On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity

Abstract

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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof