Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation

$-\Delta u={\int }_{{\mathbb{R}}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|x-y\right|}^{\alpha }}dy{\left|u\left(x\right)\right|}^{p-2}u\left(x\right)\text{in}{\mathbb{R}}^N,$

where N ≥ 3, 0 < α < {4,N}. Suppose that 2 < p < $\frac{2N-\alpha }{N-2}$, we will show that this problem does not possess nontrivial solution with finite Morse index. While for $p=\frac{2N-\alpha }{N-2}$, if i(u) < ∞ then we have${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{{\left|u\left(x\right)\right|}^p{\left|u\left(y\right)\right|}^p}{{\left|x-y\right|}^{\alpha }}\text{d}x\text{d}y<\infty$ and ${\int }_{{\mathbb{R}}^N}{\left|?u\right|}^2\text{d}x={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{{\left|u\left(x\right)\right|}^p{\left|u\left(y\right)\right|}^p}{{\left|x-y\right|}^{\alpha }}\text{d}x\text{d}y$.