Existence of a Positive Solution for a Class of Choquard Equation with Upper Critical Exponent

Abstract

In this paper, we investigate the existence of nontrivial solution for the following class of Choquard equation where $$N\in {\mathbb {N}},~N\ge 3,~\alpha \in (0,N),~I_\alpha $$ is a Riesz potential, $$\lambda >0$$ is a parameter, $$p=\frac{N+\alpha }{N-2}$$ is the upper Hardy–Littlewood–Sobolev critical exponent and $$q\in (2,\frac{2N}{N-2}).$$ We prove that there exists $$\lambda _0>0$$ such that for $$\lambda \ge \lambda _0,$$ problem (1) possesses one positive radial solution.