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Extremal solution and Liouville theorem for anisotropic elliptic equations (2101.00970v1)

Published 4 Jan 2021 in math.AP

Abstract: We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left{ \begin{aligned} -Qu&=\lambda e{u} \quad \mbox{in}\quad\Omega\ u&=0 \quad \mbox{on}\quad\partial\Omega,\ \end{aligned} \right. \end{equation} where $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}{N}$ with $N\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum_{i=1}{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $$ Here, $F_{\xi_{i}}=\frac{\partial F}{\partial\xi_{i}}$ and $F: \mathbb{R}{N}\rightarrow[0,+\infty)$ is a convex function of $ C{2}(\mathbb{R}{N}\setminus{0})$, that satisfies certain assumptions. We derive the existence of extremal solution and obtain that it's regular, if $N\leq9$. We also concern the H\'{e}non type anisotropic Liouville equation, namely, $$-Qu=(F{0}(x)){\alpha}e{u}\quad\mbox{in}\quad\mathbb{R}{N}$$ where $\alpha>-2$, $N\geq2$ and $F{0}$ is the support function of $K:={x\in\mathbb{R}{N}:F(x)<1}$ which is defined by $$F{0}(x):=\sup_{\xi\in K}\langle x,\xi\rangle.$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2\leq N<10+4\alpha$ and $3\leq N<10+4\alpha{-}$ respectively, where $\alpha{-}=\min{\alpha,0}$.

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