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On stable entire solutions of sub-elliptic system involving advection terms with negative exponents and weights

Abstract

We examine the weighted Grushin system involving advection terms given by

$$ \textstyle\begin{cases} \Delta _{G} u - a \cdot \nabla _{G} u =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v - a \cdot \nabla _{G} v =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$

where \(\Delta _{G} u= \Delta _{x} u+ |x|^{2\alpha } \Delta _{y} u\), \(\mathbf{z}=(x,y) \in \mathbb {R}^{n}:= \mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}}\) is the Grushin operator, \(\alpha \geq 0\), \(p \geq q >1\), \(\|\mathbf{z}\|^{2(\alpha +1)}= |x|^{2(\alpha +1)} + |y|^{2} \), \(\gamma \geq 0\) and a is a smooth divergence-free vector that we will specify later. Inspired by recent progress in the study of the Lane–Emden system, we establish some Liouville-type results for bounded stable positive solutions of the system. In particular, we prove the comparison principle to establish our result. As consequences, we obtain a Liouville-type theorem for the weighted Grushin equation involving advection terms

$$ \Delta _{G} u - a \cdot \nabla _{G} u =\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p} \quad \mbox{in } \mathbb {R}^{n}. $$

The main tools in the proof of the main result are the comparison principle, nonlinear integral estimates via the stability assumption and the bootstrap argument. Our results generalize and improve the previous work in (Duong et al. in Complex Var. Elliptic Equ. 64(12):2117–2129, 2019).

Introduction

In this paper, we study the Liouville-type theorem for bounded stable positive classical solutions of the weighted nonlinear degenerate elliptic system involving advection terms

$$ \textstyle\begin{cases} \Delta _{G} u - a \cdot \nabla _{G} u=(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v -a \cdot \nabla _{G} v=(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$
(1.1)

and of the scalar equation

$$ \Delta _{G} u - a \cdot \nabla _{G} u =\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p}\quad \mbox{in } \mathbb {R}^{n}, $$
(1.2)

where \(\Delta _{G} u = \Delta _{x} u + |x|^{2\alpha } \Delta _{y} u\), \(\mathbf{z}=(x,y) \in \mathbb {R}^{n}:= \mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}}\) is the Grushin operator, \(\Delta _{x}\) and \(\Delta _{y}\) are Laplace operators with respect to \(x\in \mathbb {R}^{n_{1}}\) and \(y\in \mathbb {R}^{n_{2}}\), and \(\|\mathbf{z}\|^{2(\alpha +1)}:= |x|^{2(\alpha +1)} + |y|^{2} \). Here we always assume that \(\alpha \geq 0\), \(p \geq q >1\), \(\gamma \geq 0\) and a is a smooth divergence-free vector field:

$$ \operatorname{div}_{G} a =0 \quad \mbox{and} \quad \bigl\vert a(\mathbf{z}) \bigr\vert \leq \frac{\epsilon }{1+ \Vert \mathbf{z} \Vert }\quad \mbox{for all } \mathbf{z}=(x,y)\in \mathbb {R}^{n},\ \epsilon \mbox{ small enough}, $$
(1.3)

where \(\operatorname{div}_{G}= \operatorname{div}_{x}+ |x|^{\alpha } \operatorname{div}_{y}\).

For simplicity of the presentation, we define the following parameters which play an important role in the sequel:

$$\begin{aligned}& Q:=n_{1}+(1+\alpha )n_{2} \mbox{ is the homogeneous dimension of } \mathbb {R}^{n}; \\& \tau _{0}^{+}:= \sqrt{\frac{pq(p-1)}{q-1}} + \sqrt{ \frac{pq(p-1)}{q-1}+\sqrt{\frac{pq(p-1)}{q-1}}}, \\& \tau _{0}^{-}:= \sqrt{\frac{pq(p-1)}{q-1}} - \sqrt{ \frac{pq(p-1)}{q-1}+\sqrt{\frac{pq(p-1)}{q-1}}}, \\& \sigma ^{+}:= \sqrt{\frac{pq(p+1)}{q+1}} + \sqrt{ \frac{pq(p+1}{q+1}-\sqrt{\frac{pq(p+1)}{q+1}}}, \\& \sigma ^{-}:= \sqrt{\frac{pq(p+1)}{q+1}}- \sqrt{ \frac{pq(p+1}{q+1}-\sqrt{\frac{pq(p+1)}{q+1}}}. \end{aligned}$$

We start by noting that, in the case \(a\equiv 0\) and \(\gamma =0\), the system (1.1) and Eq. (1.2) reduce to

$$ \textstyle\begin{cases} \Delta _{G} u = v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v = u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$
(1.4)

and

$$\begin{aligned} \Delta _{G} u = u^{-p}\quad \mbox{in } \mathbb {R}^{n}. \end{aligned}$$
(1.5)

In the case \(\alpha =0\), Eq. (1.5) arises in many branches of applied sciences and has been studied in a number of recent works; see [6, 20] and the references therein. The nonexistence of positive stable classical solutions of (1.5) was examined in [20]. This result was then generalized in [6] to positive stable weak solutions of a weighted equation. More precisely, the authors of [6] figured out the critical exponent and established an optimal Liouville-type theorem for this class of solutions. When \(\alpha >0\), the Liouville-type theorem for a special class of solutions of (1.4) and (1.5). “the so-called stable solutions” has been studied by Duong, Lan, Le and Nguyen [10]. We summarize here some results in [10].

Theorem 1.1

([10])

Assume that\(p\geq q>1\)and

$$ Q< 2+ \frac{8}{p+q+2} \tau _{0}^{+}, $$

then (1.4) has no bounded stable positive solution.

Corollary 1.1

([10])

Let\(p>1\)and

$$ Q< 2+ \frac{4}{p+1} \bigl(p+\sqrt{p(p+1)} \bigr), $$
(1.6)

then Eq. (1.5) has no bounded stable positive solution.

We remark that for Eq. (1.5) (with \(\alpha =0\)), the critical exponent on the right-hand side of (1.6) was first found in [20]. This exponent has been shown to be optimal in the class of positive stable weak solutions; see [6].

Similar to the celebrated Lane–Emden system in the case of positive exponents, the system (1.4) is also a natural extension of Eq. (1.5). It is worth to remark that there are many papers developing various useful tools to study the nonexistence of positive stable solutions (see for example [2, 12, 13, 17, 22, 26] and the references therein. For other results on Grushin operators, Wei et al. [25] established a Liouville-type theorem for weak stable solutions of weighted p-Laplace-type Grushin equation in the case of negative exponent nonlinearity. Some important and interesting results can be found in [21].

Recently, elliptic problems involving advection terms, i.e. \(a\neq 0\), have received considerable attention [3, 4, 8, 11]. In particular, Duong and Nguyen [11] studied the equation

$$ -\Delta _{G} u + \nabla _{G} w \cdot \nabla _{G} u = \Vert \mathbf{z} \Vert ^{s} \vert u \vert ^{p-1}u \quad \mbox{in } \mathbb {R}^{n}, s\geq 0. $$
(1.7)

Taking advantage of the variational structure, and using the approach of Farina [13], he established some Liouville-type theorems for the class of stable sign-changing weak solutions. Now, we state this result as follows.

Theorem 1.2

([11])

Suppose that there is a nonnegative constantθsuch that

$$ \vert \nabla _{G} w \vert \leq \frac{C}{ \Vert \mathbf{z} \Vert ^{\theta }+1}. $$

Assume in addition that

$$ \lim_{R \rightarrow +\infty } R^{- \frac{(1+\min (\theta ;1))(p+\beta )+s(\beta +1)}{p-1}} \int _{R< \| \mathbf{z}\| < 2R} e^{-w}=0, $$

for\(\beta \in (1, 2p+2\sqrt{p(p-1)}-1 )\). Then any stable weak solutionuto (1.7) must be the trivial one.

In the general case where \(a\neq 0\), elliptic problems with advections have no variational structure and this requires another approach to obtaining a classification of stable solutions. Recall that, in this case, see e.g. [3], a positive classical solution u of

$$ -\Delta u + a \cdot \nabla u= u^{p} \quad \mbox{in } \mathbb {R}^{n} $$
(1.8)

is called stable if there is a smooth positive function F such that

$$ -\Delta F + a.\nabla F \geq pu^{p-1} F. $$

Recently, relying on Farina’s approach [13] and the generalized Hardy inequality, Cowan [3] established a Liouville-type theorem for stable positive solution of (1.8) under the smallness condition imposed on the divergence-free a.

On the other direction, the Liouville-type theorem for the class of stable solutions for system

$$ \textstyle\begin{cases} -\Delta u + a \cdot \nabla u= v^{p} &\text{in $\mathbb {R}^{n}$}, \\ -\Delta v + a \cdot \nabla v= u^{q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$
(1.9)

was examined by Duong [8]. He established a Liouville-type result for stable positive solutions of the system in the case \(p\geq q \geq 1\) and \(pq > 1\). In particular, when \(p = q\), his result is a natural extension of Cowan [3] to the equation with advection. Furthermore, we would also like to mention that when \(pq\leq 1\), the system (1.9) has no positive supersolutions (see Theorem 1.3 [9]).

For the general equation or system with \(\gamma \neq 0\), the Liouville property is less understood and is more delicate to deal with than \(\gamma =0\). There exist many excellent papers using Farina’s approach to the Hardy–Hénon equation and the weighted nonlinear elliptic equations. We refer to [7, 22, 24] and the references therein. Inspired by the ideas in [2, 16], Hu [18] adopt the new approach of a combination of second order stability, Souplet’s inequality [23] and a bootstrap iteration to establish Liouville-type theorems for the semi-stable solutions of

$$ \textstyle\begin{cases} -\Delta u =(1+ \vert x \vert ^{2})^{\frac{\gamma }{2}} v^{p} &\text{in $\mathbb {R}^{n}$}, \\ -\Delta v =(1+ \vert x \vert ^{2})^{\frac{\gamma }{2}} u^{q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$
(1.10)

and of the scalar equation

$$ - \Delta u =\bigl(1+ \vert x \vert ^{2} \bigr)^{\frac{\gamma }{2}} u^{p} \quad \mbox{in } \mathbb {R}^{n}. $$
(1.11)

In particular, Hu [18] has obtained the following result.

Theorem 1.3

([18])

  1. 1.

    Suppose\(\gamma >0\), \(2\sigma ^{-} < p\leq q\)and

    $$ n< 2+\frac{(4+2\gamma )(q+1)}{pq-1} \sigma ^{+} . $$

    Then there is no classical positive semi-stable solution of (1.10). In particular, there is no classical positive semi-stable solution of (1.10) for any\(2\leq p\leq q\)if\(n\leq 10+4\gamma \).

  2. 2.

    Let\(p>\frac{4}{3}\), \(\gamma >0\)and

    $$ n< 2+\frac{2(2+\gamma )}{p-1} \bigl( p+\sqrt{p(p-1)} \bigr). $$

    Then there does not exist a classical positive semi-stable solution of (1.11).

In this paper, we propose to study the system (1.1) which can be regarded as a natural generalization of the scalar equation (1.2). Motivated by [8, 10, 18], we give the classification of bounded stable positive solutions of (1.1) under the assumption (1.3). Before stating our main results, let us recall the definition of such solutions motivated by [8, 10].

Definition 1.1

A positive solution \((u, v) \in C^{2}(\mathbb {R}^{n}) \times C^{2}(\mathbb {R}^{n})\) of (1.1) is called stable if there are positive smooth functions ξ, η such that

$$ \textstyle\begin{cases} -\Delta _{G} \xi + a \cdot \nabla _{G} \xi =p(1+ \Vert \mathbf{z} \Vert ^{2( \alpha +1)})^{\frac{\gamma }{2(\alpha +1)}} v^{-p-1} \eta &\text{in $\mathbb {R}^{n}$}, \\ -\Delta _{G} \eta +a \cdot \nabla _{G} \eta = q(1+ \Vert \mathbf{z} \Vert ^{2( \alpha +1)})^{\frac{\gamma }{2(\alpha +1)}} u^{-q-1} \xi &\text{in $\mathbb {R}^{n}$}. \end{cases} $$
(1.12)

The main result in this paper is the following.

Theorem 1.4

Assume that\(p\geq q>1\)and

$$ Q< 2+ \frac{4(\gamma +2)}{p+q+2} \tau _{0}^{+} , $$
(1.13)

then (1.1) has no bounded stable positive solution.

The key in our proof is the comparison principle and nonlinear integral estimates. However, the techniques used to prove the comparison principle in [14, 18] for the Laplace operator do not seem applicable to the system (1.1) because the operator \(\Delta _{G}\) no longer has symmetry and it degenerates on the manifold \(\{0 \} \times \mathbb {R}^{n_{2}}\). Then, in this paper, we establish the comparison principle for Grushin operators by developing the idea in [1, 10, 12, 15]. In addition, the \(L^{1}\)-estimate to the bootstrap iteration in [2] does not work in the case of Grushin operator, we instead switch to the \(L^{2}\)-estimate in the bootstrap argument. We also employ the idea in [1, 10, 12, 15] to prove the “inverse” comparison principle which is crucial to proving our result.

