Stable Solutions of a Class of Degenerate Elliptic Equations

Abstract

This paper deals with the second-order semi-linear degenerate elliptic equation $y{u}_{yy}+b{u}_y+{\Delta }_{x}u+{|u|}^{\alpha -1}u=0,(x,y)\in {\mathbb{R}}^n\times (0,\infty )$, where $n\ge 1,\alpha >1$. We establish a Liouville theorem of stable solution of the degenerate equation mentioned above by using the energy method. The classification results for stable solutions belonging to $C^2$ can be thought of as an analogue of the recent results of Farina for the Lane–Emden equation.

1. Introduction and Main Results

It is well known that any bounded harmonic function on ${\mathbb{R}}^n$ is constant. The Liouville theorem was immediately generalized to many other partial differential equations. This plays a great role in the classification of solutions. In this paper, we study the stable solution $u(x,y)$ of the following degenerate elliptic equation on the half space:

$$\begin{array}{cc}\hfill y{u}_{yy}+b{u}_y+{\Delta }_{x}u+{|u|}^{\alpha -1}u=0,& (x,y)\in \Omega ,\hfill \end{array}$$

(1)

where $\Omega ={\mathbb{R}}^n\times (0,\infty ),n\ge 1,{\Delta }_{x}u:=u_{x_1{x}_1}+u_{x_2{x}_2}+\dots +u_{x_n{x}_n},\alpha >1,b$ is a constant.

For the conventional Lane–Emden equation which is the simplest and most representative semi-linear elliptic equation,

$$\begin{array}{c}\hfill \Delta u+{|u|}^{\alpha -1}u=0,x\in U.\end{array}$$

(2)

There are many Liouville-type results about Equation (2), such as non-negative solutions, radial solutions, stable solutions, and finite Morse index solutions (cf. [1,2,3,4,5,6] and references therein).

When $U={\mathbb{R}}^N$, about non-negative solutions, Gidas and Spruck [4] proved through the vector field method that there is no classical positive solution to Equation (2) when $1\le \alpha <{\displaystyle \frac{N+2}{N-2}}$. Chen and Li [6] simplified the proof by using the moving plane method and obtained the same results. For $\alpha >{\displaystyle \frac{N+2}{N-2}}$, Equation (2) contains positive solutions. Regarding stable solutions (the significance of stable solutions can be referred to in Section 2), in the pioneering work of Farina [3], for Equation (2), there exists the Joseph–Lundgren exponent:

$$\begin{array}{c}\hfill {\alpha }_e:=\left\{\begin{array}{cc}+\infty \hfill & \mathrm{if}N\le 10,\hfill \\ {\displaystyle \frac{{(N-2)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}}\hfill & \mathrm{if}N>10,\hfill \end{array}\right.\end{array}$$

(3)

Especially, it was proved that when $1<\alpha <{\alpha }_e$, Equation (2) has no nontrivial stable solution; when $N\ge 11$ and $\alpha >{\alpha }_e$, by standard phase plane analysis, Equation (2) admits a positive radial stable solution.

But in the case of subdomains (half spaces, unbounded regions, etc.), the existence problem of the Lane–Emden equation mentioned above has not been fully solved. When $U=\left\{x\in {\mathbb{R}}^N:x_N>0\right\}$, Gidas and Spruck [7] proved the nonexistence of bounded positive solutions for Equation (2) with Dirichlet boundary conditions in a half space when $1<\alpha \le {\displaystyle \frac{N+2}{N-2}}$. Their proof employed the Kelvin transform in conjunction with the moving plane method, effectively reducing the original problem to a one-dimensional setting. Dancer [8] used the moving plane method to prove that any nontrivial non-negative bounded solution of Equation (2) with Dirichlet boundary conditions is monotonic in the normal vector direction, and they derived that the bounded positive solution in the half space can lead to the existence of a bounded positive solution on ${\mathbb{R}}^{N-1}$. The result means that Equation (2) does not have a boundary positive solution in the half space for $1<\alpha \le {\displaystyle \frac{N+1}{N-3}}$. In 2007, on the basis of [8], Farina [3] further extended the results of [8] by using variational estimation and stability properties, proving the nonexistence of a boundary positive solution for Equation (2) within the range of $1<\alpha <{\alpha }_e$. In 2014, Chen, Lin and Zou [1] thoroughly solved the problem of the existence of bounded positive solutions for the Dirichlet problem in half space. By using a clever auxiliary function that includes the derivative of u and considering convexity, they proved that for $\alpha >1$, there is no bounded positive solution to the equation. However, the problem of unbounded solutions for the Lane–Emden equation in the supercritical range $\alpha >{\displaystyle \frac{N+2}{N-2}}$ in half space remains unresolved. Recently, for $\alpha >1$, Dupaigne et al. [2] related monotonicity to nonexistence. Their results reveal the nonexistence of monotonic positive classical solutions in the direction of the normal vector.

Except for on the entire space and half space, when $U={\mathbb{R}}^N\backslash \left\{0\right\}$, which is the outer domain, in 2001, the study of solutions and positive solutions was further expanded by Bidaut [9], and it was proven that for $\alpha \le {\displaystyle \frac{N}{N-2}}$, Equation (2) does not have a positive solution; for $\alpha >{\displaystyle \frac{N}{N-2}}$, Equation (2) always has a singular solution $u(x)={C|x|}^{-{\displaystyle \frac{2}{\alpha -1}}}$ with $C={({\displaystyle \frac{2}{\alpha -1}}(N-2-{\displaystyle \frac{2}{\alpha -1}}))}^{{\displaystyle \frac{1}{\alpha -1}}}$.

The works referred to above are all concerned with equations which are uniformly elliptic. So, it is natural to want to consider the Liouville theorem for a class of degenerate elliptic equations. In recent years, there have been many results on the Liouville theorem of the degradation equation, especially elliptic equations with Grushin operators (cf. [10,11,12,13,14]). For the following elliptic equation,

$$\begin{array}{c}\hfill -G_{\alpha }u=f(u),(x,y)\in {\mathbb{R}}^m\times {\mathbb{R}}^k,\end{array}$$

(4)

where $G_{\alpha }u={\Delta }_{x}u+{|x|}^{2\alpha }{\Delta }_{y}u,\alpha >1$. Monticelli [13] extended some of these important results from the uniformly elliptic setting to a wider class of linear differential operators of the second order. In particular, they established the maximum principles and a similar symmetry result of Equation (4) in bounded domains. On this basis, Yu [14] proved the nonexistence of positive solutions of Equation (4) though combing the Kelvin transformation with the moving plane method under the assumption of monotonicity on the nonlinear term f. Especially, Loiudice [15] studied the following critical problems with Hardy terms

$$\begin{array}{c}\hfill -G_{\alpha }u-\mu {\displaystyle \frac{{\psi }^2}{{\rho }^2}}=K(z){|u|}^{2^{*}-2}u,z=(x,y)\in {\mathbb{R}}^m\times {\mathbb{R}}^n\end{array}$$

(5)

where $\rho$ is the gauge norm naturally associated with the Grushin operator $G_{\alpha }$, $0\le \mu <\overline{\mu }$ with $\overline{\mu }$ is the best Hardy constant for $G_{\alpha }$, $K(z)\in L^{\infty }$, $\psi :=|{\nabla }_{\alpha }\rho |$ and $2^{*}$ denotes the critical Sobolev exponent. They provided the asymptotic behavior of solutions at the singularity and at infinity. And they established some Pohozaev-type nonexistence results. Recently, Chen and Liao [16] studied the stable solutions of the Lane–Emden equation with the Baouendi–Grushin operator:

$$\begin{array}{c}\hfill -({\Delta }_{x}u+{(\alpha +1)}^2{|x|}^{2\alpha }{\Delta }_y{u)=|u|}^{p-1}u\end{array}$$

(6)

They proved that for $p>1$, when p is smaller than the Joseph–Lundgren exponent and differs from the Sobolev exponent, $u=0$ is the unique $C^2$ solution which is stable outside a compact set. More, Mtaouaa [12] studied the following weighted degenerate elliptic equation involving the Grushin operator:

$$\begin{array}{c}\hfill {\Delta }_{s}u+{\vartheta }_s(x^{\prime }){|u|}^{\alpha -1}u=0,in{\mathbb{R}}^N,N>2,\end{array}$$

(7)

where $x^{\prime }=(x_1,x_2,\dots ,x_m)\in {\mathbb{R}}^m,1<m<N,\alpha >1,{\vartheta }_s(x^{\prime })\in C({\mathbb{R}}^m,\mathbb{R})$ is a continuous positive function satisfying some conditions, and ${\Delta }_s:={\Sigma }_{i=1}^k{\partial }_{x_i}(s_i^2{\partial }_{x_i})$. They established some new Liouville-type theorems that, under some general hypotheses, the unique $C^2$ stable solution to Equation (7) is $u=0$. There is a Liouville-type theorem for many degenerate equations with weight (cf. [17,18]), and the authors in [19,20,21] futher examined some degenerate elliptic systems and established some new Liouville-type theorems for stable solutions.

For the more specific feature of degenerate elliptic Equation (1), Huang [22] transformed the singularity on the equation into a first-order derivative term through element substitution, proved some new maximal principles, and then applied the Alexandrov–Serrin moving plane method (cf. [5,23,24,25]) to obtain the Liouville theorem for nonnegative solutions of Equation (1) in a half space. Their main results included the following: for $1<\alpha <{\displaystyle \frac{N+2b+2}{N+2b-2}}$, the nonnegative solution of Equation (1) is $u=0$; for $\alpha ={\displaystyle \frac{N+2b+2}{N+2b-2}}$, $u(x,y)$ is in a two-parameter family of functions as

$$\begin{array}{c}\hfill u_{t,x_0}={\left({\displaystyle \frac{t\sqrt{(N+2b)(N+2b-2)}}{t^2+4y+{|x-x_0|}^2}}\right)}^{{\displaystyle \frac{N+2b-2}{2}}}\end{array}$$

for some $x_0\in {\mathbb{R}}^N$ and $t\ge 0$. There are many classifications of solutions to high-order equations or equations with the Laplace operator (cf. [26,27,28,29]).

The main result of this paper is

Theorem 1.

