Multiple solutions for a problem with discontinuous nonlinearity

Abstract

In this work, we use the Lusternik-Schnirelmann category to estimate the number of nontrivial solutions for a problem with discontinuous nonlinearity and subcritical growth.

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.…”

Existence and multiplicity of solutions for discontinuous elliptic problems in ℝN

“…This fact brings us some difficulties to use traditional methods to obtain the multiplicity of solutions of problem (1.1). In order to overcome this difficulty, we will use a method, called Clark's dual action principle, which was employed in [6,10] and [8].…”