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

1.   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 $L_{\alpha}u = \Delta_{x}u+(\alpha+1)^2|x|^{2\alpha}\Delta_{y}u$, $L_{\beta}u = \Delta_{x}u+(\beta+1)^2|x|^{2\beta}\Delta_{y}u$, $\alpha, \; \beta > 0$, and the right hand terms $f$ and $g$ are some continuous functions.

In the case of (1.1), $\alpha, \; \beta = 0$, 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 $0 < p, q\leq \frac{n+2}{n-2}$ and $(p, q)\ne ({\frac{n+2}{n-2}, \frac{n+2}{n-2}}).$ Guo and Liu ^[14] gave the Liouville type theorems for positive solutions of the following second order elliptic problem:

when $n\ge 3$. 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 $L_{\alpha}u = \Delta_{x}u+(\alpha+1)^2|x|^{2\alpha}\Delta_{y}u$, $\alpha > 0$, 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 (1.1), if $(u, \; v)$ satisfies

for any $\varphi_1, \varphi_2 \in C^{1}_{0}(\overline{{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}})$.

The main result in this paper is the following Liouville type theorem of nonnegative weak solution for (1.1).

Theorem 1.1. Let $(u, v)\in C({{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}})\times C({{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}})$ be a nonnegative weak solution of problem (1.1). Suppose that $f, g:[0, +\infty)\to {\mathbb{R}}$ are continuous functions satisfying

(i) $f(t)$ and $g(t)$ are nondecreasing in $(0, +\infty)$,

(ii) $h(t) = \frac{f(t)}{t^{\frac{Q+2}{P-2}}}$ and $k(t) = \frac{g(t)}{t^{\frac{P+2}{Q-2}}}$ are nonincreasing in $(0, +\infty)$, here $Q = m+(\alpha+1)k > P = m+(\beta+1)k$. 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.

For the more information of the method of moving planes of elliptic equations, see ^{[2,4,5,6,8,12,13]}, ect. There are also some results of the fractional operator and its related extension problems on Heisenberg group (see ^{[3,24,25,26,27]}, ect.) and Grushin manifold see ^{[1,11,15,20,28,29]} and the references therein.

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 $g$.

2.   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 $\mathbb{R}^d$ for some $d \in \mathbb{N}^+$, define the inner product of them as $\langle a, b\rangle = \sum_{j = 1}^{d} a_j b_j$, and $|a| = \langle a, a\rangle^{\frac{1}{2}}$ is the Euclidean norm. Let $\alpha > 0$ be a fixed constant (here we only list the case $\alpha$, it also holds for $\beta$), for $z = (x, y) \in{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$, define the Grushin norm as

It is easy to check that this Grushin norm is 1-homogeneous, that for the Lie group dilations $\delta_\lambda = (\lambda x, \lambda ^{\alpha+1}y), \lambda > 0$. Define the Grushin distance of two points $z, z_0$ in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$ by

We define the open ball which radius R and centered at $z_0$ as

A direct calculation gives that

here $Q = m+(\alpha +1)k$, and $|\cdot|$ denotes the Lebesgue measure. And the number $Q$ is the homogeneous dimension of Grushin space ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$. In order to define the Grushin gradient and operator in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$, we use Euclidean gradients $\nabla_x$ and $\nabla_y$ with respect to $x \in {\mathbb{R}}^{m}$ and $y \in {\mathbb{R}}^{k}$. And we define

as the Grushin gradient, and set

as the Grushin divergence. Then we also have $L_{\alpha}u = \mathrm{div}_{\alpha}D_{\alpha}u$. We list a Sobolev inequality in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$ (see ^[9,10]).

Lemma 2.1. Let $D^1({\mathbb{R}}^{m}\times{\mathbb{R}}^{k})$ be the completion of $C_0^{\infty}({\mathbb{R}}^{m}\times{\mathbb{R}}^{k})$ under the seminorm of

then the Sobolev inequality holds:

For the function $u\in D^1({\mathbb{R}}^{m}\times{\mathbb{R}}^{k})$ and $p = (x_0, y_0)\in {\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$, 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 ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$, we follow the Kelvin transform in ^[16,29] which centered at $p = 0$. Then we obtain that $U, V$ is continuous in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}\setminus \{0\}$ of the Kelvin transformation of $u, v$. Obviously, $U, V$ are continuous and nonnegative in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}\setminus \{0\}$. 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

