Existência de soluções para problemas elípticos não lineares com condições de Neumann e Dirichlet-Neumann via metodo variacional

Abstract

The objective of this master's dissertation is to study, under specific assumptions, the existence and nonexistence of weak, nontrivial, and nonnegative solutions to a nonlinear elliptic problem subject, initially, to Neumann boundary conditions and, subsequently, to mixed Neumann–Dirichlet boundary conditions. To this end, we employ classical variational methods from the literature, such as the Mountain Pass Theorem and Mizoguchi’s Theorem, together with fundamental tools from functional analysis. Initially, we investigate the multiplicity of solutions for the problem under Neumann boundary conditions, using the Palais–Smale condition to obtain compactness. Next, we analyze a version of the problem with slightly different assumptions and functional setting, in which compactness is ensured through the Cerami condition. Finally, we study the linear case of the same problem endowed with mixed boundary conditions, namely, Neumann conditions on part of the boundary and Dirichlet conditions on the complementary part.