Elliptic problem involving finite many critical exponents in ${\mathbb{R}}^N$

Abstract

In this paper, we consider the following problem $$-\Delta u-\zeta \frac{u}{|x{|}^2}=\sum _{i=1}^k\left({\int }_{{\mathbb{R}}^N}\frac{|u{|}^{2_{{\alpha }_i}^{\ast }}}{|x-y{|}^{{\alpha }_i}}\mathrm{d}y\right)|u{|}^{2_{{\alpha }_i}^{\ast }-2}u+|u{|}^{2^{\ast }-2}u,\mathrm{in}{\mathbb{R}}^N,$$ where $N⩾3$, $\zeta \in (0,\frac{(N-2{)}^2}{4})$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent, and $2_{{\alpha }_i}^{\ast }=\frac{2N-{\alpha }_i}{N-2}$ ($i=1,\dots ,k$) are the critical Hardy--Littlewood--Sobolev upper exponents. The parameters ${\alpha }_i$ ($i=1,\dots ,k$) satisfy some suitable assumptions. By using Coulomb--Sobolev space, endpoint refined Sobolev inequality and variational methods, we establish the existence of nontrivial solutions. Our result generalizes the result obtained by Yang and Wu [Adv. Nonlinear Stud. (2017)].