Normalized Solutions for Schrödinger–Poisson Systems Involving Critical Sobolev Exponents

Abstract

In this paper, we are concerned with the existence and properties of ground states for the Schrödinger–Poisson system with combined power nonlinearities

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u +\gamma \phi u= \lambda u+\mu |u|^{q-2}u+|u|^{4}u,&{}~~ \text{ in }~{\mathbb {R}}^3,\\ -\Delta \phi =u^2,&{}~~ \text{ in }~{\mathbb {R}}^3,\end{array}\right. } \end{aligned}$$

having prescribed mass

$$\begin{aligned} \int _{{\mathbb {R}}^3} |u|^2dx=a^2, \end{aligned}$$

in the Sobolev critical case. Here \( a>0\), and \(\gamma >0\), \(\mu >0\) are parameters, \(\lambda \in {\mathbb {R}}\) is an undetermined parameter. By using Jeanjean’ theory, Pohozaev manifold method and Brezis and Nirenberg’s technique to overcome the lack of compactness, we prove several existence results under the \(L^2\)-subcritical, \(L^2\)-critical and \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), under different assumptions imposed on the parameters \(\gamma ,\mu \) and the mass a, respectively. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions of a Sobolev critical Schrödinger–Poisson problem perturbed with a subcritical term in the whole space \({\mathbb {R}}^3\).