MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR FRACTIONAL NONLINEAR SCHRODINGER EQUATIONS WITH CRITICAL GROWTH

In this article we consider the multiplicity and concentration behavior of positive solutions for the fractional nonlinear Schriidinger equation epsilon(2s)(-Delta)(s)u + V(x)u=u(2s)*(-1) + f (u), x is an element of R-N, u is an element of H-s(R-N), u(x) > 0, where epsilon is a positive parameter, s is an element of (0, 1), N > 2s and 2(s)* = 2N/N-2s is the fractional critical exponent, and f is a C-1 function satisfying suitable assumptions. We assume that the potential V(x) is an element of C(R-N) satisfies inf(RN) V(x) > 0, and that there exits k points x(j) is an element of R-N such that for each j = 1,...,k, V(x(j)) are strictly global minimum. By using the variational method, we show that there are at least k positive solutions for a small epsilon > 0. Moreover, we establish the concentration property of solutions as E tends to zero.