Problemas de valor de fronteira envolvendo o operador de curvatura média prescrita com condições locais e globais

Abstract

In this work, we approach some boundary value problems that encompass the prescribed mean curvature operator, with the objective to verify the existence of solutions to such problems. More specifically, we study the main problems exposed in Cerda and Iturriaga [9], using as the main tool the Mountain Pass Theorem; Lorca and Ubilla [8] and Bonheure, Derlet and Valeriola [4], for which we use techniques based on Nehari manifold. Furthermore, we present some examples of these problems. We emphasize that in the articles of Cerda and Iturriaga [9] and Lorca and Ubilla [8] local conditions are considered, differently of the article of Bonheure, Derlet and Valeriola [4] that, although it presents a simpler problem provides us with a type of distinct solution of those exhibited in the other two works, i.e., nodal solutions. Another objective when developing this dissertation is to study different verification techniques of the existence of solution of boundary value problems. To reach the mentioned objectives, we utilize a variational approach and techniques based on Mountain Pass Theorem and Nehari manifold, that allowed us to ensure: the existence of at least one nonnegative solution of the problems presented in Cerda and Iturriaga [9] and Lorca and Ubilla [8]; and the existence of at least one nodal solution of the problem exposed in Bonheure, Derlet e Valeriola [4].