
Minimax Theory and its Applications 05 (2020), No. 1, 129150 Copyright Heldermann Verlag 2020 Mountain Pass Solutions for Nonsmooth Elliptic Problems Marcello Lucia Mathematics Department, City University of New York, Staten Island, NY 10314, U.S.A. marcello.lucia@csi.cuny.edu Friedemann Schuricht Fakultät Mathematik, Technische Universität, 01062 Dresden, Germany friedemann.schuricht@tudresden.de We study the existence of solutions for degenerate nonlinear elliptic equations using nonsmooth mountain pass arguments both with and without symmetry, where the weak slope is taken to define critical points. First we consider a class of problems where the leading elliptic term A(x,Du) satisfies usual pgrowth conditions (for p>1) but A is merely convex and not necessarily differentiable in the gradient. Secondly, we show the existence of a sequence of solutions for a class of problems involving the highly singular 1Laplacian as leading term. The results demonstrate that differentiablility of the leading term is not needed for typical results of elliptic problems. Keywords: Elliptic problems, critical points, mountain pass theory, nonsmooth analysis, 1Laplace operator. MSC: 35J60, 35J70, 49J35, 49J52, 49R05. [ Fulltextpdf (166 KB)] for subscribers only. 