Existence and multiplicity of solutions for the Schrödinger–Poisson equation with prescribed mass

In this paper, we study the existence and multiplicity for the following Schrödinger–Poisson equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u-\kappa (|x|^{-1}*|u|^2)u=f(u),&{}\text {in}~~{\mathbb {R}}^{3},\\ u>0,~\displaystyle \int _{{\mathbb {R}}^{3}}u^2dx=a^2, \end{array}\right. } \end{aligned}$$

where \(a>0\) is a prescribed mass, \(\kappa \in {\mathbb {R}}\setminus \{0\}\) and \(\lambda \in {\mathbb {R}}\) is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case \(\kappa <0\), we obtain the normalized ground state solution for \(a>0\) small by working on the Pohozaev manifold, where f satisfies the \(L^2\)-supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy \(c_a\) is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for \(\kappa >0\) and \(f(u)=|u|^{4}u\), we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the \(L^2\)-constraint setting.