On the Schrodinger equation in $R^N$ under the effect of a general nonlinear term

TL;DR

This paper proves the existence of positive solutions for a nonlinear Schrödinger equation in $\mathbb{R}^N$ under general conditions on the nonlinearity, and shows that a related ground state minimization problem has no solution.

Contribution

It extends the analysis of nonlinear Schrödinger equations by establishing existence results under broad nonlinear conditions and analyzing the non-existence of certain ground states.

Findings

Positive solutions exist under general nonlinear hypotheses.

A related ground state minimization problem has no solution.

The results generalize previous specific cases.

Abstract

In this paper we prove the existence of a positive solution to the equation $-\Delta u + V(x)u=g(u)$ in $R^N,$ assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution.