Solvability for a Class of Schrodinger Equations with Periodic Potential

• Autor(a) principal: Cavalcante, Marcius Petrúcio de Almeida
• Data de Publicação: 2017
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UFPB
• Texto Completo: https://repositorio.ufpb.br/jspui/handle/123456789/11234
• topic: Operador de Schrodinger; Potencial Períodico; Teoria Espectral; Teorema de Linking; Crescimento Sublinear; Crescimento Crítico; Desigualdade; Trudinger-Moser; Schrodinger Operator; Periodic Potential; Spectral Theory; Linking Theorem; Sub- linear Growth; Critical Growth; Trudinger-Moser Inequality; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

In this thesis we study the existence of solutions for a class of semilinear Schr¨odinger equations of the form −∆u + V (x)u = f¯(x, u), x ∈ RN , where N ≥ 2, the potential V is a 1-periodic continuous function. In dimension N ≥ 3, we assume that 0 lies in a spectral gap of the Schr¨odinger operator S = −∆+V and the nonlinearity is from concave and convex type. In dimension N = 2, we assume that 0 lies in a spectral gap or on the boundary of a spectral gap of S and we deal with nonlinearities having exponential growth in the Trudinger-Moser sense. We treat the case where f¯(x, t) is periodic as well as the nonperiodic one. The proofs relies on variational setting, by using linking-type theorems, some Trudinger-Moser inequalities and concentration-compactness principles.