Solutions of the fractional Schrödinger equation with a sign-changing nonlinearity

We look for solutions to a nonlinear, fractional Schrödinger equation(−Δ)α/2u+V(x)u=f(x,u)−Γ(x)|u|q−2u on RN, where the potential V is coercive or V=Vper+Vloc is a sum of a periodic in x potential Vper and a localized potential Vloc, Γ∈L∞(RN) is periodic in x, Γ(x)≥0 for a.e. x∈RN and 2