Existence of weak solutions for the Schrödinger equation and its application

Abstract

In this paper, we are concerned with the existence of weak solutions for the Schrödinger equation with sign-changing potential in a smooth cone. For solving the Dirichlet boundary-value problem with respect to the Schrödinger operator, we prove the existence of at least one weak solution using changes of Schrödingerean harmonic measure, the energy estimate method and refined inequality technique. Due to the fact that the nonlinearity is allowed to change sign in our formulation, and the novelty of the boundary conditions, these results are new for discrete and arbitrary time scales. As an application, concentration results are also investigated.