,n}-v_{n}$ is a very weak solution to

[TABLE]

By Kato’s inequality [19, Theorem 2.4] (see also [8, 9, 17]),

[TABLE]

Taking $\xi=\mathbb{G}_{B_{1}(0)}[1]$, we have that

[TABLE]

then $\varphi_{n}=0$ a.e. in $B_{1}(0)$. Then the uniqueness is obtained.   $\Box$

The next estimate plays an important role in $w_{k,n}\to w_{k}$ in $L^{p}({B_{1}(0)})$ with $p\in[1,\frac{N+1}{N-1})$.

Lemma 3.2**.**

There exists $c_{10}>0$ such that

[TABLE]

and

[TABLE]

Proof. We observe that

[TABLE]

and for $x,y\in{B_{1}(0)}$, $x\not=y$,

[TABLE]

where $\tilde{G}_{B_{1}(0)}$ is a harmonic function in $B_{1}(0)\times B_{1}(0)$. Then

[TABLE]

Therefore, we have that

[TABLE]

*Proof of (3.10). * Let $E$ be a Borel set of $B_{1}(0)$ with $|E|>0$, then there exists $r_{1}\in(0,1]$ such that

[TABLE]

We deduce that

[TABLE]

By the definition of Marcinkiewicz space, we have that

[TABLE]

*Proof of (3.11). * Let $E$ be a Borel set of $B_{1}(0)$ with $|E|>0$, then there exists $r_{2}\in(0,1]$ such that

[TABLE]

Since

[TABLE]

we deduce that

[TABLE]

This ends the proof.   $\Box$

Lemma 3.3**.**

Assume that $g:[0,\infty)\to[0,\infty)$ is continuous, nondecreasing and verifies (1.10). Then for $q\geq\frac{N+1}{N-1}$,

[TABLE]

Proof. Since

[TABLE]

and by (1.7),

[TABLE]

then

[TABLE]

The proof is complete. $\Box$

Now we are ready to prove Theorem 3.1.

Proof Of Theorem 3.1. Existence. Let $\{g_{n}\}$ be a sequence of $C^{1}$ nondecreasing functions defined by (3.2). It follows that $\{g_{n}\}$ is a sequence of odd, bounded and nondecreasing Lipschiz continuous functions.

By Proposition 3.2, problem (3.8) admits a unique $x_{N}$-odd weak solution $w_{k,n}$ such that

[TABLE]

and

[TABLE]

For $\beta\in(0,1)$, any compact set $K$ and open set $O$ in $B_{1}(0)$ satisfying $K\subset O$, $0\not\in\bar{O}$, we have that

[TABLE]

where $c_{20}>0$. Therefore, up to some subsequence, there exists $w_{k}$ such that

[TABLE]

Then $\{g_{n}(w_{k,n})\}$ converges to $g(w_{k})$ a.e. in $B_{1}(0)$. By Lemma 3.2, we have that

[TABLE]

[TABLE]

by Proposition 2.2 and $\mathbb{G}_{B_{1}(0)}[\frac{\partial\delta_{0}}{\partial x_{N}}]\in M^{\frac{N+1}{N-1}}(B_{1}(0),|x|dx)$, we have that

[TABLE]

where

[TABLE]

For any Borel set $E\subset{B_{1}(0)}$, we have that

[TABLE]

On the other hand,

[TABLE]

Thus,

[TABLE]

where $c_{26}=\frac{c_{24}}{\frac{N+1}{N-1}+1}(2k)^{2+\frac{N+1}{N-1}}$. By assumption (1.10) and Lemma 3.3, we have that $T^{-\frac{N+1}{N-1}}g(T)\to 0$ as $T\to\infty$, therefore,

[TABLE]

Notice that the quantity on the right-hand side tends to [math] when $\lambda\to\infty$. The conclusion follows: for any $\epsilon>0$, there exists $\lambda>0$ such that

[TABLE]

For $\lambda$ fixed, there exists $\delta>0$ such that

[TABLE]

which implies that $\{g_{n}\circ w_{k,n}\}$ is uniformly integrable in $L^{1}(B_{1}(0),|x|dx)$. Then $g_{n}\circ w_{k,n}\to g\circ w_{k}$ in $L^{1}(B_{1}(0),|x|dx)$ by Vitali convergence theorem, see [7].

