Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods

This paper deals with the following equation $$-\mathrm{\Delta }u=K(\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid },x^{\mathrm{\prime }\mathrm{\prime }})(\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }\ast (K(\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid },x^{\mathrm{\prime }\mathrm{\prime }})\mathrm{\mid }u{\mathrm{\mid }}^{2_{\alpha }^{\ast }}))\mathrm{\mid }u{\mathrm{\mid }}^{2_{\alpha }^{\ast }-2}u\text{quadin}{\mathbb{R}}^N,$$ where $N\ge 5$, $\alpha >5-\frac{6}{N-2}$, $2_{\alpha }^{\ast }=\frac{2N-\alpha }{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid },x^{\mathrm{\prime }\mathrm{\prime }})$, where $(x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.