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Theory and Modern Applications

Stable weak solutions to weighted Kirchhoff equations of Lane–Emden type


This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations:

$$\begin{aligned}& -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\& \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$

where \(M(t)=a+bt^{k}\), \(t\geq0\), with \(a,b,k\geq0\), \(a+b>0\), \(k=0\) if and only if \(b=0\). Let \(N=N_{1}+N_{2}\geq2\), \(p>1+2k\) and \(\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}\) be nonnegative functions such that \(\xi(z)\leq C\|z\|_{G}^{\theta}\) and \(\eta(z)\geq C'\|z\|_{G}^{d}\) for large \(\|z\|_{G}\) with \(d>\theta-2\). Here \(\alpha\geq0\) and \(\|z\|_{G}=(|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}\). \(\operatorname{div}_{G}\) (resp., \(\nabla_{G}\)) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k, θ, d, and \(N_{\alpha}=N_{1}+(1+\alpha)N_{2}\), the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function M could be zero, which implies that the above problem is degenerate.

1 Introduction and main result

In this work, we are interested in the nonexistence of stable weak solutions for the weighted Kirchhoff equations

$$\begin{aligned} &{-}M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr)\operatorname{div}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\ &\quad =\eta(z) \vert u \vert ^{p-1}u, \quad z\in\mathbb{R}^{N}= \mathbb{R}^{N_{1}}\times \mathbb{R}^{N_{2}}, \end{aligned}$$

where M is a nonnegative continuous function which will be given later. Here and thereafter, we assume that \(p>1\) and \(\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}\) are nonnegative functions. For \(z=(x,y)\in\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb {R}^{N_{2}}\) and \(\alpha\geq0\), we define the Grushin gradient \(\nabla_{G}\) and Grushin divergence \(\operatorname{div}_{G}\) as follows:

$$\begin{aligned}& \nabla_{G}u=\bigl(\nabla_{x}u,(1+\alpha) \vert x \vert ^{\alpha}\nabla_{y}u\bigr), \\& \operatorname{div}_{G}\textbf{v}=\operatorname{div}_{x} \textbf{v}+(1+\alpha ) \vert x \vert ^{\alpha} \operatorname{div}_{y} \textbf{v}. \end{aligned}$$

The Grushin operator \(\Delta_{G}\) is denoted by

$$ \Delta_{G}u=\operatorname{div}_{G}( \nabla_{G}u)=\Delta_{x}u+(1+\alpha )^{2} \vert x \vert ^{2 \alpha}\Delta_{y}u, $$

which is just the well-known Laplace operator when \(\alpha=0\).

The anisotropic dilation attached to \(\Delta_{G}\) is defined by

$$ \tau_{\delta}(z)=\bigl(\delta x,\delta^{1+\alpha}y\bigr),\quad \delta>0, z=(x,y) \in\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}. $$

It is easy to check that

$$ d\tau_{\delta}(z)=\delta^{N_{\alpha}}\,dx\,dy=\delta^{N_{\alpha}} \,dz, $$

where \(N_{\alpha}=N_{1}+(1+\alpha)N_{2}\) is a homogeneous dimension with respect to the dilation \(\tau_{\delta}\), and \(dx\,dy\) denotes the Lebesgue measure on \(\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}\). Finally, the norm of z (also known as the Grushin distance) is defined by

$$ \Vert z \Vert _{G}=\bigl( \vert x \vert ^{2(1+\alpha)}+ \vert y \vert ^{2}\bigr)^{\frac{1}{2(1+\alpha)}},\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}. $$

In the case \(\alpha=0\), \(\xi(z)\equiv1\) and \(M(t)=a+bt\), \(t\geq0\) with \(a,b>0\), (1.1) becomes the following nonlocal Kirchhoff type problem:

$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dz \biggr)\Delta u=\eta (z) \vert u \vert ^{p-1}u,\quad z\in \mathbb{R}^{N}. \end{aligned}$$

This problem is often referred to as being nonlocal because of the presence of the integral over the entire domain \(\mathbb{R}^{N}\). This phenomenon provokes many mathematical difficulties, which makes the research on the problem particularly interesting and necessary. Besides, problem (1.2) is related to the stationary analogue of the Kirchhoff equation on a bounded domain \(\Omega\subset\mathbb{R}^{N}\)

$$\begin{aligned} u_{tt}- \biggl(a+b \int_{\Omega} \vert \nabla u \vert ^{2}\,dz \biggr)\Delta u=f(z,u) \end{aligned}$$

proposed by Kirchhoff in 1883 (see [20]) as generalization of the classical D’Alembert’s wave equation for free vibration of elastic strings. It is important to note that equation (1.3) received great attention only after Lions [26] introduced a functional analysis approach. For recent interesting results concerning the various solutions of Kirchhoff equations and the stationary analogue of problem (1.3), please refer to [1, 4, 18, 19, 28, 29, 37] and the references therein.

Recently, much attention has been paid to the Kirchhoff equations in the whole space \(\mathbb{R}^{N}\). Most of recent results have been concerned with the existence and multiplicity of solutions when the nonlinearity is in subcritical case and critical case. For instance, in the paper [35], Wu considered a class of Schrödinger Kirchhoff type equations in \(\mathbb{R}^{N}\) with subcritical case, by using the symmetric mountain pass theorem, four new existence results for nontrivial solutions and a sequence of high energy solutions have been obtained. In [14], Fan studied the existence and multiplicity of positive ground state solutions for a Kirchhoff type problem involving several potentials and critical nonlinearities in \(\mathbb{R}^{N}\). In addition, in this article, the concentration phenomenon of positive solutions was also considered. For more similar works, we refer to [25, 27, 40]. However, compared with a vast number of existence results for entire solutions of Kirchhoff type equations, the nonexistence one is less investigated.

In this paper, the Kirchhoff function M is assumed to verify


\(M(t)=a+bt^{k}\), \(t\geq0\), \(a,b,k\geq0\), \(a+b>0\), and

$$\begin{aligned} k\textstyle\begin{cases} \in(0,\infty), &\text{if } b>0, \\ 0, &\text{if } b=0, \end{cases}\displaystyle \end{aligned}$$

in other words, \(k=0\) if and only if \(b=0\). Problem (1.2) is called non-degenerate if \(a>0\) and \(b\geq0\), while it is said to be degenerate if \(a=0\) and \(b>0\). The degenerate case is extremely interesting and is treated in well-known papers on Kirchhoff’s theory, for example, see [11]. For degenerate Kirchhoff type problems, we refer to [24, 27, 36, 39]. For non-degenerate Kirchhoff type problems, we refer to [14, 25, 28, 29, 35, 40].

