On the orbital stability for a class of nonautonomous NLS

TL;DR

This paper proves the existence and orbital stability of standing waves for certain nonautonomous nonlinear Schrödinger equations with specific potential conditions, extending previous approaches to more general settings.

Contribution

It introduces new conditions on potentials and applies a compactness analysis to establish stability for a class of nonautonomous NLS equations.

Findings

Existence of standing waves under new potential conditions

Orbital stability proven for the considered NLS class

Extension of Cazenave and Lions' approach to nonautonomous cases

Abstract

Following the original approach introduced by T. Cazenave and P.L. Lions in \cite{CaLi} we prove the existence and the orbital stability of standing waves for the following class of NLS: \label{intr1} i\partial_t u+ \Delta u - V(x) u + Q(x) u|u|^{p-2}=0, \hbox{} (t,x) \in \R\times \R^n, \hbox{} 2<p<2+\frac 4n and \label{intr2} i\partial_t u - \Delta^2 u - V(x) u + Q(x) u|u|^{p-2}=0, \hbox{} (t,x) \in \R\times \R^n, \hbox{} 2<p<2+\frac 8n under suitable assumptions on the potentials $V(x)$ and $Q(x)$. More precisely we assume $V(x), Q(x) \in L^\infty(\R^n)$ and $meas\{Q(x)>\lambda_0\}\in (0,\infty)$ for a suitable $\lambda_0>0$. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to \eqref{intr1} and \eqref{intr2}.