Existence, regularity and positivity of ground states for nonlocal nonlinear Schrodinger equations

Abstract

We study ground states of a nonlinear Schrödinger equation driven by the infinitesimal generator of a rotationally invariant Levy process. The equation includes many special cases such as classical Schrodinger equations, fractional Schrödinger equations and relativistic Schrödinger equations, etc. It is proved that the equation possesses ground states in a suitable space of functions, then the regularity of solutions to the equation is examined, in particular, any solution is Hölder continuous, and, if the process involves diffusion terms, any solution is twice differentiable further. Finally, we show that any ground state is either positive or negative.