Set \(\alpha=0\) and \(M(t)\equiv1\), problem (1.1) becomes the weighted Lane–Emden equation

$$\begin{aligned} -\operatorname{div}\bigl(\xi(z)\nabla u\bigr)=\eta(z) \vert u \vert ^{p-1}u \quad \text{in } \mathbb{R}^{N}. \end{aligned}$$

In recent years, much attention has been focused on studying of the nonexistence and stability of solutions to nonlinear elliptic equations like (1.4). For some physical motivation and recent developments on the topic of stable solutions, we refer to [13].

Liouville type theorems for stable solutions deal with the nonexistence of this particular type of solutions. The pioneering work in this direction is due to Farina [15], where the author established thoroughly the Liouville type theorem for stable classical solutions of problem (1.4) with \(\xi(z)\equiv1\equiv\eta(z)\). He proved that the problem does not admit a nontrivial stable \(C^{2}\) solution if and only if \(1< p< p_{c}(N)\), where

$$\begin{aligned} p_{c}(N)=\textstyle\begin{cases} +\infty, &\text{if } N\leq10, \\ \frac{(N-2)^{2}-4N+8\sqrt{N-1}}{(N-2)(N-10)}, &\text{if } N\geq11. \end{cases}\displaystyle \end{aligned}$$

Moreover, this exponent is greater than the classical critical exponent \(\frac{N+2}{N-2}\) [17] when \(N>2\). After that, the above results have been generalized to the weighted case in [7, 10, 34]. In [10], under the restriction that the solutions are locally bounded, the authors presented the nonexistence of nontrivial stable weak solutions of problem (1.4) with \(\xi(z)\equiv1\) and \(\eta(z)=|z|^{d}\). In [34], this restriction was withdrawn.

Theorem 1.1


Let u be a stable weak solution of (1.4) with \(\xi(z)\equiv1\) and \(\eta(z)=|z|^{d}\), where \(d>-2\). Then u is a trivial solution provided \(1< p< p(N,d)\). Here

$$\begin{aligned} p(N,d)=\textstyle\begin{cases} +\infty,&\textit{if } N\leq10+4d, \\ \frac{(N-2)(N-6-2d)-2(2+d)^{2}+2(2+d)\sqrt{(2+d)(2N-2+d)}}{(N-2)(N-10-4d)}, &\textit{if } N>10+4d. \end{cases}\displaystyle \end{aligned}$$

In [7], with the help of Farina’s approach, Cowan and Fazly established several Liouville type theorems for stable positive classical solutions of problem (1.4) under different assumptions on ξ and η. Later, several attempts have been made to extend Farina’s results to the weighted quasilinear case \(-\operatorname{div}(\xi(z)|\nabla u|^{p-2}\nabla u)=\eta(z)|u|^{p-1}u\), in \(\mathbb{R}^{N}\), see [5, 6, 22, 23].

We now turn to the case \(\alpha>0\). Then problem (1.1) is a nonlocal problem involving Grushin operator. It is well known that the Grushin operator belongs to the wide class of subelliptic operators studied by Franchi et al. in [16](see also [3]). Taking advantage of the Kelvin transform combined with the method of moving planes, the Liouville type theorem has been established by Monticelli [31] (resp., Yu [38]) for nonnegative classical (resp., weak) solutions of the problem \(-\Delta_{G}u=u^{p}\) in \(\mathbb{R}^{N}\), the optimal exponent is \(1< p<\frac{N_{\alpha}+2}{N_{\alpha}-2}\). Recently, Duong and Nguyen [12] studied elliptic equations involving Grushin operator and advection

$$ -\Delta_{G}u+\nabla_{G}w\cdot\nabla_{G}u= \Vert z \Vert ^{s}_{G} \vert u \vert ^{p-1}u,\quad \text{in } \mathbb{R}^{N}, s\geq0. $$

Via Farina’s approach, the authors obtained several Liouville type theorems for a class of stable sign-changing weak solutions.

Very recently, Le [21] considered the elliptic problems

$$ -\operatorname{div}_{G}(w_{1}\nabla_{G}u)=w_{2}f(u), \quad \text{in } \Omega, $$

with homogeneous Dirichlet boundary condition. By variable technique, under suitable assumptions on Ω, \(w_{1}\), \(w_{2}\), and f, nonexistence of stable weak solutions has been established. When \(\Omega=\mathbb{R}^{N}\) and f has power or exponential growth, the author also constructed some examples to show the sharpness of his results. For other results of Liouville type theorems related to Grushin operators or more general subelliptic operators, we refer the reader to [2, 9, 30, 32, 33] and the references therein.

A natural question is whether there are analogous Liouville results for Kirchhoff type equation (1.1) with \(\alpha>0\) and \(\xi,\eta\not\equiv1\). The present paper is an attempt to answer this interesting question.

Motivated by the aforementioned works, we prove the nonexistence of nontrivial stable weak solution to equation (1.1). Since solutions to elliptic equations with Hardy potentials may possess singularities, it is natural to study weak solutions of (1.1) in a suitable weighted Sobolev space. Based on this reality, we define

$$ \Vert \psi \Vert _{\xi}= \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}\psi \vert ^{2}\,dz \biggr)^{1/2} $$

for \(\psi\in C_{0}^{\infty}(\mathbb{R}^{N})\) and denote by \(H^{1,\alpha}(\mathbb{R}^{N}; \xi)\) the closure of \(C_{0}^{\infty}(\mathbb{R}^{N})\) with respect to the \(\|\cdot\|_{\xi}\)-norm. Note that, for \(\xi\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\), we have \(C_{0}^{1}(\mathbb{R}^{N})\subset H^{1,\alpha}(\mathbb{R}^{N};\xi )\). Denote also by \(H^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{N};\xi)\) the space of all functions u such that \(u\psi\in H^{1,\alpha}(\mathbb{R}^{N};\xi)\) for all \(\psi\in C^{1}_{0}(\mathbb{R}^{N})\).

Definition 1.2

Let \(X=H^{1,\alpha}(\mathbb{R}^{N};\xi)\cap H^{1,\alpha}_{\mathrm{loc}}( \mathbb{R}^{N};\xi)\), we say that \(u\in X\) is a weak solution of (1.1) if \(\eta(z)|u|^{p}\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\), and for all \(\psi\in C^{1}_{0}(\mathbb{R}^{N})\) we have

$$\begin{aligned} A \int_{\mathbb{R}^{N}}\xi(z)\nabla_{G}u\cdot \nabla_{G}\psi \,dz= \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u\psi \,dz, \end{aligned}$$

where \(A=a+b\|u\|_{\xi}^{2k}\).