Recall that a classical solution of (1.2) is called stable if

$$ p \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p-1} \phi ^{2} \,dx\,dy \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi +2 \nabla _{G} \phi \vert ^{2} \,dx\,dy, $$
(1.14)

for all \(\phi \in C^{1}_{c}(\mathbb {R}^{n})\).

When \(p=q\), by using the comparison principle below, we obtain a direct consequence of Theorem 1.4 for the scalar equation (1.2).

Corollary 1.2

Let\(p>1\)and

$$ Q< 2+ \frac{2(\gamma +2)}{p+1} \bigl(p+\sqrt{p(p+1)} \bigr). $$
(1.15)

Then Eq. (1.2) has no bounded stable positive solution.

We remark also that the method used in the present paper can be applied to study the weighted systems, and to more general class of degenerate operator, such as the \(\Delta _{\lambda }\) operator (see [19, 22]) of the form

$$ \Delta _{\lambda }:= \sum_{i=1}^{n} \partial _{x_{i}} \bigl(\lambda ^{2}_{i} \partial _{x_{i}}\bigr),\quad \lambda =(\lambda _{1}, \ldots, \lambda _{n}): \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}. $$

Here \(\lambda _{i}: \mathbb {R}^{n} \rightarrow \mathbb {R}\), \(i=1,\ldots,n\) are nonnegative continuous functions satisfying some properties such that \(\Delta _{\lambda }\) is homogeneous of degree two with respect to a group dilation in \(\mathbb {R}^{n}\).

The organization of this paper is as follows. In Sect. 2, we establish the stability inequality and the comparison principle for the system (1.1) and then prove an a priori estimate of the solutions. In Sect. 3, we give the proof of the main result.

Stability inequality and comparison principle

Stability inequality

Lemma 2.1

Assume that\((u, v)\)is a positive stable solution of the system (1.1) with (1.3) is satisfied. Then, for\(\phi \in C^{1}_{c}(\mathbb {R}^{n})\), we have

$$ \sqrt{pq} \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{\frac{\gamma }{2(\alpha +1)}} v^{\frac{-p-1}{2}} u^{ \frac{-q-1}{2}} \phi ^{2} \,dx\,dy \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi +2\nabla _{G} \phi \vert ^{2}\,dx\,dy. $$
(2.1)

Proof

We follow the idea in [2, 8]. Let \(\phi \in C^{1}_{c}(\mathbb {R}^{n})\). Multiplying the first equation in (1.12) by \(\frac{\phi ^{2}}{\xi }\), we obtain

$$ \int _{\mathbb {R}^{n}} \biggl(-\Delta _{G} \xi \frac{\phi ^{2}}{\xi } + a \cdot \nabla _{G} \xi \frac{\phi ^{2}}{\xi } \biggr)\,dx\,dy = p \int _{ \mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{-p-1} \eta \frac{\phi ^{2}}{\xi } \,dx\,dy. $$
(2.2)

Using integration by parts and Young’s inequality: \(2zz'-z^{\prime 2}\leq z^{2}\), we obtain

$$\begin{aligned}& \int _{\mathbb {R}^{n}} \biggl(-\Delta _{G} \xi \frac{\phi ^{2}}{\xi } + a \cdot \nabla _{G} \xi \frac{\phi ^{2}}{\xi } \biggr)\,dx\,dy \\& \quad = \int _{ \mathbb {R}^{n}} \biggl( 2 \frac{\phi }{\xi } \nabla _{G} \phi \nabla _{G} \xi - \vert \nabla _{G} \xi \vert ^{2} \frac{\phi ^{2}}{\xi ^{2}} + a \cdot \nabla _{G} \xi \frac{\phi ^{2}}{\xi } \biggr) \,dx\,dy \\& \quad = \int _{\mathbb {R}^{n}} \biggl( ( 2 \nabla _{G} \phi +a \phi ) \frac{\phi }{\xi } \nabla _{G} \xi - \vert \nabla _{G} \xi \vert ^{2} \frac{\phi ^{2}}{\xi ^{2}} \biggr) \,dx\,dy \\& \quad \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi +2\nabla _{G} \phi \vert ^{2} \,dx\,dy. \end{aligned}$$
(2.3)

Consequently, combining (2.2) and (2.3), it follows that

$$ p \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{-p-1} \eta \frac{\phi ^{2}}{\xi } \,dx\,dy \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi +2\nabla _{G} \phi \vert ^{2} \,dx\,dy. $$
(2.4)

By the same argument, we also have

$$ q \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q-1} \xi \frac{\phi ^{2}}{\eta } \,dx\,dy \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi +2\nabla _{G} \phi \vert ^{2} \,dx\,dy. $$
(2.5)

We now add the inequalities (2.4) and (2.5) to obtain

$$\begin{aligned}& \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} \biggl(p v^{-p-1} \eta \frac{\phi ^{2}}{\xi } +qu^{-q-1} \xi \frac{\phi ^{2}}{\eta } \biggr) \,dx \,dy \\& \quad \leq \frac{1}{2} \int _{\mathbb {R}^{n}} \vert a \phi +2\nabla _{G} \phi \vert ^{2} \,dx\,dy. \end{aligned}$$
(2.6)

Now note that

$$ 2 \sqrt{pq} v^{\frac{-p-1}{2}} u^{\frac{-q-1}{2}} \phi ^{2} \leq p v^{-p-1} \eta \frac{\phi ^{2}}{\xi }+ qu^{-q-1} \xi \frac{\phi ^{2}}{\eta }. $$

Putting this back into (2.6) gives the desired result. □

Comparison principle

In this subsection, we shall prove the comparison principle for the system (1.1) without stability assumption.