Let $N=n+1$, u be a stable solution of (1) in ${\mathbb{R}}^n\times (0,\infty )$ with

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}1<\alpha <+\infty \hfill & \mathrm{if}N\le 10,\hfill \\ 1<\alpha <{\displaystyle \frac{{(N-2)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}}\hfill & \mathrm{if}N>10.\hfill \end{array}\right.\end{array}$$

(8)

Then, $u\equiv 0$.

In our case, the main method used in solving problem (1) is the energy estimation method which is based on [3]. The key point of the method is that establish a decay estimation by selecting a suitable test function.

The present paper is organized as follows. In Section 2, we give some definitions about stable solutions of Equation (1) and some notes in order to convenient calculation. In Section 3, we prove the decay estimation and use the decay estimation to prove Liouville-type theorem. The final Section 4 contains conclusions.

2. Preliminaries

In this section, we will collect the definition and some physical connotations of the stable solution of elliptic equations.

Let us first briefly describe a common physical phenomenon: if a round stone is placed in the center of a smooth bowl and lightly tapped, the stone will return to its stable position after some back and forth rolling. On the contrary, if you flip the bowl over and carefully place the round stone at the top in the middle of the bowl, it will be in a rather unstable equilibrium state, and a gentle breeze will be enough to make it fall. Such a state is stable because it is trapped at the bottom of an energy well. Accordingly, in physics, one is interested in finding the state of a system of lowest energy E. Using professional language to describe this phenomenon, if a system can recover from disturbances, then it is in a stable state: that is, a small change will not prevent the system from returning to equilibrium. From the mathematical point of view, we obtain the following definition:

Definition 1

([30]). Let I denote an open interval of $\mathbb{R}$, $E:I\to \mathbb{R}$ is a function of class $C^2$, $t_0\in I$ is a stable critical point of E if

$$E^{\prime }(t_0)=0,E^{″}(t_0)\ge 0.$$

Remark 1.

A point $t_0$ satisfying Definition 1 is usually called semi-stable in the literature, while it is said to be stable if the strict inequality $E^{″}(t_0)>0$ holds.

Therefore, we consider the solution of a partial differential equation as the critical point of the energy functional; that is, we say a solution u is stable if the second variation at u of the energy functional is nonnegative. We shall work in a context where the energy is defined on a suitable functional space X, such as the Euclidean space. Recalling Definition 1, it is natural to define the stability of Equation (1) as follows.

Definition 2

([30]). We say a solution $u\in C^2(\Omega )$ of problem (1) is stable if

$$\begin{array}{c}\hfill Q_u(\varphi ):={\int }_{\Omega }y|{\partial }_y{\varphi |}^2+|{\nabla }_x{\varphi |}^2-\alpha {|u|}^{\alpha -1}{\varphi }^2\ge 0,\forall \varphi \in C_c^1(\Omega ).\end{array}$$

(9)

Remark 2.

The quadratic form $Q_u$ is called the second variation of the energy functional.

Remark 3.

According to the definition of a stable solution, the second variation of the energy functional $Q_u(\varphi ):={\int }_{\Omega }y|{\partial }_y{\varphi |}^2+|{\nabla }_x{\varphi |}^2+(1-b){\partial }_y\varphi \varphi -\alpha {|u|}^{\alpha -1}{\varphi }^2$. However, since ${\int }_{\Omega }{\partial }_y\varphi \varphi =0$ for $\varphi \in C_c^1(\Omega )$, we omit this term here.

Remark 4.

u is stable in Ω if and only if u is stable in every subdomain $\omega \subset \subset \Omega$. Especially, the concept of stability also applies to unbounded and nonsmooth domains. However, for given two domains ${\omega }_1,{\omega }_2$, if u is stable in these domains, u need not be stable in ${\omega }_1\cup {\omega }_2$. For more information on the stable solutions, please refer to the monograph [30].

In order to facilitate the subsequent calculations, we have the following identities:

$$\begin{array}{c}\hfill {\nabla |u|}^{\gamma +1}=(\gamma +1){|u|}^{\gamma -1}u\nabla u,\end{array}$$

(10)

$$\begin{array}{c}\hfill {\nabla (|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)={\displaystyle \frac{\gamma +1}{2}}{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}\nabla u,\end{array}$$

(11)

$$\begin{array}{c}\hfill \nabla u·{\nabla (|u|}^{\gamma -1}u)={\displaystyle \frac{4\gamma }{{(\gamma +1)}^2}}{\left(\nabla (|u{|}^{{\displaystyle \frac{\gamma -1}{2}}}u)\right)}^2,\end{array}$$

(12)

$$\begin{array}{c}\hfill {|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u\nabla (|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)={\displaystyle \frac{1}{2}}{\nabla (|u|}^{\gamma +1})\end{array}$$

(13)

3. Proof of Theorem 1

The following proposition plays a crucial role in the proof of our main result.

Proposition 1.

Let Ω be a domain (bounded or not) of ${\mathbb{R}}^N$. Let u be a stable solution of (1). For any integer $m\ge max\left\{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}},2\right\}$. Then, for any $\gamma \in [1,2\alpha +2\sqrt{\alpha (\alpha -1)}-1)$, there exits a constant $C(\alpha ,\gamma ,m)>0$ such that

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\hfill \\ \le C{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}.\hfill \end{array}$$

(14)

for every test function $\xi \in C_c^2(\Omega )$ satisfying $|\xi |\le 1$ in Ω.

