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BY 4.0 license Open Access Published by De Gruyter June 13, 2019

Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

Rainer Mandel ORCID logo

Abstract

We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

MSC 2010: 35Q60; 35Q61; 35J91

1 Introduction and Main Results

The propagation of light in nonlinear media is governed by Maxwell’s equations

× + t = 0 , div ( 𝒟 ) = 0 ,
× - t 𝒟 = 0 , div ( ) = 0 ,

for the electric respectively magnetic field ,:3×3, the displacement field 𝒟:3×3 and the magnetic induction :3×3. Here the effect of charges and currents is neglected. The nonlinearity of the medium is typically expressed through nonlinear material laws of the form 𝒟=ε(x)+𝒫 and the linear relation =μ for the permittivity function ε:3 and the magnetic permeability μ{0}. In [23, 2], it was shown that special solutions of the form (x,t)=E(x)cos(ωt), 𝒫(x,t)=P(x,E)cos(ωt) can be approximately described by solutions of the nonlinear curl-curl equation

(1.1) × × E + V ( x ) E = f ( x , E ) in 3 ,

where V(x)=-μω2ε(x)0 and f(x,E)=μω2P(x,E). We stress that V is nonpositive and it becomes a negative constant in the simplest and most relevant case of the vacuum, where ε(x)ε0>0. In particular, 0 then lies in the interior of the essential spectrum of the linear operator ××()+V(x) so that variational methods as in [23, 2, 18] do not apply, and different methods have to be used. One of our main results (Theorem 3) will provide a new existence result for solutions of this problem following a fixed point approach inspired by [17].

A simplified version of (1.1) is the nonlinear Helmholtz equation

(1.2) - Δ u + V ( x ) u = f ( x , u ) in n ,

which can be derived from (1.1) via the ansatz E(x1,x2,x3)=(0,0,u(x1,x2)) in the two-dimensional case n=2. In this paper, we are interested in solutions of (1.1), (1.2) when the potential V is a negative constant and the nonlinearity is rather general. Our principal motivation is to show that, for large classes of nonlinearities, there are uncountably many solutions of these equations sharing the same decay rate |x|1-n2 as |x| but with a different farfield pattern.

We first recall some facts about the nonlinear Helmholtz equation with constant potential

(1.3) - Δ u - λ u = f ( x , u ) in n .

We always assume λ>0 so that 0 lies in the interior of the essential spectrum of -Δ-λ. In 2004, Gutiérrez [17] set up a fixed point approach for this equation when f(x,u)=|u|2u, n{3,4} and u is complex-valued. Using an Lp-version of the limiting absorption principle for the Helmholtz operator [17, Theorem 6], she found that small nontrivial solutions of (1.3) can be obtained via the contraction mapping theorem (Banach’s fixed point theorem) on a small ball in L4(n). Around ten years later, Evéquoz and Weth started to write a series of papers [9, 10, 13, 12, 11, 14] containing new methods to prove existence results for solutions of (1.3) that, in contrast to Gutiérrez’ solutions, are large in suitable norms. Some of these results were extended by the author in [22, 21]. In each of the aforementioned papers, the nonlinearity has to satisfy quite specific conditions that allow to deal with slow decay rates of solutions at infinity.

In the case of power-type nonlinearities f(x,u)=Q(x)|u|p-2u, one of the main results in [12] is that there is an unbounded sequence of solutions in Lp(n) provided 2(n+1)n-1<p<2nn-2 and QL(n) is positive and evanescent at infinity. If Q is n-periodic and positive (for negative Q, see [22, Theorems 1.3, 1.4]), the existence of one nontrivial solution is shown. These solutions are obtained using quite sophisticated dual variational methods, and the solution at the mountain pass level of the dual functional is called a dual ground state of the equation. One of the drawbacks of this approach is that the assumption on p does not allow for cubic nonlinearities, which certainly are the most interesting ones for applications in physics. Moreover, sign-changing or non-monotone nonlinearities cannot be treated. In addition to that, solutions have to be looked for in Lp(n). This is a problem given that solutions decay slowly at infinity so that a solution theory in Lq(n) with q>p is more convenient a priori. For this reason, we will not consider dual variational methods but rather revive Gutiérrez’ fixed point approach [17].

Our refinement of Gutiérrez’ method allows to discuss nonlinear Helmholtz equations with very general nonlinearities that improve existing results even in the case of power-type nonlinearities as we will see below. In our main result dealing with (1.3), we show that, in the case f(x,u)=Q(x)|u|p-2u with QLs(n)L(n), we get solutions for all exponents

(1.4) p > max { 2 , 2 s ( n 2 + 2 n - 1 ) - 2 n ( n + 1 ) ( n 2 - 1 ) s } .

More generally, we can treat nonlinearities satisfying the following conditions:

  1. f : n × is a Carathéodory function satisfying, for some QLs(n)L(n),

    (1.5)

    | f ( x , z ) | Q ( x ) | z | p - 1 ( x n , | z | 1 ) ,
    | f ( x , z 1 ) - f ( x , z 2 ) | Q ( x ) ( | z 1 | + | z 2 | ) p - 2 | z 1 - z 2 | ( x n , | z 1 | , | z 2 | 1 ) ,

    where s[1,] and p as in (1.4).

We stress that only conditions near zero are needed since we are going to construct small solutions in Lq(n) which will turn out to be small also in L(n). Clearly, |z|1 can be replaced by |z|<z0 for any given z0>0. We mention that, in the case sn+12, all exponents p>2 in the superlinear regime are allowed.

The fundamental tools of Gutiérrez’ fixed point approach are an Lp-version of the limiting absorption principle for the Helmholtz operator -Δ-λ (λ>0) and results about the so-called Herglotz waves. As we will recall in Proposition 1, these functions are analytic solutions of the linear Helmholtz equation -Δϕ-λϕ=0 in n. They are given by the formula

h d σ λ ^ ( x ) :- 1 ( 2 π ) n / 2 S λ n - 1 h ( ξ ) e - i x , ξ d σ λ ( ξ )

for complex-valued densities hL2(Sλn-1;). Here σλ denotes the canonical surface measure of the sphere Sλn-1={ξn:|ξ|2=λ}. In order to ensure the real-valuedness and good pointwise decay properties of hdσλ^ at infinity, we will consider a smaller class of densities h belonging to the set

X λ δ :- { h C m ( S λ n - 1 ; ) : h ( ξ ) = h ( - ξ ) ¯ , h C m δ } ,

where m:-n-12+1. Here our approach differs from [17], where L2-densities are used. Theorem 1 shows that, for all hXλδ, we find a strong solution of (1.3) that resembles |x|1-n2uh(x) at infinity, where

u h ( x ) :- λ n - 3 4 π 2 Re ( e i ( n - 3 4 π - λ | x | ) ( f ( , u h ) ^ ( - λ x ^ ) + i 2 λ π h ( λ x ^ ) ) ) , x ^ :- x | x | .

More precisely, we show the following.

Theorem 1.

Assume (A) and λ>0. Then there are δ>0 and mutually different solutions (uh)hXλδ of (1.3) that form a W2,r(Rn)-continuum and satisfy uhW2,r(Rn)0 as hCm0 for any given r(2nn-1,) as well as

(1.6) lim R 1 R B R | u h ( x ) - | x | 1 - n 2 u h ( x ^ ) | 2 d x = 0 .

If additionally p>(3n-1)s-2n(n-1)s holds, then |uh(x)|Ch(1+|x|)1-n2 for all xRn.