Lemma 2.2

Suppose that\((u, v)\)is a bounded positive solution of (1.1). Assume that\(1< q\leq p\)and (1.3) hold. Then

$$ (p-1)v^{p-1} \leq (q-1)u^{q-1}. $$
(2.7)

Proof

Let \(d = \frac{q-1}{p-1}\leq 1\) and \(l= d^{\frac{1}{p-1}}\). The inequality (2.7) is equivalent to

$$ v \leq l u^{d}. $$
(2.8)

Put \(w= v-l u^{d}\). A direct calculation leads to

$$\begin{aligned} \Delta _{G} w =& \Delta _{G} v -ld u^{d-1} \Delta _{G} u - ld (d-1) u^{d-2} \vert \nabla _{G} u \vert ^{2} \\ \geq & \Delta _{G} v -ld u^{d-1} \Delta _{G} u \\ =& a \cdot \nabla _{G} v+ \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} - l d u^{d-1} \bigl( a \cdot \nabla _{G} u+ \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} \bigr) \\ =& a \cdot \nabla _{G} w +u^{d-1} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} \bigl(u^{-dp} -l^{p} v^{-p} \bigr). \end{aligned}$$

Therefore

$$\begin{aligned} \Delta _{G} w -a \cdot \nabla _{G} w \geq & u^{d-1} \bigl(1+ \Vert \mathbf{z} \Vert ^{2( \alpha +1)}\bigr)^{\frac{\gamma }{2(\alpha +1)}} \bigl(u^{-dp} -l^{p} v^{-p} \bigr) \\ \geq & u^{d-1} \bigl(u^{-dp} -l^{p} v^{-p} \bigr) \\ =& u^{d-1} \biggl(\frac{1}{u^{dp}} -\frac{l^{p}}{ v^{p}} \biggr) \\ =&u^{d-1} \frac{v^{p}-l^{p} u^{dp}}{u^{dp} v^{p}}. \end{aligned}$$
(2.9)

We now prove (2.8) by contradiction. Suppose that

$$ M= \sup_{\mathbb {R}^{n}} w >0\quad (M \leq + \infty ). $$
(2.10)

Next, we divide the proof into two cases.

Case 1. If there exists \(z^{*}\) such that \(\sup_{\mathbb {R}^{n}} w= w(z^{*})= v(z^{*})- l u ^{d} (z^{*}) >0\), then

$$ \frac{\partial w}{\partial z_{i}}\bigl(z^{*}\bigr)=0\quad \mbox{and}\quad \frac{\partial ^{2} w}{\partial z^{2}_{i}}\bigl(z^{*}\bigr) \leq 0 \quad \mbox{for } i=1, \ldots,n. $$

This implies that

$$ \nabla _{G} w \bigl(z^{*}\bigr)=0\quad \mbox{and}\quad \Delta _{G} w \bigl(z^{*}\bigr) \leq 0 . $$

However, the right-hand side of (2.9) at \(z^{*}\) is positive. This is a contradiction.

Case 2. If the supremum of w is attained at infinity.

Take a cut-off function \(\chi \in C^{\infty }_{c}( \mathbb {R}^{n}, [0,1])\) verifying \(\chi =1\) on \(B_{1} \times B_{1}\) and \(\chi =0\) outside \(B_{2} \times B_{2^{1+\alpha }}\). Put \(\phi _{R}(x)= \chi ^{m} (\frac{x}{R}, \frac{y}{R^{1+\alpha }})\). Here \(m>0\) will be chosen later. A simple calculation yields

$$ \vert \Delta _{G} \phi _{R} \vert \leq C R^{-2}\phi _{R}^{\frac{m-2}{m}},\qquad \phi _{R}^{-1} \vert \nabla _{G} \phi \vert ^{2} \leq C R^{-2}\phi _{R}^{ \frac{m-2}{m}}. $$
(2.11)

Let \(w_{R}= \phi _{R} w\) which is a compactly supported function. Then there exists \(z_{R}=(x_{R},y_{R}) \in B_{2R} \times B_{(2R)^{1+\alpha }}\) such that

$$ w_{R}(z_{R})=\sup _{\mathbb {R}^{n}} w_{R}(z) =\max_{\mathbb {R}^{n}} w_{R}(z) \rightarrow M\quad \mbox{as } R \rightarrow +\infty . $$
(2.12)

This implies that

$$ \nabla _{G} w_{R}(z_{R})=0\quad \mbox{and}\quad \Delta _{G} w_{R}(z_{R}) \leq 0. $$
(2.13)

In what follows, all the estimates are taken at the point \(z_{R}\). First, using \(\nabla _{G} w_{R} (z_{R})=0\), we have

$$ 0=\nabla _{G} w_{R}= \nabla _{G} \phi _{R} w + \phi _{R} \nabla _{G} w . $$

Hence,

$$ \nabla _{G} w = -\phi _{R}^{-1} \nabla _{G} \phi _{R} w . $$
(2.14)

Since \(\Delta _{G} w_{R} (z_{R})\leq 0\), we get

$$ 0\geq \Delta _{G} w_{R} = \Delta _{G} \phi _{R} w + 2 \nabla _{G} \phi _{R} \cdot \nabla _{G} w + \phi _{R} \Delta _{G} w . $$

Thus,

$$ \phi _{R} \Delta _{G} w \leq \bigl(2 \phi _{R}^{-1} \vert \nabla _{G} \phi _{R} \vert ^{2} - \Delta _{G} \phi _{R} \bigr) w. $$
(2.15)

Combining (2.11) and (2.15), one has

$$ \phi _{R} \Delta _{G} w \leq C R^{-2} \phi _{R}^{\frac{m-2}{m}}w. $$
(2.16)

Using (2.11), (2.14) and the fact that \(|a(\mathbf{z})| \leq \frac{\epsilon }{\|\mathbf{z}\| +1}\), one has

$$ \vert a \cdot \nabla _{G} w \phi _{R} \vert \leq C R^{-2} \phi _{R}^{ \frac{m-1}{m}}w. $$
(2.17)

Recall that \(v- l u^{d} =w\) and at \(z_{R}\), we get

$$ \frac{v^{p}}{w^{p}} - \frac{(l u^{d})^{p}}{w^{p}} \geq 1 . $$
(2.18)

Multiplying (2.9) by \(\phi _{R}\) and using (2.16), (2.17) and (2.18), we obtain

$$ \phi _{R} u^{d-1} \frac{w^{p}}{u^{dp} v^{p}} \leq C R^{-2} \phi _{R}^{ \frac{m-2}{m}} w. $$

Recall that the constant C is independent of R. Consequently,

$$ \phi _{R}^{\frac{m+2}{m}} u^{d-1}\frac{w^{p}}{u^{dp} v^{p}} \leq C R^{-2} \phi _{R} w. $$

Choosing \(m= \frac{2}{p-1}\) (or \(p=\frac{m+2}{m}\)), we get

$$ u^{d-1} \frac{w_{R}^{p}}{u^{dp} v^{p}} \leq C R^{-2} w_{R}\quad \mbox{or}\quad u^{d-1} \frac{w_{R}^{p-1}}{u^{dp} v^{p}} \leq C R^{-2}. $$
(2.19)

It follows from (2.19), the boundedness of \((u,v)\) and \(d\leq 1\) that

$$ w_{R}^{p-1}(z_{R}) \leq C R^{-2}. $$

Finally, letting \(R \rightarrow +\infty \), we get \(M=0\), which contradicts (2.10). The proof is complete. □

Combining the proof of Lemma 2.2 with the idea in [1, 10, 12, 15], we have the inverse comparison principle as follows.

