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In this paper, we mainly investigate the critical points associated tosolutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichletboundary conditions in a connected domain $\Omega$ in $\mathbb{R}^2$. Based onthe fine analysis about the distribution of connected components of asuper-level set $\{x\in \Omega: u(x)>t\}$ for any $\mathop{\min}_{\partial\Omega}u(x)<t<\mathop {\max}_{\partial\Omega}u(x)$, we obtainthe 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 ofinterior critical points $x_1,\cdots,x_k$ of $u$ and $N$ is the number ofglobal maximal points of $u$ on $\partial\Omega$. When $\Omega$ is an annulardomain 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 $\Sigma_{i = 1}^k {{m_i}}+1=N$).

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