Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

In the present paper, we consider the following singularly perturbed problem: {-epsilon(2)Delta u + V(x)u - epsilon(2)Delta(u(2))u = epsilon(-alpha)(I-alpha * G(u))g(u), x is an element of R-N; u is an element of H-1(R-N), where epsilon > 0 is a parameter, N >= 3, alpha is an element of (0, N), G(t) = integral(t)(0)g(s) ds, I-alpha : R-N -> R is the Riesz potential, and V is an element of C(R-N, R) with 0 < min(x is an element of RN) V(x) < lim(vertical bar y vertical bar ->infinity) V(y). Under the general Berestycki-Lions assumptions on g, we prove that there exists a constant epsilon(0) > 0 determined by V and g such that for e. (0, e0] the above problem admits a semiclassical ground state solution (u) over cap (epsilon) with exponential decay at infinity. We also study the asymptotic behavior of {(u) over cap (epsilon)} as epsilon -> 0.