The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schr\"odinger equation with magnetic field

Based on some ideas introduced by Benci and Cerami \cite{benci-cerami_calvar}, we obtain an abstract result that establishes a version of the Morse relations. Afterward, we use this result to prove multiplicity of solutions for a nonlinear Schr"odinger equation with an external magnetic field.