Non-Existence Results for Stable Solutions to Weighted Elliptic Systems including Advection Terms

Abstract

In this paper, we study a non-linear weighted Grushin system including advection terms. We prove some Liouville-type theorems for stable solutions of the system, based on the comparison property and the bootstrap iteration. Our results generalise and improve upon some previous works.

1. Introduction

Consider the following weighted system including advection terms:

$$\begin{array}{c}\hfill -{\mathsf{\Delta }}_{\mathbf{z}}u+\nu ·{\nabla }_{\mathbf{z}}u=\varrho \left(\mu \right)v^{q_1},-{\mathsf{\Delta }}_{\mathbf{z}}v+\nu ·{\nabla }_{\mathbf{z}}u=\varrho \left(\mu \right)u^{q_2},\\ \hfill v,u>0\mathrm{in}{\mathbb{R}}^N:={\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2},\end{array}$$

(1)

where $\varrho \left(\mu \right):={\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{\frac{\rho }{2(\mathbf{z}+1)}}$, and the scalar equation:

$$\begin{array}{c}\hfill -{\mathsf{\Delta }}_{\mathbf{z}}u+\nu ·{\nabla }_{\mathbf{z}}u=\varrho \left(\mu \right)u^{q_1},u>0\mathrm{in}{\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2},\end{array}$$

(2)

where ${\mathsf{\Delta }}_{\mathbf{z}}$ is the Grushin operator defined by

$${\mathsf{\Delta }}_{\mathbf{z}}:={\nabla }_{\mathbf{z}}·{\nabla }_{\mathbf{z}}=\sum _{i=1}^{N_1}{X}_i^2+\sum _{j=1}^{N_2}{Y}_j^2={\mathsf{\Delta }}_x+{|x|}^{2\mathbf{z}}{\mathsf{\Delta }}_y,\mathrm{with}{\nabla }_{\mathbf{z}}:=({\nabla }_x{,|x|}^{\mathbf{z}}{\nabla }_y).$$

${\mathsf{\Delta }}_x$ and ${\mathsf{\Delta }}_y$ are Laplace operators in the variables $x\in {\mathbb{R}}^{N_1}$ and $y\in {\mathbb{R}}^{N_2}$, respectively.

Here, we always assume that $\rho \ge 0,$ $\mathbf{z}\ge 0,$ $q_1\ge q_2>1$ and $\nu$ is a smooth divergence-free vector field:

$$\begin{array}{c}\hfill {\mathrm{div}}_{\mathbf{z}}\nu =0,\mathrm{and}|\nu \left(\mu \right)|\le \frac{\kappa }{1+\parallel \mu \parallel }\mathrm{for} \mathrm{all}\mu :=(x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2},\\ \hfill \kappa \mathrm{small} \mathrm{enough}.\end{array}$$

(3)

${\mathrm{div}}_{\mathbf{z}}={\mathrm{div}}_x+{|x|}^{\mathbf{z}}{\mathrm{div}}_y,$ and $\parallel \mu \parallel ={\left({|x|}^{2(\mathbf{z}+1)}+{|y|}^2\right)}^{\frac{1}{2(\mathbf{z}+1)}},$ is the norm corresponding to the Grushin distance, where $|x|$ and $|y|$ are the usual Euclidean norms in ${\mathbb{R}}^{N_1}$ and ${\mathbb{R}}^{N_2}$, respectively. It is easy to check that the $\parallel \mu \parallel$-norm is 1-homogeneous for the group of anisotropic dilations related to ${\mathsf{\Delta }}_{\mathbf{z}}.$ It is defined by

$${\sigma }_{\eta }\left(\mu \right)=(\eta x,{\eta }^{1+\mathbf{z}}y),\eta >0\mathrm{and}\mu :=(x,y)\in {\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}.$$

The change of variable formula for the Lebesgue measure gives that

$$d{\sigma }_{\eta }\left(\mu \right)={\eta }^{N_1+(1+\mathbf{z})N_2}dxdy={\eta }^{N_{\mathbf{z}}}d\mu ,\mathrm{where}{\mathbf{G}}_{\mathbf{z}}^{*}:=N_1+(1+\mathbf{z})N_2,$$

is the homogeneous dimension with respect to dilation ${\sigma }_{\eta }$ and $dxdy=d\mu$ denotes the Lebesgue measure on ${\mathbb{R}}^N.$

Recall that the Grushin operator is elliptic for $|x|\ne 0$ and degenerates on the manifold $\left\{0\right\}\times {\mathbb{R}}^{N_2}$. This operator was introduced in [1]. Problems involving the Grushin operator have been extensively studied over the years. Recall, in adition the papers [2,3,4]. In an appropriate context, the results on Grushin’s operator were obtained in the framework of Heisenberg groups [5]. The study of PDEs involving the Grushin operator has become more and more attractive in the last decades since it can serve to describe nonhomogeneous phenomena, which can occur in different branches of science as physics and astrophysics.

Recently, much attention has been focused on proving Liouville-type theorems for solutions to nonlinear degenerate elliptic systems involving advection terms such as Equations (1) and (2). This result allows us to describe qualitative properties of solutions such as existence, regularity, oscillation, asymptotic or even universal behaviour, pointwise a priori estimates of local solutions, universal and singularity estimates, decay estimates, blow-up rate of solutions of nonstationary problems, etc.; see [6,7,8,9,10,11,12,13,14] and references therein.

Firstly, we mention that, for the autonomous case, i.e., when $\varrho \left(\mu \right)\equiv 1$ and $\nu =\mathbf{z}=0$, much attention has been focused on obtaining Liouville-type theorems for stable solutions of

$$\begin{array}{c}\hfill -\mathsf{\Delta }u=v^{q_1},-\mathsf{\Delta }v=u^{q_2},\mathrm{in}{\mathbb{R}}^N.\end{array}$$

(4)

We refer to [6,15,16]. The author in [6] has first explored the nonexistence of stable solutions of (4) if $N\le 10,$ for any $q_1\ge q_2>2.$ Hu extended this result in [16], for the following systems with positive weights $\varrho \left(\tau \right):={(1+|\tau |}^2{)}^{\frac{\rho }{2}}$ with $\rho >0$:

$$\begin{array}{c}\hfill -\mathsf{\Delta }u=\varrho \left(\tau \right)v^{q_1},-\mathsf{\Delta }v=\varrho \left(\tau \right)u^{q_2},\mathrm{in}{\mathbb{R}}^N,\end{array}$$

with $N\le 10+\rho$ and $q_1\ge q_2>\frac{4}{3}$. We also mention that the previous works [6,16] were improved in [15], where the authors proved a new comparison property for $1<q_1\le \frac{4}{3}.$ Among other things, in [9], Liouville-type results for stable solutions of (4) were established, $\forall q_1,q_2>0,$ verifying

$$N<2+\alpha +\beta \mathrm{where}\alpha =\frac{2(q_1+1)}{q_1{q}_2-1},\mathrm{and}\beta =\frac{2(q_2+1)}{q_1{q}_2-1},q_1{q}_2>1.$$

In the other direction, inspired by the ideas in [6,15,16], Duong [17] proved the nonexistence of stable solutions for the following system with advection:

$$\begin{array}{c}\hfill -\mathsf{\Delta }u+\nu ·\nabla u=v^{q_1},-\mathsf{\Delta }v+\nu ·\nabla v=u^{q_2},v,u>0in{\mathbb{R}}^N.\end{array}$$

(5)

In particular, Duong [17] proved the following theorem:

Theorem 1.

1.

If $q_1\ge q_2>\frac{4}{3}$ and

$$N<2+2\beta k_0^{+},where{k}_0^{\pm }=\sqrt{\varpi }\pm \sqrt{\varpi -\sqrt{\varpi }},with\varpi =\frac{q_1{q}_2(q_1+1)}{q_2+1}.$$

There is no stable, positive solution to (5). In particular, there is no stable positive solution to (5) provided $N\le 10.$

2.

If $1<q_1\le min(\frac{4}{3},q_2)$ and

$$N<2+\left[2+\alpha +\frac{4(2-q_1)}{q_2+q_1-2}\right]k_0^{+},$$

(5) has no bounded stable positive solution. In particular, (5) does not admit bounded stable positive solution provided $N\le 6.$

It should be noticed that when $q_1=q_2$, the result is a natural extension of that in [18], for the following equation with advection:

$$\begin{array}{c}\hfill -\mathsf{\Delta }u+\nu ·\nabla u=u^{q_1},u>0\mathrm{in}{\mathbb{R}}^N.\end{array}$$

In the special case $\varrho \left(\mu \right)\equiv 1$ and $\nu =0,$ the system (1) becomes

$$\begin{array}{c}\hfill -{\mathsf{\Delta }}_{\mathbf{z}}u=v^{q_1},-{\mathsf{\Delta }}_{\mathbf{z}}v=u^{q_2},\mathrm{where}{q}_1\ge q_2>1.\end{array}$$

(6)

The main difficulty is due to the fact that ${\mathsf{\Delta }}_{\mathbf{z}}$ is not symmetric and it is generated on the manifold $\left\{0\right\}\times {\mathbb{R}}^{N_2}$, which causes some mathematical problems. Very recently, in [11], the authors have proved that if ${\mathbf{G}}_{\mathbf{z}}^{*}:=N_1+(1+\mathbf{z})N_2<2+\alpha +\beta ,$ Equation (1) has no stable solution for any $q_1,q_2>0$.

