On translation invariant constrained minimization problems with application to Schr\"odinger-Poisson equation

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} \ 2<p<3. When $2<p<3$, In particular we prove that $I$ achieves its minimum on the constraint $\{u\in H^{1}(\R^{3}): \|u\|_{2}=\rho\}$ for every $\rho>0$ 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.