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.