Infinitely many small-energy solutions of a nonlinear boundary value problem with Dirichlet-to-Neumann operator

Abstract

In this study, we investigate the following nonlinear boundary value problem{−Δu+u=|u|p−2u,inΩ∂u∂ν=Du,on∂Ω, where 2<p<+∞ if N=2 and 2<p≤2⁎ if N≥3, Ω is a bounded domain in RN (N≥2) with C∞ smooth boundary, u:Ω→R, Δu is the Laplacian operator, ν is the unit outer normal vector at ∂Ω, and D is the Dirichlet-to-Neumann operator. We prove that this equation has a sequence of solutions {um} that satisfies limm→∞⁡‖um‖Lp(Ω)=0 and liminfm→∞‖um‖L∞(Ω)≥1. To prove this, a new critical point theorem without the usual Palais-Smale condition is used.