Multi-peak semiclassical bound states for Fractional Schrodinger Equations with fast decaying potentials

We study the following fractional Schrodinger equation epsilon(2s)(-Delta)(s)u + V(x)u = f(u), x is an element of R-N, where s is an element of (0, 1). Under some conditions on f(u), we show that the problem has a family of solutions concentrating at any finite given local minima of V provided that V is an element of C(R-N, [0, +infinity)). All decay rates of V are admissible. Especially, V can be compactly supported. Different from the local case s = 1 or the case of single-peak solutions, the nonlocal effect of the operator (-Delta)(s) makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.