Furthermore, for any $\xi\in C^{1,1}_{0}(B_{1}(0))$, we know that

[TABLE]

and it follows the odd prosperity of $w_{k,n}$, $g_{n}$ and $g$, that

[TABLE]

then

[TABLE]

Then passing to the limit as $n\to+\infty$ in the identity (3.13), it implies that

[TABLE]

Thus, $w_{k}$ is a very weak solution of (1.8). The regularity results follows by Lemma 3.1.

Proof of $(i)$. Since $w_{k,n}$ is $x_{N}$-odd in $B_{1}(0)\setminus\{0\}$ and $w_{k}=\lim_{n\to+\infty}w_{k,n}$ in $B_{1}(0)\setminus\{0\}$, then it implies that $w_{k}$ is $x_{N}$-odd in $B_{1}(0)\setminus\{0\}$. By the fact that $w_{k+1,n}\geq w_{k,n}\geq 0$ in $B_{1}^{+}(0)$, it follows that

[TABLE]

Proof of $(ii)$. We observe that

[TABLE]

then $g_{n}(w_{k,n})\leq g(2k\mathbb{G}_{B_{1}(0)}[\frac{\partial\delta_{0}}{\partial x_{N}}])$ for $x\in B_{1}^{+}(0)$ and let $w_{g}$ be the unique solution of (2.3) with $f=g(2k\mathbb{G}_{B_{1}(0)}[\frac{\partial\delta_{0}}{\partial x_{N}}])$, we have that

[TABLE]

Then $w_{k}$ satisfies (3.17). From Proposition 2.1, it infers that for $x=te$ with $t\in(0,\frac{1}{2})$ $e=(e_{1},\cdots,e_{N})\in\partial B_{1}(0)$ with $e_{N}>0$,

[TABLE]

For $y\in B_{\frac{t}{2}}(te)$, we have that $|y-te|\geq\frac{t}{2}$ and $\frac{t}{2}\leq|y|\leq\frac{3t}{2}$, then

[TABLE]

where $r=t^{-\frac{1}{N-1}}$.

For $y\in B_{\frac{t}{2}}(0)$, we have that $|y-te|\geq\frac{t}{2}$ and

[TABLE]

where we have used (1.10).

For $y\in B_{1}^{+}(0)\setminus(B_{\frac{t}{2}}(0)\cup B_{\frac{t}{2}}(te))$, we have that $|y-te|\geq\frac{t}{2},|y|\geq\frac{t}{2}$ and

[TABLE]

where $\tau=2c_{13}kt^{1-N}$. Then for any $e=(e_{1},\cdots,e_{N})\in\partial B_{1}(0)$ and $e_{N}\not=0$, we have that

[TABLE]

and

[TABLE]

By (3.17),

[TABLE]

and $x_{N}$-odd property of $w_{k}$, we derive that for any $e\in\partial B_{1}(0)$,

[TABLE]

Finally, we prove the uniqueness. Let $u_{k},\,v_{k}$ be two $x_{N}-$odd solutions of (1.8), from Lemma 3.4, $u_{k},\,v_{k}$ are two solutions of

[TABLE]

Form the uniqueness of the very weak solution to (3.19), see [14, Theorem 2.1], we obtain $u_{k}=v_{k}$ in $B_{1}^{+}(0)$ and combine the $x_{N}-$odd property we obtain the uniqueness of the very weak solution of (1.8) with $j=0$. This ends the proof.  $\Box$

3.2. Nonexistence

This subsection is devoted to obtain the nonexistence of very weak solutions of (1.8) in the supercritical case.

Lemma 3.4**.**

Assume that $k>0$, the nonlinearity $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and Lipchitz continuous function. Let $u$ be an $x_{N}$-odd very weak solution of (1.8).

Then $u$ is a very weak solution of (3.19).

Proof. From Lemma 3.1 and the $x_{N}$-odd property, we have that

[TABLE]

For any $x_{N}$-odd function $\xi\in C^{1,1}_{0}(B_{1}(0))$, we deduce that

[TABLE]

where $c_{36}>0$. Since $g(u)\in L^{1}(B_{1}(0),|x|dx)$, then for $x_{N}$-odd function $\xi\in C^{1,1}_{0}(B_{1}(0))$, we have that $g(u)\xi\in L^{1}(B_{1}(0))$ and the weak solution $u$ satisfies that

[TABLE]

for $x_{N}-$odd function $\xi\ {\rm in}\ C^{1,1}_{0}(B_{1}(0))$. By $x_{N}$-odd property, we have that

