Liouville-type theorem for Kirchhoff equations involving Grushin operators

<jats:p>The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations:
<jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} \begin{aligned}[b] & -M \biggl( \int _{{\mathbb{R}} ^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)e^{u}, \\ &\quad z=(x,y) \in R^{N}=R^{N_{1}}\times R^{N_{2}} \end{aligned} \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:disp-formula> and
<jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} \begin{aligned}[b] & M \biggl( \int _{\mathbb{R}^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)u^{-q}, \\ &\quad z=(x,y) \in {\mathbb{R}} ^{N}={\mathbb{R}} ^{N_{1}}\times {\mathbb{R}} ^{N_{2}}, \end{aligned} \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                  </mml:mtable>
                </mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$M(t)=a+bt^{k}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>M</mml:mi>
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                    <mml:mi>k</mml:mi>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$t\geq 0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>t</mml:mi>
                  <mml:mo>≥</mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>, with <jats:inline-formula><jats:alternatives><jats:tex-math>$a>0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>a</mml:mi>
                  <mml:mo>></mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$b, k\geq 0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>b</mml:mi>
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                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$k=0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>k</mml:mi>
                  <mml:mo>=</mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula> if and only if <jats:inline-formula><jats:alternatives><jats:tex-math>$b=0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>b</mml:mi>
                  <mml:mo>=</mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>. <jats:inline-formula><jats:alternatives><jats:tex-math>$q>0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>q</mml:mi>
                  <mml:mo>></mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\omega (z), f(z)\in L^{1}_{\mathrm{loc}}({\mathbb{R}} ^{N})$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                    <mml:mi>R</mml:mi>
                    <mml:mi>N</mml:mi>
                  </mml:msup>
                  <mml:mo>)</mml:mo>
                </mml:math></jats:alternatives></jats:inline-formula> are nonnegative functions satisfying <jats:inline-formula><jats:alternatives><jats:tex-math>$\omega (z)\leq C_{1}\|z \|_{G}^{\theta }$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>ω</mml:mi>
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                  </mml:msub>
                  <mml:msubsup>
                    <mml:mrow>
                      <mml:mo>∥</mml:mo>
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                      <mml:mo>∥</mml:mo>
                    </mml:mrow>
                    <mml:mi>G</mml:mi>
                    <mml:mi>θ</mml:mi>
                  </mml:msubsup>
                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$f(z)\geq C_{2}\|z\|_{G}^{d}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>f</mml:mi>
                  <mml:mo>(</mml:mo>
                  <mml:mi>z</mml:mi>
                  <mml:mo>)</mml:mo>
                  <mml:mo>≥</mml:mo>
                  <mml:msub>
                    <mml:mi>C</mml:mi>
                    <mml:mn>2</mml:mn>
                  </mml:msub>
                  <mml:msubsup>
                    <mml:mrow>
                      <mml:mo>∥</mml:mo>
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                      <mml:mo>∥</mml:mo>
                    </mml:mrow>
                    <mml:mi>G</mml:mi>
                    <mml:mi>d</mml:mi>
                  </mml:msubsup>
                </mml:math></jats:alternatives></jats:inline-formula> as <jats:inline-formula><jats:alternatives><jats:tex-math>$\|z\|_{G} \geq R_{0}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mrow>
                      <mml:mo>∥</mml:mo>
                      <mml:mi>z</mml:mi>
                      <mml:mo>∥</mml:mo>
                    </mml:mrow>
                    <mml:mi>G</mml:mi>
                  </mml:msub>
                  <mml:mo>≥</mml:mo>
                  <mml:msub>
                    <mml:mi>R</mml:mi>
                    <mml:mn>0</mml:mn>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$d>\theta -2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>d</mml:mi>
                  <mml:mo>></mml:mo>
                  <mml:mi>θ</mml:mi>
                  <mml:mo>−</mml:mo>
                  <mml:mn>2</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$R_{0}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>R</mml:mi>
                    <mml:mn>0</mml:mn>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$C_{i}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>C</mml:mi>
                    <mml:mi>i</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:tex-math>$i=1,2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>i</mml:mi>
                  <mml:mo>=</mml:mo>
                  <mml:mn>1</mml:mn>
                  <mml:mo>,</mml:mo>
                  <mml:mn>2</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula>) are some positive constants, here <jats:inline-formula><jats:alternatives><jats:tex-math>$\alpha \geq 0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>α</mml:mi>
                  <mml:mo>≥</mml:mo>
                  <mml:mn>0</mml:mn>
                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\|z\|_{G}=(|x|^{2(1+ \alpha )}+|y|^{2})^{\frac{1}{2(1+\alpha )}}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mrow>
                      <mml:mo>∥</mml:mo>
                      <mml:mi>z</mml:mi>
                      <mml:mo>∥</mml:mo>
                    </mml:mrow>
                    <mml:mi>G</mml:mi>
                  </mml:msub>
                  <mml:mo>=</mml:mo>
                  <mml:msup>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:msup>
                        <mml:mrow>
                          <mml:mo>|</mml:mo>
                          <mml:mi>x</mml:mi>
                          <mml:mo>|</mml:mo>
                        </mml:mrow>
                        <mml:mrow>
                          <mml:mn>2</mml:mn>
                          <mml:mo>(</mml:mo>
                          <mml:mn>1</mml:mn>
                          <mml:mo>+</mml:mo>
                          <mml:mi>α</mml:mi>
                          <mml:mo>)</mml:mo>
                        </mml:mrow>
                      </mml:msup>
                      <mml:mo>+</mml:mo>
                      <mml:msup>
                        <mml:mrow>
                          <mml:mo>|</mml:mo>
                          <mml:mi>y</mml:mi>
                          <mml:mo>|</mml:mo>
                        </mml:mrow>
                        <mml:mn>2</mml:mn>
                      </mml:msup>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                    <mml:mfrac>
                      <mml:mn>1</mml:mn>
                      <mml:mrow>
                        <mml:mn>2</mml:mn>
                        <mml:mo>(</mml:mo>
                        <mml:mn>1</mml:mn>
                        <mml:mo>+</mml:mo>
                        <mml:mi>α</mml:mi>
                        <mml:mo>)</mml:mo>
                      </mml:mrow>
                    </mml:mfrac>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula> is the norm corresponding to the Grushin distance. <jats:inline-formula><jats:alternatives><jats:tex-math>$N_{\alpha }=N_{1}+(1+\alpha )N_{2}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>N</mml:mi>
                    <mml:mi>α</mml:mi>
                  </mml:msub>
                  <mml:mo>=</mml:mo>
                  <mml:msub>
                    <mml:mi>N</mml:mi>
                    <mml:mn>1</mml:mn>
                  </mml:msub>
                  <mml:mo>+</mml:mo>
                  <mml:mo>(</mml:mo>
                  <mml:mn>1</mml:mn>
                  <mml:mo>+</mml:mo>
                  <mml:mi>α</mml:mi>
                  <mml:mo>)</mml:mo>
                  <mml:msub>
                    <mml:mi>N</mml:mi>
                    <mml:mn>2</mml:mn>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> is the homogeneous dimension of <jats:inline-formula><jats:alternatives><jats:tex-math>${\mathbb{R}} ^{N}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mi>R</mml:mi>
                    <mml:mi>N</mml:mi>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula>. <jats:inline-formula><jats:alternatives><jats:tex-math>$\operatorname{div}_{G}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mo>div</mml:mo>
                    <mml:mi>G</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula> (resp., <jats:inline-formula><jats:alternatives><jats:tex-math>$\nabla _{G}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>∇</mml:mi>
                    <mml:mi>G</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula>) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on <jats:italic>k</jats:italic>, <jats:italic>θ</jats:italic>, <jats:italic>d</jats:italic>, and <jats:inline-formula><jats:alternatives><jats:tex-math>$N_{\alpha }$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msub>
                    <mml:mi>N</mml:mi>
                    <mml:mi>α</mml:mi>
                  </mml:msub>
                </mml:math></jats:alternatives></jats:inline-formula>, the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.</jats:p>


