Existence of Positive Ground States of Nonlocal Nonlinear Schrödinger Equations

Abstract

We investigate ground states of a (nonlocal) nonlinear Schrödinger equation which generalizes classical (fractional, relativistic, etc.) Schrödinger equations, so that we extend relevant results and study common properties of these equations in a uniform way. To obtain the existence of ground states, we first solve a minimization problem and then prove that the solution of the minimization problem is a ground state of the equation. After examining the regularity of the solutions to the equation, we demonstrate that any ground state is sign-definite.

Keywords

• nonlocal Schrödinger equation
• ground state
• infinitesimal generator
• rotationally invariant Lévy process