Existence and multiplicity of solutions for problems involving the Dirac operator

Abstract

In this thesis, we study equations that involving the Dirac operator and which have the form $-i \alpha \cdot \nabla u + a \beta u + M(x)u = F_{u}(x,u), em \mathbb{R}^{3},$ where $\alpha = (\alpha_1, \alpha_2, \alpha_3),$ with $\alpha_{j}$ and $\beta$ are complex matrices 4x4, j = 1, 2, 3 and a>0.Using variational methods and elements from critical point theory for strongly indefinite problems we obtain existence and multiplicity results of solutions $u:R^{3} \rightarrow C^{4}$ under different sets of hypothesis about the potential M and the nonlinearity F: Firstly, we consider a problem with nonperiodic potential and concave-convex type nonlinearity, nonperiodic, which contain weight functions that can present signal change. Next, using the generalized Nehari manifold, we study problems in which nonlinearity satisfies weak monotonicity conditions and may relate to the potential function. Among such problems,we consider a periodic case and, due to the assumptions, in order to obtain the multiplicity results we use the Clarke's subdifferential and Krasnoselskii genus. Finally, we approach a problem with nonlinearity asymptotically linear at infinity and matrix potential. In this case, the potential is described by a sum of a non-positive suitable matrix potential and a diagonal matrix whose elements are function in some $L^{\sigma}, \sigma >1,$ which can change signal.