Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials

Abstract

We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation (−Δ)su+V(εx)u=λu+h(εx)f(u)inRN,∫RN|u|2dx=a,, where (−Δ)s is the fractional Laplacian, s∈(0,1), a,ε>0, λ∈R is an unknown parameter that appears as a Lagrange multiplier, h:RN→[0,+∞) are bounded and continuous, and f is L2-subcritical. Under some assumptions on the potential V, we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough.