Multiple nontrivial solutions of a semilinear elliptic problem with asymmetric nonlinearity

Abstract

In this paper we study the existence and multiplicity of solutions of the boundary–value problem(1){−Δu=−λ|u|q−2u+au+b(u+)p−1, in Ω;u=0, on ∂Ω, where Δ denotes the N-dimensional Laplacian, Ω is a bounded domain with smooth boundary, ∂Ω, in RN(N⩾3), u+ denotes the positive part of u:Ω→R, 1<q<2<p<2⁎=2N/(N−2), λ>0, a∈R and b>0. Using infinite-dimensional Morse Theory, we extend the results of Paiva and Presoto [14] and establish some conditions for the existence of at least four nontrivial solutions of (1).