Let $\lambda\in \mathbb{R}$ and $z = (x, y)$. We define $T_{\lambda} = \{(x, y)\in {{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}}|\; y_1 = \lambda\},$ $\Sigma_{\lambda} = \{(x, y)\in {{\mathbb{R}}^{m}\times{\mathbb{R}}^{k}}|\; y_1 > \lambda\},$ $p_{\lambda} = (0, \cdots, 0, y_1 = 2\lambda, 0, \cdots, 0),$ $z_{\lambda} = (x_1, x_2, \cdots, x_m, 2\lambda-y_1, y_2, \cdots, y_k)$, and denote $U_{\lambda}(z) = U(z_{\lambda})$ and $V_{\lambda}(z) = V(z_{\lambda}).$ Then we can infer from problem (2.1) that $U_\lambda$ and $V_\lambda$ satisfy

Let $U^{\lambda}(z) = U(z)-U(z_{\lambda}), \; V^{\lambda}(z) = V(X)-V(z_{\lambda})$, we prove the following integral inequalities at firstly.

Lemma 2.3. For any fixed $\lambda > 0$, $U^{\lambda} \in L^{\frac{2 Q}{Q-2}}(\Sigma_\lambda)\bigcap L^{\infty}(\Sigma_\lambda), V^{\lambda} \in L^{\frac{2 P}{P -2}}(\Sigma_\lambda)\bigcap L^{\infty}(\Sigma_\lambda)$, such that

where $D_{\alpha} = (\nabla_x, (\alpha+1)|x|^{\alpha}\nabla_y)$, $D_{\beta} = (\nabla_x, (\beta+1)|x|^{\beta}\nabla_y)$, $A^{1}_{\lambda} = \{ z\in {\Sigma_\lambda}|\; {V^\lambda}\ge 0\}$, $A^{2}_{\lambda} = \{ z\in {\Sigma_\lambda}|\; {U^\lambda}\ge 0\}$, $U^{\lambda +} = \max\{U^{\lambda}, 0 \}$, $V^{\lambda +} = \max\{V^{\lambda}, 0 \}$, $C_\lambda > 0$ is a constant which is nonincreasing in $\lambda$.

Proof. We just prove the inequality (2.4), the proof of (2.5) is omitted. For any fixed $\lambda > 0$, we know that $U$ and $U^{\lambda +}\leq U \in L^{\frac{2\widetilde{Q}}{\widetilde{Q}-2}}(\Sigma_\lambda)\bigcap L^{\infty}(\Sigma_\lambda)$.

For $\varepsilon > 0$ small, choose a smooth cut-off function $\eta_{\varepsilon}\in C^{\infty}_{0}({\mathbb{R}}^{m}\times{\mathbb{R}}^{k})$ such that $0\leq \eta_{\varepsilon}\leq 1$, $\eta_{\varepsilon}(z) = 1$ for $2\varepsilon \leq{\|z-p_{\lambda} \|}\leq \varepsilon^{-1}$, $\eta_{\varepsilon} = 0$ for ${\|z-p_{\lambda} \|}\leq \varepsilon$ or ${\|z-p_{\lambda} \|}\ge 2\varepsilon^{-1}$, with that ${\left| D_{\alpha} \eta_{\varepsilon} \right|}\leq C\varepsilon^{-1}$ for $\varepsilon \leq{\| z-p_{\lambda} \|}\leq 2\varepsilon$ and ${\left| D_{\alpha}\eta_{\varepsilon} \right|}\leq C\varepsilon$ for $\varepsilon^{-1} \leq{\| z-p_{\lambda} \|}\leq 2\varepsilon^{-1}$, where the positive constant $C$ is independent of $\varepsilon$. By the assumption $h(t) = \frac{f(t)}{t^{\frac{Q+2}{P-2}}}$, for $z\in {\Sigma_\lambda}$, we notice that $U$ and $U_\lambda$ satisfy

and

Multiply the above equations by the test function $\psi = U^{\lambda +}\eta_{\varepsilon}$ and denote $\phi_{\varepsilon} = U^{\lambda +}\eta_{\varepsilon}^{2}$, we deduce that

We estimate $I_{\varepsilon}$ at first.Write $B_\varepsilon^+ = \{z\in {\Sigma_\lambda}|\varepsilon \leq{\| z-p_{\lambda} \|}\leq 2\varepsilon$ or ${\varepsilon}^{-1} \leq{\| z-p_{\lambda} \|}\leq 2{\varepsilon}^{-1} \}$, then we have

Hence, following from the Hölder inequality, we obtain

as $\varepsilon\rightarrow0$.

