Solitary Waves Solutions of a Nonlinear Schrödinger Equation

Abstract

The aim of this note is to prove the existence of standing waves solutions of the following nonlinear Schrödinger equation

$$ i\frac{{\partial \psi }} {{\partial t}} = - \Delta \psi + V(x)\psi + \varepsilon N(\psi ), $$

where NM is a nonlinear differential operator. In [8] and [9] Benci and the authors proved the existence of a finite number of solutions (µ(e), u(e)) of the eigenvalue problem

$$ - \Delta u + V\left( x \right)u + \varepsilon N\left( u \right) = \mu u $$

(Pε)

where N(u) = −Δ _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.