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On the ▫$p$▫-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity [Elektronski vir]
Zhao, Min ; Song, Yueqiang ; Repovš, Dušan, 1954-
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)(-\Delta)_{p, A}^{s} u+V(x)|u|^{p-2} u=\lambda\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{\mu, s}^{*}}}{|x-y|^{\mu}} \mathrm{d}y\right)|u|^{p_{\mu, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$▫ where ▫$0 < s < 1 < p$▫, ▫$ps < N$▫, ▫$p < q < 2p^{*}_{s,\mu}$▫, ▫$0 \mu < N$▫, ▫$\lambda$▫ and ▫$k$▫ are some positive parameters, ▫$p^{*}_{s,\mu}=\frac{pN-p\frac{\mu}{2}}{N-ps}$▫ 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.
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https://www.degruyter.com/document/doi/10.1515/dema-2023-0124/html