Proof. 

We split the proof into three steps:

Step 1. We claim that for any $\phi \in C_c^2(\Omega )$, it holds

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ ={\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2+\left({\displaystyle \frac{\gamma +1}{2\gamma }}-{\displaystyle \frac{b(\gamma +1)}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ +{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2).\hfill \end{array}$$

(15)

By some calculations, we have

$$\begin{array}{cc}\hfill {\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2& ={\int }_{\Omega }y{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\partial }_y(y{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2)\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u({\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2+y{\partial }_{yy}^2{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2+y{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\partial }_y({\phi }^2))\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2-{\int }_{\Omega }{y|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\partial }_{yy}^2{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2-{\int }_{\Omega }{y|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\partial }_y({\phi }^2)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\partial }_y{(|u|}^{\gamma +1}){\phi }^2-{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{y|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\displaystyle \frac{\partial }{\partial y}}{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{\partial }_{y}u){\phi }^2\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }y{|u|}^{\gamma -1}u{\partial }_{y}u{\partial }_y({\phi }^2)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\partial }_y{(|u|}^{\gamma +1}){\phi }^2-{\displaystyle \frac{(\gamma +1)(\gamma -1)}{4}}{\int }_{\Omega }{y|u|}^{\gamma -1}{|{\partial }_{y}u|}^2{\phi }^2\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{y|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }y{\partial }_y{(|u|}^{\gamma +1}){\partial }_y({\phi }^2)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)-{\displaystyle \frac{\gamma -1}{\gamma +1}}{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{y|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y(y{\partial }_y({\phi }^2))\hfill \\ \hfill & ={\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)-{\displaystyle \frac{\gamma -1}{\gamma +1}}{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{y|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2+{\displaystyle \frac{1}{2}}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2),\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill {\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2& ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)-{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2\hfill \\ \hfill & +{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2).\hfill \end{array}$$

(16)

Likewise, we integrate by parts to find

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\left|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)\right|}^2{\phi }^{2}dx& ={\int }_{\Omega }{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)·{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^{2}dx\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}udiv\left({\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u){\phi }^2\right)dx\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)·{\nabla }_x({\phi }^2)dx-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}udiv\left({\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)\right){\phi }^{2}dx\hfill \\ \hfill & =-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)·{\nabla }_x({\phi }^2)dx-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}udiv\left({\displaystyle \frac{\gamma +1}{2}}{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{\nabla }_{x}u\right){\phi }^{2}dx\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\nabla }_x{(|u|}^{\gamma +1})·{\nabla }_x({\phi }^2)dx-{\int }_{\Omega }{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}udiv\left({\displaystyle \frac{\gamma +1}{2}}{|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{\nabla }_{x}u\right){\phi }^{2}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)dx-{\displaystyle \frac{(\gamma +1)(\gamma -1)}{4}}{\int }_{\Omega }{|u|}^{\gamma -1}{|{\nabla }_{x}u|}^2{\phi }^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{|u|}^{\gamma -1}u{\Delta }_{x}u{\phi }^{2}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)dx-{\displaystyle \frac{\gamma -1}{\gamma +1}}{\int }_{\Omega }{\left|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u)\right|}^2{\phi }^{2}dx\hfill \\ \hfill & -{\displaystyle \frac{\gamma +1}{2}}{\int }_{\Omega }{|u|}^{\gamma -1}u{\Delta }_{x}u{\phi }^{2}dx,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill {\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2=-{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma -1}u{\Delta }_{x}u{\phi }^2+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2).\end{array}$$

(17)

Combine (17) and (16), it holds

$$\begin{array}{cc}\hfill & {\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ \hfill & =-{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma -1}u{\Delta }_{x}u{\phi }^2+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ \hfill & -{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{y|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \end{array}$$

(18)

Multiplying Equation (1) by $\varphi ={|u|}^{\gamma -1}u{\phi }^2$ and integrating by parts to find

$${\int }_{\Omega }y{\partial }_{y}u{\partial }_y{(|u|}^{\gamma -1}u{\phi }^2)+{\nabla }_{x}u·{\nabla }_x{(|u|}^{\gamma -1}u{\phi }^2{)-(b-1)|u|}^{\gamma -1}u{\partial }_{y}u{\phi }^2-{|u|}^{\alpha +\gamma }{\phi }^2=0.$$

(19)