Lemma 2.3

Suppose that\((u, v)\)is a bounded positive solution of (1.1). Assume that\(1< q\leq p\)and (1.3) hold. Then we have

$$ u\leq \Vert v \Vert ^{\frac{p-q}{q-1}}_{\infty } v, $$
(2.20)

where\(\|v\|_{\infty }= \sup_{\mathbb {R}^{n}} v\).

Proof

Let \(l=\|v\|^{\frac{p-q}{q-1}}_{\infty }\) and \(w= u-lv\). We need to show that \(w\leq 0\). Notice that

$$\begin{aligned} \Delta _{G} w - a \cdot \nabla _{G} w =& \Delta _{G} u -a \cdot \nabla _{G} u -l (\Delta _{G} v -a \cdot \nabla _{G} v) \\ = & \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{\frac{\gamma }{2(\alpha +1)}} v^{-p}- l \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} \\ =& \frac{v^{-p}}{ \Vert v \Vert ^{-p}_{\infty }}\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} \Vert v \Vert ^{-p}_{\infty }- l \bigl(1+ \Vert \mathbf{z} \Vert ^{2( \alpha +1)}\bigr)^{\frac{\gamma }{2(\alpha +1)}} u^{-q} \\ \geq & \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{ \frac{\gamma }{2(\alpha +1)}} \biggl(\frac{v^{-q}}{ \Vert v \Vert ^{-q}_{\infty }} \Vert v \Vert ^{-p}_{\infty }-l u^{-q} \biggr) \\ =& \Vert v \Vert ^{q-p}_{\infty }\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} \bigl(v^{-q} -l^{q}v^{-q} \bigr) \\ \geq & \Vert v \Vert ^{q-p}_{\infty } \bigl(v^{-q} -l^{q}v^{-q} \bigr) . \end{aligned}$$
(2.21)

In order to obtain the proof, it suffices to use the arguments as in Lemma 2.2 by noting that (2.9) is replaced by (2.21). The details are then omitted. □

In what follows, the constant C does not depend on a positive parameter R and may change from line to line.

Lemma 2.4

Suppose that\((u, v)\)be a bounded stable positive solution of (1.1). Assume that\(1< q\leq p\)and (1.3) hold. Then for\(R>0\)there exists\(C>0\)independent ofRsuch that

$$ \int _{B_{R} \times B_{R^{1+\alpha }}} u^{-q} \,dx\,dy \leq C R^{Q- \frac{2(pq-q)}{pq-1}- \frac{\gamma (pq-1)}{pq-q}} $$
(2.22)

and

$$ \int _{B_{R} \times B_{R^{1+\alpha }}} v^{-\frac{p+q+2}{2}} \,dx\,dy \leq C R^{Q-2-\gamma }. $$
(2.23)

Proof

Using Lemma 2.2, we get

$$ v^{-\frac{p+1}{2}} \geq \biggl(\frac{p-1}{q-1} \biggr)^{ \frac{p+1}{2(p-1)}} u^{-\frac{(q-1)(p+1)}{2(p-1)}}. $$
(2.24)

Take a cut-off function \(\phi \in C^{\infty }_{c}(\mathbb {R}^{n} , [0,1])\) verifying \(\phi =1 \) on \(B_{1}\times B_{1}\) and \(\phi =0\) outside \(B_{2}\times B_{2^{\alpha +1}}\). For \(R>0\), put \(\phi _{R}(x)= \phi ^{m} (\frac{x}{R}, \frac{y}{R^{1+\alpha }})\), where \(m \geq 2\) which is fixed. Then there exists \(C>0\) independent of R such that

$$ \vert \nabla _{G} \phi _{R} \vert \leq C R^{-1} \phi ^{\frac{m-1}{m}}_{R}. $$
(2.25)

By virtue of (2.1) and (2.24), we derive

$$\begin{aligned}& \biggl(\frac{p-1}{q-1} \biggr)^{\frac{p+1}{2(p-1)}} \sqrt{pq} \int _{B_{2R} \times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{\frac{\gamma }{2(\alpha +1)}} u^{\frac{-pq+1}{p-1}} \phi ^{2}_{R} \,dx \,dy \\& \quad \leq \sqrt{pq} \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{\frac{\gamma }{2(\alpha +1)}} v^{- \frac{p+1}{2}} u^{-\frac{q+1}{2}} \phi ^{2}_{R} \,dx \,dy \\& \quad \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \vert a \phi _{R} +2\nabla _{G} \phi _{R} \vert ^{2}\,dx\,dy. \end{aligned}$$
(2.26)

Recall that \(\frac{pq-1}{pq-q}>1\). Then, by combining the Hölder inequality, (2.25) and (2.26), we get

$$\begin{aligned}& \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} u^{-q} \phi ^{2}_{R} \,dx\,dy \\& \quad \leq \biggl( \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{\frac{\gamma }{2(\alpha +1)}} u^{ \frac{-pq+1}{p-1}} \phi ^{2}_{R} \,dx \,dy \biggr)^{\frac{pq-q}{pq-1}} \\& \qquad {}\times\biggl( \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{- \frac{\gamma (pq-1)^{2}}{2(\alpha +1)(q-1)(pq-q)}} \phi ^{2}_{R} \biggr)^{\frac{q-1}{pq-1}} \\& \quad \leq C \biggl( \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \vert a \phi _{R} +2 \nabla _{G} \phi _{R} \vert ^{2}\,dx \,dy \biggr)^{\frac{pq-q}{pq-1}} \\& \qquad {}\times\biggl( \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2( \alpha +1)} \bigr)^{- \frac{\gamma (pq-1)^{2}}{2(\alpha +1)(q-1)(pq-q)}} \phi ^{2}_{R} \,dx \,dy \biggr)^{\frac{q-1}{pq-1}} \\& \quad \leq C R ^{-2\frac{pq-q}{pq-1} } \biggl( \int _{B_{2R}\times B_{(2R)^{ \alpha +1}}} \phi ^{\frac{2(m-1)}{m}}_{R} \,dx \,dy \biggr)^{ \frac{pq-q}{pq-1}} \\& \qquad {}\times\biggl( \int _{B_{2R}\times B_{(2R)^{\alpha +1}}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)} \bigr)^{- \frac{\gamma (pq-1)^{2}}{2(\alpha +1)(q-1)(pq-q)}} \phi ^{2}_{R} \,dx \,dy \biggr)^{\frac{q-1}{pq-1}} \\& \quad \leq C R ^{-2\frac{pq-q}{pq-1} }R ^{Q\frac{pq-q}{pq-1}} R ^{- \frac{\gamma (pq-1)}{pq-q} } R^{Q\frac{q-1}{pq-1} }= C R^{Q- \frac{2(pq-q)}{pq-1}- \frac{\gamma (pq-1)}{pq-q}}. \end{aligned}$$
(2.27)

Hence, the desired integral estimate (2.22) follows. Finally, (2.23) follows from using the same argument as above where we use (2.20) instead of (2.7). □

Proof of the main result

Beginning of the proof

In this subsection, we give a preparation for the bootstrap iteration. Using Lemmas 2.2 and 2.1, we get the following.