Let us discuss in which way this theorem improves earlier results. Most importantly, Theorem 1 shows that nonlinear Helmholtz equations of the form (1.3) admit uncountably many solutions for a large class of nonlinearities which need not be odd, let alone of power-type. Its proof is short and elementary in the sense that it only uses the contraction mapping theorem, elliptic regularity theory and mostly well-known results about the linear Helmholtz equation. Up to now, such general nonlinearities have only been treated in the paper [11] by Evéquoz and Weth, but their additional requirement (f0) on p. 361 requires the nonlinearity to be supported in a bounded subset of n, which is quite restrictive. In our approach, such an assumption is not necessary. Given that applications often deal with power-type nonlinearities f(x,z)=Q(x)|z|p-2z, let us comment on our improvements for this particular case in more detail. In the case QL(n), we obtain solutions for all exponents p>2(n2+2n-1)n2-1. This bound is smaller than 2(n+1)n-1 so that our range of exponents is larger than in all other nonradial approaches except for [11], where Q has compact support and exponents 2<p<2nn-2 are allowed. Additionally, we need neither require Q to be periodic nor evanescent (as in [10, 12, 22]) nor compactly supported (as in [11, 13]), and the growth rate of the nonlinearity may be supercritical (i.e., p2nn-2), which is an entirely new feature. The latter fact is worth mentioning given that Evéquoz and Yesil [14] proved the nonexistence of dual ground states uLp(n) for n=3 in the critical case p=2nn-2=6 provided f(x,u)=Q(x)u5 and QL(3) is nonnegative and nontrivial. Since Theorem 1 yields solutions belonging to Lp(3), we conclude that dual ground states need not exist, while other nontrivial solutions do. Finally, we mention that, in the physically most relevant case of a cubic nonlinearity p=4, we obtain uncountably many solutions whenever n3, s[1,] or n=2, s[1,6).

Remark 1.

(a) The decay rate |x|1-n2 is best possible. This is a consequence of [20, Theorem 3], where nontrivial solutions of the elliptic PDE -Δu-λu=W(x)u in n with WLn+12(n) and λ>0 are shown to satisfy u(x)|x|-12-εL2(n) for all ε>0. In particular, decay rates like |x|1-n2-δ as |x| are impossible for any given δ>0. Notice that, in the setting of Theorem 1, the function W(x):-f(x,u(x))/u(x) satisfies WLn+12(n) because of QLs(n) and

| W ( x ) | Q ( x ) | u ( x ) | p - 2 , u r > 2 n n - 1 L r ( n ) , p > 2 s ( n 2 + 2 n - 1 ) - 2 n ( n + 1 ) ( n 2 - 1 ) s ;

see (1.4). It is remarkable that precisely this lower bound for p appears in this context. Up to now, existence and optimal decay results for nonlinear Helmholtz equations for lower exponents p are only known in the radial setting [22, 11]. For smaller p, the (nonradial) counterexample of Ionescu and Jerison [19, Theorem 2.5] has to be taken into account: for any given N, there is WLq(n) with q>n+12 and a solution of -Δu-λu=W(x)u in n with |u(x)|(1+|x|)-N for all xn.

(b) Theorem 1 yields a symmetry-breaking result: for any subgroup ΓO(n) such that Γ{id} and any Γ-invariant nonlinearity f satisfying (A) one has uncountably many solutions that are not Γ-invariant. In particular, for Γ=O(n), radial nonlinearities allow for nonradial solutions. We will prove this in Remark 2 (a).

(c) In order to construct radial solutions, we can get the same conclusions as in Theorem 1 under weaker assumptions on p. This is due to an improved version of the limiting absorption principle for the Helmholtz operator. In Remark 2 (b), we comment on the necessary modifications of the proof and show that the admissible range of exponents for the existence of radial solutions is no longer given by (1.4), but

(1.7) p > max { 2 , s ( 2 n 2 + n - 1 ) - 2 n 2 s n ( n - 1 ) } .

Notice that the resulting radial version of Theorem 1 is not covered by earlier contributions from [22, Theorems 1.2, 2.10] or [11, Theorem 4]. For instance, it provides solutions of the radial nonlinear Helmholtz equation -Δu-u=Q(x)|u|p-2u for any QLrads(n)Lrad(n) and p as in (1.7), whereas, in the above-mentioned papers, Q has to be bounded, differentiable and radially decreasing. On the other hand, our restrictions on the exponent p do not appear in [22, 11] (where all p>2 are allowed) so that (1.7) might be improved further.

Next we discuss variants of these results for related semilinear elliptic PDEs from mathematical physics. First let us mention that a nonlinearity f(,u) satisfying (A) may be without any major difficulty replaced by a nonlocal right-hand side such as Kf(,u), where KL1(n). Clearly, imposing more assumptions, K may even lead to larger ranges of exponents than (1.4). In this way, it is possible to obtain small solutions of nonlocal Helmholtz equations. Similarly, one may ask how our results are affected by changes in the linear operator. For instance, if the Helmholtz operator is perturbed to a periodic Schrödinger operator -Δ+V(x)-λ, then it should be possible to adapt the proof in such a way that it provides small solutions of -Δu+V(x)u-λu=f(x,u) in n provided λ belongs to the essential spectrum of -Δ+V(x) and the band structure of this periodic Schrödinger operator is sufficiently nice. To be more precise, one would require [21, (A1)–(A3)] to hold so that Herglotz-type waves, defined as suitable oscillatory integrals over the so-called Fermi surfaces associated with -Δ+V(x), exist and have the properties stated in Proposition 1 below. Since the technicalities (including a limiting absorption principle for such operators) are quite involved and mostly carried out in [21], we prefer not to discuss this issue further.

We now turn our attention to a fourth order version of (1.3) given by

(1.8) Δ 2 u - β Δ u + α u = f ( x , u ) in n ,

which we will briefly discuss for α,β satisfying

(1.9) (i) α < 0 , β    or    (ii) α > 0 , β < - 2 α .

Under these assumptions, 0 lies in the spectrum of Δ2-βΔ+α, and dual variational methods were employed in [6] to prove the existence of one nontrivial solution when f(x,z)=Q(x)|z|p-2z, where 2(n+1)n-1<p<2nn-2 and Q is positive and n-periodic. Notice that, in the case β2-4α<0, classical variational methods such as constrained minimization apply, and a number of papers revealed the existence of positive and sign-changing solutions uH4(n) of (1.8) again for power-type nonlinearities. We refer to [7, 8, 5] for results in this direction. Our intention is to show that, in case (i) or (ii), uncountably many solutions of (1.8) exist for all nonlinearities f satisfying (A). The main observation is that, in case (i) or (ii), there are analoga of the Herglotz waves given by densities hYδ, where

(1.10) Y δ :- { X λ δ in case (i) , where λ = - β + β 2 - 4 α 2 > 0 , X λ 1 δ × X λ 2 δ in case (ii) , where λ 1 , 2 = - β ± β 2 - 4 α 2 > 0 .

This is a consequence of the fact that the zeros of the polynomial |ξ|4+β|ξ|2+α satisfy |ξ|2=λ in case (i) and |ξ|2=λ1 or |ξ|2=λ2 in case (ii). In particular, in case (ii), two families of Herglotz waves lie in the kernel of Δ2-βΔ+α, which motivates the above definition of Yδ as a product space. The fixed point approach used in the proof of Theorem 1 may be rather easily adapted to (1.8), and we can prove the following result.

Theorem 2.

Assume (A) and (i) or (ii). Then there is δ>0 and mutually different solutions (uh)hYδ of (1.8) that form a W4,r(Rn)-continuum satisfying uhW4,r(Rn)0 as hCm0 for any given r(2nn-1,). If additionally p>(3n-1)s-2n(n-1)s holds, then |uh(x)|Ch(1+|x|)1-n2 for all xRn.

As in Theorem 1, one can say more about the asymptotics of the constructed solutions, which is different for cases (i) and (ii). They can be derived as in [6, Section 5.2]; see (5.5) for case (i) and (5.6) for case (ii). Further generalizations to more general higher order semilinear elliptic problems of the form Lu=f(x,u) in n are possible provided the linear differential operator with constant coefficients L has a Fourier symbol P(ξ) with the property that {ξn:P(ξ)=0} is a compact manifold with nonvanishing Gaussian curvature. Notice that this assumption makes the method of stationary phase work and provides pointwise decay of oscillatory integrals as demonstrated in the proof of Proposition 1. Moreover, one needs a limiting absorption principle in order to make sense of the Fourier multiplier 1/P(ξ) as a mapping between Lebesgue spaces. At least in the case P(ξ)=P0(ξ)(|ξ|2-λ1)(|ξ|2-λk) with 0<λ1<<λk and P0 positive, this can be established as in [6, Theorem 3.3]. With these tools, our fixed point approach can be adapted to find nontrivial solutions of Lu=f(x,u) in n.

Finally, we discuss nonlinear curl-curl equations of the form

(1.11) × × E - λ E = f ( x , E ) in 3

that describe the electric field E:33 of an electromagnetic wave in a nonlinear medium. This equation has been studied in the past years on bounded domains in 3 [4, 3] but also on the entire space 3, which is the situation we focus on. Up to our knowledge, there is only one result for solutions of nonlinear curl-curl equations on 3 without symmetry assumption. In [23], Mederski proves the existence of a weak solution of (1.11) by variational methods when λ is replaced by a small nonnegative potential V(x) that decays suitably fast to zero at infinity; see [23, assumption (V)]. In particular, constant functions V(x)=λ cannot be treated by this method so that our setting must be considered as entirely different from the one in [23].

In the cylindrically symmetric setting, the existence of solutions can be proved using various approaches. Here the electrical field is assumed to be of the form

(1.12) E ( x 1 , x 2 , x 3 ) = E 0 ( x 1 2 + x 2 2 , x 3 ) x 1 2 + x 2 2 ( - x 2 x 1 0 ) , where E 0 : [ 0 , ) × .

Such functions are divergence-free so that ××E=-ΔE implies that one actually has to deal with the elliptic 3×3-system

(1.13) - Δ E - λ E = f ( x , E ) in 3 ,

which may equally be expressed in terms of E0 provided the nonlinearity f(x,E) is compatible with this symmetry assumption; see [2, p. 3]. In this special case, further results [18, 2, 26] are known, but none of those applies in the case λ>0 and f(x,E)=±|E|p-2E that we are mainly interested in.

Given our earlier results for the nonlinear Helmholtz equation (1.3), it is not surprising that we obtain an existence result for (1.13) that is entirely analogous to the one from Theorem 1. Since this result fills a gap in the literature, we state it in part (i) of our theorem even though its proof is a straightforward adaptation of the fixed point approach used in the proof of Theorem 1. The corresponding assumption on the nonlinearity is the following.

  1. f : 3 × 3 3 , ( x , E ) f 0 ( x 1 2 + x 2 2 , x 3 , | E | 2 ) E is a Carathéodory function satisfying (1.5) for some QLs(3)L(3), where s[1,] and p as in (1.4).

Assumption (A’) ensures that f is compatible with cylindrical symmetry. Indeed, for E as in (1.12), one can check that f(,E) is of the form (1.12), too.

In the general non-symmetric case, the construction of solutions is more difficult since the curl-curl operator satisfies a much weaker limiting absorption principle as in the cylindrically symmetric setting (cf. Theorem 6). Moreover, well-known regularity results for elliptic problems are not available so that we have to consider a substantially smaller class of nonlinearities satisfying the following:

  1. f : 3 × 3 3 is a Carathéodory function satisfying for some QLs(3)L(3) the estimate

    | f ( x , E ) | Q ( x ) | E | p - 1 ( 1 + | E | ) p ~ - p ( x , E 3 ) ,
    | f ( x , E 1 ) - f ( x , E 2 ) | Q ( x ) ( | E 1 | + | E 2 | ) p - 2 ( 1 + | E 1 | + | E 2 | ) p ~ - p | E 1 - E 2 | ( x , E 1 , E 2 3 ) ,

    where 1s2 and p~2p<.

Additionally, we will have to require that Qs+Q is small enough in order to obtain solutions of (1.11). In the cylindrically symmetric, respectively non-symmetric, setting the counterparts of the Herglotz waves (introduced in Section 2) are parametrized by functions hZcylδ, respectively hZ, where

Z :- { h C 2 ( S λ 2 ; 3 ) : h ( ξ ) = h ( - ξ ) ¯ , h ( ξ ) , ξ = 0 for all ξ S λ 2 } ,
Z cyl δ :- { h Z : h C 2 < δ and Re ( h ) , Im ( h ) satisfy (1.12) } .

Notice that both sets are nonempty. For simplicity, we will call the vector fields hdσλ^ again Herglotz waves, and we refer to Proposition 3 for a detailed description of their properties. Let us mention that functions of the form E:-hdσλ^ appear in the literature in the context of so-called Herglotz wave pairs, which are given by (E,1iλ×E). With these definitions, we can formulate our main result for the nonlinear curl-curl equation (1.11).

Theorem 3.

The following statements hold:

  1. Assume (A’) and λ > 0 . Then there is δ > 0 and a family ( E h ) h Z cyl δ of mutually different cylindrically symmetric solutions of ( 1.11 ) that form a W 2 , r ( 3 ; 3 ) -continuum and satisfy E h W 2 , r ( 3 ; 3 ) 0 as h C 2 0 for any given r ( 3 , ) . If additionally p > 4 s - 3 s holds, then | E h ( x ) | C h ( 1 + | x | ) - 1 .

  2. Assume (B) and λ > 0 , 3<q<3s(2s-3)+. If Qs+Q is sufficiently small, then there is a family (Eh)hZ of mutually different weak solutions of (1.11) lying in Hloc(curl;3)Lq(3;3).

    1. If additionally ( p , s ) ( 2 , 2 ) holds, then E h L r ( 3 ; 3 ) for all r ( 3 , q ) .

    2. If additionally p ~ < 2 < p holds, then E h L r ( 3 ; 3 ) for all r ( q , 3 s ( p - 1 ) ( 2 s - 3 ) + ) .

As an application, we obtain uncountably many distinct weak solutions of the curl-curl equation (1.11) with saturated nonlinearities of the form

f ( x , E ) = δ | E | 2 Γ ( x ) E 1 + P ( x ) | E | 2 ,

where infP>0, ΓLs(3;3×3)L(3;3×3) and δ>0 is sufficiently small. In part (i), the precise asymptotics for Eh at infinity are given by the vectorial version of (1.6). For the solutions obtained in (ii), the asymptotics are not clear and remain to be investigated in the future.

The paper is organized as follows. In Section 2, we review the limiting absorption principles and properties of Herglotz-type waves that we will need for the proofs of our results. In Sections 35, we then prove Theorems 13. Since the proofs of Theorems 1, 2 and 3 (i) are almost identical, we carry out the first in detail and only comment on the modifications when it comes to the latter. Finally, in Appendix A, we review the method of stationary phase and prove Proposition 2. In Appendix B, we prove our limiting absorption principle for the curl-curl operator (Theorem 6). In Appendix C, we review some resolvent estimates for the Helmholtz operator due to Ruiz and Vega.

In the following, C will denote a generic constant that can change from line to line and 1r+ stands for 1r if r>0 and for if r0. The symbol f=f^ represents the Fourier transform of (the tempered distribution) fLq(n), and 1,n-1 are the Fourier transforms in 1,n-1, respectively. For R>0, the symbol BR denotes the open ball of radius R around the origin in n, and , is the inner product in n extended by bilinearity to n. Further, Lq(n),Lq,w(n) denote the classical, respectively weak, Lebesgue spaces on n equipped with the standard norms q,q,w.

2 Herglotz Waves and Limiting Absorption Principles

In this section, we review some partly well-known results on Herglotz waves and limiting absorption principles for the linear differential operators we are interested in. A classical Herglotz wave associated with the Helmholtz operator -Δ-λ is defined via the formula

( h d σ λ ) ( x ) :- h d σ λ ^ ( x ) :- 1 ( 2 π ) n / 2 S λ n - 1 h ( ξ ) e - i x , ξ d σ λ ( ξ ) ,

where hL2(Sλn-1;) and σλ denotes the canonical surface measure of Sλn-1={ξn:|ξ|2=λ}. Herglotz waves are analytic functions that solve the linear Helmholtz equation -Δϕ-λϕ=0. Their pointwise decay properties are well understood for smooth densities h and result from an application of the method of stationary phase. Unfortunately, we could not find a quantitative version of this result telling how smooth the density h needs to be in order to ensure that hdσλ^(x) decays like |x|1-n2 as |x| in the pointwise sense. In our first auxiliary result, we provide such an estimate, and its proof will be given in Appendix A. For notational convenience, we introduce the quantity

(2.1) m h ( x ) :- e i ( n - 1 4 π - λ | x | ) h ( λ x ^ ) + e - i ( n - 1 4 π - λ | x | ) h ( - λ x ^ )

so that our claim is the following.

Proposition 1.

Let nN, n2 and m:-n-12+1. Then, for all hCm(Sλn-1;C), the Herglotz wave hdσλ^ is an analytic solution of -Δϕ-λϕ=0 in Rn and satisfies the estimate |(hdσλ)^(x)|ChCm(1+|x|)1-n2 as well as

lim R 1 R B R | h d σ λ ^ ( x ) - 1 2 π ( λ | x | ) n - 1 2 m h ( x ) | 2 d x = 0 .

In particular, we have hdσλ^rCrhCm for all r>2nn-1.

While Herglotz waves solve the homogeneous Helmholtz equation, we also need to discuss the inhomogeneous equation. Since λ lies in the essential spectrum of -Δ, it is a nontrivial task to solve -Δu-λu=f in n. The method to find such solutions is to study the limit of solutions uε:-(λ+iε)fH2(n;) of -Δuε-(λ+iε)uε=f in a suitable topology. The complex-valued limit of these functions as ε0+ is denoted by (λ+i0)f, and we define λf:-Re((λ+i0))f as its real part since we are interested in real-valued solutions. These operators have the following properties.

Theorem 4 ([17, Theorem 6], [10, Theorem 2.1]).

Let nN and n2. The operator Rλ:Lt(Rn)Lq(Rn) is a bounded linear operator provided

(2.2)

1 t > n + 1 2 n , 1 q < n - 1 2 n , 2 n + 1 1 t - 1 q 2 n    ( n 3 ) ,
1 t > n + 1 2 n , 1 q < n - 1 2 n , 2 n + 1 1 t - 1 q < 2 n    ( n = 2 ) .

Moreover, for fLt(Rn), the function RλfWloc2,t(Rn) is a real-valued strong solution of -Δu-λu=f in Rn.

The last statement is actually not included in the references given above, but it is a consequence of elliptic regularity theory for distributional solutions. We refer to [12, Proposition A.1] for a similar result. Next we discuss the asymptotic behavior of the solutions λf that we will deduce from the following result.

Proposition 2.

Let nN, n2, and assume fLp(Rn) for 2(n+1)n-1p2n(n-4)+,(n,p)(4,). Then

lim R 1 R B R | ( λ + i 0 ) f ( x ) - π 2 λ ( λ | x | ) n - 1 2 e i ( n - 3 4 π - λ | x | ) f ^ ( - λ x ^ ) | 2 d x = 0 .

Proof.

The claim for n3 and λ=1 is provided in [12, Proposition 2.7] so that the general case follows from rescaling via λf(x)=1λ1(f(λ-12))(λx). The proof in the case n=2 is essentially the same. Indeed, repeating the proof of [12, Proposition 2.7], one finds that the claimed result holds true provided [12, Proposition 2.6] (the Stein–Tomas theorem) and the estimate

(2.3) sup R 1 ( 1 R B R | ( λ + i 0 ) f ( x ) | 2 d x ) 1 2 C f p

are valid in the case n=2 under our assumptions on p. For the Stein–Tomas Theorem, this is clear. Inequality (2.3) is due to Ruiz and Vega [24], but we could not find an accurate reference for it that covers our range of exponents and all space dimensions n2. We provide estimate (2.3) and further details in Appendix C so that the proof is finished. ∎

Next we recall a limiting absorption principle that we will need in the discussion of the fourth order problem (1.8). In [6, Theorem 3.3], the following extension of Theorem 4 to linear differential operators of the form Δ2-βΔ+α was proved.

Theorem 5 ([6, Theorem 3.3]).

Let nN, n2, and assume (i) or (ii). Then there is a bounded linear operator R:Lt(Rn)Lq(Rn) for

1 t > n + 1 2 n , 1 q < n - 1 2 n , 2 n + 1 1 t - 1 q 4 n 𝑖𝑓 n 5 ,
1 t > n + 1 2 n , 1 q < n - 1 2 n , 2 n + 1 1 t - 1 q < 1 𝑖𝑓 n = 4 ,
1 t > n + 1 2 n , 1 q < n - 1 2 n , 2 n + 1 1 t - 1 q 1 𝑖𝑓 n = 2 , 3

such that, for fLt(Rn), the function Rf belongs to Wloc4,t(Rn) and is a real-valued strong solution of

Δ 2 u - β Δ u + α u = 0 𝑖𝑛 n .

Finally, we provide the tools for proving Theorem 3. As for the previous results, we need a family of elements lying in the kernel of the linear operator which now is E××E-λE. These are given by vectorial variants of the Herglotz waves

h d σ λ ^ ( x ) :- 1 ( 2 π ) 3 2 S λ 2 h ( ξ ) e - i x , ξ d σ λ ( ξ )

(the integral to be understood componentwise), where h:Sλ23 is a tangential vectorfield field, i.e., h(ξ),ξ=0 for all ξSλ2. These functions are divergence-free and real-valued whenever h(ξ)=h(-ξ)¯. Applying the results from Proposition 1 in each component, we deduce the following properties.

Proposition 3.

For all hZ, the function hdσλ^ is an analytic and divergence-free solution of ××ϕ-λϕ=0 in R3 and satisfies the pointwise estimate |hdσλ^(x)|hC2(1+|x|)-1 for all xR3. In particular, we have hdσλ^rCrhC2 for all r>3. If hZcylδ, then hdσλ^ is cylindrically symmetric.

