Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent

In this paper, we first prove that each positive solution of is radially symmetric, monotone decreasing about some point and has the form where 0 < a < N if N = 3 or 4, and N - 4 = a < N if N = 5, 2 * a := N+ a N- 2 is the upper Hardy- Littlewood- Sobolev critical exponent, t > 0 is a constant and ca > 0 depends only on a and N. Based on this uniqueness result, we then study the following nonlinear Choquard equation By using Lions' Concentration- Compactness Principle, we obtain a global compactness result, i. e. wegive a complete description for the Palais- Smale sequences of the corresponding energy functional. Adopting this description, we are succeed in proving the existence of at least one positive solution if V( x) L N2 is suitable small. This result generalizes the result for semilinear Schrodinger equation by Benci and Cerami ( J Funct Anal 88: 90- 117, 1990) to Choquard equation.