| Abstract: |
| We consider the following equation
$$-\Delta u=K(|x`|,x``)\left(|x|^{-\alpha}\ast K(|x`|,x``)|u|^{2_{\alpha}^{\ast}}\right)|u|^{2_{\alpha}^{\ast}-2}u$$
in $\mathbb{R}^N$, where $N\geq 5$ and $2_{\alpha}^{\ast}$ is the so-called upper critical exponent in
the Hardy-Littlewood-Sobolev inequality and $K(|x`|, x``)$ with $(x`, x``)\in $\mathbb{R}^2\times $\mathbb{R}^{N-2}$, is bounded and non-negative. 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 Pohozaev identities. It is a remarkable fact that
the order $\alpha$ of the Riesz potential influences the existence/non-existence of solutions. |
|