Adding (19) to (18), and integrating by parts, we obtain

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ -{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }\left({y|u|}^{\gamma -1}u{\partial }_{yy}^{2}u{\phi }^2+{|u|}^{\gamma -1}u{\Delta }_{x}u{\phi }^2\right)\hfill \\ ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ +{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }\left({\partial }_{y}u{\displaystyle \frac{\partial }{{\partial }_y}}\left({y|u|}^{\gamma -1}u{\phi }^2\right)+{\nabla }_{x}u·{\nabla }_x{(|u|}^{\gamma -1}u{\phi }^2)\right)\hfill \\ ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ +{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }\left({|u|}^{\gamma -1}u{\partial }_{y}u{\phi }^2+y{\partial }_{y}u{\partial }_y{(|u|}^{\gamma -1}u{\phi }^2)+{\nabla }_{x}u·{\nabla }_x{(|u|}^{\gamma -1}u{\phi }^2)\right)\hfill \\ ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ -{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }\left({(b-1)|u|}^{\gamma -1}u{\partial }_{y}u{\phi }^2+{|u|}^{\alpha +\gamma }{\phi }^2)\right)\hfill \\ ={\displaystyle \frac{\gamma +1}{2\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ -{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+{\displaystyle \frac{(\gamma +1)}{4\gamma }}{\int }_{\Omega }(b-1){\partial }_y\left({|u|}^{\gamma +1}\right){\phi }^2+{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2\hfill \\ ={\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2+\left({\displaystyle \frac{\gamma +1}{2\gamma }}-{\displaystyle \frac{b(\gamma +1)}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ +{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2),\hfill \end{array}$$

that is the claim (15).

Step 2. For any $\phi \in C_c^2(\Omega )$,

$$\begin{array}{cc}\hfill \left(\alpha -{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2& \le {\int }_{\Omega }{|u|}^{\gamma +1}|{\nabla }_x{\phi |}^2+{\int }_{\Omega }{y|u|}^{\gamma +1}|{\partial }_y{\phi |}^2+{\displaystyle \frac{2-b(\gamma +1)}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ \hfill & +{\displaystyle \frac{1-\gamma }{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{1-\gamma }{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2).\hfill \end{array}$$

(20)

Indeed, taking the function $\varphi ={|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u\phi$ with $\phi \in C_c^2(\Omega )$ as a test function, then we have the following identities:

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|{\nabla }_x\varphi |}^2& ={\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)\phi +|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\nabla }_x{\phi |}^2\hfill \\ \hfill & ={\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }{|u|}^{\gamma +1}|{\nabla }_x{\phi |}^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2),\hfill \\ \hfill {\int }_{\Omega }y{|{\partial }_y\varphi |}^2& ={\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)\phi +|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{\partial }_y{\phi |}^2\hfill \\ & ={\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }{y|u|}^{\gamma +1}|{\partial }_y{\phi |}^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)-{\displaystyle \frac{1}{2}}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2),\hfill \\ \hfill {\int }_{\Omega }\alpha {|u|}^{\alpha -1}{\varphi }^2& =\alpha {\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2.\hfill \end{array}$$

Then, we put the upper equalities into the stability assumption on u and use identity (15),

$$\begin{array}{cc}\hfill \alpha {\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2& \le {\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }{|u|}^{\gamma +1}|{\nabla }_x{\phi |}^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)\hfill \\ \hfill & +{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }{y|u|}^{\gamma +1}|{\partial }_y{\phi |}^2-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)-{\displaystyle \frac{1}{2}}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ \hfill & ={\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2+{\int }_{\Omega }{|u|}^{\gamma +1}|{\nabla }_x{\phi |}^2+{\int }_{\Omega }{y|u|}^{\gamma +1}{|{\partial }_y\phi |}^2\hfill \\ \hfill & +\left({\displaystyle \frac{\gamma +1}{2\gamma }}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{b(\gamma +1)}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)+\left({\displaystyle \frac{\gamma +1}{4\gamma }}-{\displaystyle \frac{1}{2}}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)\hfill \\ \hfill & +\left({\displaystyle \frac{\gamma +1}{4\gamma }}-{\displaystyle \frac{1}{2}}\right){\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2),\hfill \end{array}$$

that is, we obtain (20).

Step 3. We claim that for any $\gamma \in [1,2\alpha +2\sqrt{\alpha (\alpha -1)}-1)$ and any integer $m\ge max\left\{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}},2\right\}$, there exists a positive constant C depending only on $N,\alpha ,\gamma ,m$ such that

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\hfill \\ \le C{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}.\hfill \end{array}$$

for all test functions $\xi \in C_c^2(\Omega )$, with $|\xi |\le 1$.

From (20), for any $\gamma \in [1,2\alpha +2\sqrt{\alpha (\alpha -1)}-1)$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2& \le C_0{\int }_{\Omega }{|u|}^{\gamma +1}\left(|{\nabla }_x{\phi |}^2+|y||{\partial }_y{\phi |}^2+|{\partial }_y({\phi }^2)|+|{\Delta }_x({\phi }^2)|+|y||{\partial }_{yy}^2({\phi }^2)|\right)\hfill \end{array}$$

(21)

where $C_0=max\left\{|{\displaystyle \frac{1}{\beta }}|,|{\displaystyle \frac{2-b(\gamma +1)}{4\gamma \beta }}|,|{\displaystyle \frac{1-\gamma }{4\gamma \beta }}|\right\},\beta :=\alpha -{\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}>0$.

