Normalized solutions for a fractional Schrödinger-Poisson system with critical growth
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2024
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| author | He, Xiaoming Meng, Yuxi Squassina, Marco |
| author_facet | He, Xiaoming Meng, Yuxi Squassina, Marco |
| contents | In this paper, we study the fractional critical Schrödinger-Poisson system \[\begin{cases} (-Δ)^su +λϕu= αu+μ|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-Δ)^tϕ=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $λ,μ>0$ parameters and $α\in{\mathbb R}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for a fractional Schrödinger-Poisson system with critical growth He, Xiaoming Meng, Yuxi Squassina, Marco Analysis of PDEs 35J62, 35J50, 35B65 In this paper, we study the fractional critical Schrödinger-Poisson system \[\begin{cases} (-Δ)^su +λϕu= αu+μ|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-Δ)^tϕ=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $λ,μ>0$ parameters and $α\in{\mathbb R}$ 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. |
| title | Normalized solutions for a fractional Schrödinger-Poisson system with critical growth |
| topic | Analysis of PDEs 35J62, 35J50, 35B65 |
| url | https://arxiv.org/abs/2402.00464 |