Furthermore, adopting the new approach of Cowan [6], the author in [19] established the nonexistence of stable solutions of (6) when $q_1\ge q_2>\frac{4}{3}$, and ${\mathbf{G}}_{\mathbf{z}}^{*}$ satisfies

$$\begin{array}{c}\hfill {\mathbf{G}}_{\mathbf{z}}^{*}:=N_1+(1+\mathbf{z})N_2<2+2{\beta }_0{t}_0^{+}.\end{array}$$

This result was then generalized in [10], for the system (1), i.e., $\nu =0.$

Inspired by the mentioned previous works, we classify stable positive solutions of (1) under condition (3). First of all, we need to recall the following:

Definition 1.

We say that a smooth solution $(u,v)\in C^2\left({\mathbb{R}}^N\right)\times C^2\left({\mathbb{R}}^N\right)$ of (1) is stable if there exist positive smooth functions φ, χ verifying

$$\begin{array}{c}\hfill -{\mathsf{\Delta }}_{\mathbf{z}}\phi +\nu ·{\nabla }_{\mathbf{z}}\phi =q_1\varrho \left(\mu \right)v^{q_1-1}\psi ,-{\mathsf{\Delta }}_{\mathbf{z}}\psi +\nu ·{\nabla }_{\mathbf{z}}\psi =q_2\varrho \left(\mu \right)u^{q_2-1}\phi in {\mathbb{R}}^N.\end{array}$$

This definition is motivated by [6,19,20]. Our first result concerns stable solutions:

Theorem 2.

Let ρ be positive and $m_0$ be the largest root of the polynomial

$$\begin{array}{c}\hfill Q\left(m\right)=m^4-q_1{q}_2\alpha \beta \left[4m^2-2(q_1+q_2)m+1\right].\end{array}$$

(7)

1.

If $\frac{4}{3}<q_1\le q_2$ and ${\mathbf{G}}_{\mathbf{z}}^{*}<(2+\rho )m_0+2$, then (1) does not admit any positive stable solution. Consequently, if ${\mathbf{G}}_{\mathbf{z}}^{*}\le 10+4\rho ,$ then (1) has no stable solution for all $\frac{4}{3}<q_1\le q_2$.

2.

If $1<q_1\le min(\frac{4}{3},q_2)$ and

$${\mathbf{G}}_{\mathbf{z}}^{*}<2+\frac{1}{2}\left[q_1+\frac{4(2-q_1)}{\beta (q_1+q_2-2)}\right](\rho +2)m_0,$$

then (1) has no bounded stable solution.

If in addition, ${\mathbf{G}}_{\mathbf{z}}^{*}\le 6+2\rho ,$ then (1) does not admit any bounded stable solution for all $q_2\ge q_1>1.$

If $q_1=q_2$, by means of the comparison property (see Lemmas 1 and 2 below), we get the following result.

Proposition 1.

1.

If $\frac{4}{3}<q_1$, then (2) has no stable solution if

$$\begin{array}{c}\hfill {\mathbf{G}}_{\mathbf{z}}^{*}<2+\frac{2(2+\rho )}{q_1-1}\left(q_1+\sqrt{q_1^2-q_1}\right).\end{array}$$

(8)

In particular, if ${\mathbf{G}}_{\mathbf{z}}^{*}\le 10+4\rho ,$ then (2) does not admit any stable solution for all $\frac{4}{3}<q_1$.

2.

If $1<q_1\le \frac{4}{3}$ and ${\mathbf{G}}_{\mathbf{z}}^{*}$ verifies (8) then (2) has no bounded stable solution.

Then, (2) does not admit any bounded stable solution for all $q_1>1$ if ${\mathbf{G}}_{\mathbf{z}}^{*}\le 10+4\rho .$

As in [19], the key techniques in proving Theorem 2 and Proposition 1 deal with the property of comparison and nonlinear integral estimates. Nevertheless, previous tools used in proving the comparison relation (see, e.g., [10,17,19]) do not seem to be applicable for the system (1), since the operator becomes non-symmetric and it degenerates on the manifold $\left\{0\right\}\times {\mathbb{R}}^{N_2}$ due to the advection term. This causes some principal problems in proving Theorem 2. So, we need to use other techniques motivated by [10,17]. We may also use the idea in [15] to establish the “inverse” comparison relation, which is important to treat the case $1<p\le \frac{4}{3}.$ In addition, the $L^1$-estimate to the boostrap iteration in [17] does not work in the case of Grushin operator; we instead switch to the $L^2$-estimate in the boostrap argument.

Remark 1.

• Let $m_0$ be the largest root of the polynomial Q defined in (7). It should be noticed that

  $$m_0>\beta k_0^{+}>4,for any{q}_2\ge q_1>1,$$
  (see Remark 2.1 below). Therefore, Theorem 2 enhances the bound given by Theorem 1 with $\varrho \left(\mu \right)\equiv 1$. Consequently, the range in Theorem 2, is larger than that in [17] (see Theorem 1).
• Our results can be applied also to the general class of degenerate operators (see [7,8,21,22]), namely

  $${\mathsf{\Delta }}_{\mathbf{z}}:=\sum _{j=1}^N\partial x_j\left({\mathbf{z}}_j^2\partial x_i\right)\mathbf{z}:=({\mathbf{z}}_1,\cdots ,{\mathbf{z}}_N):{\mathbb{R}}^N\to {\mathbb{R}}^{\mathbb{N}}.$$
  Here ${\mathbf{z}}_j:{\mathbb{R}}^N\to \mathbb{R},j=1,\cdots ,N,$ are nonnegative functions that are continuous and verify some properties as the homogenity of ${\mathsf{\Delta }}_{\mathbf{z}}$ of degree two with respect to a group dilation in ${\mathbb{R}}^N.$

To our knowledge, all results presented here are new. This paper proceeds as follows: In Section 2, we give some basic results. In Section 3, we prove Theorem 2 and Proposition 1.

2. Main Technical Tool

In this section, we define the following parameters: For ${\alpha }^{\prime }>0,$ set

$${\mathsf{\Omega }}_{{\alpha }^{\prime }R}:=B_{{\alpha }^{\prime }R}\times B_{{\left({\alpha }^{\prime }R\right)}^{1+\mathbf{z}}},\varrho \left(\mu \right):={\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{\frac{\rho }{2(\mathbf{z}+1)}},\mathrm{and}dxdy:=d\mu .$$

In the following, C always denotes a generic positive constant, which could be changed from one line to another.

Our proofs necessitate some technical lemmas.

2.1. Comparison Principle

Here, we establish the comparison property for system (1).

Lemma 1.

Assume that $(u,v)$ is a bounded positive solution of (1). Set $q_2\ge q_1>1$ and (3) holds. Then

$$\begin{array}{c}\hfill u^{q_2+1}\le \frac{q_2+1}{q_1+1}{v}^{q_1+1}.\end{array}$$

(9)

Proof. 

Let $\sigma =\frac{q_1+1}{q_2+1}\in (0,1],$$\lambda ={\sigma }^{\frac{-1}{q_2+1}};$ we conclude then that

$$\begin{array}{c}\hfill \mathrm{Equation}9\iff u\le \lambda v^{\sigma },\end{array}$$

(10)

we put $w=u-\lambda v^{\sigma }.$ A simple calculation gives

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{\mathbf{z}}w={\mathsf{\Delta }}_{\mathbf{z}}u-\lambda \sigma v^{\sigma -1}{\mathsf{\Delta }}_{\mathbf{z}}v-\lambda \sigma (\sigma -1){|{\nabla }_{\mathbf{z}}v|}^2{v}^{\sigma -2}& \ge {\mathsf{\Delta }}_{\mathbf{z}}u-\lambda \sigma v^{\sigma -1}{\mathsf{\Delta }}_{\mathbf{z}}v\hfill \\ \hfill & =\varrho \left(\mu \right)\left[-v^{q_1}+\lambda \sigma v^{\sigma -1}{u}^{q_2}\right]\hfill \\ \hfill & +\nu ·{\nabla }_{\mathbf{z}}u-\lambda \sigma v^{\sigma -1}a·{\nabla }_{\mathbf{z}}v\hfill \\ \hfill & =\nu ·{\nabla }_{\mathbf{z}}w+v^{\sigma -1}\left[-v^{q_1-\sigma +1}+\lambda \sigma u^{q_2}\right]\hfill \\ \hfill & =\nu ·{\nabla }_{\mathbf{z}}w+\varrho \left(\mu \right)v^{\sigma -1}\left[{\lambda }^{-q_2}{u}^{q_2}-v^{q_2\sigma }\right].\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill C{v}^{\sigma -1}\left[u^{q_2}-{\left(\lambda v^{\sigma }\right)}^{q_2}\right]\le \varrho \left(\mu \right)v^{\sigma -1}\left[\frac{u^{q_2}-{\left(\lambda v^{\sigma }\right)}^{q_2}}{{\lambda }^{q_2}}\right]\le {\mathsf{\Delta }}_{\mathbf{z}}w-\nu ·{\nabla }_{\mathbf{z}}w.\end{array}$$

(11)

Now, we use a contradiction argument to prove (10). Suppose that

$$\begin{array}{c}\hfill M=\underset{{\mathbb{R}}^N}{sup}w>0(M\le \infty ).\end{array}$$

(12)

Next, the proof splits into two cases:

Case 1: If there is ${\mu }^{*}$ verifying ${sup}_{{\mathbb{R}}^N}w=w\left({\mu }^{*}\right)=u\left({\mu }^{*}\right)-\lambda v^{\sigma }\left({\mu }^{*}\right)>0,$ we have

$$\frac{\partial w}{\partial {\mu }_i}\left({\mu }^{*}\right)=0and\frac{{\partial }^{2}w}{\partial {\mu }_i^2}\left({\mu }^{*}\right)\le 0i=1,\cdots ,n.$$

This gives

$${\nabla }_{\mathbf{z}}w\left({\mu }^{*}\right)=0\mathrm{and}{\mathsf{\Delta }}_{\mathbf{z}}w\left({\mu }^{*}\right)\le 0.$$

In addition, the left-hand side of (11) at ${\mu }^{*}$ is positive, which is a contradiction.

Case 2: The supremum of w is attained at infinity.

Choose now ${\varphi }_R(x,y)={\psi }^t(\frac{x}{R},\frac{y}{R^{1+\mathbf{z}}}),$ where $t>0,$$\psi$ is a cut-off function in $C_c^{\infty }\left({\mathbb{R}}^N\right),$ with

$$\psi =1\mathrm{on}{B}_1\times B_1,\mathrm{and}\psi =0\mathrm{outside}{B}_2\times B_{2^{1+\mathbf{z}}}.$$

So, we get

$$\begin{array}{c}\hfill \frac{|{\nabla }_{\mathbf{z}}{\varphi }_R{|}^2}{{\varphi }_R}\le \frac{C}{R^2}{\varphi }_R^{\frac{t-2}{t}}\mathrm{and}|{\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }_R\right)|\le \frac{C}{R^2}{\varphi }_R^{\frac{t-2}{t}}.\end{array}$$

(13)

Let $w_R={\varphi }_{R}w,$ a compactly supported function. Therefore, there is ${\mu }_R=(x_R,y_R)\in {\mathsf{\Omega }}_{2R},$ with

$$\begin{array}{c}\hfill w_R\left({\mu }_R\right)=\underset{{\mathbb{R}}^N}{max}{w}_R\left(\mu \right)\to \underset{{\mathbb{R}}^N}{sup}w\left(\mu \right)\mathrm{as}R\to \infty .\end{array}$$

Then

$$\begin{array}{c}\hfill {\nabla }_{\mathbf{z}}{w}_R\left({\mu }_R\right)=0\mathrm{and}{\mathsf{\Delta }}_{\mathbf{z}}{w}_R\left({\mu }_R\right)\le 0.\end{array}$$

Next, we take all the estimates at the point ${\mu }_R.$ Now, using the fact that ${\nabla }_{\mathbf{z}}{w}_R\left({\mu }_R\right)=0,$ we get

$$\begin{array}{c}\hfill {\nabla }_{\mathbf{z}}w=-{\varphi }_R^{-1}{\nabla }_{\mathbf{z}}{\varphi }_{R}w.\end{array}$$

(14)

Since ${\mathsf{\Delta }}_{\mathbf{z}}{w}_R\left({\mu }_R\right)\le 0,$ we obtain

$$\begin{array}{c}\hfill {\varphi }_R{\mathsf{\Delta }}_{\mathbf{z}}w\le 2w{\varphi }_R^{-1}{|{\nabla }_{\mathbf{z}}{\varphi }_R|}^2-w{\mathsf{\Delta }}_{\mathbf{z}}{\varphi }_R.\end{array}$$

(15)

From (13) and (15), one concludes

$$\begin{array}{c}\hfill {\varphi }_R{\mathsf{\Delta }}_{\mathbf{z}}w\le \frac{C}{R^2}{\varphi }_R^{\frac{t-2}{t}}w.\end{array}$$

(16)

Using (13) and (15), and the fact that

$$|\nu \left(\mu \right)|\le \frac{\kappa }{1+\parallel \mu \parallel },$$

we can deduce that for any $\kappa >0$, there exists a positive constant C depending only on $\kappa$ such that

$$\begin{array}{c}\hfill |\nu ·{\nabla }_{\mathbf{z}}w{\varphi }_R|\le \frac{C_{\kappa }}{R^2}{\varphi }_R^{\frac{t-1}{t}}w.\end{array}$$

(17)

Recalling now that $w=u-\lambda v^{\sigma }\ge 0,$ and at ${\mu }_R,$ it is shown that

$$\begin{array}{c}\hfill \frac{u^{q_2}}{w^{q_2}}-\frac{{\left(\lambda v^{\sigma }\right)}^{q_2}}{w^{q_2}}\ge 1,\mathrm{or} \mathrm{equivalently}{\lambda }^{-q_2}{u}^{q_2}-v^{q_2\sigma }\ge {\lambda }^{-q_2}{w}^{q_2}.\end{array}$$

(18)

Multiplying (11) by ${\varphi }_R,$ and using (16)–(18), there holds

$$\begin{array}{c}\hfill v^{\sigma -1}{w}^{q_2}{\varphi }_R^{\frac{t+2}{2}}\le \frac{C}{R^2}w{\varphi }_R.\end{array}$$

The sequence $v\left({\mu }_R\right)$ is bounded, as $\sigma \le 1$. We choose

$$\begin{array}{c}\hfill q_2-1=\frac{2}{t}\mathrm{so} \mathrm{that}t=\frac{2}{q_2-1},\end{array}$$

we then get

$$\begin{array}{c}\hfill w_R^{q_2-1}\left({\mu }_R\right)\le \frac{C}{R^2}.\end{array}$$

Letting $R\to \infty$, we get ${sup}_{{\mathbb{R}}^N}w=0,$ which is a contradiction with (12). The proof is completed. □

We proceed, like for the proof of the above lemma, to establish an inverse comparison property.

Lemma 2.

Let $q_2\ge q_1>1.$ Assume that $(u,v)$ is a bounded positive solution of (1), which satisfies Equation (3); we have

$$\begin{array}{c}\hfill v\le {\parallel u\parallel }_{\infty }^{\frac{q_2-q_1}{q_1+1}}u.\end{array}$$

(19)

Put $ł={\parallel u\parallel }_{\infty }^{\frac{q_2-q_1}{q_1+1}}$ and $w=v-łu,$. As $q_2\ge q_1$, we get

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{s}w-\nu ·{\nabla }_{s}w=\varrho \left(\mu \right)\left(łv^{q_1}-u^{q_2}\right)& =\varrho \left(\mu \right)\left[łv^{q_1}-{\left(\frac{u}{{\parallel u\parallel }_{\infty }}\right)}^{q_2}{\parallel u\parallel }_{\infty }^{q_2}\right]\hfill \\ & \ge \varrho \left(\mu \right)\left[łv^{q_1}-{\left(\frac{u}{{\parallel u\parallel }_{\infty }}\right)}^{q_1}{\parallel u\parallel }_{\infty }^{q_2}\right]\hfill \\ \hfill & ={\varrho \left(\mu \right)\parallel u\parallel }_{\infty }^{q_2-q_1}\left(\frac{łv^{q_1}}{{\parallel u\parallel }_{\infty }^{q_2-q_1}}-u^{q_1}\right)\hfill \\ \hfill & ={\varrho \left(\mu \right)\parallel u\parallel }_{\infty }^{q_2-q_1}\left({ł}^{-q_1}{v}^{q_1}-u^{q_1}\right).\hfill \end{array}$$

The rest of the proof is then obtained by Lemma 1. We then omit the details. □

2.2. Integral Estimates

We now point out the following useful lemma.

Lemma 3.

Assume that $(u,v)$ is a positive stable solution of (1) with (3) verified. Then, for $\gamma \in C_c^1\left({\mathbb{R}}^N\right)$, we obtain

$$\sqrt{q_1{q}_2}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{\gamma }^{2}dxdy\le \frac{1}{4}{\int }_{{\mathbb{R}}^N}|\nu \gamma +2{\nabla }_{\mathbf{z}}\gamma {|}^{2}dxdy.$$

(20)

Proof. 

