Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

Bartsch, Thomas; Mederski, Jarosław

doi:10.1016/j.jfa.2017.02.019

Cited by 31 publications

(76 citation statements)

References 32 publications

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“…We recall the abstract setting from [4,5]. Let X be a reflexive Banach space with the norm · and a topological direct sum decomposition X = X + ⊕ X, where X + is a Hilbert space with a scalar product .…”

Section: Critical Point Theorymentioning

confidence: 99%

“…Note that m needs not be C 1 , and M needs not be a differentiable manifold because I ′ is only required to be continuous. Recall from [5] that (u n ) is called a (P S) c -sequence for J if J ′ (u n ) → 0 and J (u n ) → c, and J satisfies the (P S) T c -condition on M if each (P S) c -sequence (u n ) ⊂ M has a subsequence converging in the T -topology. In order to apply classical critical point theory like the mountain pass theorem to J • m : X + → R we need some additional assumptions.…”

Section: Critical Point Theorymentioning

confidence: 99%

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Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Mederski

Schino

Szulkin

2019

Arch Rational Mech Anal

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We look for ground states and bound states E : R 3 → R 3 to the curl-curl problemwhich originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of ∇ × (∇ × ·).The growth of the nonlinearity f is controlled by an N -function Φ :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 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 Schrödinger equations. 2 2010 Mathematics Subject Classification. Primary: 35Q60; Secondary: 35J20, 78A25.In general J ′ is not (sequentially) weak-to-weak * continuous, however we show the weakto-weak * continuity of J ′ for sequences on the topological manifold M. Obviously, the same regularity holds for E ′ and M E .for any (φ, ψ) ∈ V × W.

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Section: Critical Point Theorymentioning

confidence: 99%

Section: Critical Point Theorymentioning

confidence: 99%

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Mederski

Schino

Szulkin

2019

Arch Rational Mech Anal

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“…The following abstract setting is recalled from [4,5,22], where super-quadratic problems have been considered. Our aim is to refine this theory for partially super-quadratic problems as (1.1).…”

Section: Critical Point Theorymentioning

confidence: 99%

“…Note that in the previous works [5,22], instead of (I7), the following stronger condition has been assumed:…”

Section: Critical Point Theorymentioning

confidence: 99%

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Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Bieganowski¹,

Mederski²

2019

Applied Mathematics Letters

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We prove the existence results for the Schrödinger equation of the formwhere g is superlinear and subcritical in some periodic set K and linear in R N \ K for sufficiently large |u|. The periodic potential V is such that 0 lies in a spectral gap of −∆ + V . We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that g is odd in u.Recently it has been shown that materials with large range of prescribed properties can be created ([12, 20, 24, 26, 27]) with different linear and nonlinear effects. Our aim is to model a wide range of nonlinear phenomena that allow to consider a composite of materials with different nonlinear polarization. In our case, the polarization g(x, ·) may be linear for some x ∈ R N \K (for sufficiently large |u|) and nonlinear outside of it, where K is a given Z N -periodic subset of R N . We admit Date: May 14, 2019.Lemma 6.1. Let β ≥ c N and suppose that K has a finite number of distinct orbits. If (u n ), (v n ) ⊂ X + are two Cerami sequences for J • m such thatand lim inf n→∞ u n − v n < κ, then lim n→∞ u n − v n = 0.

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Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

Qin

Tang

2017

Math Methods in App Sciences

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This paper is focused on following time‐harmonic Maxwell equation: ∇×false(μ−1false(xfalse)∇×ufalse)−ω2εfalse(xfalse)u=ffalse(x,ufalse),2emin.4emnormalΩ,ν×u=0,2em2emon.4em∂normalΩ, where normalΩ⊂R3 is a bounded Lipschitz domain, ν:∂normalΩ→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as false|ufalse|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor ε∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

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Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

Abstract: We find solutions E : Ω → R 3 of the problem

Cited by 31 publications

References 32 publications

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

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