Lemma 3.1

Under the same assumptions of Lemma 2.4, suppose that

$$ \tau _{0}^{-}< t < \tau _{0}^{+}. $$

Then we have

$$\begin{aligned}& \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} v^{-2t-1}\phi ^{2} \,dx\,dy \\& \quad \leq C \int _{\mathbb {R}^{n}} v^{-2t} \bigl( \vert \nabla _{G} \phi \vert ^{2} + \phi \vert \Delta _{G} \phi \vert + \vert a \vert \bigl\vert \nabla _{G} \bigl(\phi ^{2}\bigr) \bigr\vert \bigr) \,dx \,dy, \end{aligned}$$

for all\(\phi \in C^{2}_{c}(\mathbb {R}^{n})\)satisfying\(0\leq \phi \leq 1\). HereCdoes not depend on\((u, v)\).

Proof

Using Lemma 2.1 with the test function \(v^{-t} \phi \) we have

$$\begin{aligned}& \sqrt{pq} \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{\frac{-p-1}{2}} u^{\frac{-q-1}{2}} v^{-2t} \phi ^{2} \,dx\,dy \\& \quad \leq \frac{1}{4} \int _{\mathbb {R}^{n}} \bigl\vert a v^{-t} \phi +2 \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigr\vert ^{2}\,dx\,dy \\& \quad = \frac{1}{4} \int _{\mathbb {R}^{n}} \bigl( \bigl\vert a v^{-t} \phi \bigr\vert ^{2} +4 a \cdot \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigl(v^{-t} \phi \bigr) +4 \bigl\vert \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigr\vert ^{2} \bigr) \,dx\,dy \\& \quad = \frac{1}{4} \int _{\mathbb {R}^{n}} \bigl( \bigl\vert a v^{-t} \phi \bigr\vert ^{2} +4 \bigl\vert \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigr\vert ^{2} \bigr) \,dx \,dy, \end{aligned}$$

where in the last equality, we have used \(\operatorname{div}_{G} a=0\).

Applying the Hardy inequality related to Grushin type operators, see e.g. [5], one obtains

$$ \int _{\mathbb {R}^{n}} \bigl\vert a (\mathbf{z}) v^{-t} \phi \bigr\vert ^{2} \,dx\,dy \leq \epsilon ^{2} \int _{\mathbb {R}^{n}} \frac{ \vert v^{-t} \phi \vert ^{2}}{ \Vert \mathbf{z} \Vert ^{2}}\,dx\,dy \leq \frac{4\epsilon ^{2}}{(Q-2)^{2}} \int _{\mathbb {R}^{n}} \bigl\vert \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigr\vert ^{2} \,dx\,dy. $$

Combining the above two estimates, we derive that

$$\begin{aligned}& \sqrt{pq} \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{\frac{-p-1}{2}} u^{\frac{-q-1}{2}} v^{-2t} \phi ^{2} \,dx \,dy \\& \quad \leq \biggl( 1+ \frac{\epsilon ^{2}}{(Q-2)^{2}} \biggr) \int _{\mathbb {R}^{n}} \bigl\vert \nabla _{G} \bigl(v^{-t} \phi \bigr) \bigr\vert ^{2} \,dx\,dy \\& \quad = \biggl( 1+ \frac{\epsilon ^{2}}{(Q-2)^{2}} \biggr) \biggl( t^{2} \int _{\mathbb {R}^{n}} \vert \nabla _{G} v \vert ^{2} v^{-2t-2} \phi ^{2} \,dx \,dy \\& \qquad {}+ \int _{ \mathbb {R}^{n}} v^{-2t} \vert \nabla _{G} \phi \vert ^{2} \,dx\,dy -\frac{1}{2} \int _{\mathbb {R}^{n}} v^{-2t}\Delta _{G} \bigl(\phi ^{2}\bigr) \,dx\,dy \biggr) . \end{aligned}$$
(3.1)

Multiplying the second equation in (1.1) by \(v^{-2t-1} \phi ^{2}\) and integrating by parts we arrive at

$$\begin{aligned}& (2t+1) \int _{\mathbb {R}^{n}} \vert \nabla _{G} v \vert ^{2} v^{-2t-2} \phi ^{2} \,dx \,dy - \frac{1}{2t} \int _{\mathbb {R}^{n}} v^{-2t}\Delta _{G} \bigl(\phi ^{2}\bigr) \,dx\,dy \\& \qquad {}- \frac{1}{2t} \int _{\mathbb {R}^{n}} v^{-2t} a \cdot \nabla _{G} \bigl(\phi ^{2}\bigr) \,dx\,dy \\& \quad = \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} v^{-2t-1}\phi ^{2} \,dx \,dy. \end{aligned}$$
(3.2)

Combining (3.1) and (3.2), we obtain

$$\begin{aligned}& \sqrt{pq} \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} v^{\frac{-p-1}{2}} u^{\frac{-q-1}{2}} v^{-2t} \phi ^{2} \,dx \,dy \\& \quad \leq \frac{t^{2}}{2t+1} \biggl( 1+ \frac{\epsilon ^{2}}{(Q-2)^{2}} \biggr) \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} v^{-2t-1}\phi ^{2} \,dx\,dy \\& \qquad {}+ C \int _{\mathbb {R}^{n}} v^{-2t} \bigl( \vert \nabla _{G} \phi \vert ^{2} + \phi \vert \Delta _{G} \phi \vert + \vert a \vert \bigl\vert \nabla _{G} \bigl(\phi ^{2}\bigr) \bigr\vert \bigr) \,dx \,dy. \end{aligned}$$
(3.3)