Having described the analoga of the Herglotz waves, we finally discuss a limiting absorption principle for the curl-curl operator. So let (λ+iε) denote the resolvent of E××E-(λ+iε)E, which we will prove to exist in Proposition 6. As for the Helmholtz operator, one is interested in the (complex-valued) limit (λ+i0)G for GL2(3;3). It turns out that G decomposes into two parts behaving quite differently. So we split G into a curl-free (gradient-like) part G1:33 and a divergence-free remainder G2:33 of G which are defined via

(2.4) G 1 :- - 1 ( G ^ ( ξ ) , ξ | ξ | ξ | ξ | ) , G 2 :- - 1 ( G ^ ( ξ ) - G ^ ( ξ ) , ξ | ξ | ξ | ξ | ) .

This splitting corresponds to a Helmholtz decomposition of a vector field in 3.

Theorem 6.

Let λ>0, and assume that t,q(1,) satisfy (2.2) for n=3. Then there is a bounded linear operator Rλ:Lt(R3;R3)Lq(R3;R3)Lq(R3;R3) such that RλGHloc(curl;R3) is a weak solution of ××E-λE=G provided GLt(R3;R3)Lq(R3;R3). Moreover, we have

R λ G q C ( G 1 q + G 2 t ) C ( G q + G t ) 𝑎𝑛𝑑 R λ G = - 1 λ G 1 + λ G 2

for Rλ from Theorem 4 (applied componentwise). If GLt(R3;R3) is cylindrically symmetric, then so is RλG, and RλGWloc2,q(R3;R3) is a strong solution satisfying RλGqCGt.

The proof of Theorem 6 will be given in Appendix B. With these technical preparations, we have all the tools to prove our main results in the following sections.

3 Proof of Theorem 1

We prove Theorem 1 with the aid of Banach’s fixed point theorem following the approach by Gutiérrez [17]. We consider the map T(,h):Lq(n)Lq(n) given by

(3.1) T ( u , h ) :- h d σ λ ^ + λ ( f ( , χ ( u ) ) ) ,

where hXλδ as in Proposition 1 and χ is a smooth function such that |χ(z)|min{|z|,1} and χ(z)=z for |z|12. In view of the properties of the Herglotz waves and λ mentioned earlier, a fixed point of T(,h) is a strong solution of the equation -Δu-λu=f(x,χ(u)) in n. Since that fixed point u will belong to a small ball in Lq(n), we will be able to show χ(u)=u so that a solution of (1.3) is found. The choice of the exponent q is delicate; it is the major technical issue in our approach. Our assumption (1.4) implies that the set

Ξ s , p :- { q ( 2 n n - 1 , 2 n ( n - 3 ) + ) : q < min { s ( n + 1 ) ( p - 2 ) ( 2 s - ( n + 1 ) ) + , 2 n s ( p - 1 ) ( s ( n + 1 ) - 2 n ) + } }

is non-empty, so we may choose some arbitrary but fixed qΞs,p throughout this section.

Proposition 4.

Assume (A) and λ>0, hXλδ. Then the map T(,h):Lq(Rn)Lq(Rn) from (3.1) is well-defined, and we have

(3.2) T ( u , h ) q C ( u q α + h C m ) , T ( u , h ) - T ( v , h ) q C u - v q ( u q + v q ) α - 1

for some α>1 and all u,vLq(Rn).

Proof.

Using (A), Hölder’s inequality and |χ(u)|min{|u|,1}, for all uLq(n), we get

(3.3) f ( , χ ( u ) ) t | Q | | χ ( u ) | p - 1 t Q s ~ χ ( u ) p - 1 t s ~ s ~ - t Q s ~ u q q t - q s ~ < , provided t * ( q , s ~ ) :- max { 1 , s ~ q s ~ ( p - 1 ) + q } t s ~ , s ~ [ s , ] .

In particular, we find f(,χ(u))Lt(n) for all t[t*(q,s),]. For any such t we choose

s ~ :- t q ( q - t ( p - 1 ) ) + [ s , ] (largest possible)

so that t*(q,s~)ts~ holds. So the previous estimate gives, for αt:-qt-qs~=min{qt,p-1},

(3.4) f ( , χ ( u ) ) t C u q α t < for all t [ t * ( q , s ) , ] .

Now we have to choose t[t*(q,s),] in such a way that the mapping properties of λ from Theorem 4 ensure λ(f(,χ(u)))Lq(n). In view of (2.2), we have to require

(3.5) n q n + 2 q t < ( n + 1 ) q n + 1 + 2 q and t < 2 n n + 1 .

Since qΞs,p implies q<2n(n-3)+ and hence nqn+2q<2nn+1, we can find such t if and only if

(3.6) t * ( q , s ) < ( n + 1 ) q n + 1 + 2 q and t * ( q , s ) < 2 n n + 1 .

These two inequalities hold due to qΞs,p. From this, Proposition 1 and Theorem 4, we get

T ( u , h ) q λ ( f ( , χ ( u ) ) ) q + h d σ λ ^ q C ( f ( , χ ( u ) ) t + h C m ) ( 3.4 ) C ( u q α t + h C m ) .

Since t was chosen according to (3.5), we have t<q and thus αt=min{p-1,qt}>1. Moreover, from (A) and Hölder’s inequality (1t=1s~+1q+αt-1q), we get

(3.7) f ( , χ ( u ) ) - f ( , χ ( v ) ) t C | Q | | χ ( u ) - χ ( v ) | ( | χ ( u ) | + | χ ( v ) | ) p - 2 t C Q s ~ χ ( u ) - χ ( v ) q ( | χ ( u ) | + | χ ( v ) | ) p - 2 q α t - 1 C u - v q ( | u | + | v | ) α t - 1 q α t - 1 C u - v q ( u q + v q ) α t - 1 .

Here p-2αt-1>0 was used. Hence we get

T ( u , h ) - T ( v , h ) q λ ( f ( , χ ( u ) ) ) - λ ( f ( , χ ( v ) ) ) q C u - v q ( u q + v q ) α t - 1 .

Proof of Theorem 1.

The proof is divided into the following steps.

Step 1: Existence of a solution continuum (uh) in Lq(Rn). We apply Banach’s fixed point theorem to T(,h) on a closed small ball around zero Bρ¯Lq(n), qΞs,p and hXλδ with δ>0 sufficiently small. From (3.2), we get that T(,h):Bρ¯Bρ¯ is a contraction provided ρ,δ>0 are chosen sufficiently small. It is even a uniform contraction since its Lipschitz constant is independent of h. Moreover, T is continuous with respect to the topology of Lq(n)×Cm(Sλn-1;), which follows from Propositions 1 and 4. Hence Banach’s fixed point theorem for continuous uniform contractions yields a continuum of (uniquely determined) fixed points uhBρ¯ of T(,h) for hXδ provided δ>0 is small enough.

Step 2: The continuum property in Lr(Rn) for r(q,]. As a fixed point of T(,h), the function uh solves -Δu-λu=f(x,χ(u)) in the strong sense on n (see Theorem 4). Since f(,χ(uh)) is bounded, we even have uhWloc2,r(n) for all r[1,). Fixing now q~q such that q~>n2, from [15, Theorem 8.17] (Moser iteration), we get

u h 1 - u h 2 u h 1 - u h 2 q + f ( , χ ( u h 1 ) ) - f ( , χ ( u h 2 ) ) q ~ u h 1 - u h 2 q + Q ( | χ ( u h 1 ) | p - 2 + | χ ( u h 2 ) | p - 2 ) ( χ ( u h 1 ) - χ ( u h 2 ) ) q ~ u h 1 - u h 2 q + 2 p - 2 + q ~ - q q Q | u h 1 - u h 2 | q q ~ q ~ C ( u h 1 - u h 2 q + u h 1 - u h 2 q q / q ~ )

so that (uh) is also a continuum in L(n) and hence in Lr(n) for all r(q,].

