ABSTRACT

In this paper, we investigate the existence of ground state solutions for fractional Schrödinger–Choquard–Kirchhoff type equations with critical growth $\left\{\begin{array}{l}\left(a+b{\int }_{{\mathbb{R}}^N}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u{|}^2\mathrm{d}x\right)(-\mathrm{\Delta }{)}^{s}u+V(x)u\\ =\lambda f(x,u)+[|x{|}^{-\mu }\ast |u{|}^{2_{\mu ,s}^{\ast }}]|u{|}^{2_{\mu ,s}^{\ast }-2}u,x\in {\mathbb{R}}^N,\\ u\in H^s\left({\mathbb{R}}^N\right),\end{array}\right.$ where a, b>0 are constants, $\lambda >0$ is a parameter, 0<s<1, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order s, N>2s, $0<\mu <2s$ and $2_{\mu ,s}^{\ast }=\frac{2N-\mu }{N-2s}$. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.

中文翻译：

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具有临界增长的分数 Schrödinger-Choquard-Kirchhoff 型方程的基态解

摘要

在本文中，我们研究了具有临界增长的分数 Schrödinger-Choquard-Kirchhoff 型方程的基态解的存在性$\left\{\begin{array}{l}\left(\mathrm{一个}+b{\int }_{{\mathbb{R}}^{ñ}}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}你{|}^2\mathrm{d}X\right)(-\mathrm{\Delta }{)}^s你+五(X)你\\ =\lambda F(X,你)+[|X{|}^{-\mu }*|你{|}^{2_{\mu ,s}^{*}}]|你{|}^{2_{\mu ,s}^{*}-2}你,X\in {\mathbb{R}}^{ñ},\\ 你\in H^s\left({\mathbb{R}}^{ñ}\right),\end{array}\right.$其中a , b >0 是常数，$\lambda >0$是一个参数，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$2_{\mu ,s}^{*}=\frac{2ñ-\mu }{ñ-2s}$. 当V和f在x中渐近周期性时，我们通过 Nehari 方法证明了该方程对于大λ具有基态解。