Electron. J. Diff. Eqns., Vol. 2005(2005), No. 05, pp. 1-20.

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

Daniela Visetti

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 $p$ greater than $n$. 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.

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Daniela Visetti
Dipartimento di Matematica Applicata
"U. Dini", Università degli studi di Pisa
via Bonanno Pisano 25/B, 56126 Pisa, Italy
email: visetti@mail.dm.unipi.it

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