Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions

Abstract

In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $\Omega$ in ${\mathbb{R}}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in \Omega :u(x)>t\}$ for any ${\min }_{\partial \Omega }u(x)<t<{\max }_{\partial \Omega }u(x)$, we obtain the geometric structure of interior critical points of $u$. Precisely, when $\Omega$ is simply connected, we develop a new method to prove ${\Sigma }_{i=1}^k{m}_i+1=N$, where $m_1,\cdots ,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots ,x_k$ of $u$ and $N$ is the number of global maximal points of $u$ on $\partial \Omega$. When $\Omega$ is an annular domain with the interior boundary ${\gamma }_I$ and the external boundary ${\gamma }_E$, where $u{|}_{{\gamma }_I}=H,u{|}_{{\gamma }_E}=\psi (x)$ and $\psi (x)$ has $N$ local (global) maximal points on ${\gamma }_E$. For the case $\psi (x)\ge H$ or $\psi (x)\le H$ or $\underset{{\gamma }_E}{\min }\psi (x)<H<\underset{{\gamma }_E}{\max }\psi (x)$, we show that ${\Sigma }_{i=1}^k{m}_i\le N$ (either ${\Sigma }_{i=1}^k{m}_i=N$ or ${\Sigma }_{i=1}^k{m}_i+1=N$).

arXiv comment21pages, 13figures

References in corpus (2)

• Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains
• Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions

Cited by in corpus (1)

• Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains