ABSTRACT

In this paper, we study the following fractional Choquard equation with critical or supercritical growth $(-\mathrm{\Delta }{)}^{s}u+V(x)u=[|x{|}^{-\mu }\ast F\left(u\right)\left]f\right(u)+\lambda [|x{|}^{-\mu }\ast |u{|}^p]p|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ where 0<s<1, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order s, N>2s, $0<\mu <2s$ and $p\ge 2_{\mu ,s}^{\ast }:=(2N-\mu )/(N-2s)$. Under some suitable conditions, we prove that the equation has a nontrivial solution for small $\lambda >0$ by the variational method.

中文翻译：

---

具有临界或超临界增长的分数 Choquard 方程的非平凡解的存在性

摘要

在本文中，我们研究了以下具有临界或超临界增长的分数 Choquard 方程$(-\mathrm{\Delta }{)}^s你+五(X)你=[|X{|}^{-\mu }*F\left(你\right)\left]F\right(你)+\lambda [|X{|}^{-\mu }*|你{|}^p]p|你{|}^{p-2}你,X\in {\mathbb{R}}^{ñ},$其中 0< s <1，$(-\mathrm{\Delta }{)}^s$表示s阶的分数拉普拉斯算子，N >2 s，$0<\mu <2s$和$p\ge 2_{\mu ,s}^{*}:=(2ñ-\mu )/(ñ-2s)$. 在一些合适的条件下，我们证明了该方程对于小$\lambda >0$通过变分法。