On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
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| Format: | Preprint |
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2024
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| _version_ | 1866909068872908800 |
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| author | Zhao, Min Song, Yueqiang Repovš, Dušan D. |
| author_facet | Zhao, Min Song, Yueqiang Repovš, Dušan D. |
| contents | In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-Δ)_{p, A}^{s} u+V(x)|u|^{p-2} u=λ\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{μ, s}^{*}}}{|x-y|^μ} \mathrm{d}y\right)|u|^{p_{μ, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$$ where $0<s<1<p$, $ps < N$, $p<q<2p^{*}_{s,μ}$, $0<μ<N$, $λ$ and $k$ are some positive parameters, $p^{*}_{s,μ}=\frac{pN-p\fracμ{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity Zhao, Min Song, Yueqiang Repovš, Dušan D. Analysis of PDEs 35J10, 35B99, 35J60, 47G20 In this article, we deal with the following $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-Δ)_{p, A}^{s} u+V(x)|u|^{p-2} u=λ\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{μ, s}^{*}}}{|x-y|^μ} \mathrm{d}y\right)|u|^{p_{μ, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$$ where $0<s<1<p$, $ps < N$, $p<q<2p^{*}_{s,μ}$, $0<μ<N$, $λ$ and $k$ are some positive parameters, $p^{*}_{s,μ}=\frac{pN-p\fracμ{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem. |
| title | On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity |
| topic | Analysis of PDEs 35J10, 35B99, 35J60, 47G20 |
| url | https://arxiv.org/abs/2401.05755 |