具有倒数可积位势的临界非线性Choquard方程基态解的存在性

Existence of Ground State Solutions for Critical Nonlinear Choquard Equations with Reciprocally Integrable Potentials

摘要: 本文致力于研究Choquard方程$-\Delta u+V\left(x\right)u=\left(I_{\alpha }\ast \left({\left|u\right|}^{\overline{p}}+G\left(u\right)\right)\right)\left({\left|u\right|}^{\overline{p}-2}u+\frac{1}{\overline{p}}g\left(u\right)\right),x\in {ℝ}^N,$ 其中$N\ge 3$ ，$\alpha \in \left(0,N\right)$ ，$G\left(u\right)={\displaystyle {\int }_0^{u}g\left(s\right)ds}$ ，$\overline{p}=\frac{N+\alpha }{N-2}$ 为Hardy-Littlewood-Sobolev上临界指数，$I_{\alpha }$ 是Riesz势。当正位势$V$ 的倒数可积且扰动项$g$ 满足适当条件时，我们利用变分方法证明上述问题存在Nehari型基态解。

Abstract: This paper is devoted to the study of the Choquard equation $-\Delta u+V\left(x\right)u=\left(I_{\alpha }\ast \left({\left|u\right|}^{\overline{p}}+G\left(u\right)\right)\right)\left({\left|u\right|}^{\overline{p}-2}u+\frac{1}{\overline{p}}g\left(u\right)\right),x\in {ℝ}^N,$where $N\ge 3$ , $\alpha \in \left(0,N\right)$ , $G\left(u\right)={\displaystyle {\int }_0^{u}g\left(s\right)ds}$ , $\overline{p}=\frac{N+\alpha }{N-2}$ is the upper critical Hardy-Littlewood-Sobolev exponent, and $I_{\alpha }$ is the Riesz potential. When the positive potential $V$ is reciprocally integrable and the perturbed term $g$ satisfies suitable conditions, we prove the existence of Nehari-type ground state solutions to the above problem by variational methods.