The sharp existence of constrained minimizers for the $L^2$-critical Schrödinger-Poisson system and Schrödinger equations

Abstract

In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schrödinger-Poisson equations: $$-\Delta u+V(x)u+(|x{|}^{-1}\ast u^2)u-|u{|}^{\frac{4}{3}}u=\lambda u,x\in {\mathbb{R}}^3$$ on the $L^2$-spheres $\tilde{S}(c)=\{u\in H^1({\mathbb{R}}^3)|{\int }_{{\mathbb{R}}^3}V(x)u^{2}dx<+\infty ,|u{|}_2^2=c>0\}$. If $V(x)\equiv 0$, 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 $0\le V(x)\in L_{loc}^{\infty }({\mathbb{R}}^3)$ and $\underset{|x|\to +\infty }{\lim }V(x)=+\infty$, then a minimizer exists if and only if $0<c\le c^{\ast }=|Q{|}_2^2$, where $Q$ is the unique positive radial solution of $-\Delta u+u=|u{|}^{\frac{4}{3}}u,$ $x\in {\mathbb{R}}^3$. Our results are sharp. We also extend some results to constrained minimization problems on $\tilde{S}(c)$ derived from Schrödinger operators: $$F_μ(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2-\fracμ2\ds\int_{\R^N}V(x)u^2-\frac{N}{2N+4}|u|^\frac{2N+4}{N}$$ where $0\le V(x)\in L_{loc}^{\infty }({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }V(x)=0$. We show that if $\mu >{\mu }_1$ for some ${\mu }_1>0$, then a minimizer exists for each $c\in (0,c^{\ast })$.