Five nontrivial solutions of superlinear elliptic problem

Abstract

In this paper, we consider the following superlinear elliptic problem(P){−Δu=λ|u|p−2u+f(x,u),inΩ,u=0,on∂Ω, where λ>0 and 2<p<2+δ for some δ>0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u=0 is a local minimizer of the associated energy functional of equation (P).Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ=1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ.