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

Abstract

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 Ω in R2. Based on the fine analysis about the distribution of connected components of a super-level set {x∈Ω:u(x)>t} for any min∂Ωu(x)<t<max∂Ωu(x), we obtain the geometric structure of interior critical points of u. Precisely, when Ω is simply connected, we develop a new method to prove Σi=1kmi+1=N, where m1,⋯,mk are the respective multiplicities of interior critical points x1,⋯,xk of u and N is the number of global maximal points of u on ∂Ω. When Ω is an annular domain with the interior boundary γI and the external boundary γE, where u|γI=H,u|γE=ψ(x) and ψ(x) has N local (global) maximal points on γE. For the case ψ(x)≥H or ψ(x)≤H or minγEψ(x)<H<maxγEψ(x), we show that Σi=1kmi≤N (either Σi=1kmi=N or Σi=1kmi+1=N).