Now, we give the stability definition. First, we consider that the energy functional \(I: X\rightarrow\mathbb{R}\) associated with (1.1) is

$$\begin{aligned} I(u)=\frac{a}{2} \Vert u \Vert ^{2}_{\xi}+ \frac{b}{2(k+1)} \Vert u \Vert _{\xi}^{2(k+1)}- \frac{1}{p+1} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+1}\,dz. \end{aligned}$$

Set \(E(t)=I(u+t\psi)\) for \(\psi\in C_{0}^{1}(\mathbb{R}^{N})\). Then, by routine calculations, we show \(E''(0)\geq0\) is equivalent to

$$\begin{aligned} \begin{aligned} p \int_{\mathbb{R}^{N}} \eta(z) \vert u \vert ^{p-1} \psi^{2}\,dz \leq{}&2bk \Vert u \Vert _{\xi}^{2(k-1)} \biggl( \int_{\mathbb{R}^{N}} \xi(z) \nabla_{G}u\cdot \nabla_{G}\psi \,dz \biggr)^{2} \\ &{} +A \int_{\mathbb{R}^{N}} \xi(z) \vert \nabla_{G}\psi \vert ^{2}\,dz. \end{aligned} \end{aligned}$$

A weak solution u of (1.1) is called stable if (1.8) holds for all \(\psi\in C_{0}^{1}(\mathbb{R}^{N})\). Therefore, if u is a stable weak solution of (1.1), by Hölder’s inequality and (1.8), it follows that

$$\begin{aligned} p \int_{\mathbb{R}^{N}} \eta(z) \vert u \vert ^{p-1} \psi^{2}\,dz\leq B \int_{ \mathbb{R}^{N}} \xi(z) \vert \nabla_{G}\psi \vert ^{2}\,dz,\quad \forall\psi\in C_{0}^{1} \bigl( \mathbb{R}^{N}\bigr), \end{aligned}$$

where \(B=a+b(1+2k)\|u\|_{\xi}^{2k}\). Remark that (1.7)–(1.9) hold for all \(\psi\in H^{1,\alpha}(\mathbb{R}^{N};\xi)\) by density arguments.

Throughout this paper, we assume that the functions \(\xi(z)\), \(\eta(z)\) satisfy the following assumption:


\(\xi(z), \eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}\) are nonnegative functions. In addition, there exist \(d>\theta-2\), \(C, C'>0\), and \(R_{0}>0\) such that

$$ \xi(z)\leq C \Vert z \Vert _{G}^{\theta},\qquad \eta(z) \geq C' \Vert z \Vert _{G}^{d},\quad \forall \Vert z \Vert _{G}\geq R_{0}. $$

To facilitate the writing, we denote

$$\begin{aligned}& \begin{aligned} &\lambda_{0}= \lambda_{0}(k,\theta,d)= \textstyle\begin{cases} +\infty, &k=0, \\ 2-\theta+\frac{2-\theta+d}{k}, &k>0, \end{cases}\displaystyle \\ &\lambda_{1}=\lambda_{1}(k,\theta,d)=2- \theta+ \frac{4(2-\theta+d)}{1+2k}, \\ &\lambda_{2}=\lambda_{2}(k,\theta,d)=2-\theta+ \frac{(1+\sqrt{1+2k})(2-\theta+d)}{2k}\quad (k>0). \end{aligned} \end{aligned}$$

Let us now state the main result of this work.

Theorem 1.3

Let \(u\in X\) be a stable weak solution of problem (1.1) with \(p>1+2k\). Assume that \((M)\) and \((H)\) hold. We further suppose that one of the following conditions occurs:


\(k\geq0\), \(N_{\alpha}\leq\min\{\lambda_{0},\lambda_{1}\}\), and \(p>1+2k\);


\(0\leq k<\frac{1}{2}\), \(\lambda_{1}< N_{\alpha}\leq\lambda_{0}\), and \(1+2k< p< p_{c}\);


\(0< k\leq\frac{1}{2}\), \(\lambda_{0}< N_{\alpha}<\lambda_{2}\), and \(\hat{p}_{c}< p< p_{c}\);


\(k>\frac{1}{2}\), \(\lambda_{0}< N_{\alpha}<\lambda_{1}\), and \(p>\hat{{p}}_{c}\);


\(\frac{1}{2}< k<\frac{3}{2}\), \(N_{\alpha}= \lambda_{1}\), and \(p>\frac{4}{3-2k}\);


\(\frac{1}{2}< k<\frac{3}{2}\), \(\lambda_{1}< N_{\alpha}< \lambda_{2}\), and \(\hat{p}_{c}< p< p_{c}\),

with critical exponents

$$\begin{aligned}& \begin{aligned} &\hat{p}_{c}= \hat{p}_{c}(k,\theta,d,N_{\alpha}) \\ &\hphantom{\hat{p}_{c}}=1+ \frac{2(2-\theta+d) (N_{\alpha}-4+2\theta-d-\sqrt{(N_{\alpha }+d)^{2}-(1+2k)(N_{\alpha}-2+\theta)^{2}} )}{(N_{\alpha }-2+\theta)(1+2k)(N_{\alpha}-\lambda_{1})}; \\ &p_{c}=p_{c}(k,\theta,d,N_{\alpha}) \\ &\hphantom{p_{c}}=1+ \frac{2(2-\theta+d) (N_{\alpha}-4+2\theta-d+\sqrt{(N_{\alpha }+d)^{2}-(1+2k)(N_{\alpha}-2+\theta)^{2}} )}{(N_{\alpha }-2+\theta)(1+2k)(N_{\alpha}-\lambda_{1})}. \end{aligned} \end{aligned}$$

Then \(u\equiv0\) in \(\mathbb{R}^{N}\).

Remark 1.4

Indeed, the assumption on p in Theorem 1.3 is equivalent to

$$ N_{\alpha}< 2-\theta+ \frac{2(2-\theta+d) (p+\sqrt{p(p-1-2k)} )}{(1+2k)(p-1)}. $$

Remark 1.5

If \(k=0\) (i.e., \(b=0\)), then Theorem 1.3 recovers the known result for the Grushin operator in Le [21, Proposition 3]. Furthermore, if \(\alpha=k=0\), then our result recovers the previous result for elliptic problems with weights in Cowan and Fazly [7, Theorem 3]. If \(\alpha=k=\theta=0\), then we have

$$\begin{aligned} \begin{aligned} p_{c}(0,0,d,N) =1+ \frac{2(2+d) (N-4-d+\sqrt{(2+d)(2N-2+d)} )}{(N-2)(N-10-4d)}. \end{aligned} \end{aligned}$$

Note that the exponent \(p_{c}(0,0,d, N)\) equals \(p(N,d)\) of (1.6) when \(N>10+4d\) and is the critical exponent \(\bar{p}(d)\) in [10]. When \(\alpha=k=\theta=d=0\), we obtain

$$\begin{aligned} \begin{aligned} p_{c}(0,0,0,N) =1+ \frac{4 (N-4+2\sqrt{N-1} )}{(N-2)(N-10)}, \end{aligned} \end{aligned}$$

which is the critical exponent \(p_{c}(N)\) in [15]. Finally, when \(\alpha=\theta=0\) and \(k=1\), it is not difficult to verify that \(\hat{p}_{c}(1,0,d,N)\) (resp., \(p_{c}(1,0,d,N)\)) equals \(q_{1}(N,d)\) (resp., \(q_{2}(N,d)\)) in [24].

