Ground state solutions for semilinear time-harmonic Maxwell equations

Abstract

This paper is concerned with the time-harmonic semilinear Maxwell equation: ∇ × (∇ × u) + λu = f(x, u) in Ω with the boundary condition ν × u = 0 on ∂Ω, where Ω ⊂ ℝ3 is a simply connected, smooth, bounded domain with connected boundary and ν : ∂Ω → ℝ3 is the exterior normal. Here ∇ × denotes the curl operator in ℝ3 and the boundary condition holds when Ω is surrounded by a perfect conductor. By using the generalized Nehari manifold method due to Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (International Press, Somerville, 2010), pp. 597–632] and some new techniques, existence of ground state solutions for above equation is established under some generic conditions on f.

Nonlinear time-harmonic Maxwell equations in domains

“…The proof was based on the Nehari-Pankov manifold approach. A closely related result using the same approach is due to [50]. In these papers one can find several classes of functions that satisfy the hypotheses, in particular (F9).…”

“…In this section we consider the curl-curl equation (1.4) on a bounded Lipschitz domain Ω, and we present results from [11,12,38,50]. Recall the hypotheses (L1), (F1)-(F3) from Section 2 which imply that the functional…”

“…Then in Section 3 we present a theorem in a simpler symmetric setting where standard methods from critical point theory can be applied, which can be found in the seminal paper [1] by Ambrosetti and Rabinowitz and the book [39] by Rabinowitz. In order to treat the full strongly indefinite functional we recall some critical point theory developed for (1.4). Then in Section 5 we discuss results from [11,12,38,50] about (1.4) on bounded domains. The case Ω = R 3 will be discussed in Section 6.…”

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

“…For the isotropic case (ie, ε and μ are scalar functions of position), Bartsch et al considered on the whole space with μ =1, F ( x , u )=Γ( x )| u | p , and V , F being cylindrically symmetric, see also Mederski where ground state solutions were obtained provided that F ( x , u ) is periodic in x and , 1< p <5< q . Without cylindrical symmetry assumption on F , Equation was studied in Bartsch and Mederski and Tang and Qin for case μ =1 and V >0. When the material is anisotropic (ie, electric or magnetic properties of the materials depends on the direction of the field in such case ), ground state solutions as well as symmetric solutions for Equation were established in Bartsch and Mederski .…”

“…When the polarization is asymptotically linear, Equation in nonsymmetric case and on a bounded domain has first been studied by authors in Qin and Tang where a homeomorphism between a subset of unit sphere S + in E + and Nehari‐Pankov manifold , defined later in , was constructed. Motivated by the works, we are going to consider further in an anisotropic medium. In the present paper, we allow μ and V to be nonisotropic tensors and do not need the restrictions on the topology of domain.…”