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

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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  • 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.

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


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