Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent

We consider nonlinear Choquard equation -Delta u + Vu = (I-alpha * vertical bar u vertical bar(alpha/N + 1))vertical bar u vertical bar(alpha/N - 1)u in R-N, where N >= 3, V is an element of L-infinity(R-N) is an external potential and I-alpha(x) is the Riesz potential of order alpha is an element of (0, N). The power alpha/N + 1 in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if lim inf(vertical bar x vertical bar ->infinity)(1 - V (x))vertical bar x vertical bar(2) > N-2(N-2)/4(N+1) then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.