Let $\gamma ,\phi \in C_c^1\left({\mathbb{R}}^N\right).$ We multiply the first equation in Definition 1 of stability by $\frac{{\gamma }^2}{\phi }$ and integrate over ${\mathbb{R}}^N,$ we get

$$\begin{array}{c}\hfill q_1{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^{q_1-1}\frac{\psi }{\phi }{\gamma }^{2}dxdy=-{\int }_{{\mathbb{R}}^N}\left(\frac{{\mathsf{\Delta }}_{\mathbf{z}}\phi }{\phi }{\gamma }^2+\nu ·{\nabla }_{\mathbf{z}}\phi \frac{{\gamma }^2}{\phi }\right)dxdy.\end{array}$$

A direct calculation gives

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\left(-\frac{{\mathsf{\Delta }}_{\mathbf{z}}\phi }{\phi }{\gamma }^2+\nu ·{\nabla }_{\mathbf{z}}\phi \frac{{\gamma }^2}{\phi }\right)dxdy& ={\int }_{{\mathbb{R}}^N}\left({\nabla }_{\mathbf{z}}\phi ·{\nabla }_{\mathbf{z}}\left({\gamma }^2{\phi }^{-1}\right)+\nu ·{\nabla }_{\mathbf{z}}\phi \frac{{\gamma }^2}{\phi }\right)dxdy\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(-{\phi }^{-2}{|{\nabla }_{\mathbf{z}}\phi |}^2{\gamma }^2+2{\phi }^{-1}\gamma {\nabla }_{\mathbf{z}}\phi ·{\nabla }_{\mathbf{z}}\gamma +\nu ·{\nabla }_{\mathbf{z}}\phi \frac{{\gamma }^2}{\phi }\right)dxdy\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(+{\phi }^{-1}\gamma {\nabla }_{\mathbf{z}}\phi \left(\nu \gamma +2{\nabla }_{\mathbf{z}}\gamma \right)-{\phi }^{-2}{|{\nabla }_{\mathbf{z}}\phi |}^2{\gamma }^2\right)dxdy.\hfill \end{array}$$

Using the inequality $2ab-a^2\le b^2,$ we deduce that

$$\begin{array}{c}\hfill 4q_1{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^{q_1-1}\frac{\psi }{\phi }{\gamma }^{2}dxdy\le {\int }_{{\mathbb{R}}^N}|\nu \gamma +2{\nabla }_{\mathbf{z}}\gamma {|}^{2}dxdy.\end{array}$$

(21)

By the same argument, we also have

$$\begin{array}{c}\hfill 4q_2{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{q_2-1}\frac{\phi }{\psi }{\gamma }^{2}dxdy\le {\int }_{{\mathbb{R}}^N}|\nu \gamma +2{\nabla }_{\mathbf{z}}\gamma {|}^{2}dxdy.\end{array}$$

(22)

Adding the inequalities (21) and (22), we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)\left(q_1{v}^{q_1-1}\frac{\psi }{\phi }{\gamma }^2+q_2{u}^{q_2-1}\frac{\phi }{\psi }{\gamma }^2\right)dxdy\le \frac{1}{2}{\int }_{{\mathbb{R}}^N}|\nu \gamma +2{\nabla }_{\mathbf{z}}\gamma {|}^{2}dxdy.\end{array}$$

(23)

Denote

$$\begin{array}{c}\hfill 2\varrho \left(\mu \right)\sqrt{q_1{q}_2{v}^{q_1-1}{u}^{q_2-1}{\gamma }^4}\le \varrho \left(\mu \right)\left(q_1{v}^{q_1-1}\frac{\psi }{\phi }{\gamma }^2+q_2{u}^{q_2-1}\frac{\phi }{\psi }{\gamma }^2\right).\end{array}$$

(24)

Combining (24) with (23), we readily get the estimate (20). □

We will use also the following integral estimates for all solutions of (1), where (3) is satisfied.

Lemma 4.

Assume that $q_2\ge q_1>1$ and (3) holds. There exists $C>0$ for any solution $(u,v)$ of (1) such that

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_R}\varrho \left(\mu \right)v^{q_1}dxdy\le C{R}^{{\gamma }_1},where{\gamma }_1:={\mathbf{G}}_{\mathbf{z}}^{*}-q_1\beta -\frac{\alpha }{2}\rho andR\ge 1.\end{array}$$

(25)

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_R}\varrho \left(\mu \right)u^{q_2}dxdy\le C{R}^{{\gamma }_2},where{\gamma }_2:={\mathbf{G}}_{\mathbf{z}}^{*}-q_2\alpha -\frac{\beta }{2}\rho ,\end{array}$$

(26)

where

$${\mathsf{\Omega }}_R:=B_R\times B_{R^{1+\mathbf{z}}},\varrho \left(\mu \right):={\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{\frac{\rho }{2(\mathbf{z}+1)}},and\alpha =\frac{2(q_1+1)}{q_1{q}_2-1},\beta =\frac{2(q_2+1)}{q_1{q}_2-1},q_1{q}_2>1.$$

Proof. 

We use the cut-off function $0\le {\xi }_k\in C_c^{\infty }\left(\mathbb{R}\right)\le 1,$ satisfying

$${\xi }_k=1\mathrm{on}[-1,1],\mathrm{and}{\xi }_k=0\mathrm{outside}[-2^{1+(k-1)\mathbf{z}},2^{1+(k-1)\mathbf{z}}]\mathrm{where}k=1,2.$$

For $R\ge 1,$ put ${\psi }_R(x,y)={\xi }_1\left(\frac{x}{R}\right){\xi }_2\left(\frac{y}{R^{1+\mathbf{z}}}\right),$ we can easily see that

$$|{\nabla }_x{\psi }_R|\le \frac{C}{R}\mathrm{and}|{\nabla }_y{\psi }_R|\le \frac{C}{R^{1+\mathbf{z}}},$$

$$|{\mathsf{\Delta }}_x{\psi }_R|\le \frac{C}{R^2}\mathrm{and}|{\mathsf{\Delta }}_y{\psi }_R|\le \frac{C}{R^{2(1+\mathbf{z})}}.$$

Multiplying

$$-{\mathsf{\Delta }}_{\mathbf{z}}u+\nu ·{\nabla }_{\mathbf{z}}u=\varrho \left(\mu \right)v^{q_1},$$

By ${\psi }_R^m,$ and integrating by parts, there holds

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{m}dxdy=-{\int }_{{\mathsf{\Omega }}_{2R}}u\left({\mathsf{\Delta }}_{\mathbf{z}}\left({\psi }_R^m\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\psi }_R^m\right)\right)dxdy\le \frac{C}{R^2}{\int }_{{\mathsf{\Omega }}_{2R}}u{\psi }_R^{m-2}dxdy,\end{array}$$

where

$${\mathsf{\Omega }}_{2R}:=B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}.$$

Let $\frac{1}{q_2}+\frac{1}{q_2^{\prime }}=1.$ Applying Hölder’s inequality, we get then

$$\begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{m}dxdy\le & \frac{C}{R^2}{\left[{\int }_{{\mathsf{\Omega }}_{2R}}{\left(\varrho \left(\mu \right)\right)}^{-\frac{q_2^{\prime }}{q_2}}dxdy\right]}^{\frac{1}{q_2^{\prime }}}\times {\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)u^{q_2}{\psi }_R^{(m-2)q_2}dxdy\right)}^{\frac{1}{q_2}}\hfill \\ \hfill \le & C{R}^{\frac{N_1+(1+\mathbf{z})N_2}{q_2^{\prime }}-\frac{\rho }{q_2}-2}{\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)u^{q_2}{\psi }_R^{(m-2)q_2}dxdy\right)}^{\frac{1}{q_2}}.\hfill \end{array}$$

Now, we multiply

$$-{\mathsf{\Delta }}_{\mathbf{z}}v+\nu ·{\nabla }_{\mathbf{z}}u=\varrho \left(\mu \right)u^{q_2},$$

by ${\psi }_R^k$ with $k\ge 2$ and we integrate by parts. By Hölder’s inequality, we have

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)u^{q_2}{\psi }_R^{k}dxdy\le C{R}^{\frac{N_1+(1+\mathbf{z})N_2}{q_1^{\prime }}-\frac{\rho }{q_1}-2}{\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{(k-2)q_1}dxdy\right)}^{\frac{1}{q_1}},\end{array}$$

where $\frac{1}{q_1}+\frac{1}{q_1^{\prime }}=1.$ Taking large k and m such that $m\le (k-2)q_1$ and $k\le (m-2)q_2$, in view of the two above inequalities, we get

$$\begin{array}{cc}\hfill & {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{m}dxdy\hfill \\ \hfill \le & C{R}^{\frac{N_1+(1+\mathbf{z})N_2}{q_2^{\prime }}-\frac{\rho }{q_2}-2}{R}^{\left(\frac{N_1+(1+\mathbf{z})N_2}{q_1^{\prime }}-\frac{\rho }{q_1}-2\right)\frac{1}{q_2}}{\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{(k-2)q_1}dxdy\right)}^{\frac{1}{q_1{q}_2}}\hfill \\ \hfill \le & C{R}^{N_1+(1+\mathbf{z})N_2-\frac{N_1+(1+\mathbf{z})N_2}{q_1{q}_2}-\frac{\rho (q_1+1)}{q_1{q}_2}-\frac{2(q_2+1)}{q_2}}{\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{m}dxdy\right)}^{\frac{1}{q_1{q}_2}}.\hfill \end{array}$$

So, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}dxdy\le {\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}{\psi }_R^{m}dxdy\le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}-q_1\beta -\frac{\alpha }{2}\rho }.\end{array}$$

Finally, using the same argument as above, we obtain the estimate (26). □

We need the following integral estimate for $u,$ which is crucial to deal with the case $1<q_1\le \frac{4}{3}.$

Lemma 5.

