Existence and multiplicity of solutions for discontinuous elliptic problems in ℝ^N

Abstract

This paper concerns with the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity. By using dual variational methods, properties of the Nehari manifolds and Ekeland's variational principle, we show how the ‘shape’ of the graph of the function A affects the number of nontrivial solutions.

On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity

“…Several techniques have been developed or applied to study this kind of problem, such as variational methods for nondifferentiable functionals, lower and upper solutions, dual variational principle, global branching, Palais principle of symmetric criticality for locally Lipschitz functional and the theory of multivalued mappings. See for instance, Alves, Yuan and Huang [1], Alves, Santos and Nemer [2], Ambrosetti and Badiale [3], Ambrosetti, Calahorrano and Dobarro [4], Ambrosetti and Turner [5], Anmin and Chang [11], Arcoya and Calahorrano [13], Cerami [14], Chang [15][16][17], Clarke [20,21], Gazzola and Rǎdulescu [24], Krawcewicz and Marzantowicz [28], Molica Bisci and Repovš [30], Rǎdulescu [34], dos Santos and Figueiredo [23] and their references.…”