Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities. (February 2016)

• Record Type: Journal Article
• Authors: Barletta, Giuseppina; Chinnì, Antonia; O'Regan, Donal
• Is Part Of: Nonlinear analysis. Volume 27(2016)
• Page Start: 312
• Page End: 325
• Publication Date: 2016-02
• Subjects: p(x)-Laplacian -- Variable exponent Sobolev spaces -- Critical points of non-smooth functions; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.08.002
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper the existence of a nontrivial solution to a parametric Neumann problem for a class of nonlinear elliptic equations involving the p ( x ) -Laplacian and a discontinuous nonlinear term is established. Under a suitable condition on the behavior of the potential at 0 +, we obtain an interval ] 0, λ ∗ ], such that, for any λ ∈ ] 0, λ ∗ ] our problem admits at least one nontrivial weak solution. The solution is obtained as a critical point of a locally Lipschitz functional. In addition to providing a new conclusion on the existence of a solution even for λ = λ ∗, our theorem also includes other results in the literature for regular problems.