Using Lemma 2.2 and (3.3), we get

$$\begin{aligned}& \biggl(\sqrt{\frac{pq(p-1)}{q-1}}- \frac{t^{2}}{2t+1} \biggl( 1+ \frac{\epsilon ^{2}}{(Q-2)^{2}} \biggr) \biggr) \int _{\mathbb {R}^{n}} \bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{\frac{\gamma }{2(\alpha +1)}} u^{-q} v^{-2t-1} \phi ^{2} \,dx\,dy \\& \quad \leq C \int _{\mathbb {R}^{n}} v^{-2t} \bigl( \vert \nabla _{G} \phi \vert ^{2} + \phi \vert \Delta _{G} \phi \vert + \vert a \vert \bigl\vert \nabla _{G} \bigl(\phi ^{2}\bigr) \bigr\vert \bigr) \,dx \,dy. \end{aligned}$$

Since \(\tau _{0}^{-}< t <\tau _{0}^{+} \), we have \(\sqrt{\frac{pq(p-1)}{q-1}}- \frac{t^{2}}{2t+1} ( 1+ \frac{\epsilon ^{2}}{(Q-2)^{2}} )>0\) provided ϵ is sufficiently small, which completes the proof. □

End of the proof

The bootstrap argument in this subsection is quite similar to that in [10, 12]. For completeness, we present the details.

Take a cut-off function \(\phi \in C^{\infty }_{c} (\mathbb {R}^{n}, [0,1])\) verifying

$$ \phi =1\quad \mbox{on } B_{1} \times B_{1}\quad \mbox{and} \quad \phi =0 \quad \mbox{outside } B_{2} \times B_{2^{\alpha +1}}. $$
(3.4)

Let w be a smooth function and let \(\kappa = \frac{Q}{Q -2}\) if \(Q>2\). For \(Q>2\), put \(w_{R}(x)= w(R x, R^{1+\alpha } y)\), then, by using the Sobolev inequality (see [26]) and integration by parts, we have

$$\begin{aligned}& \biggl( \int _{B_{1} \times B_{1} } w_{R}^{2\kappa } \,dx \,dy \biggr)^{ \frac{1}{2 \kappa }} \\& \quad \leq \biggl( \int _{B_{2} \times B_{2^{\alpha +1}}} (w_{R} \phi )^{2\kappa } \,dx \,dy \biggr)^{\frac{1}{2 \kappa }} \\& \quad \leq C \biggl( \int _{B_{2} \times B_{2^{\alpha +1}}} \bigl\vert \nabla _{G} (w_{R} \phi ) \bigr\vert ^{2} \,dx \,dy \biggr)^{\frac{1}{2}} \\& \quad = C \biggl( \int _{B_{2} \times B_{2^{\alpha +1}} } \vert \nabla _{G} w_{R} \vert ^{2} \phi ^{2} + w_{R}^{2} \vert \nabla _{G} \phi \vert ^{2} + \frac{1}{2} \nabla _{G} \bigl(w_{R}^{2}\bigr) \nabla _{G}\bigl( \phi ^{2}\bigr) \,dx \,dy \biggr)^{ \frac{1}{2}} \\& \quad = C \biggl( \int _{B_{2} \times B_{2^{\alpha +1}}} \vert \nabla _{G} w_{R} \vert ^{2} \phi ^{2} + w_{R}^{2} \vert \nabla _{G} \phi \vert ^{2} + \frac{1}{2} w_{R}^{2} \bigl(- \Delta _{G}\bigl(\phi ^{2}\bigr)\bigr) \,dx \,dy \biggr)^{\frac{1}{2}} \\& \quad \leq C \biggl( \int _{B_{2} \times B_{2^{\alpha +1}} } \bigl(R^{2} \vert \nabla _{G} w \vert ^{2} + w^{2} \bigr) \bigl(R x,\ldots, R^{1+\alpha } y\bigr) \,dx \,dy \biggr)^{\frac{1}{2}}. \end{aligned}$$

So we get

$$ \biggl( \int _{B_{1} \times B_{1}} w_{R}^{2\kappa } \,dx \,dy \biggr)^{ \frac{1}{2 \kappa }} \leq C \biggl( \int _{B_{2} \times B_{2^{\alpha +1}} } \bigl(R^{2} \vert \nabla _{G} w \vert ^{2} + w^{2} \bigr) \bigl(R x, R^{1+ \alpha } y\bigr) \,dx \,dy \biggr)^{\frac{1}{2}}. $$

From a scaling argument it follows that

$$\begin{aligned}& \biggl( \int _{B_{R} \times B_{R^{\alpha +1}}} w^{2\kappa } \,dx \,dy \biggr)^{\frac{1}{\kappa }} \\& \quad \leq C R^{2+Q(\frac{1}{\kappa }-1)} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}}} \vert \nabla _{G} w \vert ^{2} \,dx \,dy + C R^{Q( \frac{1}{\kappa }-1)} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}}} w^{2} \,dx \,dy. \end{aligned}$$
(3.5)

Suppose that \((u, v)\) is a positive stable solution of (1.1). Set

$$ w=v^{-t}\quad \mbox{for } \tau _{0}^{-}< t< \tau _{0}^{+}. $$

A simple calculation gives

$$ \vert \nabla _{G} w \vert ^{2}= t^{2} \vert \nabla _{G} v \vert ^{2} v^{-2t-2}. $$

Let \(\phi _{R}= \phi (\frac{x}{R},\frac{y}{R^{1+\alpha }})\) where ϕ is given in (3.4). Then

$$\begin{aligned} \int _{B_{R} \times B_{R^{\alpha +1}}} \vert \nabla _{G} w \vert ^{2} \,dx \,dy =& C \int _{B_{R}} \vert \nabla _{G} v \vert ^{2} v^{-2t-2} \,dx \,dy \\ \leq & C \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}} } \vert \nabla _{G} v \vert ^{2} v^{-2t-2} \phi _{R}^{2} \,dx \,dy. \end{aligned}$$
(3.6)

Multiplying the second equation in (1.1) by \(v^{-2t-1} \phi _{R}^{2}\) and using integration by parts, we obtain

$$\begin{aligned}& \int _{B_{2R} \times B_{(2R)^{\alpha +1}}} \vert \nabla _{G} v \vert ^{2} v^{-2t-2} \phi _{R}^{2} \,dx \,dy \\& \quad = \frac{1}{2t+1} \int _{{B_{2R} \times B_{(2R)^{ \alpha +1}}}}\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} v^{-2t-1} \phi _{R}^{2} \,dx \,dy \\& \qquad {}+ \frac{1}{2t(2t+1)} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}}} v^{-2t} \Delta _{G} \bigl(\phi _{R}^{2}\bigr) \,dx \,dy \\& \qquad {}+ \frac{1}{2t(2t+1)} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}} } v^{-2t} a \cdot \nabla _{G}\bigl(\phi _{R}^{2}\bigr) \,dx\,dy . \end{aligned}$$

