Nontrivial Solutions for Fractional Schrödinger Equations with Electromagnetic Fields and Critical or Supercritical Growth

In this paper, we study the following fractional Schr & ouml;dinger equation with electromagnetic fields and critical or supercritical growth(-Delta)(s)(A)u + V (X) u = lambda|u|(p-2) u + f (x,|u|(2)) u, x is an element of R-N,where (-Delta)(s)(A) is the fractional magnetic operator with 0< s < 1, N > 2s, 2(s)(& lowast;)=2N/N-2s, lambda > 0, V is an element of C (R-N, R) and A is an element of C (R-N,R-N) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, and f is a continuous function and there exists 2 < q < 2(s)(& lowast;) such that |f (x,t)| <= C(1+|t|q-2/2) for all (x, t), for 2(s)(& lowast; )<= p < 22(s)(& lowast;)-q. For any D >0 fixed, if lambda is an element of (0,D] we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior D-infinity-estimate.