[TABLE]

which, combined with (3.20), implies that

[TABLE]

So we have that $u$ is a weak solution of (3.19).  $\Box$

Proof of Theorem 1.2. When $g(s)=|s|^{p-1}s$ with $p\geq\frac{N+1}{N-1}$ and $k>0$, [14, Theorem 3.1] shows that the nonnegative solution of

[TABLE]

has removable singularity at origin, so problem (3.19) has no very weak solution, which contradicts Lemma 3.4. Then (1.8) has no $x_{N}$-odd weak solution when $g(s)=|s|^{p-1}s$ with $p\geq\frac{N+1}{N-1}$.  $\Box$

4. Strongly anisotropic singularity for $p\in(1,\frac{N+1}{N-1})$

In this section, we consider the limit of weak solutions $w_{k}$ as $k\to\infty$ to

[TABLE]

where $p\in(1,\frac{N+1}{N-1})$. By Theorem 1.1, we observe that the mapping $k\mapsto w_{k}$ is nondecreasing in $B_{1}^{+}(0)$, then $\lim_{k\to+\infty}w_{k}(x)$ exists for any $x\in B_{1}(0)\setminus\{0\}$, denoting

[TABLE]

For $w_{\infty}$, we have the following result.

Proposition 4.1**.**

Let $p\in(1,\frac{N+1}{N-1})$, then $w_{\infty}$ is $x_{N}$-odd,

[TABLE]

for some $\lambda_{0}>0$ and $w_{\infty}$ is a classical solution of

[TABLE]

Proof. In order to obtain the asymptotic behavior of $w_{\infty}$ near the origin, we construct the function

[TABLE]

For $p\in(1,\frac{N+1}{N-1})$, there exists $\lambda_{0}>0$ such that $\lambda_{0}v_{p}$ is a super solution of (4.3) and $-\lambda_{0}v_{p}$ is a sub solution of (4.3).

It follows by Theorem 1.1 that $w_{k}$ is a classical solution of (4.3) satisfying (1.12), hence by Comparison Principle, for any $k$, there exists $r_{k}\in(0,1)$ small enough such that

[TABLE]

By Comparison Principle, we have that

[TABLE]

Since $k$ is arbitrary, we deduce that

[TABLE]

Therefore, from standard Stability Theorem, we derive that $w_{\infty}$ is a classical solution of (4.3).   $\Box$

We next do a precise bound for $w_{\infty}$ to prove (1.18).

Lemma 4.1**.**

Let $p\in(1,\frac{N+1}{N-1})$ and $w_{\infty}$ be defined by (4.2), then $w_{\infty}$ satisfies (1.18).

Proof. We claim that

[TABLE]

We have that

[TABLE]

where $\varphi_{p}:=k^{p}\mathbb{G}_{\mathbb{R}^{N}}[\mathbb{G}_{\mathbb{R}^{N}}[\frac{\partial\delta_{0}}{\partial x_{N}}]^{p}]\geq\mathbb{G}_{B_{1}(0)}[w_{k}^{p}]$, and $\varphi_{p}$ is a $x_{N}$-odd solution of

[TABLE]

Indeed, $\mathbb{G}_{\mathbb{R}^{N}}[\mathbb{G}_{\mathbb{R}^{N}}[\frac{\partial\delta_{0}}{\partial x_{N}}]^{p}]$ is $x_{N}-$odd and

[TABLE]

where $e_{z}=\frac{z}{|z|}$.

We observe that $\varphi_{p}$ satisfies

[TABLE]

and then it follows by Hopf’s Lemma (see [6]) that

[TABLE]

Therefore,

[TABLE]

Proof of lower bound in (4.4). It follows by (4.5) that

[TABLE]

Set $\rho_{k}=(2^{(N-1)(p-1)-3}\frac{c_{39}}{c_{40}}k^{p-1})^{\frac{1}{(N-1)(p-1)-2}}$, then

[TABLE]

and

[TABLE]

where $c_{41}>0$ independent of $k$. Thus, for $t\in(\frac{\rho_{k}}{2},\rho_{k}),\ e\in\partial B_{1}(0),\ e\cdot e_{N}>0,$

[TABLE]

Now we can choose a sequence $\{k_{n}\}\subset[1,\infty)$ such that

[TABLE]

and for any $x\in B_{\frac{1}{2}}^{+}(0)\setminus\{0\}$, there exists $k_{n}$ such that $x\in B_{\rho_{k}}^{+}(0)\setminus B_{\frac{\rho_{k}}{2}}^{+}(0)$ and

[TABLE]

Together with $w_{k_{n+1}}\geq w_{k_{n}}$ in $B_{1}^{+}(0)$, we have that

[TABLE]

Proof of the upper bound in (4.4). Let $\bar{w}_{p}=|x|^{-\frac{2}{p-1}-1}x_{N}$, then

[TABLE]

where $(\frac{2}{p-1}+1)(\frac{2}{p-1}+1-N)>0$ for $p<\frac{N+1}{N-1}$. By Comparison Principle, there exists $t_{0}>0$ independent of $k$ such that

[TABLE]

which implies that

[TABLE]

Proof of (1.18). We observe there exists $t_{00}\in(0,t_{0})$ such that

[TABLE]

Therefore,

[TABLE]

admits a unique solution $u_{p}$ and by scaling property, we have that

[TABLE]

By Comparison Principle, we have that

[TABLE]

which implies (1.18) with $\varphi(e)=u_{p}(e)$.   $\Box$

Proof of Theorem 1.3. From Proposition 4.1 and Lemma 4.1, one has that $u_{\infty}$ is a classical solution of (1.17) satisfying (1.18).  $\Box$

5. Non $x_{N}$-odd solutions

5.1. Existence

Under the assumptions on $g$ in Theorem 1.1, it shows from [19] that the problem

[TABLE]

admits a unique weak solution, denoting by $u_{0,j}$. In the approaching the weak solution of problem (1.8) with $j>0$, a barrier will be constructed by $u_{0,j}$ and $u_{k}$, where $u_{k}$ is the unique $x_{N}-$odd weak solution of (1.8) with $j=0$.

**Proof of Theorem 1.1 with $j>0$. ** Step 1. We observe that $g_{n}$ is bounded, Lipschitz continuous and nondecreasing, where $g_{n}$ is defined by (3.2), then it follows from [19, Theorem 3.7] and the Kato’s inequality that

[TABLE]

admits a unique weak solution $v_{k,j,n,t}$, which is a classical solution of

[TABLE]

Moreover, $v_{k,j,n,t}$ could be approximated by the classical solutions $\{v_{n,t,m}\}$ to

[TABLE]

where

[TABLE]

and $\{\sigma_{m}\}$ is a sequence of radially symmetric, nondecreasing smooth functions converging to $\delta_{0}$ in the distribution sense. Furthermore,

[TABLE]

Since $k\mu_{t,m}+j\sigma_{m}\geq k\mu_{t,m}$, it implies by Comparison Prinsiple that

[TABLE]

where $w_{k,n,t,m}$ is the weak solution of (3.5) and $\upsilon_{j,n,m}$ is the unique solution of the equation

[TABLE]

Therefore, $v_{k,j,n,t}$ satisfies

[TABLE]

and

[TABLE]

Step 2. From Step 1, problem (5.2) admits a unique weak solution $v_{k,j,n,t}$, that is,

[TABLE]

On the one hand, by Lemma 2.1, we have that

[TABLE]

On the other hand, by the fact that

[TABLE]

we have that

[TABLE]

By interior regularity results, see [15], for $\sigma\in(0,1)$ and any compact set $K$ and open set $O$ in $B_{1}(0)$ such that $K\subset O$, $\bar{O}\cap\{te_{N}:\ t\in(-\frac{1}{2},\frac{1}{2})\}=\emptyset$, there exist $c_{43},c_{44}>0$ independent of $t$ such that

[TABLE]

Moreover, by [4, Lemma 3.6] $\{\mathbb{G}_{B_{1}(0)}[\mu_{t}]\}$ is uniformly bounded in $M^{\frac{N}{N-1}}(B_{1}(0),dx)$ if $N\geq 3$ and in $M^{\frac{2}{1+\sigma}}(B_{1}(0),dx)$ for any $\sigma\in(0,\frac{1}{2})$ if $N=2$, $\{\mathbb{G}_{B_{1}(0)}[\delta_{0}]\}$ is uniformly bounded in $M^{\frac{N}{N-2}}(B_{1}(0))$ if $N\geq 3$ and $\{\mathbb{G}_{B_{1}(0)}[\delta_{0}]\}$, is uniformly bounded in $M^{q}(B_{1}(0),dx)$ for any $q>0$. Therefore, $\{v_{k,j,n,t}\}_{t}$ is relatively compact in $L^{p}({B_{1}(0)})$ for any $p\in[1,\frac{N}{N-1})$. Then there exists $v_{k,j,n}\in L^{1}(B_{1}(0))$ such that