The rest of the paper is devoted to the proof of Theorem 1.3. In the following, C stands for a generic positive constant which may vary even in the same line. If this constant depends on an arbitrary small number ε, then we denote it by \(C_{\varepsilon}\).

2 Proof of Theorem 1.3

We first give the following proposition, which plays a crucial role in arriving at Theorem 1.3.

Proposition 2.1

Let \(u\in X\) be a stable weak solution of (1.1) with \(p>1+2k\). Then, for every \(s\in(1,h(p))\), where

$$\begin{aligned} \begin{aligned} h(t) =-1+ \frac{2 (t+\sqrt{t(t-1-2k)} )}{1+2k},\quad t>1+2k, \end{aligned} \end{aligned}$$

and for any constant \(m\geq\frac{p+s}{p-1}\), there exists a constant \(C>0\) depending only on p, s, m, and k such that

$$\begin{aligned}& \begin{aligned} & \int_{\mathbb{R}^{N}} \bigl(\eta(z) \vert u \vert ^{p+s}+ \xi(z) \vert \nabla_{G}u \vert ^{2} \vert u \vert ^{s-1} \bigr)\varphi^{2m}\,dz \\ &\quad \leq CB^{\frac{1+s}{p-1}}(1+B) \int_{\mathbb{R}^{N}}\xi(z)^{ \frac{p+s}{p-1}}\eta(z)^{\frac{1+s}{1-p}} \vert \nabla_{G}\varphi \vert ^{ \frac{2(p+s)}{p-1}}\,dz \end{aligned} \end{aligned}$$

holds for all functions \(\varphi\in C_{0}^{1}(\mathbb{R}^{N})\) verifying \(0\leq\varphi\leq1\) and \(\nabla_{G}\varphi=0\) in a neighborhood of \(\{z\in\mathbb{R}^{N}: \eta(z)=0\}\). Here, B is given by (1.9).


Some ideas in this proof are inspired by [8, 12, 15, 21]. Since the solution u is not necessary locally bounded, to overcome this difficulty, we will construct a sequence of suitable cut-off functions. Let n be a positive integer, we denote

$$\begin{aligned} \delta_{n}(t)=\textstyle\begin{cases} \vert t \vert ^{\frac{s-1}{2}}t, & \vert t \vert \leq n, \\ n^{\frac{s-1}{2}}t, & \vert t \vert >n, \end{cases}\displaystyle \qquad \nu_{n}(t)=\textstyle\begin{cases} \vert t \vert ^{s-1}t, & \vert t \vert \leq n, \\ n^{s-1}t, & \vert t \vert >n. \end{cases}\displaystyle \end{aligned}$$

By a direct computation, we obtain that, for any \(t\in\mathbb{R}\), there exists a positive constant C depending only on s such that

$$\begin{aligned} \begin{aligned} \delta_{n}^{2}(t)=t \nu_{n}(t),\qquad \delta'_{n}(t)^{2} \leq\frac{(s+1)^{2}}{4s}\nu'_{n}(t),\qquad \delta_{n}^{2}(t)+\nu^{2}_{n}(t) \bigl(\nu'_{n}(t) \bigr)^{-1}\leq C \vert t \vert ^{s+1}. \end{aligned} \end{aligned}$$

Moreover, since \(u\in H^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{N},\xi )\), we deduce that \(\delta_{n}(u), \nu_{n}(u)\in H^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{N}, \xi)\) for any \(n\in\mathbb{Z}^{+}\).

For any nonnegative function, \(\phi\in C_{0}^{1}(\mathbb{R}^{N})\) satisfies \(0\leq\phi\leq1\). Setting \(\psi=\nu_{n}(u)\phi^{2}\) as a test function in (1.7), we have

$$\begin{aligned}& \begin{aligned} &A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\nu'_{n}(u) \phi^{2} \,dz+2A \int_{\mathbb{R}^{N}}\xi(z)\phi\nu_{n}(u) \nabla_{G}u \cdot\nabla_{G}\phi \,dz \\ &\quad = \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u)\phi^{2}\,dz. \end{aligned} \end{aligned}$$

Applying Young’s inequality, for any \(\varepsilon>0\),

$$\begin{aligned} &A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\nu'_{n}(u) \phi^{2}\,dz \\ &\quad \leq2A \int_{\mathbb{R}^{N}}\xi(z) \bigl\vert \nu_{n}(u) \bigr\vert \vert \nabla_{G}u \vert \vert \nabla_{G} \phi \vert \phi \,dz+ \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u) \phi^{2}\,dz \\ &\quad \leq\varepsilon A \int_{\mathbb{R}^{N}} \bigl(\xi(z)^{1/2} \vert \nabla_{G}u \vert \nu'_{n}(u)^{1/2} \phi \bigr)^{2}\,dz \\ &\qquad {}+C_{\varepsilon}A \int_{\mathbb{R}^{N}} \bigl(\xi(z)^{1/2} \bigl\vert \nu_{n}(u) \bigr\vert \nu'_{n}(u)^{-1/2} \vert \nabla_{G}\phi \vert \bigr)^{2}\,dz \\ &\qquad {} + \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u)\phi^{2}\,dz \\ &\quad =\varepsilon A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\nu'_{n}(u) \phi^{2}\,dz +C_{\varepsilon}A \int_{\mathbb{R}^{N}}\xi(z)\nu_{n}^{2}(u) \nu'_{n}(u)^{-1} \vert \nabla_{G}\phi \vert ^{2}\,dz \\ &\qquad {} + \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u)\phi^{2}\,dz, \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned} &(1-\varepsilon)A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2} \nu'_{n}(u)\phi^{2}\,dz \\ & \quad \leq C_{\varepsilon}A \int_{\mathbb{R}^{N}}\xi(z)\nu _{n}^{2}(u) \nu'_{n}(u)^{-1} \vert \nabla_{G}\phi \vert ^{2}\,dz+ \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u) \phi^{2}\,dz. \end{aligned} \end{aligned}$$

