Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator

Phuong Le

doi:10.3934/cpaa.2020025

Article Contents

2020, Volume 19, Issue 1: 511-525. Doi: 10.3934/cpaa.2020025

This issue Previous Article The weak maximum principle for second-order elliptic and parabolic conormal derivative problems Next Article Symmetry of singular solutions for a weighted Choquard equation involving the fractional

$ p $

-Laplacian

Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator

Phuong Le

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received: January 2019

Revised: May 2019

Published: July 2019

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.307

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Abstract

We consider the boundary value problem

$\begin{equation*} \begin{cases} -{\rm div}_G(w_1\nabla_G u) = w_2f(u) &\text{ in } \Omega,\\ u=0 &\text{ on } \partial\Omega, \end{cases}\end{equation*}$

where $\Omega$ is a bounded or unbounded $C^1$ domain of $\mathbb{R}^N$, $w_1, w_2 \in L^1_{\rm loc}(\Omega)\setminus\{0\}$ are nonnegative functions, $f$ is an increasing function, $\nabla_G$ and ${\rm div}_G$ are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on $\Omega$, $w_1$, $w_2$ and $f$. We also show the sharpness of our results when $\Omega=\mathbb{R}^N$ and $f$ has power or exponential growth.

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Mathematics Subject Classification: Primary: 35J25, 35H20; Secondary: 35B53, 35B35.

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