Let $(u,v)$ be a stable solution of (1), $1<q_1\le min(\frac{4}{3},q_2)$, and (3) is satisfied. Assume that u is bounded; we have

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_R}\varrho \left(\mu \right)v^{2}dxdy\le C{R}^{{\gamma }_3},\end{array}$$

(27)

where

$${\mathsf{\Omega }}_R:=B_R\times B_{R^{1+\mathbf{z}}},\varrho \left(\mu \right):={\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{\frac{\rho }{2(\mathbf{z}+1)}},and{\gamma }_3:={\gamma }_1-\frac{2(2+\rho )(2-q_1)}{q_1+q_2-2},{\gamma }_1:={\mathbf{G}}_{\mathbf{z}}^{*}-q_1\beta -\frac{\alpha }{2}\rho .$$

Proof. 

Take ${\eta }_R(x,y)=\xi (\frac{x}{R},\frac{y}{R^{1+\mathbf{z}}}),$ where $\xi \in C_c^{\infty }\left({\mathbb{R}}^N\right)$ is a cut-off function satisfying

$$\xi =1\mathrm{on}{B}_1\times B_1,\mathrm{and}\xi =0\mathrm{outside}{B}_2\times B_{2^{1+\mathbf{z}}}.$$

Multiplying

$$-{\mathsf{\Delta }}_{\mathbf{z}}v+\nu ·{\nabla }_{\mathbf{z}}v=\varrho \left(\mu \right)u^{q_2},$$

by $v{\eta }_R^2$ and integrating parts, we get

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy={\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{q_2}vs.{\eta }_R^{2}dxdy+\frac{1}{2}{\int }_{{\mathbb{R}}^N}{v}^2\left({\mathsf{\Delta }}_{\mathbf{z}}\left({\eta }_R^2\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\eta }_R^2\right)\right)dxdy.\end{array}$$

Using Lemma 1, we get

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy\le \sqrt{\frac{q_2+1}{q_1+1}}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1+1}{2}}v{\eta }_R^{2}dxdy+\frac{1}{2}{\int }_{{\mathbb{R}}^N}{v}^2\left({\mathsf{\Delta }}_{\mathbf{z}}\left({\eta }_R^2\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\eta }_R^2\right)\right)dxdy.\end{array}$$

Set $\gamma =vs.{\eta }_R$ in (20) and integrating by parts; we obtain

$$\begin{array}{cc}\hfill & \sqrt{q_2{q}_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{v}^2{\eta }_R^{2}dxdy\hfill \\ \hfill & \le \frac{1}{4}{\int }_{{\mathbb{R}}^N}|\nu vs.{\eta }_R+2{\nabla }_{\mathbf{z}}\left(v{\eta }_R\right){|}^{2}dxdy\hfill \\ \hfill & =\frac{1}{4}\left({\int }_{{\mathbb{R}}^N}|{\nu }^2{v}^2{\eta }_R^2+4{\left({\nabla }_{\mathbf{z}}\left(v{\eta }_R\right)\right)}^2|\right)dxdy+{\int }_{{\mathbb{R}}^N}|\nu ·{\nabla }_{\mathbf{z}}\left(v{\eta }_R\right)v{\eta }_R|dxdy\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^N}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy+{\int }_{{\mathbb{R}}^N}{v}^2\left(\frac{1}{4}{\nu }^2{v}^2{\eta }_R^2+{|{\nabla }_{\mathbf{z}}{\eta }_R|}^2\right)dxdy-\frac{1}{2}{\int }_{{\mathbb{R}}^N}{v}^2{\mathsf{\Delta }}_{\mathbf{z}}\left({\eta }_R^2\right)dxdy\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\nu ·{\nabla }_{\mathbf{z}}\left(v^2{\eta }_R^2\right)|dxdy.\hfill \end{array}$$

Combining the two last inequalities, we obtain

$$\begin{array}{c}\hfill \left(\sqrt{q_2{q}_1}-\sqrt{\frac{q_2+1}{q_1+1}}\right){\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1+3}{2}}{\eta }_R^{2}dxdy\le \frac{1}{2}{\int }_{{\mathbb{R}}^N}{v}^2\left(\frac{1}{2}{\nu }^2{v}^2{\eta }_R^2+2{|{\nabla }_{\mathbf{z}}{\eta }_R|}^2+\nu ·{\nabla }_{\mathbf{z}}\left({\eta }_R^2\right)\right)dxdy.\end{array}$$

We assume that ${\eta }_R={\phi }_R^m$ with $m>2.$ Using Lemma 2, we get

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^{q_0}{\phi }_R^{2m}dxdy\le \frac{C}{R^{2+\rho }}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^2{\phi }_R^{2m-2}dxdy,\end{array}\end{array}$$

(28)

where $q_0=\frac{q_2+q_1+2}{2}.$ Denote

$$\begin{array}{c}\hfill J_1:={\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^{q_0}{\phi }_R^{2m}dxdy,J_2:={\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^2{\phi }_R^{2m-2}dxdy.\end{array}$$

As $q_1\ge q_2,$ we observe that $q_1<2<q_0$ for $1<q_1\le \frac{4}{3}$. A simple calculation yields

$$\begin{array}{c}\hfill 2=q_1\lambda +q_0(1-\lambda )\mathrm{with} \lambda =\frac{q_2+q_1-2}{q_2-q_1+2}\in (0,1).\end{array}$$

Since we assume that m large with $m\lambda >1$, from Hölder’s inequality, Lemma 4, and according to inequality (28), we have

$$\begin{array}{cc}\hfill J_2\le J_1^{1-\lambda }{\left({\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)v^{q_1}{\phi }_R^{2m\lambda -2}dxdy\right)}^{\lambda }& \le {\left(\frac{C{J}_2}{R^{2+\rho }}\right)}^{1-\lambda }{\left({\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{q_1}dxdy\right)}^{\lambda }\hfill \\ \hfill & \le C^{\prime }{J}_2^{1-\lambda }{R}^{-(2+\rho )(1-\lambda )}{R}^{\lambda {\gamma }_1},\hfill \end{array}$$

which gives

$$\begin{array}{c}\hfill J_2\le C{R}^{{\gamma }_1-\frac{2(2+\rho )(2-q_1)}{q_1+q_2-2}},\end{array}$$

where ${\gamma }_1$ is given in Lemma 4. So we are done. □

We need also the following technical lemma, which plays a crucial role in establishing Theorem 2 and Proposition 1.

Lemma 6.

Assume that $(u,v)$ is a stable solution to (1) with $\rho \ge 0,$ and (3) is satisfied. There exists $C<\infty$ such that for any $n>\frac{q_1+1}{2}$ verifying $P\left(n\right)<0$, we have

$${\int }_{B_R\times B_{R^{1+\mathbf{z}}}}\varrho \left(\mu \right)u^{q_2}{v}^{n-1}dxdy\le \frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{v}^{n}dxdy,$$

(29)

where

$$\begin{array}{c}\hfill P\left(n\right):=n^4-16q_1{q}_2\frac{\alpha }{\beta }\left(n^2+\frac{q_1+q_2+2}{q_2+1}n-\frac{\alpha }{\beta }\right).\end{array}$$

(30)

Proof. 

Set $\varphi \in C_0^2\left({\mathbb{R}}^N\right).$ Let $(u,v)$ be a stable solution of (1); we integrate by parts to get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}{|{\nabla }_{\mathbf{z}}{u}^{\frac{s+1}{2}}|}^2{\varphi }^{2}dxdy& =\frac{{(s+1)}^2}{4}{\int }_{{\mathbb{R}}^N}{u}^{s-1}{|{\nabla }_{\mathbf{z}}u|}^2{\varphi }^{2}dxdy\hfill \\ & =\frac{{(s+1)}^2}{4s}{\int }_{{\mathbb{R}}^N}{\varphi }^2{\nabla }_{\mathbf{z}}\left(u^s\right){\nabla }_{\mathbf{z}}udxdy\hfill \\ \hfill & =\frac{{(s+1)}^2}{4s}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy\hfill \\ \hfill & +\frac{s+1}{4s}{\int }_{{\mathbb{R}}^N}{u}^{s+1}\left({\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }^2\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\varphi }^2\right)\right)dxdy,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill (s+1){\int }_{{\mathbb{R}}^N}{u}^s\varphi {\nabla }_{\mathbf{z}}u\nabla \varphi dxdy=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\nabla \left(u^{s+1}\right){\nabla }_{\mathbf{z}}\left({\varphi }^2\right)dxdy=-\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^{s+1}{\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }^2\right)dxdy,\end{array}$$

and

$$\begin{array}{c}\hfill (s+1){\int }_{{\mathbb{R}}^N}\nu ·{\nabla }_{\mathbf{z}}u{u}^s{\varphi }^{2}dxdy={\int }_{{\mathbb{R}}^N}\nu ·{\nabla }_{\mathbf{z}}\left(u^{s+1}\right){\varphi }^{2}dxdy=-{\int }_{{\mathbb{R}}^N}{u}^{s+1}\nu ·{\nabla }_{\mathbf{z}}\left({\varphi }^2\right)dxdy.\end{array}$$