On the other hand, by virtue of the stability definition, we take \(\psi=\delta_{n}(u)\phi\) in (1.9) and get

$$\begin{aligned} \begin{aligned} &p \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u) \phi^{2}\,dz \\ &\quad \leq B \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\delta'_{n}(u)^{2} \phi^{2}\,dz+2B \int_{\mathbb{R}^{N}}\xi(z)\delta'_{n}(u) \bigl\vert \delta_{n}(u) \bigr\vert \vert \nabla_{G}u \vert \vert \nabla_{G}\phi \vert \phi \,dz \\ & \qquad {}+B \int_{\mathbb{R}^{N}}\xi(z)\delta^{2}_{n}(u) \vert \nabla_{G} \phi \vert ^{2}\,dz. \end{aligned} \end{aligned}$$

We use Young’s inequality to estimate the middle term of the right-hand side of (2.5):

$$\begin{aligned} \begin{aligned} &2B \int_{\mathbb{R}^{N}}\xi(z)\delta'_{n}(u) \bigl\vert \delta_{n}(u) \bigr\vert \vert \nabla_{G}u \vert \vert \nabla_{G}\phi \vert \phi \,dz \\ &\quad \leq\varepsilon B \int_{\mathbb{R}^{N}} \bigl(\xi(z)^{1/2} \vert \nabla_{G}u \vert \delta'_{n}(u)\phi \bigr)^{2}\,dz+C_{\varepsilon}B \int_{\mathbb{R}^{N}} \bigl(\xi(z)^{1/2} \bigl\vert \delta_{n}(u) \bigr\vert \vert \nabla_{G}\phi \vert \bigr)^{2}\,dz \\ &\quad =\varepsilon B \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\delta '_{n}(u)^{2} \phi^{2}\,dz+C_{\varepsilon}B \int_{\mathbb{R}^{N}}\xi(z)\delta_{n}^{2}(u) \vert \nabla_{G}\phi \vert ^{2}\,dz. \end{aligned} \end{aligned}$$

Substituting this inequality into (2.5), one has

$$\begin{aligned} \begin{aligned} &p \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u) \phi^{2}\,dz \\ &\quad \leq(1+\varepsilon)B \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2} \delta'_{n}(u)^{2} \phi^{2}\,dz +C_{\varepsilon}B \int_{\mathbb{R}^{N}} \xi(z)\delta^{2}_{n}(u) \vert \nabla_{G}\phi \vert ^{2}\,dz. \end{aligned} \end{aligned}$$

With the help of (2.3), it follows from (2.4) and (2.6) that

$$\begin{aligned} p \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u) \phi^{2}\,dz \leq& \frac{(1+\varepsilon)(1+s)^{2}B}{4s} \int_{\mathbb{R}^{N}} \xi(z) \vert \nabla_{G}u \vert ^{2}\nu'_{n}(u)\phi^{2}\,dz \\ &{} +C_{\varepsilon}B \int_{\mathbb{R}^{N}}\xi(z)\delta^{2}_{n}(u) \vert \nabla_{G}\phi \vert ^{2}\,dz \\ \leq&\frac{(1+\varepsilon)(1+s)^{2}B}{4s(1-\varepsilon)A} \int_{ \mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u)\phi^{2}\,dz \\ &{} +C_{\varepsilon}B \int_{\mathbb{R}^{N}}\xi(z) \bigl(\delta^{2}_{n}(u)+ \nu_{n}^{2}(u)\nu'_{n}(u)^{-1} \bigr) \vert \nabla_{G}\phi \vert ^{2}\,dz \\ \leq&\frac{(1+\varepsilon)(1+s)^{2}(1+2k)}{4s(1-\varepsilon)} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u)\phi^{2}\,dz \\ &{} +C_{\varepsilon}B \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G} \phi \vert ^{2}\,dz, \end{aligned}$$

that is,

$$\begin{aligned} q_{\varepsilon} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u)\phi^{2}\,dz \leq& C_{\varepsilon}B \int_{\mathbb{R}^{N}} \xi(z) \vert u \vert ^{1+s} \vert \nabla_{G}\phi \vert ^{2}\,dz, \end{aligned}$$

where \(q_{\varepsilon}=p- \frac{(1+\varepsilon)(1+s)^{2}(1+2k)}{4s(1-\varepsilon)}\). Since \(\lim_{\varepsilon\rightarrow0^{+}}q_{\varepsilon}=q_{0}=p- \frac{(1+s)^{2}(1+2k)}{4s}\), we have \(q_{0}>0\) under assumption on \(s\in(1,h(p))\). Thus, choose small \(\varepsilon>0\) such that \(q_{\varepsilon}>0\). Consequently,

$$\begin{aligned} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u) \phi^{2}\,dz \leq& CB \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G} \phi \vert ^{2}\,dz, \end{aligned}$$

where positive constant C depends only on p, k, and s.

From (2.8) and Fatou’s lemma, we derive, as \(n\rightarrow+\infty\),

$$\begin{aligned} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+s} \phi^{2}\,dz \leq& CB \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G}\phi \vert ^{2}\,dz. \end{aligned}$$

On the other hand, choosing \(\varepsilon=1/2\) in (2.4) and combining (2.3) with (2.8), we can find

$$\begin{aligned} &A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\nu'_{n}(u) \phi^{2}\,dz \\ & \quad \leq CA \int_{\mathbb{R}^{N}}\xi(z)\nu_{n}^{2}(u) \nu'_{n}(u)^{-1} \vert \nabla_{G}\phi \vert ^{2}\,dz+2 \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1}u \nu_{n}(u) \phi^{2}\,dz \\ &\quad \leq CA \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G}\phi \vert ^{2}\,dz+2 \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p-1} \delta^{2}_{n}(u)\phi^{2}\,dz \\ & \quad \leq C(A+B) \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G}\phi \vert ^{2}\,dz. \end{aligned}$$

Letting \(n\rightarrow+\infty\) in the above inequality, we have from Fatou’s lemma that

$$\begin{aligned} A \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2} \vert u \vert ^{s-1} \phi^{2}\,dz \leq& C(A+B) \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G} \phi \vert ^{2}\,dz. \end{aligned}$$


$$\begin{aligned} \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2} \vert u \vert ^{s-1} \phi^{2}\,dz \leq& C \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \vert \nabla_{G} \phi \vert ^{2}\,dz. \end{aligned}$$

Now, we assert that (2.2) holds true. In fact, we can select some positive constant \(m\gg1\) such that

$$ \frac{(m-1)(p+s)}{1+s}\geq m, \quad \text{or}\quad m\geq \frac{p+s}{p-1}. $$

Recalling \(0\leq\phi(z)\leq1\) in \(\mathbb{R}^{N}\), we obtain

$$ \bigl(\phi(z) \bigr)^{\frac{2(m-1)(p+s)}{1+s}}\leq \bigl(\phi(z) \bigr)^{2m}, \quad \forall z\in\mathbb{R}^{N}. $$

