The sharp existence of constrained minimizers for the $L^2$-critical Schr\"{o}dinger-Poisson system and Schr\"{o}dinger equations

Abstract

In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schr\"{o}dinger-Poisson equations: $$-\Delta u+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=\lambda u,~~x\in\R^3$$ on the $L^2$-spheres $\widetilde{S}(c)=\{u\in H^1(\R^3)|~\int_{\R^3}V(x)u^2dx<+\infty,~|u|_2^2=c>0\}$. If $V(x)\equiv0$, then by a different method from Jeanjean and Luo [Z. Angrew. Math. Phys. 64 (2013), 937-954], we show that there is no minimizer for all $c>0$; If $0\leq V(x)\in L^{\infty}_{loc}(\R^3)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$, then a minimizer exists if and only if $0<c\leq c^*=|Q|_2^2$, where $Q$ is the unique positive radial solution of $-\Delta u+u=|u|^{\frac{4}{3}}u,$ $x\in\R^3$. Our results are sharp. We also extend some results to constrained minimization problems on $\widetilde{S}(c)$ derived from Schr\"{o}dinger operators: $$F_\mu(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2-\frac{\mu}2\ds\int_{\R^N}V(x)u^2-\frac{N}{2N+4}|u|^\frac{2N+4}{N}$$ where $0\leq V(x)\in L^{\infty}_{loc}(\R^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=0$. We show that if $\mu>\mu_1$ for some $\mu_1>0$, then a minimizer exists for each $c\in(0,c^*)$.