Uncountably many solutions for nonlinear Helmholtz and curl-curl equations with general nonlinearities
aa r X i v : . [ m a t h . A P ] N ov UNCOUNTABLY MANY SOLUTIONS FOR NONLINEAR HELMHOLTZAND CURL-CURL EQUATIONS WITH GENERAL NONLINEARITIES
RAINER MANDEL
Abstract.
We obtain uncountably many solutions of nonlinear Helmholtz and curl-curlequations on the entire space using a fixed point approach. As an auxiliary tool a LimitingAbsorption Principle for the curl-curl operator is proved. Introduction and main results
The propagation of light in nonlinear media is governed by Maxwell’s equations ∇ × E + ∂ t B = , div ( D ) = , ∇ × H − ∂ t D = , div ( B ) = E , H ∶ R × R → R , the displacement field D ∶ R × R → R and the magnetic induction B ∶ R × R → R . Here, the effect of chargesand currents is neglected. The nonlinearity of the medium is typically expressed throughnonlinear material laws of the form D = ε ( x )E + P and the linear relation H = µ B for thepermittivity function ε ∶ R → R and the magnetic permeability µ ∈ R ∖ { } . In [2, 23] it wasshown that special solutions of the form E ( 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 equations(1) ∇ × ∇ × E + V ( x ) E = f ( x, E ) in R where V ( x ) = − µω ε ( x ) ≤ f ( x, E ) = µω P ( x, E ) . We stress that V is nonpositive andit becomes a negative constant in the simplest and most relevant case of the vacuum where ε ( x ) ≡ ε >
0. We refer to Section 1.3 in [2] for further details concerning the modellingaspect of (1). One of our main results (Theorem 3) will provide a new existence result forsolutions of this problem. A simplified version of (1) is the nonlinear Helmholtz equation(2) − ∆ u + V ( x ) u = f ( x, u ) in R n , which in the two-dimensional case n = E ( x , x , x ) =( , , u ( x , x )) . In this paper we are interested in solutions of (1),(2) when the potential V is a negative constant and the nonlinearity is rather general. Our principal motivation is toshow that for large classes of nonlinearities there are uncountably many solutions of theseequations sharing the same decay rate ∣ x ∣ − n as ∣ x ∣ → ∞ but with a different farfield pattern. Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary: 35Q60, 35Q61, 35J91.
Key words and phrases.
Nonlinear Helmholtz equations, Curl-Curl equations, Limiting Absorption Prin-ciples, Herglotz waves.
We first recall some facts about the nonlinear Helmholtz equation with constant potential(3) − ∆ u − λu = f ( x, u ) in R n . In 2004 Guti´errez [17] set up a fixed point approach for this equation when f ( x, u ) = ∣ u ∣ u , n ∈ { , } and u is complex-valued. Using an L p -version of the Limiting Absorption Principlefor the Helmholtz operator (Theorem 6 in [17]) she found that small nontrivial solutions of (3)can be obtained via the Contraction Mapping Theorem (Banach’s Fixed Point Theorem) ona small ball in L ( R n ) . Around ten years later Ev´equoz and Weth started to write a seriesof papers [9–14] containing new methods to prove existence results for solutions of (3) that,in contrast to Guti´errez’ solutions, are large in suitable norms. Some of these results wereextended by the author in [21, 22]. In each of the aforementioned papers the nonlinearity hasto satisfy quite specific conditions that allow to deal with slow decay rates of solutions atinfinity.In the case of power-type nonlinearities f ( x, u ) = Q ( x )∣ u ∣ p − u one of the main results in [10]is that there is an unbounded sequence of solutions in L p ( R n ) provided ( n + ) n − < p < nn − and Q ∈ L ∞ ( R n ) is positive and evanescent at infinity. If Q is Z n -periodic and positive (fornegative Q see Theorem 1.3, 1.4 in [21]) the existence of one nontrivial solution is shown.These solutions are obtained using quite sophisticated dual variational methods and thesolution at the mountain pass level of the dual functional is called a dual ground state ofthe equation. One of the drawbacks of this approach is that the assumption on p does notallow for cubic nonlinearities, which certainly are the most interesting ones for applicationsin physics. Moreover, sign-changing or non-monotone nonlinearities can not be treated. Inaddition to that, solutions have to be looked for in L p ( R n ) . This is a problem given thatsolutions decay slowly at infinity so that a solution theory in L q ( R n ) with q > p is moreconvenient a priori. For this reason we will not consider dual variational methods but ratherrevive Guti´errez’ fixed point approach [17].Our refinement of Guti´errez’ method allows to discuss nonlinear Helmholtz equations withvery general nonlinearities that improve existing results even in the case of power-type non-linearities as we will see below. In our main result dealing with (3) we show that in the case f ( x, u ) = Q ( x )∣ u ∣ p − u with Q ∈ L s ( R n ) ∩ L ∞ ( R n ) we get solutions for all exponents(4) p > max { , s ( n + n − ) − n ( n + )( n − ) s } . More generally, we can treat nonlinearities satisfying the following conditions:(A) f ∶ R n × R → R is a Carath´eodory function satisfying for some Q ∈ L s ( R n ) ∩ L ∞ ( R n )∣ f ( x, z )∣ ≤ Q ( x )∣ z ∣ p − ( x ∈ R n , ∣ z ∣ ≤ )∣ f ( x, z ) − f ( x, z )∣ ≤ Q ( x )(∣ z ∣ + ∣ z ∣) p − ∣ z − z ∣ ( x ∈ R n , ∣ z ∣ , ∣ z ∣ ≤ ) . (5) where s ∈ [ , ∞ ] and p as in (4).We stress that only conditions near zero are needed since we are going to construct smallsolutions in L q ( R n ) which will turn out to be small also in L ∞ ( R n ) . Clearly, ∣ z ∣ ≤ OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 3 replaced by ∣ z ∣ < z for any given z >
0. We mention that in the case s ≤ n + all exponents p > L p -version of the LimitingAbsorption Principle for the Helmholtz operator − ∆ − λ ( λ >
0) and results about the so-calledHerglotz waves. As we will recall in Proposition 1, these functions are analytic solutions ofthe linear Helmholtz equation − ∆ φ − λφ = R n . They are given by the formula ̂ h dσ λ ( x ) ∶= ( π ) n / ∫ S n − λ h ( ξ ) e − i ⟨ x,ξ ⟩ dσ λ ( ξ ) for complex-valued densities h ∈ L ( S n − λ ; C ) . Here, σ λ denotes the canonical surface measureof the sphere S n − λ = { ξ ∈ R n ∶ ∣ ξ ∣ = λ } . In order to ensure the real-valuedness and goodpoinwise decay properties of ̂ h dσ λ at infinity, we will consider a smaller class of densities h belonging to the set X δλ ∶= { h ∈ C m ( S n − λ ; C ) ∶ h ( ξ ) = h ( − ξ ) , ∥ h ∥ C m ≤ δ } where m ∶= ⌊ n − ⌋ +
1. Here, our approach differs from [17] where L -densities are used.Theorem 1 shows that for all h ∈ X δλ we find a strong solution of (3) that resembles ∣ x ∣ − n u ∞ h ( x ) at infinity where u ∞ h ( x ) ∶= λ n − √ π ( e i ( n − π −√ λ ∣ x ∣) ( ̂ f ( ⋅ , u h )( − √ λ ˆ x ) + i ⋅ √ λπ h (√ λ ˆ x ))) , ˆ x ∶ = x ∣ x ∣ . More precisely, we show the following.
Theorem 1.
