Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

This paper deals with the following Choquard equation with a local nonlinear perturbation: {-Delta u + u = (I-a* vertical bar u vertical bar(a/2)-1u + f(u), x is an element of R-2; u is an element of H-1(R-2), where alpha is an element of (0, 2), I-alpha : R-2 -> R is the Riesz potential and f is an element of C(R, R) is of critical exponential growth in the sense of Trudinger-Moser. The exponent alpha/2 + 1 is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent alpha/2 + 1 and the critical exponential growth of f (u).