Soluções nodais para problemas elípticos semilineares com crescimento crítico exponencial.

Abstract

In this work, we study existence, non-existence and multiplicity results of nodal solutions for the nonlinear Schrödinger equation in a smooth domain of R^2 not necessarily bounded, f is a continuous function which has exponential critical growth and V is a continuous and non-negative potential. In the first part, we prove the existence of least enegy nodal solutions in both cases, bounded and unbounded domain. Morover, we also prove a non-existence result of least energy nodal solution for the autonomous case in whole R^2. In the second part, we established multiplicity of multi-bump type nodal solutions. Finally, for V=0, we prove a result of infinitaly many nodal solutions on a ball. The main tools used are variation methods, Lions'Lemma, penalizations methods and a process of anti-symmetric continuation.