Existence of solutions for an asymptotically linear Dirichlet problem viaLazer-Solimini results

摘要

In this paper we prove that an asymptotically linear Dirichlet problem has at least threenontrivial solutions when the range of the derivative of the nonlinearity includes at leastthe first k eigenvalues of minus Laplacian, without any restriction about nondegeneracyof solutions. A pair is of one sign (positive and negative, respectively). We construct athird 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-maxtype, Nonlinear Anal. 12 (8) (1988) 761-7751). This gives a partial answer to a conjecturestated in [J. Cossio, S. Herrón, Nontrivial solutions for a semilinear Dirichlet problem withnonlinearity crossing multiple eigenvalues, J. Dyn. Differential Equations 16 (3) (2004)795-8031. Moreover, in the particular case of nondegenerate critical points, we prove thatthere are at least four nontrivial solutions, the one sign solutions are of Morse index equalto 1, the third solution has Morse index k, and there is a fourth solution. For this case, weuse the Leray-Schauder degree and Lazer-Solimini results.