On the number of solutions of NLS equations with magnetics fields in expanding domains

In this paper we look for multiple weak solutions u:Ωλ→C for the complex equation in Ωλ=λΩ. The set Ω⊂RN is a smooth bounded domain, λ>0 is a parameter, A is a regular magnetic field and f is a superlinear function with subcritical growth. Our main result relates, for large values of λ, the number of solutions with the topology of the set Ω. In the proof we apply minimax methods and Ljusternik–Schnirelmann theory.