IUMJ

Title: On the orbital stability for a class of nonautonomous NLS

Authors: Jacopo Bellazzini and Nicola Visciglia

Issue: Volume 59 (2010), Issue 3, 1211-1230

Abstract:

Following the original approach introduced by T. Cazenave and P.L. Lions (\emph{Orbital stability of standing waves for some nonlinear Schroedinger equations}, Comm. Math. Phys. \textbf{85} (1982), 549--561), we prove the existence and the orbital stability of standing waves for the following class of NLS:\begin{align}\MoveEqLeft[5] i \partial_t u + \Delta u - V(x)u + Q(x)u |u|^{p-2} = 0, \label{intr1}\\ &(t,x) \in \mathbb{R} \times \mathbb{R}^n,\ 2 < p < 2 + \frac{4}{n} \notag\\ \shortintertext{and} &i \partial_t u - \Delta^2 u - V(x)u + Q(x)u |u|^{p-2} = 0,\label{intr2}\\ &(t,x) \in \mathbb{R} \times \mathbb{R}^n,\ 2 < p < 2 + \frac{8}{n} \notag\end{align} under suitable assumptions on the potentials $V(x)$ and $Q(x)$.\par More precisely, we assume $V(x)$, $Q(x) \in L^{\infty}(\mathbb{R}^n)$ and $\mathrm{meas} \{Q(x) > \lambda_0 \} \in (0,\infty)$ for a suitable $\lambda_0 > 0$. The main point is the analysis of the compactness of minimizing sequences to suitable constrained minimization problems related to \eqref{intr1} and \eqref{intr2}.