Introduction and main result
In this paper, we study the nonexistence of stable weak solutions for the weighted Kirchhoff equations where M(t) = a + bt k , t ≥ 0, with a > 0, b, k ≥ 0, k = 0 if and only if b = 0. q > 0 and ω(z), f (z) ∈ L 1 loc (R N ) are nonnegative functions verifying ω(z) ≤ C 1 z θ G and f (z) ≥ C 2 z d G as z G ≥ R 0 with d > θ -2, where R 0 , C i (i = 1, 2) are some positive constants. Here α ≥ 0 and z G = |x| 2(1+α) + |y| 2 1 2(1+α) , z = (x, y) ∈ R N = R N 1 × R N 2 is the norm corresponding to the Grushin distance, where |x| and |y| are the usual Euclidean norms in R N 1 and R N 2 , respectively.
Set ∇ x and ∇ y as Euclidean gradients with respect to the variables x ∈ R N 1 and y ∈ R N 2 , respectively. The Grushin gradient is defined by ∂h j ∂y j = div x g + (1 + α)|x| α div y h, (g, h) ∈ C 1 R N , R N 1 × R N 2 as the Grushin divergence. Then the Grushin operator is given by where x and y represent the usual Laplacians on R N 1 and R N 2 respectively. This operator is uniformly elliptic for x = 0 and degenerate when x = (x 1 , x 2 , . . . , x N 1 ) goes to 0. The anisotropic dilation attached to G is defined by It is not hard to see that where N α = N 1 + (1 + α)N 2 is usually called the homogeneous dimension of R N , dx dy denotes the Lebesgue measure in R N . For more details about Grushin operators and their basic properties, we refer the reader for instance to [34].
In the case α = 0 and ω(z) ≡ 1, problems (1.1) and (1.2) are related to the stationary analogue of the following Kirchhoff model: which was proposed by Kirchhoff in 1883 as a generalization of the well-known D' Alembert wave equation for free vibrations of elastic strings, see [26], where ρ, p 0 , h, E, L are constants which represent some physical meanings respectively. Indeed, Kirchhoff 's model considers the changes in length of the string produced by transverse vibrations. Up to now, a great attention has been paid to the study of the Kirchhoff-type problems involving nonlocal operators, because nonlinear equations with nonlocal operators have a broad application background and play an important role in physics, probability, biology, finance, etc. With the help of variational calculus, some important and interesting results for this direction, especially those concerning the existence and multiplicity of solutions, have been established, we refer the interested reader to [17-19, 24, 31, 37] and the references therein.
On the other hand, the nonexistence and stability of solutions to nonlinear elliptic equations have drawn much attention in the last decades. For some physical motivation and recent development on the topic of stable solutions, we refer to [13]. Also, see [2,33] for related problems.
The motivation of writing this article is to prove a Liouville-type theorem for stable solutions of equations (1.1) and (1.2). We recall that Liouville-type theorem focuses on the nonexistence of nontrivial solution in the entire space R N . In 1981, in their pioneering article [22], Gidas and Spruck established the optimal Liouville-type result for positive solutions to the equation They proved that (1.3) has no positive solution if and only if 1 < q < q s = N+2 [14,16] also considered problem (1.3). He proved that there is no nontrivial stable solution if 1 < q < q c (N), where q c (N) is explicitly given and is always greater than the classical critical exponent N+2 N-2 . It is worth pointing out that his proof makes a delicate application of the classical Moser iterative method. Later, these results were extended to the quasilinear casep u = |u| q-1 u in [7] and the weighted quasilinear case p u = f (z)|u| q-1 u in [3].
Obviously, equation (1.1) becomes the following Laplace equation with exponential nonlinearity: for the case M(t) ≡ 1, α = 0, and ω(z) ≡ 1 ≡ f (z). Problem (1.4) has been studied by several experts; for example, Farina [15] proved that all stable C 2 solutions of (1.4) must be zero if 2 ≤ N ≤ 9; Dancer and Farina [9] proved that equation ( For the case of negative exponent nonlinearity, the authors [32] obtained the following. then there are no positive stable solutions to (1.2) in R N . Very recently, by Farina's approach, Cowan and Fazly [6] established the nonexistence of nontrivial stable solution of the weighted elliptic equation with positive smooth weights ω i (z), i = 1, 2, where the nonlinearity g(u) = e u , |u| p-1 u with p > 1 and -u -p with p > 0. After that, these results were extended to the quasilinear casep u = f (z)g(u) in [4,27] and the weighted quasilinear case -div(ω(z)|∇u| p-2 ∇u) = f (z)g(u) in [28,29], where g(u) = e u or g(u) = -u -q , q > 0. Similar works can be found in [5,21,23,25,41]. We now turn to the case where α > 0, equations (1.1) and (1.2) become nonlinear elliptic equations involving Grushin operator. It is well known that the Grushin operator belongs to the wide class of subelliptic operators studied by Franchi et al. [20] (also see [1]). The Liouville-type theorem has been recently proved by Monticelli [35] for nonnegative classical solutions and by Yu [40] for nonnegative weak solutions of the problem The optimal exponent is τ < N α +2 N α -2 , where N α = N 1 + (1 + α)N 2 is the homogeneous dimension. The main tool they used [35,40] is the Kelvin transform combined with the moving planes technique. On the other hand, Monti and Morbidelli [34] considered the classification results for equation the main tool they used is the moving spheres technique, which is a variant of the moving plane technique and was widely used in elliptic equations such as [30]. For other results of Liouville-type theorem related to Grushin operators, we refer the reader to [8,11,12,36] and the references therein.
However, as far as we know, there are few results on the Liouville-type theorem for problem (1.1) or (1.2) with α = 0 and M(t), ω(z), f (z) ≡ 1. Motivated by the above works, in the present paper, we try to establish the Liouville property for the class of stable weak solutions of (1.1) and (1.2).
Since solutions to elliptic equations with Hardy potentials may possess singularities, it is natural to study weak solutions of (1.1) and (1.2) in a suitable weighted Sobolev space. Based on this reality, we define Here and in the following C k c (R N ) denotes the set of C k functions with compact support in R N . To facilitate the writing, we unify equations (1.1) and (1.2) into the following equation: where g(u) = e u or g(u) = -u -q .

Definition 1.4 Let
The energy functional J : X → R corresponding to (1.7) is Obviously, if u ∈ X is a weak solution of (1.7), then for any ϕ ∈ C 1 c (R N ), the function E(t) := J (u + tϕ) satisfies E (0) = 0. As in [7], we say that the solution u of (1.7) is stable if we obtain We are ready to state the stability as follows.
Definition 1. 5 We say that a weak solution u of (1 Remark 1.6 If u is a stable weak solution of (1.1), in view of (1.9) with g(u) = e u , it can be deduced that Similarly, if u is a positive stable weak solution of (1.2), by virtue of (1.9) with g(u) = -u -q , it follows that where A is given in (1.11). Note that (1.8)-(1.10) and (1.12) hold for all ϕ ∈ H 1 0 (R N , ω) by density arguments.
Throughout this paper, the functions ω(z), f (z) satisfy the following assumptions: To simplify the notations, we denote Our results can be stated as follows.
Then there is no stable weak solution u ∈ X to (1.1).
This paper is organized as follows. In Sect. 2, we give the proof of Theorem 1.7. The proof of Theorem 1.9 is finally finished in Sect. 3. In the sequel, we denote by C some constant, which may vary from line to line. If this constant depends on an arbitrary small number ε, then we denote it by C ε .

