Solitary waves solutions of a nonlinear Schrodinger equation

The aim of this note is to prove the existence of standing waves solutions of the following nonlinear Schrodinger equation iphipsi/phit = -Deltapsi + V(x)psi + epsilonN(psi), where N(psi) is a nonlinear differential operator. In [8] and [9] Benci and the authors proved the existence of a finite number of solutions (mu(epsilon), u(epsilon)) of the eigenvalue problem (Pepsilon) -Deltau + V(x)u + epsilonN(u) = muu where N(u) = -Delta(p)u + W'(u). The number of solutions can be as large as one wants. Since W is singular in a point these solutions are characterized by a topological invariant, the topological charge. A min-max argument is used.