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

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.