A unified approach for $ p $-Laplacian differential inclusions depending on a parameter

Abstract

A unified approach was developed to investigate the existence of at least one smooth nontrivial solution for p-Laplacian differential inclusions with upper/lower semicontinuos set-valued righthand side and depending on a positive parameter. The main idea was to find a solution of the given inclusion as a solution of an auxiliary differential inclusion associated with the generalized gradient (in Clarke’s sense) of a primitive of a selection of the multivalued nonlinearity. Since we also dealt with discontinuous selections, variational methods for locally Lipschitz functionals provided the right tools to achieve our goals.