Nonlinear Schrodinger Equation With Neumann Boundary Condition In Presence Of A Magnetic Field

In this PhD thesis, we mainly study the existence and multiplicity of solutions for nonlinear Schrodinger equation with Neumann boundary condition in presence of a magnetic field. More specifically, we study the following Schrodinger equations on bounded domainΩ when the nonlinear term g is subcritical, under suitable assumptions, we prove that it has at least one least energy solution, infinite many pairs of solutions, and at least cat((?)Ω) distinct nonzero solutions. When g=|u|2*-2u+f(|u|2)u,f satisfies some suitable conditions, we show that it possesses at least one least energy solution, and at least cat((?)Ω) distinct nonzero solutions.Our main methods are variational methods and Ljusternik-Schnirelmann theory.