Existence and multiplicity of semiclassical solutions for a Schrِdinger equation

We establish the existence and multiplicity of solutions for the semiclassical nonlinear Schrödinger equation { − ε 2 Δ u + V ( x ) u = g ( x , u ) for x ∈ R N , u ( x ) → 0 as | x | → ∞ , where ε > 0 is a small parameter, and the potential V changes sign and may not be bounded from below, and g ( x , u ) is asymptotically linear in u as | u | → ∞ .