一类非自治NLS的轨道稳定性
On the orbital stability for a class of nonautonomous NLS
摘要：
根据T. Cazenave和P.L.在\cite{CaLi}中，我们证明了下列NLS类驻波的存在性和轨道稳定性：\label{intr1} i \partial_t u+ Δ u-V（x）u + Q（x）u |u| ^{p-2}=0，\hbox{}（t，x）\in \R \times \R^n，\hbox{} 2<p<2+ \frac 4 n且\label{intr2} i \partial_t u - Δ^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。更精确地，我们假设$V(x), Q(x) \in L^\infty(\R^n)$和$meas\{Q(x)>λ_0\}\in (0,\infty)$为合适的$λ_0>0$。本文的重点是分析极小化序列对于与\eqref{intr1}和\eqref{intr2}有关的适当约束极小化问题的紧性。
英文摘要：
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+ Δu - 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 - Δ^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 $meas\{Q(x)>λ_0\}\in (0,\infty)$ for a suitable $λ_0>0$. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to \eqref{intr1} and \eqref{intr2}.
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