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The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.

We study the field equation −∆u + V (x)u + ε r (−∆pu + W (u)) = µu on R n , with ε positive parameter. The function W is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for ε sufficiently small, there exists a finite number of solutions (µ(ε), u(ε)) of the eigenvalue problem for any given charge q ∈ Z \ {0}.