Translator Disclaimer
May/June 2011 Magnetostatic solutions for a semilinear perturbation of the Maxwellequations
Teresa D'Aprile, Gaetano Siciliano
Adv. Differential Equations 16(5/6): 435-466 (May/June 2011).

Abstract

In this paper we consider a model introduced in [3] which describes the interaction between the matter and the electromagnetic field from a unitarian standpoint. This model is based on a semilinear perturbation of the Maxwell equations and, in the magnetostatic case, reduces to the following nonlinear elliptic degenerate equation: $$\nabla \times (\nabla \times {\bf A})=W'(|{\bf A}|^2){\bf A},$$ where "$\nabla\times$" is the curl operator, $W:\mathbb R\to\mathbb R$ is a suitable nonlinear term, and ${\bf A}:\mathbb R^3\to\mathbb R^3$ is the gauge potential associated with the magnetic field ${\bf H}$. We prove the existence of a nontrivial finite energy solution with a kind of cylindrical symmetry. The proof is carried out by using a suitable variational framework based on the Hodge decomposition, which is crucial in order to handle the strong degeneracy of the equation. Moreover, the use of a natural constraint and a concentration-compactness argument are also required.

Citation

Download Citation

Teresa D'Aprile. Gaetano Siciliano. "Magnetostatic solutions for a semilinear perturbation of the Maxwellequations." Adv. Differential Equations 16 (5/6) 435 - 466, May/June 2011.

Information

Published: May/June 2011
First available in Project Euclid: 17 December 2012

zbMATH: 1260.35224
MathSciNet: MR2816112

Subjects:
Primary: 35B40, 35B45, 92C15

Rights: Copyright © 2011 Khayyam Publishing, Inc.

JOURNAL ARTICLE
32 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.16 • No. 5/6 • May/June 2011
Back to Top