We now apply the stability inequality (20) for $\gamma =u^{\frac{s+1}{2}}\varphi$ with $s>0$ and $\varphi \in C_c^{\infty }\left({\mathbb{R}}^N,[0,1]\right)$. In view of the above equalities, we deduce that

$$\begin{array}{cc}\hfill & \sqrt{q_1{q}_2}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{s+1}{\varphi }^{2}dxdy\hfill \\ \hfill & \le \frac{1}{4}{\int }_{{\mathbb{R}}^N}|\nu u^{\frac{s+1}{2}}\varphi +2{\nabla }_{\mathbf{z}}\left(u^{\frac{s+1}{2}}\varphi \right){|}^{2}dxdy\hfill \\ \hfill & \le \frac{1}{4}{\int }_{{\mathbb{R}}^N}{\nu }^2{u}^{s+1}{\varphi }^{2}dxdy+{\int }_{{\mathbb{R}}^N}{|{\nabla }_{\mathbf{z}}\left(u^{\frac{s+1}{2}}\varphi \right)|}^{2}dxdy+\frac{1}{2}{\int }_{{\mathbb{R}}^N}|\nu ·{\nabla }_{\mathbf{z}}\left(u^{s+1}{\varphi }^2\right)|dxdy\hfill \\ \hfill & \le \frac{{(s+1)}^2}{4s}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy+C{\int }_{{\mathbb{R}}^N}{u}^{s+1}\left[{\nu }^2{\varphi }^2+{|{\nabla }_{\mathbf{z}}\varphi |}^2+{\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }^2\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\varphi }^2\right)\right]dxdy.\hfill \end{array}$$

So we get

$$\begin{array}{cc}\hfill & b_1{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{s+1}{\varphi }^{2}dxdy\hfill \\ \hfill \le & {\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy+C{\int }_{{\mathbb{R}}^N}{u}^{s+1}\left[{\nu }^2{\varphi }^2+{|{\nabla }_{\mathbf{z}}\varphi |}^2+{\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }^2\right)+\nu ·{\nabla }_{\mathbf{z}}\left({\varphi }^2\right)\right]dxdy,\hfill \end{array}$$

where $b_1=\frac{4s\sqrt{q_1{q}_2}}{{(s+1)}^2}$. We choose $\varphi (x,y)=\xi (\frac{x}{R},\frac{y}{R^{1+\mathbf{z}}}),$ where $0\le \xi \in C_c^{\infty }\left({\mathbb{R}}^{\mathbb{N}}\right)\le 1,$ is a cut-off function satisfying

$$\xi =1\mathrm{on}{B}_1\times B_1,\mathrm{and}\xi =0\mathrm{outside}{B}_2\times B_{2^{1+\mathbf{z}}}.$$

A direct calculation gives

$$|{\nabla }_{\mathbf{z}}\varphi |\le \frac{C}{R}\mathrm{and}|{\mathsf{\Delta }}_{\mathbf{z}}\left({\varphi }^2\right)|\le \frac{C}{R^2}.$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill I_1:={\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{s+1}{\varphi }^{2}dxdy\le \frac{1}{b_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy+\frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{u}^{s+1}dxdy.\end{array}\end{array}$$

If we now invoke (20) with $\gamma =v^{\frac{t+1}{2}}\varphi ,$$t>0,$ it follows from $I_1$ that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill I_2:={\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{v}^{t+1}{\varphi }^{2}dxdy\le \frac{1}{b_2}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{q_2}{v}^t{\varphi }^{2}dxdy+\frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{v}^{t+1}dxdy,\end{array}\end{array}$$

with $b_2=\frac{4t\sqrt{q_1{q}_2}}{{(t+1)}^2}$. Combining the two last inequalities, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & I_1+{b_2}^{\frac{2(t+1)}{q_1+1}}{I}_2\hfill \\ \hfill & \le \frac{1}{b_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy+{b_2}^{\frac{2t+1-q_1}{q_1+1}}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{q_2}{v}^t{\varphi }^{2}dxdy\hfill \\ \hfill & +\frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}\left(u^{s+1}+v^{t+1}\right)dxdy.\hfill \end{array}\end{array}$$

(31)

Fix

$$s=\frac{(q_2+1)p}{q_1+1}+\frac{q_2-q_1}{q_1+1},\mathrm{or} \mathrm{equivalently}s+1=\frac{(q_2+1)(p+1)}{q_1+1}.$$

(32)

Let $t>\frac{q_1-1}{2}.$ We apply Young’s inequality, and from (32), we get then

$$\begin{array}{cc}\hfill \frac{1}{b_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^s{v}^{q_1}{\varphi }^{2}dxdy=& \frac{1}{b_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{\frac{(q_2+1)t}{q_1+1}+\frac{q_2+1}{q_1+1}\left(\frac{1-q_1}{2}\right)}{v}^{\frac{q_1+1}{2}}{\varphi }^{2}dxdy\hfill \\ \hfill =& \frac{1}{b_1}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{(s+1)\frac{2t+1-q_1}{2(t+1)}}{v}^{\frac{q_1+1}{2}}{\varphi }^{2}dxdy\hfill \\ \hfill \le & \frac{2t+1-q_1}{2(t+1)}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{u}^{s+1}{\varphi }^{2}dxdy\hfill \\ \hfill & +\frac{q_1+1}{2(t+1)}{b}_1^{-\frac{2(t+1)}{q_1+1}}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{v}^{t+1}{\varphi }^{2}dxdy\hfill \\ \hfill =& \frac{2t+1-q_1}{2(t+1)}{I}_1+\frac{q_1+1}{2(t+1)}{b}_1^{-\frac{2(t+1)}{q_1+1}}{I}_2,\hfill \end{array}$$

and similarly

$$\begin{array}{c}\hfill {b_2}^{\frac{2t+1-q_1}{q_1+1}}{\int }_{{\mathbb{R}}^N}\varrho \left(\mu \right)u^{q_2}{v}^t{\varphi }^{2}dxdy\le \frac{q_1+1}{2(t+1)}{I}_1+\frac{2t+1-q_1}{2(t+1)}{b_2}^{\frac{2(t+1)}{q_1+1}}{I}_2.\end{array}$$

We insert the two above estimates in (31), and we get

$$\begin{array}{c}\hfill {b_2}^{\frac{2(t+1)}{q_1+1}}{I}_2\le \left[\frac{2t+1-q_1}{2(t+1)}{b_2}^{\frac{2(t+1)}{q_1+1}}+\frac{q_1+1}{2(t+1)}{b_1}^{\frac{-2(t+1)}{q_1+1}}\right]I_2+\frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}\left(u^{s+1}+v^{t+1}\right)dxdy.\end{array}$$

Combining (32) and (9), we obtain

$$u^{s+1}\le C{v}^{t+1}\mathrm{and}{u}^{\frac{q_2-1}{2}}{v}^{\frac{q_1-1}{2}}{v}^{t+1}\ge u^{q_2}{v}^t.$$

We get then

$$\begin{array}{c}\hfill \frac{q_1+1}{2(t+1)}\left[{\left(b_1{b}_2\right)}^{\frac{2(t+1)}{q_1+1}}-1\right]{\int }_{{\mathbb{R}}^N}{u}^{q_2}{v}^t{\varphi }^{2}dxdy\le C{R}^{-2}{b}_1^{\frac{2(t+1)}{q_1+1}}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{v}^{t+1}dxdy.\end{array}$$

Hence, if $b_1{b}_2>1$, we conclude then

$$\begin{array}{c}\hfill {\int }_{B_R\times B_{{\left(R\right)}^{1+\mathbf{z}}}}\varrho \left(\mu \right)u^{q_2}{v}^{t}dxdy\le {\int }_{{\mathbb{R}}^N}{u}^{q_2}{v}^t{\varphi }^{2}dxdy\le \frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{v}^{t+1}dxdy.\end{array}$$

Denote $n-1=t$; we deduce that if $b_1{b}_2>1$ and $n>\frac{q_1+1}{2}$,

$$\begin{array}{c}\hfill {\int }_{B_R\times B_{{\left(R\right)}^{1+\mathbf{z}}}}\varrho \left(\mu \right)u^{q_2}{v}^{n-1}dxdy\le \frac{C}{R^2}{\int }_{B_{2R}\times B_{{\left(2R\right)}^{1+\mathbf{z}}}}{v}^{n}dxdy.\end{array}$$

Furthermore, we can verify the equivalence between $b_1{b}_2>1$ and $P\left(n\right)<0$. So we are done. □

We change the variables $m=\left(\frac{\beta }{2}\right)n$ in (7); a direct calculation gives

$$Q\left(m\right)={\left(\frac{\beta }{2}\right)}^{4}P\left(n\right),\mathrm{where}\beta =\frac{2(q_2+1)}{q_1{q}_2-1},$$

and P is given by (30). Hence $Q\left(m\right)<0$ if and only if $P\left(n\right)<0$. Moreover, using Lemma 6 in [15], we have the following remark.

