Multiple positive solutions for a~Schr\"odinger--Poisson system with critical and supercritical growths

J. Lei ; H. Suo

Izvestiya. Mathematics , Tome 87 (2023) no. 1, pp. 29-44

Résumé

In this paper, we are concerned with the following Schrödinger–Poisson system $$ \begin{cases} -\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu \dfrac{|x|^\beta}{1+|x|^\beta}|u|^{q-2}u\text{in }\mathbb{R}^3, \\ -\Delta \phi=u^{2} \text{in }\mathbb{R}^3, \end{cases} $$ where $0 \beta3$,  $6$, $Q(x)$ is a positive continuous function on $\mathbb{R}^3$, $\lambda,\mu>0$ are real parameters. By the variational method and the Nehari method, we obtain that the system has $k$ positive solutions.

Détail

Keywords: Schrödinger–Poisson system, critical exponent, supercritical growth.