Nonlinear hemivariational inequalities at resonance

Abstract

In this paper we consider nonlinear hemivariational inequalities involving the pLaplacian at resonance. We prove the existence of a nontrivial solution. Our approach is variational based on the critical point theory for nonsmooth, locally Lipschitz functionals due to Chang.

An existence theorem for nonlinear hemivariational inequalities at resonance

“…In a recent paper (see Gasinski and Papageorgiou [9]), we examined a nonlinear hemivariational inequality at resonance. We proved an existence theorem under the assumptions that the Clarke subdifferential of the generalised potential is bounded and that it has nonzero limits at ±oo.…”

Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities

“…The next example is in this direction. Owing to Theorem 4.3, for each λ > 56 3 √ 4 there is δ > 0 such that for all μ ∈ ]0, δ] the problem , are actually weak solutions; on the contrary, in most of the papers that investigate Dirichlet problems for elliptic equations with discontinuous nonlinearities, the solutions are multi-valued solutions, namely solutions for the corresponding differential inclusion obtained by filling the gaps at the discontinuity points (see, for instance, [20,22] and the references therein). This is due to the assumption (m 4 ) that allows us to apply a classical lemma of [18].…”

“…Problems of this type have been studied by looking for solutions of the corresponding differential inclusion obtained by filling the gaps at the discontinuity points of f and g with respect to u (see, for instance, [20,22] and the references therein). On the contrary, Theorem 4.2 and its consequence, that is, Theorem 4.3, ensure solutions that are actually weak solutions for problem (D λ,μ ); nevertheless, the set of discontinuity points may also be uncountable (see Remark 4.4 and Example 4.2).…”

Positive solutions for nonlinear hemivariational inequalities

“…The study of positive solutions for nonlinear equations is lagging behind. Some of the research in this direction includes Allegretto-Huang [1], Citti-Uguzzoni [6], Guo [11], for smooth problems monitored by the p-Laplacian; and Gasinski-Papageorgiou [9], KyritsiPapageorgiou [13], Motreanu-Papageorgiou [16], for nonsmooth problems monitored by the p-Laplacian.…”