Step 3: The continuum property in Lr(Rn) for r(2nn-1,q). From uhLq(n), we deduce (see (3.3))

f ( , χ ( u h ) ) L t * ( q , s ) ( n ) L ( n ) .

We set

q ~ :- max { r , ( n + 1 ) t * ( q , s ) n + 1 - 2 t * ( q , s ) }

so that Theorem 4 gives uhLq~(n) since the tuple of exponents (t*(q,s),q~) satisfies inequalities (2.2). Notice that q<s(n+1)(p-2)(2s-(n+1))+ (because of qΞs,p) implies q~<q. Moreover, we have

u h 1 - u h 2 q ~ ( ( h 1 - h 2 ) d σ λ ) q ~ + λ ( f ( , χ ( u h 1 ) ) - f ( , χ ( u h 2 ) ) ) q ~ C h 1 - h 2 C m + C f ( , χ ( u h 1 ) ) - f ( , χ ( u h 2 ) ) t * ( q , s ) ( 3.7 ) C ( h 1 - h 2 C m + u h 1 - u h 2 q ( u h 1 q + u h 2 q ) p - 2 ) C ( h 1 - h 2 C m + u h 1 - u h 2 q ) .

Taking now q~ as the new q and repeating the above arguments, we get after finitely many steps uhLr(n) as well as uh1-uh2rC(h1-h2Cm+uh1-uh2q). Hence (uh) is a continuum in Lr(n) for all r(2nn-1,q).

Step 4: The continuum property in W2,r(Rn) for r(2nn-1,) and (1.3). From steps 2 and 3, we get

- Δ u h + u h = ( 1 + λ ) u h + f ( , χ ( u h ) ) L r ( n )

because of |f(x,χ(uh))|Q|uh|Lr(n). Bessel potential estimates imply uhW2,r(n) and, as above, one finds uh1-uh2W2,r(n)Cuh1-uh2r so that the continuum property is proved. In particular, hCm0 implies uh=uh-u00, and thus χ(uh)=uh is a W2,r(n)-solution of (1.3) for all hXλδ provided δ>0 is sufficiently small.

Step 5: Asymptotics of uh. From the previous steps, we get uhLr(n) for all r(2nn-1,], and this implies |f(,uh)|Q|χ(uh)|p-1Lt(n) for t>2ns2n+s(n-1)(p-1). In particular, we have f(,uh)L2(n+1)n+3(n) because

p > ( 1.4 ) 2 s ( n 2 + 2 n - 1 ) - 2 n ( n + 1 ) ( n 2 - 1 ) s > s ( 2 n 2 + 3 n - 1 ) - 2 n ( n + 1 ) ( n 2 - 1 ) s .

Hence Proposition 2 yields

lim R 1 R B R | λ ( f ( , u h ) ) ( x ) - π 2 λ ( λ | x | ) n - 1 2 Re ( e i ( n - 3 4 π - λ | x | ) f ( , u h ) ^ ( - λ x ^ ) ) | 2 d x = 0 .

Using h(-ξ)¯=h(ξ) and Proposition 1, we moreover get

lim R 1 R B R | h d σ λ ^ ( x ) - 2 π ( λ | x | ) n - 1 2 Re ( e i ( n - 1 4 π - λ | x | ) h ( λ x ^ ) ) | 2 d x = 0 .

So we have uh=T(uh,h)=hdσλ^+λ(f(,uh)), and the above asymptotics imply (1.6). Finally, in the case p>s(3n-1)-2n(n-1)s, we have V:-Q|u|p-2Lt(n) for some t<2nn+1 because of QLs(n) and uLr(n) for all r>2nn-1. Hence [12, Lemma 2.9] yields the pointwise bounds if n3 and [10, Lemma 2.3] in the case n=2.

Step 6: Distinguishing uh1,uh2. We show that uh1=uh2 implies h1=h2. Indeed, assuming uh1=uh2, we get T(uh1,h1)=T(uh2,h2) and thus ((h1-h2)dσλ)=0. Using the scaling law

( h d σ λ ) ( x ) = λ n - 1 2 ( h ( λ ) d σ 1 ) ( λ x )

as well as [1, Corollary 4.6], we infer

0 = lim R 1 R B R | ( ( h 1 - h 2 ) d σ λ ) | 2 d x = λ n - 1 lim R 1 R B R | ( ( h 1 - h 2 ) ( λ ) d σ 1 ) ( λ x ) | 2 d x = λ n - 1 2 lim R 1 λ R B λ R | ( ( h 1 - h 2 ) ( λ ) d σ 1 ) ( x ) | 2 d x = 2 ( 2 π ) n - 1 λ n - 1 2 S n - 1 | ( h 1 - h 2 ) ( λ x ) | 2 d σ 1 ( x ) ,

which implies h1=h2. ∎

Remark 2.

(a) Let us describe how Theorem 1 provides non-symmetric solutions of symmetric nonlinear Helmholtz equations as mentioned in Remark 1 (b). We assume f(γx,z)=f(x,z) for almost all xn and all z, γΓ, where ΓO(n), Γ{id} is a subgroup. Since Γ{id}, we can find hXλδ and γΓ satisfying hhγ, and our claim is that this implies uhuhγ. Indeed, otherwise, we would have

h d σ λ ^ + λ ( f ( , χ ( u h ) ) ) = T ( u h , h ) = u h = u h γ = T ( u h , h ) γ = h d σ λ ^ γ + λ ( f ( , χ ( u h ) ) ) γ = h γ d σ λ ^ + λ ( f ( , χ ( u h ) ) γ ) = h γ d σ λ ^ + λ ( f ( , χ ( u h γ ) ) ) = h γ d σ λ ^ + λ ( f ( , χ ( u h ) ) ) .

From the second to the third line, we used that λ is a convolution operator with a radially symmetric and hence Γ-symmetric kernel, and from the third to the fourth line, we used that f is Γ-invariant. So we conclude hdσλ^=hγdσλ^, which implies h=hγ as in step 6 above, a contradiction. Hence uhuhγ so that uh is not Γ-symmetric.

(b) In Remark 1 (c), we claimed that Theorem 1 provides radial solutions assuming the weaker condition (1.7) instead of (1.4). This is due to an improved version of the resolvent estimates from (2.2), where, in both lines, 2n+11t-1q can be replaced by 3n-12n2<1t-1q. This was demonstrated in [6, Remark 3.1]. Let us explain how these improved resolvent estimates allow to obtain radial solutions for a larger range of exponents. The only radially symmetric Herglotz waves hdσλ^ are given by real-valued and constant densities h. So, for h, we get that T(,h) maps Lradq(n) into itself provided f(x,u)=f0(|x|,u) is radially symmetric. Here the exponent q may be chosen from

Ξ s , p rad :- { q ( 2 n n - 1 , 2 n ( n - 3 ) + ) : q < min { 2 n 2 s ( p - 2 ) ( ( 3 n - 1 ) s - 2 n 2 ) + , 2 n s ( p - 1 ) ( s ( n + 1 ) - 2 n ) + } } ,

which now is nonempty due to (1.7). Replacing in Proposition 4 the first inequality in (3.6) by

t * ( q , s ) < 2 n 2 q 2 n 2 + ( 3 n - 1 ) q

and redefining q~ in the proof of Theorem 1 accordingly, we get the desired existence result again from Banach’s fixed point theorem.

(c) Under severe restrictions on the nonlinearity, our result may also be proved using dual variational methods originally developed by Evéquoz and Weth [12]. To demonstrate this, we consider the special case f(x,z)=|z|p-2z with 2(n+1)n-1<p<2nn-2. The dual functional Jh:Lp(n) is then given by

J h ( v ) :- 1 p n | v | p - n v ( h d σ λ ) - 1 2 p . v . n | v ^ ( ξ ) | 2 | ξ | 2 - λ d ξ ,

and a local minimizer of Jh lying in the interior of a small ball may be shown to exist for hXλδ for δ>0 sufficiently small using Ekeland’s variational principle. Since every critical point vh of Jh provides a fixed point of T(,h) via uh:-|vh|p-2vh (see [12, Section 4]), we rediscover the solutions found in Theorem 1.

4 Proof of Theorem 2

In this section, we discuss how the above approach needs to be modified in order to get solutions of the fourth order problem (1.8).

We first consider case (i) in (1.9). From Theorem 5, we know that there is a resolvent-type operator 𝐑 associated with Δ2-βΔ+α which is linear and bounded between the same (and even more) pairs of Lebesgue spaces as λ. So all estimates in Proposition 4 involving λ equally hold for 𝐑. As a replacement for the Herglotz wave of the Helmholtz operator, we take again a Herglotz wave hdσλ^, where now hYδ=Xλδ and λ was defined in (1.10) in dependence of α,β. This definition of λ was made in such a way that hdσλ^ satisfies the homogeneous equation Δ2ϕ-βΔϕ+αϕ=0 because of λ2-βλ+α=0. As a consequence, also the Herglotz-wave part of the map

T ( u , h ) :- h d σ λ ^ + 𝐑 ( f ( , χ ( u ) ) )

may be estimated as in Proposition 4. So we can find a fixed point of T(,h) in a small ball of Lq(n) for qΞs,p exactly as in the proof of Theorem 1. The qualitative properties can also be proved the same way; see also [6, Section 5], where uhW4,r(n) for all r(2nn-1,) as well as its pointwise decay rate was proved in the special case f(x,z)=Γ(x)|z|p-2z, ΓL(n).

In case (ii), the proof is essentially the same. The only difference is that the fixed point operator now reads

T ( u , h ) :- h 1 d σ λ 1 ^ + h 2 d σ λ 2 ^ + 𝐑 ( f ( , χ ( u ) ) )

for h=(h1,h2)Yδ=Xλ1δ×Xλ2δ. Besides that, all arguments are identical, and we conclude as above. ∎

5 Proof of Theorem 3

Theorem 3 will be proved via the contraction mapping theorem on a small ball in Lcylq(3;3) (part (i)) or on the entire space Lq(3;3) (part (ii)). The reason for this is that the limiting absorption principle in the latter case is much weaker and forces us to consider nonlinear curl-curl equations with nonlinearities that grow sublinearly at infinity. Notice that the growth of the nonlinearity with respect to E cannot be ignored by using a truncation as in our results proved above. In fact, an equivalent of local elliptic regularity theory and in particular (Lr,L)-estimates for the curl-curl-operator are not known and may even be false.

We start with a few words on the proof of part (i), which is very similar to the proof of Theorems 1 and 2. So let f satisfy (A’). For qΞs,p defined in (1.4) (for n=3), we consider the map

T ( , h ) : L cyl q ( 3 ; 3 ) L cyl q ( 3 ; 3 ) ,

where

(5.1) T ( E , h ) :- h d σ λ ^ + R λ ( f ( , χ ( | E | ) E / | E | ) ) .

Here Lcylq(3;3) denotes the Banach space of cylindrically symmetric functions lying in Lq(3;3), and the function χC() is chosen as before, i.e., it satisfies |χ(z)|min{|z|,1} as well as χ(z)=z provided |z|12. The map T(,h) is well-defined for hZcylδ, δ>0 since the nonlinearity is compatible with cylindrical symmetry by (A’). By Proposition 3, the functions hdσλ^ are smooth cylindrically symmetric solutions of ××E-λE=0 and satisfy the same estimates as their scalar counterparts used in the proof of Theorem 1. Likewise, Theorem 6 implies that Rλ restricted to the space of cylindrically symmetric functions has the same Lp-Lq mapping properties as λ. Moreover, not only f but also the function (x,E)f(x,χ(|E|)E/|E|) satisfies (A’) because zχ(z)/z is smooth. So the operator T defined in (5.1) also satisfies the estimates from Proposition 4, and one obtains a unique fixed point Eh of T(,h) on a small ball in Lcylq(3;3) via the contraction mapping theorem. As a cylindrically symmetric and hence divergence-free solution of (1.11), the vector field Eh even solves the elliptic system (1.13) so that elliptic regularity theory implies χ(Eh)=Eh provided the ball in Lcylq(3;3) and hZcylδ,δ>0 are chosen small enough. Also, the estimate |Eh(x)|Ch(1+|x|)1-n2 is proved as in step 5 of the proof of Theorem 1 under the assumption p>s(3n-1)-2n(n-1)s.

From now on, we prove part (ii), so let f satisfy assumption (B). We fix an exponent q such that 3<q<3s(2s-3)+, which is possible due to s[1,2]. For hZ, we set

(5.2) T ( E , h ) :- h d σ λ ^ + R λ ( f ( , E ) ) .

We first verify that T(,h):Lq(3;3)Lq(3;3) is well-defined and Lipschitz continuous. In the proof of these estimates, we use the number

α p , p ~ :- sup z | z | p - 2 ( 1 + | z | ) p ~ - p = ( p - 2 ) p - 2 ( 2 - p ~ ) 2 - p ~ ( p - p ~ ) p - p ~ where  0 0 :- 1 and p ~ 2 p .

Proposition 5.

Assume (B) and λ>0, hZ. Then the map T(,h):Lq(R3;R3)Lq(R3;R3) from (5.2) is well-defined with

(5.3) T ( E 1 , h ) - T ( E 2 , h ) q C α p , p ~ ( Q s + Q ) E 1 - E 2 q ,

where C only depends on q and s.

Proof.

By Proposition 3, the functions hdσλ^ belong to Lq(3;3) for all hZ. So the definition of T from (5.2) and the limiting absorption principle for the curl-curl operator (Theorem 6) imply that T(,h) is well-defined if we can show f(,E)Lq(3;3)Lt(3;3) for some t(1,) satisfying (2.2). To verify this, we set s~:-max{s,32} for s[1,2] as in assumption (B) and choose t:-s~qs~+q. This implies 1<t<32, 32s~2 so that (t,q) indeed satisfies (2.2). So we infer from assumption (B)

f ( , E ) q Q | E | p - 1 ( 1 + | E | ) p ~ - p q α p , p ~ Q | E | q α p , p ~ Q E q < ,
f ( , E ) t Q | E | p - 1 ( 1 + | E | ) p ~ - p t α p , p ~ Q | E | t α p , p ~ Q s ~ E q < .

This implies that T(,h) is well-defined. Moreover, we have

f ( , E 1 ) - f ( , E 2 ) q α p , p ~ Q E 1 - E 2 q ,
f ( , E 1 ) - f ( , E 2 ) t α p , p ~ Q s ~ E 1 - E 2 q α p , p ~ ( Q s + Q ) E 1 - E 2 q .

Combining these estimates with Theorem 6, one gets (5.3) from (5.2). ∎

Proof of Theorem 3 (ii).

From the previous proposition, we get that T(,h) maps Lq(3;3) into itself, and it is a contraction provided αp,p~(Qs+Q) is small enough, which is guaranteed by the assumptions of Theorem 3. So, for any given hZ, the contraction mapping theorem and Theorem 6 provide a unique weak solution EhHloc(curl;3)Lq(3;3) of (1.11). It remains to discuss the integrability properties of Eh. In this discussion, we will, w.l.o.g., assume that assumption (B) holds with p~(1,2] because otherwise we may simply increase p~.

Proof of (ii) (a). Under the additional assumption (p,s)(2,2), we want to show EhLr(3;3) for all r(3,q). To achieve this iteratively, we use EhLq(3;3) and hence, by assumption (B),

f ( , E h ) L t ( 3 ; 3 ) L q ~ ( 3 ; 3 ) for all t , q ~ [ t * ( q , s ) , q p ~ - 1 ] .

This follows as in (3.3), where also t*(q,s) is defined. In order to prove EhLq~(3;3) with q~<q, we use Eh=T(Eh,h)=hdσλ^+Rλ(f(,Eh)). In view of Proposition 3 and the mapping properties of Rλ from Theorem 6, we obtain EhLq~(3;3) with q~<q provided the pair (t,q~) satisfies 1<t,q~< as well as the inequalities from (2.2) in the three-dimensional case n=3. In other words, we require

(5.4) 1 < t < 3 2 , 3 < q ~ < q , 1 2 1 t - 1 q ~ 2 3 , t * ( q , s ) t , q ~ q p ~ - 1 .

So we set

q ~ :- max { r , 2 t 2 - t , t * ( q , s ) } = max { r , 2 t 2 - t } r > 3 .

(Notice that our choice for t will ensure t*(q,s)<t<32<3<r.) Plugging in the definition of t*(q,s) from (3.3), we obtain after some calculations that inequalities (5.4) hold if

max { 1 , 3 r 3 + 2 r , q s q + s ( p - 1 ) } < t < min { 3 2 , 2 q q + 2 , q p ~ - 1 } .

Here non-strict inequalities in (5.4) were sharpened to strict inequalities for notational convenience. Such a choice for t is possible because of 1s2p, (p,s)(2,2) and 3<r<q<3s(2s-3)+<3s(p-1)(2s-3)+. For instance, we may choose

t = t q :- 1 2 ( max { 1 , 3 r 3 + 2 r , q s q + s ( p - 1 ) } + min { 3 2 , 2 q q + 2 , q p ~ - 1 } ) ,

and Theorem 6 implies EhLq~(3;3). In the case q~=r, we are done. Otherwise, we may repeat this argument replacing q by q~ so that r(3,q~). The corresponding iteration yields EhLr(3;3) after finitely many steps.

Proof of (ii) (b). Using p~<2<p, we now prove EhLr(3;3) for all r(q,3s(p-1)(2s-3)+). In view of (B) and EhLq(3;3), we now have to choose t,q~[t*(q,s),qp~-1] with q~>q such that the pair (t,q~) satisfies 1<t,q~< as well as

1 < t < 3 2 , q < q ~ < , 1 2 1 t - 1 q ~ 2 3 , t * ( q , s ) t , q ~ q p ~ - 1 .

So we set q~:-min{3t3-2t,qp~-1}, and due to 1<p~<2, it remains to choose t such that

max { 1 , 3 q 3 + 2 q , q s q + s ( p - 1 ) } < t < min { 3 2 , 2 q q + 2 ( p ~ - 1 ) , q p ~ - 1 } .

Such a choice is possible thanks to q<3s(p-1)(2s-3)+ and 1s2, 1<p~<2. For instance, we may choose

t = t q :- 1 2 ( max { 1 , 3 q 3 + 2 q , q s q + s ( p - 1 ) } + min { 3 2 , 2 q q + 2 ( p ~ - 1 ) , q p ~ - 1 } ) .

Then Theorem 6 implies EhLq~(3;3), and we have q~>q. We may repeat this argument as long as q~<3s(p-1)(2s-3)+ and thereby obtain uLr(3;3) for all r(q,3s(p-1)(2s-3)+), which finishes the proof. ∎


Communicated by Aldo Pratelli


Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: CRC 1173

Funding statement: The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 “Wave phenomena: analysis and numerics”.

A Proof of Proposition 1

In this section we give the proof of Proposition 1 which we repeat for convenience.

Proposition.

Let nN, n2 and m:-n-12+1. Then, for all hCm(Sλn-1;C), the Herglotz wave hdσλ^ is an analytic solution of -Δϕ-λϕ=0 in Rn and satisfies the estimate |(hdσλ)^(x)|ChCm(1+|x|)1-n2 as well as

lim R 1 R B R | h d σ λ ^ ( x ) - 1 2 π ( λ | x | ) n - 1 2 m h ( x ) | 2 d x = 0 .

In particular, we have hdσλ^rCrhCm for all r>2nn-1.

The asymptotics of the functions hdσλ^ are proved using the method of stationary phase, but typically it is assumed that h is smooth; see, for instance, [25, Propositions 4–6 in Chapter VIII, § 2] or [17, pp. 6–7]. In spirit, the above result is not new, but we could not find a reference for it covering all space dimensions n2 with an explicit value for m. For this reason, we decided to present a proof here.

Proof of Proposition 1.

We consider the Herglotz wave

h d σ λ ^ ( x ) = 1 ( 2 π ) n / 2 S λ n - 1 h ( ξ ) e - i x , ξ d σ λ ( ξ ) .

To investigate its pointwise behavior as |x|, let Q=QxO(n) satisfy QTx=|x|en, so

h d σ λ ^ ( x ) = 1 ( 2 π ) n / 2 S λ n - 1 h ( Q ξ ) e - i | x | ξ n d σ λ ( ξ ) .

Now we choose η1,,η2nC(n) such that η1++η2n=1 on Sλn-1 and, for k=1,,n,

supp ( η 2 k - 1 ) { ξ = ( ξ 1 , , ξ n ) n : ξ k > + λ δ } ,
supp ( η 2 k ) { ξ = ( ξ 1 , , ξ n ) n : ξ k < - λ δ } ,

where δ(0,1n) is fixed. Notice that such a partition of unity exists since the open sets on the right-hand side cover the sphere Sλn-1. We define hj(ξ):-(2π)-n2h(Qξ)ηj(ξ) so that we have to investigate the integrals

I j :- S λ n - 1 h j ( ξ ) e - i | x | ξ n d σ λ ( ξ ) ( j = 1 , , 2 n ) .

We first estimate the oscillatory integrals I1,,I2n-2, where the resonant poles ±en are cut out by the choice of η1,,η2n-2. To estimate Ij, we use the local parametrization given by

ψ j ( ξ ) :- { λ ( ξ 1 , , ξ k - 1 , + 1 - | ξ | 2 , ξ k , , ξ n - 1 ) S λ n - 1 if j = 2 k - 1 , λ ( ξ 1 , , ξ k - 1 , - 1 - | ξ | 2 , ξ k , , ξ n - 1 ) S λ n - 1 if j = 2 k ,

for ξ:-(ξ1,,ξn-1) belonging to the unit ball Bn-1. The function

H j ( ξ ) :- λ n - 1 2 h j ( ψ j ( ξ ) ) ( 1 - | ξ | 2 ) - 1 2

satisfies supp(Hj)B so that HjC0m(n) by trivial extension. Integration by parts yields, for all |x|1 and j=1,,2n-2,

| I j | = | B H j ( ξ ) e - i λ | x | ξ n - 1 d ξ | = | n - 1 H j ( ξ ) m ( ξ n - 1 ) m ( e - i λ