For any $\xi \in C_c^2(\Omega )$, with $|\xi |\le 1$ in $\Omega$, we set $\phi ={\xi }^m\in C_c^2(\Omega )$ where $m\ge 2$. By a computation,

$$\begin{array}{cc}\hfill {\nabla }_x\phi & =m{\xi }^{m-1}{\nabla }_x\xi ,\hfill \\ \hfill {\partial }_y\phi & =m{\xi }^{m-1}{\partial }_y\xi ,\hfill \\ \hfill {\nabla }_x{\phi }^2& =2m{\xi }^{2m-1}{\nabla }_x\xi ,\hfill \\ \hfill {\Delta }_x{\phi }^2& =2m{\xi }^{2m-1}{\Delta }_x\xi +2m(2m-1){\xi }^{2m-2}{|{\nabla }_x\xi |}^2,\hfill \\ \hfill {\partial }_y{\phi }^2& =2m{\xi }^{2m-1}{\partial }_y\xi ,\hfill \\ \hfill {\partial }_{yy}^2{\phi }^2& =2m{\xi }^{2m-1}{\partial }_{yy}^2+2m(2m-1){\xi }^{2m-2}{|{\partial }_y\xi |}^2.\hfill \end{array}$$

We obtain that

$$\begin{array}{cc}\hfill {\int }_{\Omega }{|u|}^{\alpha +\gamma }{\xi }^{2m}& \le C_0{\int }_{\Omega }{|u|}^{\gamma +1}(m^2{\xi }^{2m-2}|{\nabla }_x{\xi |}^2+m^2{\xi }^{2m-2}|y||{\partial }_y{\xi |}^2+|2m{\xi }^{2m-1}||{\partial }_y\xi |\hfill \\ \hfill & +|2m(2m-1){\xi }^{2m-2}||{\nabla }_x{\xi |}^2+|2m{\xi }^{2m-1}||{\Delta }_x\xi |+|y||2m(2m-1){\xi }^{2m-2}||{\partial }_y{\xi |}^2\hfill \\ \hfill & +|y||2m{\xi }^{2m-1}||{\partial }_{yy}^2\xi |)\hfill \\ \hfill & =C_0{\int }_{\Omega }{|u|}^{\gamma +1}{\xi }^{2m-2}((m^2+2m(2m-1))|{\nabla }_x{\xi |}^2+(m^2+2m(2m-1))|y||{\partial }_y{\xi |}^2\hfill \\ \hfill & +|2m\xi ||{\partial }_y\xi |+|2m\xi ||{\Delta }_x\xi |+|y||2m\xi ||{\partial }_{yy}^2\xi |),\hfill \end{array}$$

thus

$$\begin{array}{c}\hfill {\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\le C_1{\int }_{\Omega }{|u|}^{\gamma +1}{|\xi |}^{2m-2}\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right),\end{array}$$

(22)

with $C_1=max(|(m^2+2m(2m-1))|C_0,2m{C}_0)>0$.

We notice that $m\ge max\left\{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}},2\right\}$ implies $(2m-2){\displaystyle \frac{\alpha +\gamma }{\gamma +1}}\ge 2m$, then ${|\xi |}^{(2m-2){\displaystyle \frac{\alpha +\gamma }{\gamma +1}}}\le {|\xi |}^{2m}$ thanks to $|\xi |\le 1$. Hence, by applying Holder’s inequality, we obtain that

$$\begin{array}{c}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\hfill \\ \le C_2{\left\{{\int }_{\Omega }{(|u|}^{\gamma +1}{|\xi |}^{2m-2}{)}^{{\displaystyle \frac{\alpha +\gamma }{\gamma +1}}}\right\}}^{{\displaystyle \frac{\gamma +1}{\alpha +\gamma }}}{\left\{{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}\right\}}^{{\displaystyle \frac{\alpha -1}{\alpha +\gamma }}}\hfill \\ \le C_2{\left\{{\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\right\}}^{{\displaystyle \frac{\gamma +1}{\alpha +\gamma }}}{\left\{{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}\right\}}^{{\displaystyle \frac{\alpha -1}{\alpha +\gamma }}}.\hfill \end{array}$$

where $C_2$ is a positive constant. This implies

$$\begin{array}{c}\hfill {\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\le {C_2}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}.\end{array}$$

(23)

Meanwhile, we combine (15) and (21) to obtain

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{\phi }^2+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}u{)|}^2{\phi }^2\hfill \\ ={\displaystyle \frac{{(\gamma +1)}^2}{4\gamma }}{\int }_{\Omega }{|u|}^{\alpha +\gamma }{\phi }^2+\left({\displaystyle \frac{\gamma +1}{2\gamma }}-{\displaystyle \frac{b(\gamma +1)}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ +{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ \le C_0{\int }_{\Omega }{|u|}^{\gamma +1}\left(|{\nabla }_x{\phi |}^2+|y||{\partial }_y{\phi |}^2+|{\partial }_y({\phi }^2)|+|{\Delta }_x({\phi }^2)|+|y||{\partial }_{yy}^2({\phi }^2)|\right)+\left({\displaystyle \frac{\gamma +1}{2\gamma }}-{\displaystyle \frac{b(\gamma +1)}{4\gamma }}\right){\int }_{\Omega }{|u|}^{\gamma +1}{\partial }_y({\phi }^2)\hfill \\ +{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }{|u|}^{\gamma +1}{\Delta }_x({\phi }^2)+{\displaystyle \frac{\gamma +1}{4\gamma }}{\int }_{\Omega }y{|u|}^{\gamma +1}{\partial }_{yy}^2({\phi }^2)\hfill \\ \le C_3{\int }_{\Omega }{|u|}^{\gamma +1}\left(|{\nabla }_x{\phi |}^2+|y||{\partial }_y{\phi |}^2+|{\partial }_y({\phi }^2)|+|{\Delta }_x({\phi }^2)|+|y||{\partial }_{yy}^2({\phi }^2)|\right),\hfill \end{array}$$

where $C_3$ is a positive constant.

