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 $\mathrm{\Omega }$ in ${\mathbb{R}}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in \mathrm{\Omega }:u(x)>t\}$ 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 $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$. When $\mathrm{\Omega }$ is an annular domain with the interior boundary ${\gamma }_I$ and the external boundary ${\gamma }_E$, where $u{\mathrm{\mid }}_{{\gamma }_I}=H,u{\mathrm{\mid }}_{{\gamma }_E}=\psi (x)$ 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 $\mathop {\min}\limits_{\gamma_E}\psi(x)<H<\mathop {\max}\limits_{\gamma_E}\psi(x)$,weshowthat$\Sigma_{i = 1}^k {{m_i}} \le N$(either$\Sigma_{i = 1}^k {{m_i}}=N$or$\Sigma_{i = 1}^k {{m_i}}+1=N$).