Existence of solutions for an asymptotically linear Dirichlet problem via Lazer–Solimini results

Abstract

In this paper we prove that an asymptotically linear Dirichlet problem has at least three nontrivial solutions when the range of the derivative of the nonlinearity includes at least the first k eigenvalues of minus Laplacian, without any restriction about nondegeneracy of solutions. A pair is of one sign (positive and negative, respectively). We construct a third solution using arguments of the type Lazer–Solimini (see [A.C. Lazer, S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min–max type, Nonlinear Anal. 12 (8) (1988) 761–775]). This gives a partial answer to a conjecture stated in [J. Cossio, S. Herrón, Nontrivial solutions for a semilinear Dirichlet problem with nonlinearity crossing multiple eigenvalues, J. Dyn. Differential Equations 16 (3) (2004) 795–803]. Moreover, in the particular case of nondegenerate critical points, we prove that there are at least four nontrivial solutions, the one sign solutions are of Morse index equal to 1, the third solution has Morse index k , and there is a fourth solution. For this case, we use the Leray–Schauder degree and Lazer–Solimini results.