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 R 2. Based on the fine analysis about the distribution of connected components of a super-level set { x ∈ Ω : u (x) > t } for any min ∂ Ω u (x) < t < max ∂ Ω u (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 Σ i = 1 k 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 ∂Ω. When Ω is an annular domain with the interior boundary γ I and the external boundary γ E , where u | γ I = H , u | γ E = ψ (x) and ψ (x) has N local (global) maximal points on γ E. For the case ψ (x) ≥ H or ψ (x) ≤ H or min γ E ψ (x) < H < max γ E ψ (x) , we show that Σ i = 1 k m i ≤ N (either Σ i = 1 k m i = N or Σ i = 1 k m i + 1 = N). [ABSTRACT FROM AUTHOR]