[TABLE]

which implies that

[TABLE]

Therefore, up to some subsequence, passing to the limit as $t\to 0^{+}$ in the identity (5.3), it infers that $v_{k,j,n}$ is the unique very weak solution of

[TABLE]

Here the uniqueness follows by the Kato’s inequality. It follows by (5.4), (3.12) and $x_{N}$-odd property of $w_{k,n}$ that

[TABLE]

where $w_{k,n}$ is the unique $x_{N}$-odd solution of (3.8). From the proof of Theorem 3.1, we known that for any $\xi\in C^{1.1}_{0}(B_{1}(0))$,

[TABLE]

*Step 3. * It follows by (5.7) that

[TABLE]

For $\beta\in(0,1)$, any compact set $K$ and open set $O$ in $B_{1}(0)$ satisfying $K\subset O$, $0\not\in\bar{O}$, we have that

[TABLE]

Therefore, up to some subsequence, there exists $v_{k,j}$ such that

[TABLE]

Then $\{g_{n}(v_{k,j,n})\}$ converges to $g(v_{k,j})$ a.e. in $B_{1}(0)$.

We observe that $\tilde{v}_{n}:=v_{k,j,n}-w_{k,n}$ is the very weak solution of

[TABLE]

Note that $0\leq\tilde{v}_{n}\leq j\mathbb{G}_{B_{1}(0)}[\delta_{0}]$ and by (1.11), it follows that

[TABLE]

Thus, $\{g_{n}(w_{k,n}+\tilde{v}_{n})-g_{n}(w_{k,n})\}$ converges to $g(w_{k}+\tilde{v})-g(w_{k})$ a.e. in $B_{1}(0)$, where $\tilde{v}=v_{k,j}-w_{k}$.

Let $\tilde{g}_{n}(s)=g_{n}(w_{k,n}+s)-g_{n}(w_{k})$, we see that $\tilde{g}_{n}(0)=0$ and function $\tilde{g}_{n}$ is nondecreasing and verifies (1.10). Then it follows by Theorem 3.7 in [19] that

[TABLE]

and

[TABLE]

Thus, it follows by (1.11) that

[TABLE]

Let $S_{\lambda}=\{x\in B_{1}(0):|x|>\lambda^{-1}\}$, then

[TABLE]

For any Borel set $E\subset{B_{1}(0)}$, we have that

[TABLE]

Since the critical index in (1.10) is $\frac{N+1}{N-1}$, we have that

[TABLE]

Thus,

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

by the assumption (1.10) and Lemma 3.3.

Therefore,

[TABLE]

and

[TABLE]

Notice that the quantities on the right-hand side tends to [math] when $\lambda\to\infty$. The conclusion follows: for any $\epsilon>0$, there exists $\lambda>0$ such that

[TABLE]

and

[TABLE]

For $\lambda$ fixed, there exists $\delta>0$ such that

[TABLE]

which implies that $\{\tilde{g}_{n}\circ\tilde{v}_{n}\}$ is uniformly integrable in $L^{1}(B_{1}(0))$. Then

[TABLE]

by Vitali convergence theorem.

Then passing to the limit as $n\to+\infty$ in the identity (3.13), it implies that for any $\xi\in C^{1,1}_{0}(B_{1}(0))$,

[TABLE]

Thus, $v_{k,j}$ is a very weak solution of (1.8). The regularity results follows by Lemma 3.1.

*Proof of $(ii)$. * It follows from (1.11) and (5.7) that

[TABLE]

and

[TABLE]

where $w_{g}$ is the unique solution of (2.3) and

[TABLE]

We see that for any $e=(e_{1},\cdots,e_{N})\in\partial B_{1}(0)$ with $e_{N}\not=0$,

[TABLE]

which, together with (3.18), implies that

[TABLE]

Then

[TABLE]

Thus,

[TABLE]

and by $x_{N}$-odd property of $w_{k}$, we derive that

[TABLE]

This ends the proof.  $\Box$

Acknowledgements: H. Chen is supported by NSFC, No: 11401270, 11661045 and by the Jiangxi Provincial Natural Science Foundation, No: 20161ACB20007 and the Project-sponsored by SRF for ROCS, SEM.