Elliptic problems with nonhomogeneous differential operators and multiple solutions

Abstract

The present survey aims to report on recent advances in the study of non-linear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δ p with. More general operators can be considered, possibly having completely different properties, for instance not satisfying any homogeneity requirement. A major objective of our work is to provide existence theorems of multiple solutions for boundary value problems governed by such general operators. In this direction, a three nontrivial solutions theorem is presented. In the case of problems determined by the p-Laplacian, we give a theorem ensuring the existence of at least four nontrivial solutions. Moreover, a complete sign information is available: two positive solutions, a negative solution and a nodal (sign-changing) solution. Finally, we provide a theorem guaranteeing the existence of a positive solution for a problem involving the -Laplacian operator, with, and a nonlinearity depending on the solution and its gradient.