Stationary solutions of semilinear Schr\"odinger equations with trapping potentials in supercritical dimensions

TL;DR

This paper introduces a shooting method approach to prove the existence of ground states for semilinear Schrödinger equations with trapping potentials in critical and supercritical dimensions, extending beyond traditional variational methods.

Contribution

It presents a novel shooting method framework for establishing ground state solutions in supercritical regimes, applicable to Schrödinger-Newton and Gross-Pitaevskii equations.

Findings

Proves existence of ground states in supercritical dimensions

Applies method to Schrödinger-Newton and Gross-Pitaevskii equations

Provides sufficient conditions for the shooting method to work

Abstract

Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schr\"odinger-Newton and Gross-Pitaevskii equations with harmonic potentials.