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 R-2. Based on the fine analysis about the distribution of connected components of a super-level set {x is an element of Omega : u(x) > t} for any min(partial derivative Omega) u(x) < t < max(partial derivative 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(k)(i=1) m(i) + 1 = N, where m(1), ..., m(k) are the respective multiplicities of interior critical points x(1), ..., x(k) of u and N is the number of global maximal points of u on partial derivative Omega. When Omega is an annular domain with the interior boundary gamma(I) and the external boundary gamma(E), where u vertical bar gamma(1) = H, u vertical bar gamma(E) = psi(x) and psi(x) has N local (global) maximal points on gamma(E) . For the case psi(x) >= H or psi(x) <= H or min(gamma E) psi(x) < H < max(gamma E) psi(x), we show that Sigma(k)(i=1) m(i)( )<= N (either Sigma(k)(i=1) or Sigma(k)(i=1) m(i) + 1 = N). (C) 2018 Elsevier Inc. All rights reserved.