Advances in Differential Equations

Magnetostatic solutions for a semilinear perturbation of the Maxwell equations

Teresa D'Aprile and Gaetano Siciliano

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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.

Article information

Adv. Differential Equations Volume 16, Number 5/6 (2011), 435-466.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B40: Asymptotic behavior of solutions 35B45: A priori estimates 92C15: Developmental biology, pattern formation


D'Aprile, Teresa; Siciliano, Gaetano. Magnetostatic solutions for a semilinear perturbation of the Maxwell equations. Adv. Differential Equations 16 (2011), no. 5/6, 435--466.

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