Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916912054665216 |
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| author | Luo, Jiao Yang, Zhipeng |
| author_facet | Luo, Jiao Yang, Zhipeng |
| contents | This paper concerns the existence of multiple solutions for the fractional logarithmic Schrödinger-Possion system of the form
\begin{equation*}
\begin{cases}
{\varepsilon}^{2α} (-Δ)^αu+V(x) u+ϕu=u \log u^{2}+u^{q-1}, & \text{in}\quad \mathbb{R}^{3},
{\varepsilon}^{2α} (-Δ)^αϕ=u^2, & \text{in}\quad \mathbb{R}^{3}.
\end{cases}
\end{equation*} where $\varepsilon>0$ is a small parameter, $q \in (4, 2_α^*)$ with $α\in(\frac{3}{4},1)$, $V: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is a continuous function that satisfies some local potential hypothesis. By introducing a new Banach space, the energy functional become $C^{1}$, which create the conditions for studying the multiplicity of solutions involving Lusternik-Schnirelmann category. We prove that for $\varepsilon>0$ small enough, the system has a positive ground state solution and each positive solution concentrates around a local minimum point of $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method Luo, Jiao Yang, Zhipeng Analysis of PDEs This paper concerns the existence of multiple solutions for the fractional logarithmic Schrödinger-Possion system of the form \begin{equation*} \begin{cases} {\varepsilon}^{2α} (-Δ)^αu+V(x) u+ϕu=u \log u^{2}+u^{q-1}, & \text{in}\quad \mathbb{R}^{3}, {\varepsilon}^{2α} (-Δ)^αϕ=u^2, & \text{in}\quad \mathbb{R}^{3}. \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $q \in (4, 2_α^*)$ with $α\in(\frac{3}{4},1)$, $V: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is a continuous function that satisfies some local potential hypothesis. By introducing a new Banach space, the energy functional become $C^{1}$, which create the conditions for studying the multiplicity of solutions involving Lusternik-Schnirelmann category. We prove that for $\varepsilon>0$ small enough, the system has a positive ground state solution and each positive solution concentrates around a local minimum point of $V$. |
| title | Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.16229 |