Inserting this into (3.6), using Lemma 3.1, we obtain

$$ \int _{B_{R} \times B_{R^{\alpha +1}} } \vert \nabla _{G} w \vert ^{2} \,dx \,dy \leq C R^{-2} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}}} w^{2} \,dx \,dy. $$

Substituting the above inequality into (3.5), we have

$$ \biggl( \int _{B_{R} \times B_{R^{\alpha +1}} } w^{2\kappa } \,dx \,dy \biggr)^{\frac{1}{ \kappa }} \leq C R^{Q(\frac{1}{\kappa }-1)} \int _{{B_{2R} \times B_{(2R)^{\alpha +1}}}} w^{2} \,dx \,dy. $$
(3.7)

We fix a real positive number \(\delta = \frac{p+q+2}{4}\) and recall the fact that

$$ 2\tau _{0}^{-} < 0< 2 \delta = \frac{p+q+2}{2}. $$
(3.8)

Let m be a nonnegative integer satisfying \(\delta \kappa ^{m-1}< \tau _{0}^{+}\leq \delta \kappa ^{m}\). We construct an increasing geometric sequence.

$$ \tau _{0}^{-}< t_{1}< t_{2}< \cdots< t_{m}< \tau _{0}^{+}, $$

given by

$$ 2t_{1}= 2 \delta r ,\qquad 2t_{2}= 2\delta r \kappa ,\qquad \ldots, \qquad 2t_{m}= 2\delta r \kappa ^{m-1}, $$

where \(r\in [1, \kappa ]\) is chosen such that \(t_{m}\) is arbitrarily close to \(\tau _{0}^{+}\).

To simplify notations below, we use \(R_{n}= 2^{n} R\). By using (3.7) and an induction argument, we obtain

$$\begin{aligned} \biggl( \int _{B_{R} \times B_{R^{\alpha +1}}} v^{-2t_{m}\kappa } \,dx \,dy \biggr)^{\frac{1}{ t_{m} \kappa }} & \leq C \bigl(R^{Q( \frac{1}{\kappa }-1)} \bigr)^{\frac{1}{t_{m}}} \biggl( \int _{B_{R_{1}} \times B_{R_{1}^{\alpha +1}}} v^{-2t_{m}} \,dx \biggr)^{ \frac{1}{ t_{m} }} \\ & = C R^{Q(\frac{1}{t_{m} \kappa }-\frac{1}{t_{m}})} \biggl( \int _{B_{R_{1}} \times B_{R_{1}^{\alpha +1}}} v^{-2t_{m-1}\kappa } \,dx \biggr)^{ \frac{1}{ t_{m-1} \kappa }} \\ &\leq C R^{Q(\frac{1}{t_{m} \kappa }- \frac{1}{t_{1}})} \biggl( \int _{B_{R_{m}} \times B_{R_{m}^{\alpha +1}}} v^{-2t_{1}} \,dx \,dy \biggr)^{\frac{1}{ t_{1} }} \\ & = C R^{Q(\frac{1}{t_{m} \kappa }-\frac{1}{\delta r})} \biggl( \int _{B_{R_{m}} \times B_{R_{m}^{\alpha +1}}} v^{-2\delta r} \,dx \,dy \biggr)^{ \frac{1}{ \delta r }}. \end{aligned}$$
(3.9)

For the last integral, we shall use Hölder’s inequality, (3.7) and Lemma 2.4 to obtain

$$\begin{aligned} \int _{B_{R_{m}} \times B_{R_{m}^{\alpha +1}} } v^{-2\delta r} \,dx \,dy \leq & \biggl( \int _{B_{R_{m}} \times B_{R_{m}^{\alpha +1}} } v^{-2 \delta \kappa } \,dx \,dy \biggr)^{\frac{r}{\kappa }} \biggl( \int _{B_{R_{m}} \times B_{R_{m}}^{\alpha +1}} 1 \,dx \,dy \biggr)^{1-\frac{r}{\kappa }} \\ \leq & C \biggl(R^{Q(\frac{1}{\kappa }-1)} \int _{B_{R_{m+1}} \times B_{R_{m+1}^{\alpha +1}} } v^{-2\delta } \,dx \,dy \biggr)^{r} R^{Q(1- \frac{r}{\kappa })} \\ =& C R^{Q(1-r)} \biggl( \int _{B_{R_{m+1}} \times B_{R_{m+1}^{ \alpha +1}} } v^{-\frac{p+q+2}{2}} \,dx \,dy \biggr)^{r} \\ \leq & C R^{Q(1-r)} R^{r(Q-2-\gamma )} =C R^{Q-r(\gamma +2)}. \end{aligned}$$

Consequently,

$$ \biggl( \int _{B_{R_{m}} \times B_{R_{m}^{\alpha +1}}} v^{-2\delta r} \,dx \biggr)^{\frac{1}{\delta r}} \leq C R^{\frac{Q}{\delta r}- \frac{\gamma +2}{\delta }}. $$
(3.10)

Substituting (3.10) into the last inequality of (3.9), one has

$$ \biggl( \int _{B_{R} \times B_{R^{\alpha +1}}} v^{-2t_{m}\kappa } \,dx \biggr)^{\frac{1}{ t_{m} \kappa }} \leq C R^{ \frac{Q}{\kappa t_{m}}-\frac{\gamma +2}{\delta }}= C R^{ \frac{Q}{\kappa t_{m}}-\frac{4(\gamma +2)}{p+q+2}}. $$
(3.11)

Since \(r\in [1, \kappa ]\) is chosen such that \(t_{m}\) is close to \(\tau _{0}^{+}\), the exponent in the right-hand side of (3.11) is negative. Letting \(R \rightarrow + \infty \), we obtain a contradiction.

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Acknowledgements

We would like to express our gratitude to the anonymous referee for the careful reading and helpful suggestions.

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Rahal, B. On stable entire solutions of sub-elliptic system involving advection terms with negative exponents and weights. J Inequal Appl 2020, 119 (2020). https://doi.org/10.1186/s13660-020-02385-x

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MSC

  • 35J55
  • 35J65
  • 35B65

Keywords

  • Liouville-type theorem
  • Grushin operator
  • Stable solutions
  • Elliptic system
  • Weighted Grushin system
  • Weighted Grushin equation