Likewise, we insert the test function $\phi ={\xi }^m$ in the latter inequality to find

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}\hfill \\ \le C_4{\int }_{\Omega }{|u|}^{\gamma +1}{|\xi |}^{2m-2}\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)\hfill \\ \le C_5{\left\{{\int }_{\Omega }{(|u|}^{\gamma +1}{|\xi |}^{2m-2}{)}^{{\displaystyle \frac{\alpha +\gamma }{\gamma +1}}}\right\}}^{{\displaystyle \frac{\gamma +1}{\alpha +\gamma }}}{\left\{{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}\right\}}^{{\displaystyle \frac{\alpha -1}{\alpha +\gamma }}}\hfill \\ \le C_5{\left\{{\int }_{\Omega }{|u|}^{\alpha +\gamma }{|\xi |}^{2m}\right\}}^{{\displaystyle \frac{\gamma +1}{\alpha +\gamma }}}{\left\{{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}\right\}}^{{\displaystyle \frac{\alpha -1}{\alpha +\gamma }}}.\hfill \end{array}$$

with some positive constants $C_4$ and $C_5$. It implies

$$\begin{array}{c}{\int }_{\Omega }|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}+{\int }_{\Omega }y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2{|\xi |}^{2m}\hfill \\ \le C_6{\int }_{\Omega }{\left(|{\nabla }_x{\xi |}^2+|y||{\partial }_y{\xi |}^2+|\xi ||{\partial }_y\xi |+|\xi ||{\Delta }_x\xi |+|y||\xi ||{\partial }_{yy}^2\xi |\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}},\hfill \end{array}$$

(24)

with some positive constants $C_6$.

By adding inequality (23) to the inequality (24), we immediately yield the desired inequality (14).  □

Next, we prove the classification of stable solutions.

Proof of Theorem 1.

We consider ${\xi }_R(x)={\chi }_1({\displaystyle \frac{x}{R}}){\chi }_2({\displaystyle \frac{y}{R^{\frac{3}{2}}}})$ for every $R>1$, where ${\chi }_1\in ({\mathbb{R}}^n;[0,1])$ and ${\chi }_2\in ((0,\infty );[0,1])$ with $|\xi |\le 1$ and

$$\begin{array}{c}\hfill {\chi }_1(x)=\left\{\begin{array}{c}1\mathrm{if}|x|\le 1,\hfill \\ 0\mathrm{if}|x|\ge 2\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\chi }_2(y)=\left\{\begin{array}{c}1\mathrm{if}|y|\le 1,\hfill \\ 0\mathrm{if}|y|\ge 2.\hfill \end{array}\right.\end{array}$$

We insert the test function ${\xi }_R(x)\in \left({\mathbb{R}}^n\times (0,\infty );[0,1]\right)$ into inequality (14) of Proposition 1. Then, it is easy to see

$$\begin{array}{c}\hfill |{\nabla }_x{\xi }_R(x)|={\displaystyle \frac{1}{R}}|{\nabla }_x{\chi }_1({\displaystyle \frac{x}{R}}){\chi }_2({\displaystyle \frac{y}{R^{\frac{3}{2}}}})|,\\ \hfill |{\partial }_y{\xi }_R(x)|={\displaystyle \frac{1}{R^{\frac{3}{2}}}}|{\chi }_1({\displaystyle \frac{x}{R}}){\partial }_y{\chi }_2({\displaystyle \frac{y}{R^{\frac{3}{2}}}})|,\\ \hfill |{\Delta }_x{\xi }_R(x)|={\displaystyle \frac{1}{R^2}}|{\Delta }_x{\chi }_1({\displaystyle \frac{x}{R}}){\chi }_2({\displaystyle \frac{y}{R^{\frac{3}{2}}}})|,\\ \hfill |{\partial }_{yy}^2{\xi }_R(x)|={\displaystyle \frac{1}{R^3}}|{\chi }_1({\displaystyle \frac{x}{R}}){\partial }_{yy}^2{\chi }_2({\displaystyle \frac{y}{R^{\frac{3}{2}}}})|.\end{array}$$

Let us fix $\alpha >1$. We first observe that for any $\gamma \in [1,2\alpha +2\sqrt{\alpha (\alpha -1)}-1)$ and any integer $m\ge max\left\{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}},2\right\}$, it holds that

$$\begin{array}{c}{\int }_{B_R^{+}}|{\nabla }_x{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2+y|{\partial }_y{(|u|}^{{\displaystyle \frac{\gamma -1}{2}}}{u)|}^2+{|u|}^{\alpha +\gamma }\hfill \\ \le C{\int }_{B_{2R}^{+}\backslash B_R^{+}}{\left(|{\nabla }_x{\xi }_R{|}^2+|y||{\partial }_y{\xi }_R{|}^2+|{\xi }_R||{\partial }_y{\xi }_R|+|{\xi }_R||{\Delta }_x{\xi }_R|+|y||{\xi }_R||{\partial }_{yy}^2{\xi }_R|\right)}^{{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}}\hfill \\ \le C(\alpha ,\gamma ,m)R^{N-2{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}},\hfill \end{array}$$

(25)

where $B_R^{+}=\left\{(x,y)\in \Omega \mid |x|+|y|<R\right\}$ denotes the open upper ball centered at the origin and with radius R, $C(\alpha ,\gamma ,m)$ are positive constants independent of radius R.

