Five nontrivial solutions of superlinear elliptic problem

In this paper, we consider the following superlinear elliptic problem {-Delta u = lambda vertical bar u vertical bar(p-2) u + f(x, u), in Omega, (P) u = 0, on partial derivative Omega, where lambda > 0and 2 < p < 2 + delta for some delta > 0 small. The nonlinearity fsatisfies 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 fat infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator - Delta. 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 lambda = 1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five