Proof of Theorem 1.7
We first give the following proposition which plays a crucial role in the proof of Theorem 1.7. that u is a stable weak solution of (1.1). Then, for any s ∈ (0, 4 1+2k ), there exists a constant C = C(k, s) > 0 such that

Proposition 2.1 Assume
Proof We will use some of the ideas in [23,28] to complete the proof. For each i ∈ N, we define It is not difficult to verify that β i , γ i are increasing positive C 1 (R) functions and for all t ∈ R, where C > 0 depends only on s. Owing to u ∈ H 1 loc (R N , ω), we conclude β i (u), γ i (u) ∈ H 1 loc (R N , ω) for any i ∈ N. Let ε ∈ (0, 1) and ψ ∈ C 1 c (R N ) be a nonnegative function. Set ϕ = γ i (u)ψ 2 as a test function in (1.8) with g(u) = e u , we have On the other hand, according to the stability assumption, we take ϕ = β i (u)ψ in (1.10) and get We use Young's inequality to estimate the middle term of the right-hand side of the above inequality: Substituting this estimate into (2.4), there holds Together with (2.2), (2.3), (2.5), we obtain
Letting i → ∞ in (2.6), by Fatou's lemma, we have for some constant C > 0 depending only on k and s. On the other hand, replacing ψ by ψ 1+s in (2.7) and using Hölder's inequality, we derive , are open balls centered at 0, the radii are 2R and 2R 1+α , respectively. Let By a series of calculations, one can verify that there exists a constant C > 0 independent of R such that (2.9) Proof of Theorem 1.7 Arguing by contradiction, we assume that u is a stable weak solution of (1.1). Let ψ = ψ R (x, y) = ψ 1,R (x)ψ 2,R (y) with R ≥ R 0 in (2.1), then there exists a positive constant C independent of R such that where assumption (H 1 ) and (2.9) have been used. Since N α < μ 0 (k, θ , d) and we may choose some s ∈ (0, 4 1+2k ) suitably near 4 1+2k such that N α + θ (1 + s)-ds -2(1+s) < 0. Letting R → +∞ in (2.10), we have R N f (z)e (1+s)u dz = 0.
A contradiction! The proof is completed.

Proof of Theorem 1.9
In this section, we give the proof of Theorem 1.9, which mainly relies on the following a priori estimate.

1)
and for any constant τ ≥ q-s q+1 , there exists a constant C > 0 depending on q, s, τ , a, and k such that holds for all functions ψ ∈ C 1 c (R N ) verifying 0 ≤ ψ ≤ 1 and ∇ G ψ = 0 in a neighborhood of {z ∈ R N | f (z) = 0}.