SEMICLASSICAL STATES FOR FRACTIONAL CHOQUARD EQUATIONS WITH CRITICAL GROWTH

In this paper, we are concerned with fractional Choquard equation epsilon(2 alpha)(-Delta)(alpha)u + V (x) u =epsilon(mu-3) (integral(3)(R) vertical bar u(y)vertical bar(2)*(mu,alpha) + F(u(y)) / vertical bar x - y vertical bar(mu) dy) (vertical bar u vertical bar(2)*(mu,alpha) -2 u + 1/2*(mu,alpha) f(u)) in R-3 where epsilon > 0 is a parameter, 0 < alpha < 1, 0 < mu < 3, 2*(mu,alpha) = 6 - mu / 3- 2 alpha is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator, f is a continuous subcritical term, and F is the primitive function of f. By virtue of the method of Nehari manifold and Ljusternik-Schnirelmann category theory, we prove that the equation has a ground state for epsilon small enough and investigate the relation between the number of solutions and the topology of the set where V attains its global minimum for small epsilon. We also obtain sufficient conditions for the nonexistence of ground states.