Article Contents
Article Contents

# Extremal solution and Liouville theorem for anisotropic elliptic equations

• We study the quasilinear Dirichlet boundary problem

$$$\nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases}$$$

where $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^{N}$ ($N\geq2$) is a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by

$Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)).$

Here, $F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi)$ and $F: \mathbb{R}^{N}\rightarrow [0, +\infty)$ is a convex function of $C^{2}(\mathbb{R}^{N}\setminus\{0\})$, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if $N\leq9$.

We also concern the Hénon type anisotropic Liouville equation,

$-Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N},$

where $\alpha>-2$, $N\geq2$ and $F^{0}$ is the support function of $K: = \{x\in\mathbb{R}^{N}:F(x)<1\}$. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for $2\leq N<10+4\alpha$ and $3\leq N<10+4\alpha^{-}$ respectively, where $\alpha^{-} = \min\{\alpha, 0\}$.

Mathematics Subject Classification: Primary: 35B53, 35B65; Secondary: 35J62.

 Citation:

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