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

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 $\mathop {\min}_{\partial\Omega}u(x)<t<\mathop {\max}_{\partial\Omega}u(x)$, 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 $\psi(x)\geq H$ or $\psi(x)\leq H$ or $\mathop {\min}\limits_{\gamma_E}\psi(x)<H<\mathop {\max}\limits_{\gamma_E}\psi(x)$, we show that $\Sigma_{i = 1}^k {{m_i}} \le N$ (either $\Sigma_{i = 1}^k {{m_i}}=N$ or $\Sigma_{i = 1}^k {{m_i}}+1=N$).