Now, we derive the range of $\alpha$ such that

$$\begin{array}{c}\hfill N-2{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}<0,\end{array}$$

(26)

The proof in this case is like [3] so we prove it briefly. Let ${\gamma }_M=2\alpha +2\sqrt{\alpha (\alpha -1)}-1$. Firstly, we consider the case $N\le 10$ and the case $N>10$ separately.

Case 1: $N\le 10$.

$$\begin{array}{c}\hfill \alpha +{\gamma }_M(\alpha )=3\alpha +2\sqrt{\alpha (\alpha -1)}-1>5(\alpha -1)\end{array}$$

So, we have

$$\begin{array}{c}\hfill N-2{\displaystyle \frac{\alpha +{\gamma }_M(\alpha )}{\alpha -1}}<N-10\le 0,\end{array}$$

Again, $F(x):=N-2{\displaystyle \frac{\alpha +x}{\alpha -1}}$ is continuous, which implies that $\gamma$ satisfies (26).

Case 2: $N>10$.

We define the function $f(\alpha ):(1,+\infty )\to \mathbb{R}$,

$$\begin{array}{cc}\hfill f(\alpha )& :={\displaystyle \frac{\alpha +{\gamma }_M(\alpha )}{\alpha -1}}\hfill \\ \hfill & ={\displaystyle \frac{\alpha +2\alpha +2\sqrt{\alpha (\alpha -1)}-1}{\alpha -1}}\hfill \\ \hfill & =3+{\displaystyle \frac{2}{\alpha -1}}+2\sqrt{1+{\displaystyle \frac{1}{\alpha -1}}}.\hfill \end{array}$$

Since $f(\alpha )$ is a strictly decreasing function, and $f(\alpha )$ satisfies

$$\begin{array}{c}\hfill \underset{\alpha \to +\infty }{lim}f(\alpha )=5\mathrm{and}\underset{\alpha \to 1^{+}}{lim}f(\alpha )=+\infty .\end{array}$$

Then, there exists a unique ${\alpha }_0$, we have $N-2f({\alpha }_0)=0$. Taking ${\alpha }_c={\displaystyle \frac{{(N-2)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}}$. Now, we prove that

$${\alpha }_0={\alpha }_c.$$

Indeed,

$$\begin{array}{c}\hfill N=2{\displaystyle \frac{{\alpha }_0+{\gamma }_M({\alpha }_0)}{{\alpha }_0-1}}\iff (N-2)({\alpha }_0-1)-4{\alpha }_0=4\sqrt{{\alpha }_0({\alpha }_0-1)}.\end{array}$$

From this, we can easily see

$$\begin{array}{c}\hfill (N-2)({\alpha }_0-1)-4{\alpha }_0>4({\alpha }_0-1),\end{array}$$

(27)

and

$$\begin{array}{c}\hfill (N-2)(N-10){\alpha }_0^2-2(N^2-8N+4){\alpha }_0+{(N-2)}^2=0.\end{array}$$

(28)

By solving this quadratic equation, we obtain that the roots are

$$\begin{array}{c}\hfill {\alpha }_1={\displaystyle \frac{{(N-2)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}}={\alpha }_c,\\ \hfill {\alpha }_2={\displaystyle \frac{{(N-2)}^2-4N-8\sqrt{N-1}}{(N-2)(N-10)}}<{\alpha }_c,\end{array}$$

and (27) shows

$$\begin{array}{c}\hfill {\alpha }_0>{\displaystyle \frac{N-6}{N-10}}={\displaystyle \frac{(N-6)(N-2)}{(N-10)(N-2)}}={\displaystyle \frac{{(N-2)}^2-4N+8}{(N-2)(N-10)}}>{\alpha }_2.\end{array}$$

Then, we can easily obtain that ${\alpha }_0={\alpha }_c$.

Since the continuity of $x\to N-2{\displaystyle \frac{\alpha +x}{\alpha -1}}$ and the monotony of $f(\alpha )$, we obtain

$$\begin{array}{c}\forall 1<\alpha <{\displaystyle \frac{{(N-2)}^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}},\exists \gamma \in [1,2\alpha +2\sqrt{\alpha (\alpha -1)}-1),\hfill \\ N-2{\displaystyle \frac{\alpha +\gamma }{\alpha -1}}<0.\hfill \end{array}$$

Finally, letting $R\to +\infty$, we obtain $u\equiv 0$. The proof of Theorem 1 is completed.  □

4. Conclusions

In this work, we discuss the classification results for stable solutions of the degenerate equation using the energy method. Based on the pioneering work of Farina [3], we derive a critical exponent that depends only on the dimension N. Specifically, when the exponent of the nonlinear term in the degenerate equation is smaller than this critical exponent, the stable solution of the equation is trivial. Our results extend the critical exponent of Farina to this degenerate equation.