Bound states of fractional Choquard equations with Hardy-Littlewood-Sobolev critical exponent.

We deal with the following fractional Choquard equation (− Δ) s u + V (x) u = (I μ ⁎ | u | 2 μ , s ⁎ ) | u | 2 μ , s ⁎ − 2 u , x ∈ R N , where I μ (x) is the Riesz potential, s ∈ (0 , 1) , 2 s < N ≠ 4 s , 0 < μ < min ⁡ { N , 4 s } and 2 μ , s ⁎ = 2 N − μ N − 2 s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V (x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V (x) vanishes at infinity. [ABSTRACT FROM AUTHOR]