一类非线性Schroedinger-泊松方程的稳定驻波
Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
摘要：
本文证明了Schrödinger-Poisson型方程$L^2$ %{ll} i_t}+ Δ τ-（|x| ^{-1}*| ψ| ^{2}）+| ψ| ^{p-2} Δ ε =0 \text{in} \R^{3}，%-Δ ε =| ψ| ^{2}和\text{in} \R^{3}，%。当$p\in \{8/3\}\cup (3,10/3)$。在$3<p<10/3$的情况下，我们证明了存在性和稳定性，只有足够大的$L^2$ -范数。在$p=8/3$的情况下，我们的方法恢复了Sanchez和Soler \cite{SS} %关于足够小的电荷的存在性和稳定性的结果。重点是对相关的约束极小化问题的极小化序列的紧性进行分析。在最后一节的进一步应用薛定谔方程涉及的双调和算子。
英文摘要：
We prove the existence of orbitally stable standing waves with prescribed $L^2$-norm for the following Schrödinger-Poisson type equation \label{intro} %{%{ll} iψ_{t}+ Δψ- (|x|^{-1}*|ψ|^{2}) ψ+|ψ|^{p-2}ψ=0 \text{in} \R^{3}, %-Δϕ= |ψ|^{2}& \text{in} \R^{3},%. when $p\in \{8/3\}\cup (3,10/3)$. In the case $3<p<10/3$ we prove the existence and stability only for sufficiently large $L^2$-norm. In case $p=8/3$ our approach recovers the result of Sanchez and Soler \cite{SS} %concerning the existence and stability for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In a final section a further application to the Schrödinger equation involving the biharmonic operator is given.
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