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Multiple solutions to a nonlinear curl-curl problem in R^3
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We look for ground states and bound states  $E:\mathbb{R}^3\to\mathbb{R}^3$ to the  curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\textnormal{ in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations.

The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is controlled by an $N$-function $\Phi:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}\Phi(s)/s^6=\lim_{s\to+\infty}\Phi(s)/s^6=0$. We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in $\mathbb{R}^3$ and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schr\"odinger equations.

Keywords [en]
Time-harmonic Maxwell equations, ground state, variational methods, strongly indefinite functional, curl-curl problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-164806OAI: oai:DiVA.org:su-164806DiVA, id: diva2:1280341
Available from: 2019-01-18 Created: 2019-01-18 Last updated: 2019-02-04

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arXiv:1901.05776

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