Existence and multiplicity results for nonlinear elliptic problems

In this work of thesis, we investigate existence and multiplicity results for a class of nonlinear elliptic problems. First, we deal with problems involving the p-Laplacian operator on bounded smooth domains, where a diffusion term appears into the nonlinearity. For this reason, variational methods cannot be used. Secondly, we treat existence and multiplicity of weak solutions for (p; q)- Laplacian equations, as well as for singular p-Laplacian Schrodinger equations, in the entire R^N whose nonlinearity combines a power-type term at critical level with a subcritical term, involving also nontrivial weights and a positive parameter. This latter case, considered also in a symmetric setting, allows us to use variational methods, but in the delicate situation of lack of compactness, so that classical results cannot be directly used, they need to be adapted.