Abstract

In this paper, we study a nonlinear elliptic problem at resonance driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions due to Chang. We prove a theorem guaranteeing the existence of one solution which is smooth and strictly positive. Then by strengthening the assumptions, we establish a multiplicity result providing the existence of at least two distinct solutions. Our hypotheses permit resonance and asymmetric behavior at +∞ and −∞. As a byproduct of our analysis we obtain an nonlinear and nonsmooth generalization of a result of Brézis–Nirenberg about H₀¹ versus C₀¹ minimizers of a smooth functional.

Author Keywords: Clarke subdifferential; Nonsmooth Palais–Smale condition; Ekeland variational principle; Nonlinear regularity; Nonsmooth Mountain Pass Theorem; Resonant problem; p-Laplacian; Principal eigenvalue

Mathematical subject codes: 35J20; 35J85; 35R70

Article Outline

1. Introduction

2. Mathematical preliminaries

3. One smooth and strictly positive solution

4. Multiple solutions

References