Remark 2.

• Let $1<q_1\le q_2$, then $P\left(2\right)<0$, and P has a unique root $n_0$ in $(2,\infty )$ and $2k_0^{+}<n_0$.
• If $q_1>\frac{4}{3}$, then $P\left(q_1\right)<0$ and $n_0$ is the unique root of P in $(q_1,\infty ),$ hence $m_0=\left(\frac{\beta }{2}\right)n_0$.
• From Remark 3 in [6], we get

  $$m_0>\beta k_0^{+}>4,\forall q_2\ge q_1>1.$$
• Obviously $2k_0^{-}<q_1$ if $q_1>\frac{4}{3}$. Indeed, if $q_1>\frac{4}{3}$ then $q_2\ge q_1>\frac{4}{3}$ and $\varpi :=\frac{q_1{q}_2(q_1+1)}{q_2+1}>\frac{16}{9}.$ Since $g\left(\varpi \right):=\sqrt{\varpi }-\sqrt{\varpi -\sqrt{\varpi }}$ is decreasing in ϖ; there holds $2k_0^{-}=2g\left(\varpi \right)<2f\left(\frac{16}{9}\right)=\frac{4}{3}<q_1.$

3. Proofs of Main Results

Let $0\le \eta \in C_c^{\infty }\left({\mathbb{R}}^{\mathbb{N}}\right)\le 1$ be a cut-off function satisfying

$$\begin{array}{c}\hfill \eta =1\mathrm{on}{B}_1\times B_1,\mathrm{and}\eta =0\mathrm{outside}{B}_2\times B_{2^{1+\mathbf{z}}}.\end{array}$$

(33)

The proof may be divided in three parts.

Step $\mathbf{1}.$ There exists a constant $C>0$ such that for any smooth function $h>0,$ and ${\theta }_{\mathbf{z}}=\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{{\mathbf{G}}_{\mathbf{z}}^{*}-2},$ we have

$$\begin{array}{c}\hfill {\left({\int }_{{\mathsf{\Omega }}_R}{h}^{2{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{{\theta }_{\mathbf{z}}}}\le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{{\theta }_{\mathbf{z}}}-1\right)+2}{\int }_{{\mathsf{\Omega }}_{2R}}{|{\nabla }_{\mathbf{z}}h|}^{2}dxdy+C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{{\theta }_{\mathbf{z}}}-1\right)}{\int }_{{\mathsf{\Omega }}_{2R}}{h}^{2}dxdy,\end{array}$$

(34)

where

$${\mathsf{\Omega }}_{{\alpha }^{\prime }R}:=B_{{\alpha }^{\prime }R}\times B_{{\left({\alpha }^{\prime }R\right)}^{1+\mathbf{z}}}with,{\alpha }^{\prime }>0.$$

Indeed, employing Sobolev inequality [14] and integrating by parts, we get

$$\begin{array}{cc}\hfill {\left({\int }_{B_1\times B_1}{h}^{2{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{2{\theta }_{\mathbf{z}}}}& \le {\left({\int }_{B_2\times B_{2^{1+\mathbf{z}}}}{\left(h\eta \right)}^{2{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{2{\theta }_{\mathbf{z}}}}\hfill \\ \hfill & \le C{\left({\int }_{B_2\times B_{2^{1+\mathbf{z}}}}{|{\nabla }_{\mathbf{z}}\left(h\eta \right)|}^{2}dxdy\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le C{\left[{\int }_{B_2\times B_{2^{1+\mathbf{z}}}}\left(|{\nabla }_{\mathbf{z}}{h|}^2{\eta }^2+h^2{|{\nabla }_{\mathbf{z}}\eta |}^2-\frac{h^2}{2}{\mathsf{\Delta }}_{\mathbf{z}}\left(\eta \right)\right)dxdy\right]}^{\frac{1}{2}}.\hfill \end{array}$$

So, we obtain

$$\begin{array}{c}\hfill {\left({\int }_{B_1\times B_1}{h}^{2{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{{\theta }_{\mathbf{z}}}}\le C{\int }_{B_2\times B_{2^{1+\mathbf{z}}}}\left(|{\nabla }_{\mathbf{z}}{h|}^2+h^2\right)dxdy.\end{array}$$

Making use of scaling argument, we get the inequality (34).

Step $\mathbf{2}.$ There exists a positive constant $C>0$ such that for any ${\theta }_{\mathbf{z}}=\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{{\mathbf{G}}_{\mathbf{z}}^{*}-2}$ and $2k_0^{-}<n_0,$ there holds

$${\left({\int }_{{\mathsf{\Omega }}_R}{v}^{n_0{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{{\theta }_{\mathbf{z}}}}\le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{{\theta }_{\mathbf{z}}}-1\right)}{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0}dxdy.$$

(35)

In order to prove this, for $2k_0^{-}<n_0,$ in what follows, taking

$$h=v^{\frac{n_0}{2}}.$$

Set ${\eta }_R(x,y)=\eta (\frac{x}{R},\frac{y}{R^{1+\mathbf{z}}}),$ where $\eta$ is given in (33). By a simple calculation, we obtain readily

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_R}|{\nabla }_{\mathbf{z}}{h|}^{2}dxdy\le C{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0-2}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy.\end{array}$$

(36)

Multiplying

$$-{\mathsf{\Delta }}_{\mathbf{z}}v+\nu ·{\nabla }_{\mathbf{z}}v=\varrho \left(\mu \right)u^{q_2},$$

by $v^{n_0-1}{\eta }_R^2$ and integrating by parts, we derive

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & (n_0-1){\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0-2}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy\hfill \\ \hfill & ={\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{n_0-1}{u}^{q_2}{\eta }_R^{2}dxdy-\frac{1}{n_0}{\int }_{{\mathsf{\Omega }}_{2R}}\nu ·{\nabla }_{\mathbf{z}}\left(v^{n_0}\right){\eta }_R^2-2{\int }_{{\mathsf{\Omega }}_{2R}}{\eta }_R{v}^{n_0-1}{\nabla }_{\mathbf{z}}v·{\nabla }_{\mathbf{z}}{\eta }_{R}dxdy.\hfill \end{array}\end{array}$$

(37)

Using Young’s inequality, we obtain

$$\begin{array}{c}\hfill 2{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0-1}|{\nabla }_{\mathbf{z}}v||{\nabla }_{\mathbf{z}}{\eta }_R|{\eta }_{R}dxdy\le \frac{n_0-1}{2}{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0-2}|{\nabla }_{\mathbf{z}}{v|}^2{\eta }_R^{2}dxdy+C{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0}{|{\nabla }_{\mathbf{z}}{\eta }_R|}^{2}dxdy.\end{array}$$

Substituting this in (37), and by inequality (36), we get

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}_R}|{\nabla }_{\mathbf{z}}{h|}^{2}dxdy\le {\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0-2}{|{\nabla }_{\mathbf{z}}v|}^2{\eta }_R^{2}dxdy\le C{\int }_{{\mathsf{\Omega }}_{2R}}\varrho \left(\mu \right)v^{n_0-1}{u}^{q_2}{\eta }_R^{2}dxdy+\frac{C}{R^2}{\int }_{{\mathsf{\Omega }}_{2R}}{v}^{n_0}dxdy.\end{array}$$

The inequality (34) and Lemma 6 give the estimate (35).

Step $\mathbf{3}.$ Let $q\in (2k_0^{-},n_0),$ then for any ${\theta }_{\mathbf{z}}=\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{{\mathbf{G}}_{\mathbf{z}}^{*}-2}$ and $q<n_m{\theta }_{\mathbf{z}},$ we deduce that there exists a positive constant $C>0$ with

$$\begin{array}{c}\hfill {\left({\int }_{{\mathsf{\Omega }}_R}{v}^{n_m{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{n_m{\theta }_{\mathbf{z}}}}\le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{n_m{\theta }_{\mathbf{z}}}-\frac{1}{q}\right)}{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{q}dxdy\right)}^{\frac{1}{q}},\end{array}$$

(38)

where

$${\mathsf{\Omega }}_{R_m}:=B_{R_m}\times B_{{\left(R_m\right)}^{1+\mathbf{z}}}\mathrm{with},m>0.$$

