Positive solutions for Schrödinger-Poisson type systems

• Autor(a) principal: Rodriguez, Edwin Gonzalo Murcia
• Data de Publicação: 2017
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-17122017-094108/
• topic: Asymptotically cubic nonlinearity; Categoria de Ljusternick-Schnirelmann; Ljusternick-Schnirelmann category; Métodos variacionais; Não linearidade assintoticamente cúbica; Schrödinger-Poisson system; Sistema Schrödinger-Poisson; Variational methods

In this thesis we study Schrödinger-Poisson systems and we look for positive solutions. Our work consists in three chapters. Chapter 1 includes some basic facts on critical point theory. In Chapter 2 we consider a fractional Schrödinger-Poisson system in the whole space R^N in presence of a positive potential and depending on a small positive parameter . We show that, for suitably small (i.e. in the \"semiclassical limit\") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential. Finally, in Chapter 3, we analyze a Schrödinger-Poisson system in R^3 under an asymptotically cubic nonlinearity. We prove the existence of positive, radial solutions inside a ball and in an exterior domain.