Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913564416016384 |
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| author | Sbai, Abdelaaziz hadfi, Youssef El Ouardy, Mounim El |
| author_facet | Sbai, Abdelaaziz hadfi, Youssef El Ouardy, Mounim El |
| contents | In this paper we are going to prove existence for positive solutions of the following Schrödinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(Ω):-\operatorname{div}\left(a(x) \nabla u\right)+ψ|u|^{r-2} u=\frac{f(x)}{u^θ}, ψ\in W_{0}^{1,2}(Ω):-\operatorname{div}(M(x) \nabla ψ)=|u|^{r} \end{array}\right. \end{equation} where $Ω$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $ψ>0,$ $0 < θ<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_08899 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity Sbai, Abdelaaziz hadfi, Youssef El Ouardy, Mounim El Analysis of PDEs In this paper we are going to prove existence for positive solutions of the following Schrödinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(Ω):-\operatorname{div}\left(a(x) \nabla u\right)+ψ|u|^{r-2} u=\frac{f(x)}{u^θ}, ψ\in W_{0}^{1,2}(Ω):-\operatorname{div}(M(x) \nabla ψ)=|u|^{r} \end{array}\right. \end{equation} where $Ω$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $ψ>0,$ $0 < θ<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions. |
| title | Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2110.08899 |