Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities

We study the existence of standing waves of fractional Schrodinger equations with a potential term and a general nonlinear term: iut-(-triangle)su-V(x)u+f(u) =0,(t,x)is an element of R+xR(N) where s is an element of(0, 1),N>2sis an integer and V(x)<= 0 is radial. More precisely, we investigate the minimizing problem withL2-constraint E(alpha) =inf{12 integral R-N|(-triangle)s2u|(2)+V(x)|u|(2)-2F(|u|)divided by divided by divided by divided by u is an element of H-s(R-N),& Vert;u & Vert;2L2(R-N)=alpha} Under general assumptions on the nonlinearity termf(u)and the potential termV(x),we prove that there exists a constant alpha 0 >= 0 such that E(alpha)can be achieved for all alpha>alpha 0, and there is no global minimizer with respect toE(alpha)for all 0<alpha<alpha 0.Moreover, we propose some criteria determining alpha 0=0 or alpha 0>0