Then, by (2.9) with \(\phi=\varphi^{m}\) and Hölder’s inequality, one sees

$$\begin{aligned} \begin{aligned} & \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+s} \varphi^{2m}\,dz \\ &\quad \leq CB \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \varphi^{2(m-1)} \vert \nabla_{G} \varphi \vert ^{2}\,dz \\ & \quad \leq CB \biggl( \int_{\mathbb{R}^{N}} \bigl(\eta(z)^{\frac{1+s}{p+s}} \vert u \vert ^{1+s} \varphi^{2(m-1)} \bigr)^{\frac{p+s}{1+s}}\,dz \biggr)^{\frac{1+s}{p+s}} \\ & \qquad {}\times \biggl( \int_{\mathbb{R}^{N}} \bigl(\xi(z)\eta(z)^{- \frac{1+s}{p+s}} \vert \nabla_{G}\varphi \vert ^{2} \bigr)^{\frac{p+s}{p-1}} \,dz \biggr)^{\frac{p-1}{p+s}} \\ &\quad = CB \biggl( \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+s} \varphi^{ \frac{2(m-1)(p+s)}{1+s}}\,dz \biggr)^{\frac{1+s}{p+s}} \\ & \qquad {}\times \biggl( \int_{\mathbb{R}^{N}}\xi(z)^{\frac{p+s}{p-1}} \eta(z)^{\frac{1+s}{1-p}} \vert \nabla_{G}\varphi \vert ^{\frac{2(p+s)}{p-1}}\,dz \biggr)^{\frac{p-1}{p+s}} \\ & \quad \leq CB \biggl( \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+s} \varphi^{2m}\,dz \biggr)^{\frac{1+s}{p+s}} \biggl( \int_{\mathbb{R}^{N}}\xi(z)^{ \frac{p+s}{p-1}}\eta(z)^{\frac{1+s}{1-p}} \vert \nabla_{G}\varphi \vert ^{ \frac{2(p+s)}{p-1}}\,dz \biggr)^{\frac{p-1}{p+s}}. \end{aligned} \end{aligned}$$


$$\begin{aligned} \int_{\mathbb{R}^{N}}\eta(z) \vert u \vert ^{p+s} \varphi^{2m}\,dz \leq& CB^{\frac{p+s}{p-1}} \int_{\mathbb{R}^{N}}\xi(z)^{\frac{p+s}{p-1}} \eta(z)^{\frac{1+s}{1-p}} \vert \nabla_{G}\varphi \vert ^{\frac{2(p+s)}{p-1}}\,dz. \end{aligned}$$

Analogously, take \(\phi=\varphi^{m}\) in (2.10) and combining (2.11) with (2.12), one can achieve

$$\begin{aligned} & \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2} \vert u \vert ^{s-1} \varphi^{2m}\,dz \\ &\quad \leq C \int_{\mathbb{R}^{N}}\xi(z) \vert u \vert ^{1+s} \varphi^{2(m-1)} \vert \nabla_{G} \varphi \vert ^{2}\,dz \\ & \quad \leq CB^{\frac{1+s}{p-1}} \int_{\mathbb{R}^{N}}\xi(z)^{ \frac{p+s}{p-1}}\eta(z)^{\frac{1+s}{1-p}} \vert \nabla_{G}\varphi \vert ^{ \frac{2(p+s)}{p-1}}\,dz. \end{aligned}$$

Therefore, combining this with (2.12), (2.2) is obtained immediately. This completes the proof. □

Let \(R>0\), \(\Omega_{2R}=B_{1}(0,2R)\times B_{2}(0,2R^{1+\alpha})\), where \(B_{i}\subset\mathbb{R}^{N_{i}}\), with \(i=1,2\), are open balls centered at 0, the radii are 2R and \(2R^{1+\alpha}\), respectively. We consider a cut-off function \(\kappa(t)\in C_{0}^{\infty}([0,+\infty);[0,1])\) satisfying

$$\begin{aligned} \kappa(t)=\textstyle\begin{cases} 1, & 0\leq t\leq1, \\ 0, & t\geq2. \end{cases}\displaystyle \end{aligned}$$

Moreover, we define

$$\begin{aligned} \varphi_{1,R}(x) =&\kappa\biggl(\frac{ \vert x \vert }{R}\biggr),\quad x \in\mathbb{R}^{N_{1}},\qquad \varphi_{2,R}(y)=\kappa\biggl( \frac{ \vert y \vert }{R^{1+\alpha}}\biggr),\quad y\in \mathbb{R}^{N_{2}}, \end{aligned}$$


$$\begin{aligned} \varphi_{R}(x,y) =& \varphi_{1,R}(x)\varphi_{2,R}(y),\quad (x,y)\in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}. \end{aligned}$$

The direct calculations yield

$$\begin{aligned} \begin{aligned} & \vert \nabla_{x}\varphi_{1,R} \vert \leq CR^{-1}, \qquad \vert \nabla_{y} \varphi_{2,R} \vert \leq CR^{-(1+\alpha)}, \\ & \vert \Delta_{x}\varphi_{1,R} \vert \leq CR^{-2},\qquad \vert \Delta_{y}\varphi_{2,R} \vert \leq CR^{-2(1+\alpha)}, \\ & \vert \nabla_{G}\varphi_{R} \vert ^{2}+ \vert \Delta_{G}\varphi_{R} \vert \leq CR^{-2},\quad \forall x\in\mathbb{R}^{N_{1}}, y\in \mathbb{R}^{N_{2}}, \\ &R\leq \Vert z \Vert _{G}\leq CR,\quad \forall z=(x,y)\in \Omega_{2R}\setminus \Omega_{R}, \end{aligned} \end{aligned}$$

where positive constant C is independent of R.

Proof of Theorem 1.3

By contradiction, we assume that (1.1) admits a nontrivial stable weak solution u. Applying (2.2) for a test function \(\varphi_{R}(x,y)\), which is given by (2.13), we derive that, for all \(R\geq R_{0}\), there exists a constant \(C>0\) independent of R such that

$$\begin{aligned} \begin{aligned} & \int_{\Omega_{R}}\bigl(\eta(z) \vert u \vert ^{p+s}+ \xi(z) \vert \nabla _{G}u \vert ^{2} \vert u \vert ^{s-1}\bigr)\,dz \\ & \quad \leq CB^{\frac{1+s}{p-1}}(1+B)R^{-\frac{2(p+s)}{p-1}} \int_{ \Omega_{2R}\setminus\Omega_{R}} \Vert z \Vert _{G}^{ \frac{(p+s)\theta-(1+s)d}{p-1}} \,dz \\ &\quad \leq CB^{\frac{1+s}{p-1}}(1+B)R^{\mu} \end{aligned} \end{aligned}$$


$$\begin{aligned} \mu=N_{\alpha}-\frac{(2-\theta)(p+s)+(1+s)d}{p-1}. \end{aligned}$$

Here, we have utilized assumption \((H)\) and (2.14).

