CONCENTRATION PHENOMENON FOR FRACTIONAL NONLINEAR SCHRODINGER EQUATIONS

We study the concentration phenomenon for solutions of the fractional nonlinear Schrodinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation (-epsilon(2)Delta)(epsilon) v + V-v - vertical bar v vertical bar(alpha)v = 0 in R-n, (1) where n = 1, 2, 3, max{1/2, n/4} < s < 1, 1 <= alpha < alpha(*)(s, n), V is an element of C-b(3)(R-n). Here the exponent alpha(*)(s, n) = 4s/n-2s for 0 < s < and n/2 and alpha(*)(s, n) = infinity for s >= n/2. Then for each non-degenerate critical point z(0) of V, there is a nontrivial solution of equation (1) concentrating to z(0) as epsilon -> 0.