Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910105181618176 |
|---|---|
| author | Luo, Jiao Yang, Zhipeng |
| author_facet | Luo, Jiao Yang, Zhipeng |
| contents | We study a logarithmic fractional Schrödinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αϕ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here \(α\in\bigl(\frac34,1\bigr)\), \(4<p<2_α^{*}=\frac{6}{3-2α}\), and \(V\) satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a \(C^{1}\) variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed \(δ>0\) and all sufficiently small \(\varepsilon>0\), the system admits at least \(\operatorname{cat}_{M_δ}(M)\) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of \(V\) as \(\varepsilon\to0\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04148 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System Luo, Jiao Yang, Zhipeng Analysis of PDEs 35J20, 35A01, 58E05 We study a logarithmic fractional Schrödinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αϕ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here \(α\in\bigl(\frac34,1\bigr)\), \(4<p<2_α^{*}=\frac{6}{3-2α}\), and \(V\) satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a \(C^{1}\) variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed \(δ>0\) and all sufficiently small \(\varepsilon>0\), the system admits at least \(\operatorname{cat}_{M_δ}(M)\) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of \(V\) as \(\varepsilon\to0\). |
| title | Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System |
| topic | Analysis of PDEs 35J20, 35A01, 58E05 |
| url | https://arxiv.org/abs/2604.04148 |