Existence and multiplicity of solutions for problems involving the Dirac operator

• Ano de defesa: 2019
• Autor(a) principal: Somavilla, Fernanda
• Orientador(a): Paiva, Francisco Odair Vieira de
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: Dirac equation; Existence and multiplicity results; Linking arguments; Strongly indefinite functionals; Operador de Dirac; Funcionais fortemente indefinidos
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE::EQUACOES DIFERENCIAIS PARCIAIS; CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufscar.br/handle/20.500.14289/11856

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.