Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 05, pp. 1-20.

Title: Multiplicity of symmetric solutions for a nonlinear eigenvalue
       problem in $R^n$

Author: Daniela Visetti (Univ. degli studi di Pisa, Italy)

Abstract: 
  In this paper, we study the nonlinear  eigenvalue field equation
  $$
  -\Delta u+V(|x|)u+\varepsilon(-\Delta_p u+W'(u))=\mu u
  $$
  where $u$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$
  with $n\geq 3$,  $\varepsilon$ is a positive parameter and
  $p>n$.   We find a multiplicity of solutions, symmetric with 
  respect to an action of the orthogonal group $O(n)$:
  For any $q\in\mathbb{Z}$ we prove the existence of finitely many
  pairs $(u,\mu)$ solutions for $\varepsilon$ sufficiently small,
  where $u$ is   symmetric and has topological charge $q$.
  The multiplicity of our solutions can be as large as desired,
  provided that the singular point   of $W$ and $\varepsilon$ are
  chosen accordingly.

Submitted October 22, 2004. Published January 2, 2005.
Math Subject Classifications: 35Q55, 45C05.
Key Words: Nonlinear Schrodinger equations; nonlinear eigenvalue problems.