On translation invariant constrained minimization problems with application to Schrödinger-Poisson equation

Details On translation invariant constrained minimization problems with application to Schrödinger-Poisson equation On translation invariant constrained minimization problems with application to Schrödinger-Poisson equation

2010-07-23

arxiv.org/abs/1007.4139v2

Mathematics

Analysis of PDEs

This paper has been withdrawn by the author due to an error in the Application section

Scientific paper

In this paper we study the existence of minimizers for a class of constrained minimization problems that are invariant under translations. We call $$I_{\rho^{2}}:=\inf_{B_{\rho}}I(u) \ $$ where $B_{\rho}=\{u\in H^{m}(\R^{N}):\|u\|_{2}=\rho\},$ and $I(u)=1/2\|u\|^{2}_{D^{m,2}}+T(u)$, $T$ fulfilling general assumptions. We show that the regularity of the function $$(0,\infty)\ni s \mapsto I_{s^2}\,$$ and the behaviour of $\frac{I_{s^2}}{s^2}$ in the neighborhood of zero allows to prove the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional $I$ associated to the Schr\"odinger-Poisson equation in $\R^{3}$ orbitally stable standing waves with arbitray charge for the following Schr\"odinger-Poisson type equation \label{SP} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \text{in} \R^{3}, \text{with} \ 20$ and, as a consequence, the set of minimizers is orbitally stable. This covers the physically relevant case, $p=8/3$, the so called Schr\"odinger-Poisson-Slater system.

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