Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations

TL;DR

This paper develops new methods in nonsmooth critical point theory to establish existence and multiplicity of solutions for Born-Infeld type equations without relying on the Palais-Smale condition.

Contribution

It introduces novel existence and multiplicity results for critical points of nonsmooth functionals, extending the theory to broader classes of equations.

Findings

Existence of positive solutions for Born-Infeld equations

Infinitely many solutions in symmetric and non-symmetric classes

Solutions with finite energy under minimal conditions

Abstract

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.