Problemas elípticos quasilineares envolvendo não-linearidades interagindo com autovalores de alta energia

Abstract

In this thesis, using variational methods, we obtain nontrivial solutions for the quasilinear elliptic equations whose the nonlinear term interacts with high eigenvalues for the linear problem. Here we consider also the cases where the nonlinearity is subcritical or critical. Due to the presence of the quasilinear term in order to treat the problems proposed in the present work which becomes infeasible using directly the functional energy due the fact that it does not well defined in the Sobolev space $H^1(\mathbb(R)^N)$. Hence we apply a change variable obtaining an auxiliary energy functional associated with a semilinear elliptic problem. For this functional is possible to show the existence of a Linking structure taking into account several behaviors on the potential. Furthermore, critical points for that functional are weak solutions of the semilinear problem.