Evidently, if \(\mu<0\) for some certain \(s\in(1,h(p))\), it implies from (2.15) that

$$ \int_{\mathbb{R}^{N}}\bigl(\eta(z) \vert u \vert ^{p+s}+ \xi(z) \vert \nabla _{G}u \vert ^{2} \vert u \vert ^{s-1}\bigr)\,dz=0 $$

as \(R\rightarrow+\infty\), i.e., \(u\equiv0\) in \(\mathbb{R}^{N}\), which contradicts the assumption of u. Therefore, we obtain the desired conclusion.

Now, we consider the cases in which \(\mu<0\). Set

$$\begin{aligned} g(t)=\frac{(2-\theta) (t+h(t) )+ (1+h(t) )d}{t-1},\quad t>1+2k, \end{aligned}$$

where \(h(t)\) is given by (2.1). Elementary calculations lead to

$$\begin{aligned} \lim_{t\rightarrow(1+2k)^{+}}h(t)=1, \qquad \lim_{t\rightarrow+\infty}h(t)=+ \infty,\qquad h'(t)>0,\quad t>1+2k \end{aligned}$$


$$\begin{aligned}& \lim_{t\rightarrow(1+2k)^{+}}g(t)=\lambda_{0}, \qquad \lim _{t \rightarrow+\infty}g(t)=\lambda_{1}, \\& g'(t)= \frac{(2-\theta+d) (-2\sqrt{t(t-1-2k)}+t(2k-1)+1+2k )}{(1+2k)(t-1)^{2}\sqrt{t(t-1-2k)}},\quad t>1+2k, \end{aligned}$$

where \(\lambda_{0}\), \(\lambda_{1}\) are given by (1.10). By routine computations we show that if \(k=0\), then \(g(t)\) is strictly decreasing on \((1,+\infty)\). If \(0< k<\frac{3}{2}\), then \(g(t)\) is strictly increasing on \((1+2k,\frac{1+2k+2\sqrt{1+2k}}{3-2k})\) and strictly decreasing on \((\frac{1+2k+2\sqrt{1+2k}}{3-2k},+\infty)\), in this case, \(g(t)\) achieves its maximum at \(t=\frac{1+2k+2\sqrt{1+2k}}{3-2k}\) and \(g(\frac{1+2k+2\sqrt{1+2k}}{3-2k})=\lambda_{2}\), where \(\lambda_{2}\) is given by (1.10). If \(k\geq\frac{3}{2}\), then \(g(t)\) is strictly increasing on \((1+2k,+\infty)\). Moreover, when \(0\leq k<\frac{1}{2}\), \(\lambda_{1}<\lambda_{0}\), when \(k=\frac{1}{2}\), \(\lambda_{1}=\lambda_{0}\), when \(k>\frac{1}{2}\), \(\lambda_{1}>\lambda_{0}\). We also have \(g(\frac{4}{3-2k})=\lambda_{1}(k,\theta,d)\).

When \(k\geq0\) and \(N_{\alpha}\leq\min\{\lambda_{0},\lambda_{1}\}\), we have \(N_{\alpha}< g(t)\) for \(t>1+2k\). Thus, if we fix \(s\in(1,h(p))\) sufficiently near to \(h(p)\), we see that

$$\begin{aligned} \mu =&N_{\alpha}-\frac{(2-\theta)(p+s)+(1+s)d}{p-1}< 0,\quad p>1+2k. \end{aligned}$$

It is a contradiction by letting \(R\rightarrow+\infty\) in (2.15).

When \(0\leq k<\frac{1}{2}\) and \(\lambda_{1}< N_{\alpha}\leq\lambda_{0}\). By the monotonicity of \(g(t)\), there exists a unique critical value \(p_{c}>1+2k\) such that \(N_{\alpha}< g(t)\) for \(1+2k< t< p_{c}\). So if we choose \(s\in(1,h(p))\) sufficiently near to \(h(p)\), we get

$$\begin{aligned} \mu &=N_{\alpha}-\frac{(2-\theta)(p+s)+(1+s)d}{p-1}< 0,\quad 1+2k< p< p_{c} , \end{aligned}$$

which is a contradiction by taking advantage of the previous similar argument.

Assume now \(0< k\leq\frac{1}{2}\) and \(\lambda_{0}< N_{\alpha}<\lambda_{2}\). Make use of the monotonicity of \(g(t)\), there are critical exponents \(\hat{p}_{c}\) and \(p_{c}\) such that \(N_{\alpha}< g(t)\) for \(\hat{p}_{c}< t< p_{c}\). Similar to the above argument, we get a contradiction.

Combining \(g(\frac{4}{3-2k})=\lambda_{1}(k,\theta,d)\) (\(\frac{1}{2}< k< \frac{3}{2}\)) with the monotonicity of \(g(t)\), using the similar argument above, we can always deduce a contradiction provided that one of conditions \((H_{4})\), \((H_{5})\), and \((H_{6})\) is satisfied.

In the above statements, \(\hat{p}_{c}\) and \(p_{c}\) can be derived from the equation \(N_{\alpha}=h(p)\), which is given by (1.11). The proof is finished. □

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  1. Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MathSciNet  Google Scholar 

  2. Anh, C.T., Lee, J., My, B.K.: On the classification of solutions to an elliptic equation involving the Grushin operator. Complex Var. Elliptic Equ. 63, 671–688 (2018)

    Article  MathSciNet  Google Scholar 

  3. Birindelli, I., Capuzzo Dolcetta, I., Cutrì, A.: Liouville theorems for semilinear equations on the Heisenberg group. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 295–308 (1997)

    Article  MathSciNet  Google Scholar 

  4. Chen, C.-Y., Kuo, Y.-C., Wu, T.-F.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)

    Article  MathSciNet  Google Scholar 

  5. Chen, C.S.: Liouville type theorem for stable solutions of p-Laplace equation in \(\mathbb{R}^{N}\). Appl. Math. Lett. 68, 62–67 (2017)

    Article  MathSciNet  Google Scholar 

  6. Chen, C.S., Song, H.X., Yang, H.W.: Liouville-type theorems for stable solutions of singular quasilinear in \(\mathbb{R}^{N}\). Electron. J. Differ. Equ. 2018, 81 (2018)

    Article  Google Scholar 

  7. Cowan, C., Fazly, M.: On stable entire solutions of semi-linear elliptic equations with weights. Proc. Am. Math. Soc. 140, 2003–2012 (2012)

