Existence and multiplicity of semiclassical solutions for a Schrödinger equation

We establish the existence and multiplicity of solutions for the semiclassical nonlinear Schrödinger equation{−ε2Δu+V(x)u=g(x,u)for x∈RN,u(x)→0as |x|→∞, where ε>0ε>0 is a small parameter, and the potential V   changes sign and may not be bounded from below, and g(x,u)g(x,u) is asymptotically linear in u   as |u|→∞|u|→∞.