We estimate $I$ at first. Since ${\| z \|} > {\| z_\lambda \|}$, and $h$ is nonincreasing. If $v(z)\ge v(z_\lambda)\ge 0$, then $-h({\| z \|^{Q-2}}V)\ge -h({\| z_{\lambda} \|^{Q-2}}V_{\lambda})$. By (2.6) and (2.7), we have

By Lemma 2.1, Sobolev trace inequality on extended Grushin manifolds, letting $\varepsilon\to 0$ in (2.8), we get

Hence, (2.4) is holds. □

3.   Proof of Theorem 1.1

We start moving planes from some place.

Lemma 3.1. There exist $\lambda_0 > 0$ such that for all $\lambda\ge \lambda_0$, $U^{\lambda}(z)\leq 0$ and $V^{\lambda}(z)\leq 0$ for any $z\in{\Sigma_\lambda}$.

Proof. If $\lambda > 0$ large enough, then

we have

and

By Lemma 2.3, for $U^{\lambda} \in L^{\frac{2 Q}{Q-2}}(\Sigma_\lambda)\bigcap L^{\infty}(\Sigma_\lambda), V^{\lambda} \in L^{\frac{2 P}{P -2}}(\Sigma_\lambda)\bigcap L^{\infty}(\Sigma_\lambda)$, we deduce

and

for all $\lambda\ge \lambda_0$. Thus, for $\lambda > 0$ large enough, we obtain that $U^{\lambda}(X)\leq 0$ and $V^{\lambda}(X)\leq 0$, for all $z\in{\Sigma_\lambda}$. □

The above process provides a starting point of moving planes, now we can go on moving the planes. Let

Lemma 3.2. If $\Lambda > 0$, then $U^{\Lambda}(z)\equiv 0$ and $V^{\Lambda}(z)\equiv 0$ for any $z\in{\Sigma_\Lambda}$.

Proof. By the continuity of $u$ and $v$, we have $U^{\Lambda}(z)\leq 0$ and $V^{\Lambda}(z)\leq 0$ for any $z\in{\Sigma_\Lambda}$.

Suppose on the contrary that $V^{\Lambda}(z)\not\equiv 0$ in ${\Sigma_\Lambda}$, then we have

Using the Lemma 2.1 and the strong maximum principle to $V^{\Lambda}(z)$, we obtain that $V^{\Lambda}(z)\leq 0$, and then $V^{\Lambda}(z) < 0$ in ${\Sigma_\Lambda}$. This strict inequality shows that the characteristic function $\mathcal{X}_{A^{2}_{\lambda}} \to 0$ a.e. in ${\mathbb{R}}^{m}\times{\mathbb{R}}^{k}$, when ${\lambda} \to{\Lambda}$. By the dominated convergence theorem, we know

therefore, for ${\lambda} \in({\Lambda-\delta}, {\Lambda})$, we see

where $\delta > 0$ is a sufficiently small constant. Using the previous argument, we obtain that $V^{\lambda}(z)\leq 0$ and $U^{\lambda}(z)\leq 0$ for any $z\in{\Sigma_\lambda}$, which contradicts with the definition of $\Lambda$. □

If $\Lambda = 0$, for any $(x, y_1, y_2, \cdots, y_n)\in \Sigma_0$, we get $U(x, y_1, y_2, \cdots, y_n)\leq U(x, -y_1, y_2, \cdots, y_n)$ and $V(x, y_1, y_2, \cdots, y_n)\leq V(x, -y_1, y_2, \cdots, y_n).$ One can move the planes in the contrary direction, which will show that $U(x, y_1, y_2, \cdots, y_n)\ge U(x, -y_1, y_2, \cdots, y_n)$ and $V(x, y_1, y_2, \cdots, y_n)\ge V(x, -y_1, y_2, \cdots, y_n).$ Hence, we have

and

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 $y_2, \cdots, y_n$. Then we obtain that $U, V$ are independent of $y$. However, this claim implies that $U$ and $V$ satisfy the system

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.

4.   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.

Acknowledgments

This work was supported by the Key Scientific Research Project of the Colleges and Universities in Henan Province (No. 22A110013), the Key Specialized Research and Development Breakthrough Program in Henan Province (No. 222102310265), and the Natural Science Foundation of Henan Province of China (No. 222300420499).

Conflict of interest

The author declares that he has no conflict of interest.