Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem

Aixia Qian

doi:10.1155/bvp.2005.329

Abstract

We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions. The study of such a problem is based on the variational methods and critical point theory. We prove the conclusion by using the symmetric mountain-pass theorem under the Cerami condition.

Highlights

Consider the Neumann boundary value problem:− u + αu = f (x, u), x ∈ Ω, ∂u ∂ν = 0, x ∈ ∂Ω, (1.1)where Ω ⊂ RN (N ≥ 1) is a bounded domain with smooth boundary ∂Ω and α > 0 is a constant
We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions
330 Infinitely many nodal solutions for a Neumann problem ( f4) There exist θ ≥ 1, s ∈ [0, 1] such that θG(x, t) ≥ G(x, st), (x, u) ∈ Ω × R

Summary

Introduction

330 Infinitely many nodal solutions for a Neumann problem ( f4) There exist θ ≥ 1, s ∈ [0, 1] such that θG(x, t) ≥ G(x, st), (x, u) ∈ Ω × R. [8, Theorem 3.2] obtained infinitely many solutions under ( f1)–( f5) and ( f3) lim|u|→∞ inf( f (x, u)u)/|u|μ ≥ c > 0 uniformly for x ∈ Ω, where μ > 2. It turns out that ( f3) and ( f4) are stronger than ( f3) and ( f4), respectively, the function (1.6) does not satisfy ( f3) , Theorem 1.1 applied to Dirichlet boundary value problem improves [8, Theorem 3.2]. [1, Theorem 7.3] got infinitely many nodal solutions for (1.1) under assumption that the functional is of C2. Let M ⊂ X be an invariant set under σ.

Assume for each k there exist ρk

We see that

Ei and