The sharp existence of constrained minimizers for the $L^2$-critical Schrödinger-Poisson system and Schrödinger equations
In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schrödinger-Poisson equations: $$-Δu+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=λu,~~x\in\R^3$$ 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 $0 μ_1$ for some $μ_1>0$, then a minimizer exists for each $c\in(0,c^*)$.
math.AP↗