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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.12294 |
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| _version_ | 1866908469087436800 |
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| author | de Miranda, Luís Henrique Santana, Ayana Pinheiro de Castro |
| author_facet | de Miranda, Luís Henrique Santana, Ayana Pinheiro de Castro |
| contents | In this paper we prove existence and regularity of weak solutions for the following system
\begin{align*}
\begin{cases}
&-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla u|^{p-2}\nabla u\Bigg) + g(x,u,v)=f \ \ \ \mbox{in} \ Ω;
&-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla v|^{p-2}\nabla v\Bigg) = h(x,u,v) \ \ \ \ \mbox{in} \ Ω;
&u=v=0 \ \mbox{on} \ \partialΩ.
\end{cases}
\end{align*}
where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $N>2$, $f\in L^m(Ω)$, where $m>1$ and $g$, $h$ are two Carathéodory functions, which may be non monotone. We prove that under appropriate conditions on $g$ and $h$, there is gain of Sobolev and Lebesgue regularity for the solutions of this system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularizing Effect for a Nonlocal Maxwell-Schrödinger System de Miranda, Luís Henrique Santana, Ayana Pinheiro de Castro Analysis of PDEs 35B65, 35D30, 35B45, 35D99 In this paper we prove existence and regularity of weak solutions for the following system \begin{align*} \begin{cases} &-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla u|^{p-2}\nabla u\Bigg) + g(x,u,v)=f \ \ \ \mbox{in} \ Ω; &-\mbox{div}\Bigg(\bigg(\|\nabla u\|^{p}_{L^{p}}+\|\nabla v\|^{p}_{L^{p}}\bigg)|\nabla v|^{p-2}\nabla v\Bigg) = h(x,u,v) \ \ \ \ \mbox{in} \ Ω; &u=v=0 \ \mbox{on} \ \partialΩ. \end{cases} \end{align*} where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $N>2$, $f\in L^m(Ω)$, where $m>1$ and $g$, $h$ are two Carathéodory functions, which may be non monotone. We prove that under appropriate conditions on $g$ and $h$, there is gain of Sobolev and Lebesgue regularity for the solutions of this system. |
| title | Regularizing Effect for a Nonlocal Maxwell-Schrödinger System |
| topic | Analysis of PDEs 35B65, 35D30, 35B45, 35D99 |
| url | https://arxiv.org/abs/2507.12294 |