Nontrivial Solutions for a Semilinear Dirichlet Problem with Nonlinearity Crossing Multiple Eigenvalues

abstract

In this paper we prove that a semilinear elliptic boundary value problem has at least three nontrivial solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues of the Laplacian and all solutions are nondegenerate. A pair are of one sign (positive and negative, respectively). The one sign solutions are of Morse index less than or equal to 1 and the third solution has Morse index greater than or equal to 2. Extensive use is made of the mountain pass theorem, and Morse index arguments of the type Lazer–Solimini (see Lazer and Solimini, Nonlinear Anal. 12(8), 761–775, 1988). Our result extends and complements a theorem of Cossio and Veléz, Rev. Colombiana Mat. 37(1), 25–36, 2003.