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Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions. (arXiv:1712.08458v1 [math.AP])
来源于:arXiv
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 maximum 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) maxim 查看全文>>