Gaps of consecutive eigenvalues of Laplace operator and the existence of multiple solutions for superlinear elliptic problem

Abstract

This paper is mainly related to multiple nontrivial solutions of the elliptic boundary value problem{−Δu=|u|p−2u+f(x,u),x∈Ω,u=0,x∈∂Ω, for p∈(2,2⁎). It is reasonable to guess that for dim⁡Ω≥2 above problem possesses infinitely many distinct solutions since this is proved to be true for ODE. However, so far one does not even know if there exists a fourth nontrivial solution. By using a new homological linking theorem, Morse theory, and some precise estimates we disclose the relationship among the gaps of consecutive eigenvalues of Laplace operator, growth trend of nonlinear terms and the existence of multiple solutions of superlinear elliptic boundary value problem. Moreover, as p is close to 2, we get the fourth nontrivial solution under appropriate hypotheses, where f(x,u) satisfies Ambrosetti–Rabinowitz condition.