We know that $2k_0^{-}<q_1,$ from Remark 2. We choose a real number $q>0$ such as

$$2k_0^{-}<q<q_1.$$

Let m be a non-negative integer satisfying

$$q{\theta }_{\mathbf{z}}^{m-1}<n_0<q{\theta }_{\mathbf{z}}^m.$$

We construct an increasing geometric sequence given by

$$n_1=qk,n_2=qk{\theta }_{\mathbf{z}},....,n_m=qk{\theta }_{\mathbf{z}}^{m-1},$$

where

$$2k_0^{-}<n_1<n_2<,....,<n_m<n_0.$$

Here we choose the constant $k\in [1,{\theta }_{\mathbf{z}}],$ such that $n_m$ is arbitrarily close to $n_0.$ Set $R_n=2^{n}R.$ From the inequality (35) and by using an induction argument, we get then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\int }_{{\mathsf{\Omega }}_R}{v}^{n_m{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{n_m{\theta }_{\mathbf{z}}}}& \le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{n_m{\theta }_{\mathbf{z}}}-\frac{1}{n_m}\right)}{\left({\int }_{{\mathsf{\Omega }}_1}{v}^{n_m}dxdy\right)}^{\frac{1}{n_m}}\hfill \\ \hfill & =C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{n_m{\theta }_{\mathbf{z}}}-\frac{1}{n_m}\right)}{\left({\int }_{{\mathsf{\Omega }}_1}{v}^{n_{m-1}{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{n_{m-1}{\theta }_{\mathbf{z}}}}\hfill \\ \hfill & \le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{n_m{\theta }_{\mathbf{z}}}-\frac{1}{n_1}\right)}{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{n_1}dxdy\right)}^{\frac{1}{n_1}}\hfill \\ \hfill & \le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{n_m{\theta }_{\mathbf{z}}}-\frac{1}{qk}\right)}{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{qk}dxdy\right)}^{\frac{1}{qk}},\hfill \end{array}\end{array}$$

(39)

where ${\mathsf{\Omega }}_1:=B_1\times B_{{\left(1\right)}^{1+\mathbf{z}}}.$ Now, using Hölder’s inequality, there holds

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{qk}dxdy\right)}^{\frac{1}{qk}}& \le {\left[{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{q{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{k}{{\theta }_{\mathbf{z}}}}{\left({\int }_{B_{R_m}\times B_{{\left(R_m\right)}^{1+s}}}dxdy\right)}^{1-\frac{k}{{\theta }_{\mathbf{z}}}}\right]}^{\frac{1}{qk}}\hfill \\ \hfill & \le C{\left[{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{q{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{k}{{\theta }_{\mathbf{z}}}}C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(1-\frac{k}{{\theta }_{\mathbf{z}}}\right)}\right]}^{\frac{1}{qk}}\hfill \\ \hfill & \le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{kq}-\frac{1}{q{\theta }_{\mathbf{z}}}\right)}{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{q{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{q{\theta }_{\mathbf{z}}}}\hfill \\ \hfill & \le C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{kq}-\frac{1}{q{\theta }_{\mathbf{z}}}\right)}{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}\left(\frac{1}{q{\theta }_{\mathbf{z}}}-\frac{1}{q}\right)}{\left({\int }_{{\mathsf{\Omega }}_{R_m}}{v}^{q}dxdy\right)}^{\frac{1}{q}}.\hfill \end{array}\end{array}$$

(40)

Then, we combine the last tow inequalities to get the result.

We are now ready to prove Theorem 2.

3.1. Proof of Theorem 2

Let $1<q_1\le q_2$ and $(u,v)$ be a stable solution to (1), where (3) is satisfied. We divide the proof into two cases:

Case 1: $q_1>\frac{4}{3}$. Let $q_1>q>0.$ From (25), Hölder’s inequality implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}_R}{v}^{q}dxdy& \le {\left({\int }_{{\mathsf{\Omega }}_R}\varrho \left(\mu \right)v^{q_1}dxdy\right)}^{\frac{q}{q_1}}\times {\left({\int }_{{\mathsf{\Omega }}_R}{\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{-\frac{\rho q_1}{2(\mathbf{z}+1)(q_1-q)}}dxdy\right)}^{\frac{q_1-q}{q_1}}\hfill \\ \hfill & \le C{R}^{\left[{\mathbf{G}}_{\mathbf{z}}^{*}-\frac{2(q_2+1)q_1}{q_1{q}_2-1}-\frac{(q_1+1)\rho }{q_1{q}_2-1}\right]\frac{q}{q_1}+\left({\mathbf{G}}_{\mathbf{z}}^{*}-\frac{\rho q}{q_1-q}\right)\frac{q_1-q}{q_1}}=C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}-\frac{(2+\rho )(q_2+1)}{q_1{q}_2-1}q}.\hfill \end{array}\end{array}$$

(41)

Substituting this in (38), we then get

$$\begin{array}{c}\hfill {\left({\int }_{{\mathsf{\Omega }}_R}{v}^{n_m{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{n_m{\theta }_{\mathbf{z}}}}\le C{R}^{\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{n_m{\theta }_{\mathbf{z}}}-\frac{\beta (2+\rho )}{2}}.\end{array}$$

(42)

Recall that ${\theta }_{\mathbf{z}}=\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{{\mathbf{G}}_{\mathbf{z}}^{*}-2}.$ Since

$${\mathbf{G}}_{\mathbf{z}}^{*}<2+\left(\frac{\beta }{2}\right)(2+\rho )n_0,$$

We choose $k\in [1,{\theta }_{\mathbf{z}}],$ such that $n_m$ is close to $n_0.$ Then, (42) implies that

$${\parallel v\parallel }_{L^{n_m{\theta }_{\mathbf{z}}}\left({\mathbb{R}}^N\right)}=0,\mathrm{as},R\to \infty$$

i.e., $v\equiv 0$ in ${\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2}.$ This is a contradiction. Then, we deduce that (1) does not admit any stable solution if

$${\mathbf{G}}_{\mathbf{z}}^{*}<2+(2+\rho )m_0\mathrm{where},m_0=\left(\frac{\beta }{2}\right)n_0\mathrm{and}\beta =\frac{2(q_2+1)}{q_1{q}_2-1}.$$

Finally, Remark 2 implies that if ${\mathbf{G}}_{\mathbf{z}}^{*}\le 10+4\rho$, (1) has no stable solution for any $q_2\ge q_1>\frac{4}{3}.$

Case 2:$1<q_1\le \frac{4}{3}$ and u is bounded. Let $0<q<2$ and from (27), we derive

$$\begin{array}{cc}\hfill {\int }_{{\mathsf{\Omega }}_R}{v}^{q}dxdy& \le {\left({\int }_{{\mathsf{\Omega }}_R}\varrho \left(\mu \right)v^{2}dxdy\right)}^{\frac{q}{2}}\times {\left({\int }_{{\mathsf{\Omega }}_R}{\left(1+{\parallel \mu \parallel }^{2(\mathbf{z}+1)}\right)}^{-\frac{\rho q}{2(\mathbf{z}+1)(2-q)}}dxdy\right)}^{\frac{2-q}{2}}\hfill \\ \hfill & \le C{R}^{{\gamma }_3\frac{q}{2}+\left({\mathbf{G}}_{\mathbf{z}}^{*}-\frac{\rho q}{2-q}\right)\frac{2-q}{2}}=C{R}^{{\mathbf{G}}_{\mathbf{z}}^{*}-\left[\left(\frac{\beta }{2}\right)q_1+\frac{(2+\rho )(2-q_1)}{q_2+q_1-2}+\left(\frac{\beta }{4}\right)q_1\rho \right]q}.\hfill \end{array}$$

Substituting this in (38), we get

$$\begin{array}{c}\hfill {\left({\int }_{{\mathsf{\Omega }}_R}{v}^{n_m{\theta }_{\mathbf{z}}}dxdy\right)}^{\frac{1}{n_m{\theta }_{\mathbf{z}}}}\le C{R}^{\frac{{\mathbf{G}}_{\mathbf{z}}^{*}}{n_m{\theta }_{\mathbf{z}}}-\left[\left(\frac{\beta }{2}\right)q_1+\frac{(2+\rho )(2-q_1)}{q_2+q_1-2}+\left(\frac{\beta }{4}\right)q_1\rho \right]}.\end{array}$$

Proceeding as Case 1, we get the desired result. □

3.2. Proof of Proposition 1

Let u be a stable solution of (2) with $q_1=q_2>1.$ We can proceed like for the proof of Theorem 2. By Remark 2, we can easily show that if $k_0^{+}>q_1>1$ then $2k_0^{+}$ is the largest root of

$$\begin{array}{c}\hfill P\left(n\right)=n^4-16q_1^2{n}^2+32q_1^{2}n-16q_1^2=(n^2+4q_1(n-1))(n-2k_0^{-})(n-2k_0^{+}),\end{array}$$

with $k_0^{\pm }=q_1\pm \sqrt{q_1^2-q_1}$.

So, we get

$$m_0=\frac{2q_1+2\sqrt{q_1^2-q_1}}{q_1-1}>4\mathrm{for} \mathrm{all} q_1>1.$$

The result follows directly by relying on Theorem 2. □

4. Conclusions

In this paper, we consider a class of weighted Grushin system involving the advection term. Relying on Mtiri’s approach [10] and using the techniques developed in [17,19], we gave a Liouville-type theorem for the class of stable positive solution under some assumptions. Therefore, our conclusion of Theorem 2 and Proposition 1 can be viewed as an expansion of previous works, which is therefore interesting and meaningful. For future works, giving attention to [23], we believe that Theorem 2 can be generalised for systems including advection terms with negative exponents and for fractional Grushin systems involving advection term with exponential nonlinearity.