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

Abstract

We consider the boundary value problem \begin{document}$\begin{equation*}\begin{cases}-{\rm div}_G(w_1\nabla_G u) = w_2f(u) &\text{ in } \Omega,\\u=0 &\text{ on } \partial\Omega,\end{cases}\end{equation*}$ \end{document} 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.