The existence and the concentration behavior of normalized solutions for the $L^2$-critical Schrödinger–Poisson system

Abstract

In this paper, we study the existence and the concentration behavior of critical points for the following functional derived from the Schrödinger–Poisson system:

$$E\left(u\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^3}{\left|\nabla u\right|}^2+\frac{1}{4}{\int }_{{\mathbb{R}}^3}({\left|x\right|}^{-1}\ast u^2)u^2-\frac{3}{10}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{\frac{10}{3}}$$

constrained on the $L^2$-spheres $S\left(c\right)=\{u\in H^1\left({\mathbb{R}}^3\right)|{\left|u\right|}_2=c\}$ when $c>c_{\ast }={\left|Q\right|}_2$, where $Q$ is up to translations, the unique positive of $-\Delta Q+\frac{2}{3}Q={\left|Q\right|}^{\frac{4}{3}}Q$ in ${\mathbb{R}}^3$.

As such constrained problem is $L^2$-critical, $E\left(u\right)$ is unbounded from below on $S\left(c\right)$ when $c>c_{\ast }$ and the existence of critical points constrained on $S\left(c\right)$ is obtained by a mountain pass argument on $S\left(c\right)$. We show that there exists $c_1>{\left(\frac{9}{7}\right)}^{\frac{3}{4}}{c}_{\ast }$ such that $E\left(u\right)$ has at least one positive critical point restricted to $S\left(c\right)$ for $c_{\ast }<c\le c_1$. As $c$ approaches $c_{\ast }$ from above, we obtain that the critical point $u_c$ behaves like

$${\left(\frac{1}{{\left|\nabla u_c\right|}_2}\right)}^{\frac{3}{2}}{u}_c\left(\frac{1}{{\left|\nabla u_c\right|}_2}(x+y_c)\right)\approx {\left(\frac{1}{c_{\ast }}\right)}^{\frac{3}{2}}Q\left(\left(\frac{1}{c_{\ast }}\right)x\right)$$

for some $y_c\in {\mathbb{R}}^3$.