    Article  MathSciNet  Google Scholar 

  8. Damascelli, L., Farina, A., Sciunzi, B., Valdinoci, E.: Liouville results for m-Laplace equations of Lane–Emden–Fowler type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, 1099–1119 (2009)

    Article  MathSciNet  Google Scholar 

  9. D’Ambrosio, L., Lucente, S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differ. Equ. 193, 511–541 (2003)

    Article  MathSciNet  Google Scholar 

  10. Dancer, E.N., Du, Y.H., Guo, Z.M.: Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differ. Equ. 250, 3281–3310 (2011)

    Article  MathSciNet  Google Scholar 

  11. D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)

    Article  MathSciNet  Google Scholar 

  12. Duong, A.T., Nguyen, N.T.: Liouville type theorems for elliptic equations involving Grushin operator and advection. Electron. J. Differ. Equ. 2017, 108 (2017)

    Article  MathSciNet  Google Scholar 

  13. Dupaigne, L.: Stable Solutions of Elliptic Partial Differential Equations. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 143. Chapman & Hall, Boca Raton (2011)

    Book  Google Scholar 

  14. Fan, H.N.: Positive solutions for a Kirchhoff-type problem involving multiple competitive potentials and critical Sobolev exponent. Nonlinear Anal. 198, 111869 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. Farina, A.: On the classification of solutions of the Lane–Emden equation on unbounded domain of \(\mathbb{R}^{N}\). J. Math. Pures Appl. 87, 537–561 (2007)

    Article  MathSciNet  Google Scholar 

  16. Franchi, B., Gutiérrez, C.E., Wheeden, R.L.: Weighted Sobolev–Poincaré inequalities for Grushin type operators. Commun. Partial Differ. Equ. 19, 523–604 (1994)

    Article  Google Scholar 

  17. Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)

    Article  MathSciNet  Google Scholar 

  18. Han, Y.Z., Li, Q.W.: Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput. Math. Appl. 75, 3283–3297 (2018)

    Article  MathSciNet  Google Scholar 

  19. Huynh, N.V., Le, P.: Instability of solutions to Kirchhoff type problems in low dimension. Ann. Pol. Math. 124, 75–91 (2020)

    Article  MathSciNet  Google Scholar 

  20. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  21. Le, P.: Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Commun. Pure Appl. Anal. 19, 511–525 (2020)

    Article  MathSciNet  Google Scholar 

  22. Le, P., Ho, V.: Stable solutions to weighted quasilinear problems of Lane–Emden type. Electron. J. Differ. Equ. 2018, 71 (2018)

    Article  MathSciNet  Google Scholar 

  23. Le, P., Ho, V.: Liouville results for stable solutions of quasilinear equations with weights. Acta Math. Sci. Ser. B Engl. Ed. 39, 357–368 (2019)

    Article  MathSciNet  Google Scholar 

  24. Le, P., Huynh, N.V., Ho, V.: Classification results for Kirchhoff equations in \(\mathbb{R}^{N}\). Complex Var. Elliptic Equ. 64, 1146–1157 (2019)

    Article  MathSciNet  Google Scholar 

  25. Li, Q.Q., Teng, K.M., Wang, W.B., Zhang, J.: Concentration phenomenon of solutions for a class of Kirchhoff-type equations with critical growth. J. Math. Anal. Appl. 491, 124355 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  26. Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) NorthHolland Math. Stud., vol. 30, pp. 284–346. North-Holland, Amsterdam (1978)

    Google Scholar 

  27. Liu, J., Liao, J.F., Tang, C.L.: Positive solutions for Kirchhoff-type equations with critical exponent in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 429, 1153–1172 (2015)

    Article  MathSciNet  Google Scholar 

  28. Liu, W.J., Li, G., Hong, L.H.: General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping. J. Funct. Spaces 2014, 284809 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, W.J., Zhu, B.Q., Li, G., Wang, D.H.: General decay for a viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping, dynamic boundary conditions and a time-varying delay term. Evol. Equ. Control Theory 6, 239–260 (2017)

    Article  MathSciNet  Google Scholar 

  30. Monti, R., Morbidelli, D.: Kelvin transform for Grushin operators and critical semilinear equations. Duke Math. J. 131, 167–202 (2006)

    Article  MathSciNet  Google Scholar 

  31. Monticelli, D.D.: Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J. Eur. Math. Soc. 12, 611–654 (2010)

    Article  MathSciNet  Google Scholar 

  32. Rahal, B.: Liouville-type theorems with finite Morse index for semilinear \(\Delta_{\lambda}\)-Laplace operators. NoDEA Nonlinear Differ. Equ. Appl. 25, 21 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  33. Rahal, B.: On stale entire solutions of sub-elliptic system involving advection terms with negative exponents and weights. J. Inequal. Appl. 2020, 119 (2020).

    Article  MathSciNet  Google Scholar 

  34. Wang, C., Ye, D.: Some Liouville theorems for Hénon type elliptic equations. J. Funct. Anal. 262, 1705–1727 (2012)

    Article  MathSciNet  Google Scholar 

  35. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 12, 1278–1287 (2011)

    Article  MathSciNet  Google Scholar 

  36. Xie, Q.L.: Bounded state solution of degenerate Kirchhoff type problem with a critical exponent. J. Math. Anal. Appl. 479, 1–24 (2019)

    Article  MathSciNet  Google Scholar 

  37. Ye, Y.J.: Blow-up of solutions for a system of higher-order nonlinear Kirchhoff-type equations. Bull. Malays. Math. Sci. Soc. 40, 665–677 (2017)

    Article  MathSciNet  Google Scholar 

  38. Yu, X.H.: Liouville type theorem for nonlinear elliptic equation involving Grushin operators. Commun. Contemp. Math. 17, 1450050 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, C.H., Liu, Z.S.: Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem. Appl. Math. Lett. 69, 87–93 (2017)

    Article  MathSciNet  Google Scholar 

  40. Zhang, F.B., Du, M.: Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well. J. Differ. Equ. 269, 10085–10106 (2020)

    Article  MathSciNet  Google Scholar 

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The authors would like to express their sincere gratitude to the anonymous reviewer for their valuable comments and suggestions which improved the presentation of the paper.


This work was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No. 19KJD100002), the Natural Science Foundation of Shandong Province (Grant No. ZR2018MA017), and the China Postdoctoral Science Foundation (Grant No. 2017M610436).

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Each of the authors contributed to each part of this study equally. All authors read and approved the final manuscript.

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Correspondence to Yunfeng Wei.

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Wei, Y., Yang, H. & Yu, H. Stable weak solutions to weighted Kirchhoff equations of Lane–Emden type. Adv Differ Equ 2021, 27 (2021).

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