SEMICLASSICAL GROUND STATE SOLUTIONS FOR A CHOQUARD TYPE EQUATION IN R2 WITH CRITICAL EXPONENTIAL GROWTH

In this paper we study a nonlocal singularly perturbed Choquard type equation -epsilon(2)Delta u + V(x)u = epsilon(mu-2) [1/vertical bar x vertical bar(mu) * (P(x) G(u))] P(x)g(u) in R-2, where epsilon is a positive parameter, 1/vertical bar x vertical bar(mu) with 0 < mu < 2 is the Riesz potential, * is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in R-2 in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.