Liouville type theorem for weak solutions of nonlinear system for Grushin operator

: In this paper, we prove Liouville type theorem of positive weak solution of nonlinear system for Grushin operator. We give some integral inequalities, which combine the method of moving plane with the integral inequality to get the result for nonlinear system


Introduction
In this paper, we study the Liouville type theorem of positive weak solution of the nonlinear system for Grushin operator where we denote the Grushin operators L α u = ∆ x u + (α + 1) 2 |x| 2α ∆ y u, L β u = ∆ x u + (β + 1) 2 |x| 2β ∆ y u, α, β > 0, and the right hand terms f and g are some continuous functions. In the case of (1.1), α, β = 0, it should be the Laplacian problem here, n = m + k. De Figueiredo and Felmer in [7] conjectured that the hyperbola curve is the dividing curve between existence and nonexistence of the solution to the Laplacian problem. The conjecture is right for the radial solutions of the above problem, see Serrin and Zou [18,19]. The authors in [7] proved that the above Laplacian problem has no positive solutions, when 0 < p, q ≤ n+2 n−2 and (p, q) ( n+2 n−2 , n+2 n−2 ). Guo and Liu [14] gave the Liouville type theorems for positive solutions of the following second order elliptic problem: when n ≥ 3. The key tool in [14] is the method of moving planes which combined with integral inequalities.
For the Grushin operator, Yu [29] gave the nonexistence of positive solutions for the degenerate equation where L α u = ∆ x u + (α + 1) 2 |x| 2α ∆ y u, α > 0, and f satisfies some assumptions. We investigate the nonexistence result of the elliptic system involving of the Grushin operator and with general nonlinear terms. We call (u, v) is a weak solution of system ( . The main result in this paper is the following Liouville type theorem of nonnegative weak solution for (1.1). Then, (u, v) = (c 1 , c 2 ) for some constants c 1 , c 2 satisfying f (c 2 ) = g(c 1 ) = 0.
In this paper, a key tool is some integral inequality, which was used in Terracini [21,22]. And then there were some related results in the references [14,23], etc. Quaas and Xia in [17] gave the Liouville theorem for fractional Lane-Emden system, which used the monotonicity argument for some suitable transformed functions with the method of moving planes, where some maximum principles which obtained by suitable barrier functions and a coupling argument for fractional Sobolev trace inequality in an infinity half cylinder be used. For this paper, since the extended Grushin operator is degenerated, and not conclude that (u, v) belongs to C 2 class. To handle the weak solution and the degeneracy, we also borrow the idea that the integral inequality can play the role to the maximum principles when moving planes.
This paper is organized as follows. In Section 2, we collect some well known results, and give the proof of key integral inequalities. In Section 3, Theorem 1.1 will be proved with some conditions of f and g.

Preliminaries
We list some basic information about the Grushin operator. For the details can refer to [16]. Let a and b be two vectors in R d for some d ∈ N + , define the inner product of them as ⟨a, b⟩ = d j=1 a j b j , and |a| = ⟨a, a⟩ 1 2 is the Euclidean norm. Let α > 0 be a fixed constant (here we only list the case α, it also holds for β), for z = (x, y) ∈ R m × R k , define the Grushin norm as It is easy to check that this Grushin norm is 1-homogeneous, that for the Lie group dilations δ λ = (λx, λ α+1 y), λ > 0. Define the Grushin distance of two points z, We define the open ball which radius R and centered at z 0 as here Q = m + (α + 1)k, and | · | denotes the Lebesgue measure. And the number Q is the homogeneous dimension of Grushin space R m × R k . In order to define the Grushin gradient and operator in R m × R k , we use Euclidean gradients ∇ x and ∇ y with respect to x ∈ R m and y ∈ R k . And we define as the Grushin gradient, and set as the Grushin divergence. Then we also have L α u = div α D α u. We list a Sobolev inequality in R m × R k (see [9,10]).
then the Sobolev inequality holds: For the function u ∈ D 1 (R m × R k ) and p = (x 0 , y 0 ) ∈ R m × R k , let U be the Kelvin transformation of u with respect to point p, define as In this paper, let (u, v) be nonnegative in the Grushin manifolds R m × R k , we follow the Kelvin transform in [16,29] which centered at p = 0. Then we obtain that U, V is continuous in R m × R k \ {0} of the Kelvin transformation of u, v. Obviously, U, V are continuous and nonnegative in R m × R k \ {0}. And a direct computation shows that: Lemma 2.2. Let (u, v) be a nonnegative weak solution of system (1.1). Then (U, V) weakly satisfies the following system: Moreover, (U, V) has decay at infinity as x m , 2λ − y 1 , y 2 , · · · , y k ), and denote U λ (z) = U(z λ ) and V λ (z) = V(z λ ). Then we can infer from problem (2.1) that U λ and V λ satisfy , we prove the following integral inequalities at firstly.

Proof of Theorem 1.1
We start moving planes from some place.
For any point can be chosen as the center of Kelvin transform, then U and V must be independent of the variable y 1 . Similarly, we can process the above procedure of the directions y 2 , · · · , y n . Then we obtain that U, V are independent of y. However, this claim implies that U and V satisfy the system in R m , By [7,14], we have that (u, v) = (c 1 , c 2 ) for some constants c 1 , c 2 with f (c 2 ) = g(c 1 ) = 0. This completes the proof of Theorem 1.1.

Conclusions
In this paper, the Liouville type theorem of positive weak solution of nonlinear system for Grushin operator be given by the method of moving plane. In stead of the maximum principles, we introduce some integral inequality, which be used in the processes of moving planes.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.