Calculus of Variations and Geometric Measure Theory

X. He - Y. Meng - M. Squassina

Normalized solutions for a fractional Schrodinger-Poisson system with critical growth

created by squassina on 01 Feb 2024



Inserted: 1 feb 2024
Last Updated: 1 feb 2024

Pages: 43
Year: 2024


In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu
^{2^*_s-2}u,&~~ \mbox{in}~\mathbb R^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~\mathbb R^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3}
^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $\lambda,\mu>0$ parameters and $\alpha\in\mathbb R$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.