Semiclassical stationary states of Nonlinear Schrödinger equations with an external magnetic field

AbstractIn this paper we obtain multiple solutions u:RN→C of the nonlinear Schrödinger equation with an external magnetic fieldhi∇−A(x)2u+(U(x)−E)u=f(x,u),x∈RN,where N⩾2, A is a real-valued vector magnetic potential, U is a real electric potential function and the nonlinear term f(x,t) grows subcritically in t. The number of solutions to the equation is shown to be bounded below by some number which depends on the category of a set defined by some properties of V and the coefficients of the nonlinear term. We perform appropriate changes of gauges which are made on functions which are concentrated around points lying in some well-defined manifold