Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the $p$-Laplacian

Abstract

For a certain range of the eigenvalue parameter we prove a new multiple and sign-changing solutions theorem. The novelties of our paper are twofold. First, unlike recent papers in the field we do not assume jumping nonlinearities and allow a rather general growth condition on the nonlinearity involved. Second, our approach strongly relies on a combined use of variational and topological arguments (e.g. critical points, mountain--pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p-Laplacian) on the one hand, and comparison principles for nonlinear differential inequalities, in particular, the existence of extremal constant-sign solutions, on the other hand.