Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schr\"odinger equations

TL;DR

This paper investigates the properties of standing wave solutions for pseudo-relativistic nonlinear Schrödinger equations, focusing on their ground states, symmetry, and behavior in the nonrelativistic limit.

Contribution

It establishes existence, symmetry, and sign properties of ground states, and analyzes their nonrelativistic limit in the Sobolev space.

Findings

Ground states are positive and radially symmetric.

Existence of ground state solutions is proven.

Ground states converge in $H^1$ as the nonrelativistic limit is taken.

Abstract

In this paper we study standing waves for pseudo-relativistic nonlinear Schr\"odinger equations. In the first part we find ground state solutions. We also prove that they have one sign and are radially symmetric. The second part is devoted to take nonrelativistic limit of the ground state solutions in $H^1 (\mathbb{R}^n)$ space.