A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality

In this paper, we are concerned with the following nonlinear Choquard equation -Delta u + V(x)u = (integral(RN) G(y,u)/vertical bar x-y vertical bar(mu)dy) g(x, u) in R-N, where N >= 4, 0 < mu < N and G(x, u) = integral(u)(0) g(x, s)ds. If 0 lies in a gap of the spectrum of -Delta+V and g(x, u) is of critical growth due to the Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrodinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423-443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179-186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557-6579].