Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance

Abstract

In this paper we study a class of nonlinear elliptic eigenvalue problems driven by the p-Laplacian and having a nonsmooth locally Lipschitz potential. We show that as the parameter $\lambda$ approaches $\lambda_1$ (= the principal eigenvalue of $(-\triangle_p, W_0^{1,p}(Z))$ ) from the right, the problem has three nontrivial solutions of constant sign. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions. In the process of the proof we also establish a generalization of a recent result of Brezis and Nirenberg for C 0 1 versus W 0 1,p minimizers of a locally Lipschitz functional. In addition we prove a result of independent interest on the existence of an additional critical point in the presence of a local minimizer of constant sign. Finally by restricting further the asymptotic behavior of the potential at infinity, we show that for all $\lambda > \lambda_1$ the problem has two solutions one strictly positive and the other strictly negative.