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Minimax Theory and its Applications 05 (2020), No. 1, 129--150
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.

Friedemann Schuricht
Fakultšt Mathematik, Technische Universitšt, 01062 Dresden, Germany

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 p-growth 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 1-Laplacian 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, 1-Laplace operator.

MSC: 35J60, 35J70, 49J35, 49J52, 49R05.

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