Existence of ground states for a modified nonlinear Schrodinger equation

TL;DR

This paper proves the existence of ground state solutions for a modified nonlinear Schrödinger equation with specific potential conditions, extending previous results to include exponents p in (1,3) using minimization techniques.

Contribution

It establishes the existence of ground states for a modified nonlinear Schrödinger equation with exponents p in (1,3), a novel extension over prior work.

Findings

Existence of ground state solutions under certain potential conditions.

Extension of results to exponents p in (1,3).

Use of minimization under a constraint to prove existence.

Abstract

In this paper we prove existence of ground state solutions of the modified nonlinear Schrodinger equation: $$ -\Delta u+V(x)u-{1/2}u \Delta u^{2}=|u|^{p-1}u, x \in \R^N, N \geq 3, $$ under some hypotheses on $V(x)$. This model has been proposed in the theory of superfluid films in plasma physics. As a main novelty with respect to some previous results, we are able to deal with exponents $p\in(1,3)$. The proof is accomplished by minimization under a convenient constraint.