Assume (A) and λ > . Then there are δ > and mutually different solutions ( u h ) h ∈ X δλ of (3) that form a W ,r ( R n ) -continuum and satisfy ∥ u h ∥ W ,r ( R n ) → as ∥ h ∥ C m → for any given r ∈ ( nn − , ∞ ) as well as lim R →∞ R ∫ B R ∣ u h ( x ) − ∣ x ∣ − n u ∞ h ( ˆ x )∣ dx = . If additionally p > ( n − ) s − n ( n − ) s holds, then ∣ u h ( x )∣ ≤ C h ( + ∣ x ∣) − n for all x ∈ R n . Let us discuss in which way this theorem improves earlier results. Most importantly,Theorem 1 shows that nonlinear Helmholtz equations of the form (3) admit uncountablymany 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 MappingTheorem, elliptic regularity theory and mostly well-known results about the linear Helmholtzequation. Up to now such general nonlinearities have only been treated in the paper [12] byEv´equoz and Weth, but their additional requirement ( f ) on p.361 requires the nonlinearityto be supported in a bounded subset of R n , which is quite restrictive. In our approachsuch an assumption is not necessary. Given that applications often deal with power-typenonlinearities f ( x, z ) = Q ( x )∣ z ∣ p − z let us comment on our improvements for this particularcase in more detail. In the case Q ∈ L ∞ ( R n ) we obtain solutions for all exponents p > ( n + n − ) n − . This bound is smaller than ( n + ) n − so that our range of exponents is larger than in RAINER MANDEL all other nonradial approaches except for [12] where Q has compact support and exponents2 < p < nn − are allowed. Additionally, we need not require Q to be periodic nor evanescent (asin [9, 10, 21]) nor compactly supported (as in [12, 13]) and the growth rate of the nonlinearitymay be supercritical (i.e. p ≥ nn − ) which is an entirely new feature. The latter fact isworth mentioning given that Ev´equoz and Yesil [14] proved the nonexistence of dual groundstates u ∈ L p ( R n ) for n = p = nn − = f ( x, u ) = Q ( x ) u and Q ∈ L ∞ ( R ) is nonnegative and nontrivial. Since Theorem 1 yields solutions belonging to L p ( R ) we conclude that dual ground states need not exist while other nontrivial solutionsdo. Finally we mention that in the physically most relevant case of a cubic nonlinearity p = n ≥ , s ∈ [ , ∞ ] or n = , s ∈ [ , ) . Remark 1. (a) The decay rate ∣ x ∣ − n is best possible. This is a consequence of Theorem 3in [20] where nontrivial solutions of the elliptic PDE − ∆ u − λu = W ( x ) u in R n with W ∈ L n + ( R n ) and λ > are shown to satisfy u ( x )∣ x ∣ − − ε ∉ L ( R n ) for all ε > . Inparticular, better decay rates than ∣ x ∣ − n as ∣ x ∣ → ∞ are excluded. Notice that in thesetting of Theorem 1 the function W ( x ) ∶ = f ( x, u ( x ))/ u ( x ) satisfies W ∈ L n + ( R n ) because of Q ∈ L s ( R n ) and ∣ W ( x )∣ ≤ Q ( x )∣ u ( x )∣ p − , u ∈ ⋂ r > nn − L r ( R n ) , p > s ( n + n − ) − n ( n + )( n − ) s , see (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 forlower exponents p are only known in the radial setting [12,21]. Notice that for smaller p the (nonradial) counterexample of Ionescu and Jerison from Theorem 2.5 in [19]has to be taken into account: For any given N ∈ N there is W ∈ L q ( R n ) with q > n + and a solution of − ∆ u − λu = W ( x ) u in R n with ∣ u ( x )∣ ≤ ( + ∣ x ∣) − N for all x ∈ R n .(b) Theorem 1 yields a symmetry-breaking result: For any subgroup Γ ⊂ O ( n ) such that Γ ≠ { id } and any Γ -invariant nonlinearity f satisfying (A) one has uncountably manysolutions that are not Γ -invariant. In particular, for Γ = O ( n ) , radial nonlinearitiesallow 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 1under weaker assumptions on p . This is due to an improved version of the LimitingAbsorption Principle for the Helmholtz operator. In Remark 2(b) we comment on thenecessary modifications of the proof and show that the admissible range of exponentsfor the existence of radial solutions is no longer given by (4) , but (6) p > max { , s ( n + n − ) − n sn ( n − ) } . Notice that the resulting radial version of Theorem 1 is not covered by earlier contri-butions from [21] (Theorem 1.2, Theorem 2.10) or [12] (Theorem 4). For instance, itprovides solutions of the radial nonlinear Helmholtz equation − ∆ u − u = Q ( x )∣ u ∣ p − u for any Q ∈ L srad ( R n ) ∩ L ∞ rad ( R n ) and p as in (6) whereas in the above-mentioned papers Q has to be bounded, differentiable and radially decreasing. On the other hand, our OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 5 restrictions on the exponent p do not appear in [12, 21] (where all p > are allowed)so that (6) might be improved further. Next we discuss variants of these results for related semilinear elliptic PDEs from math-ematical physics. First let us mention that a nonlinearity f ( ⋅ , u ) satisfying (A) may bewithout any major difficulty be replaced by a nonlocal right hand side such as K ∗ f ( ⋅ , u ) where K ∈ L ( R n ) . Clearly, imposing more assumptions K may even lead to larger ranges ofexponents than (4). In this way it is possible to obtain small solutions of nonlocal Helmholtzequations. Similarly, one may ask how our results are affected by changes in the linearoperator. For instance, if the Helmholtz operator is perturbed to a periodic Schr¨odingeroperator − ∆ + V ( x ) − λ then it should be possible to adapt the proof in such a way thatit provides small solutions of − ∆ u + V ( x ) u − λu = f ( x, u ) in R n provided λ belongs to theessential spectrum of − ∆ + V ( x ) and the band structure of this periodic Schr¨odinger operatoris sufficiently nice. To be more precise, one would require (A1),(A2),(A3) from [22] to holdso that Herglotz-type waves, defined as suitable oscillatory integrals over the so-called Fermisurfaces associated with − ∆ + V ( x ) , exist and have the properties stated in Proposition 1below. Since the technicalities (including a Limiting Absorption Principle for such operators)are quite involved and mostly carried out in [22], we prefer not to discuss this issue further.We now turn our attention to a fourth order version of (3) given by(7) ∆ u − β ∆ u + αu = f ( x, u ) in R n , which we will briefly discuss for α, β satisfying(8) (i) α < , β ∈ R or (ii) α > , β < − √ α. Under these assumptions dual variational methods were employed in [7] to prove the existenceof one nontrivial solution when f ( x, z ) = Q ( x )∣ z ∣ p − z where ( n + ) n − < p < nn − and Q is positiveand Z n -periodic. Notice that in the case β − α < u ∈ H ( R n ) of (7) again for power-type nonlinearities. We referto [5, 6, 8] for results in this direction. Our intention is to show that in the case (i) or(ii) uncountably many solutions of (7) exist for all nonlinearities f satisfying (A). The mainobservation is that in case (i) or (ii) there are analoga of the Herglotz waves given by densities h ∈ Y δ where In case (i) : Y δ ∶ = X δλ , where λ = − β + √ β − α > , In case (ii): Y δ ∶ = X δλ × X δλ where λ , = − β ± √ β − α > . (9)The fixed point approach used in the proof of Theorem 1 may be rather easily adapted to (7)and we can prove the following result. Theorem 2.
Assume (A) and (i) or (ii). Then there is δ > and mutually different solutions ( u h ) h ∈ Y δ of (7) that form a W ,r ( R n ) -continuum satisfying ∥ u h ∥ W ,r ( R n ) → as ∥ h ∥ C m → RAINER MANDEL for any given r ∈ ( nn − , ∞ ) . If additionally p > ( n − ) s − n ( n − ) s holds, then ∣ u h ( x )∣ ≤ C h ( + ∣ x ∣) − n for all x ∈ R n . As in Theorem 1 one can say more about the asymptotics of the constructed solutions; werefer to Section 5.2 in [7] for a related discussion. Further generalizations to more generalhigher order semilinear elliptic problems of the form Lu = f ( x, u ) in R n are possible providedthe linear differential operator with constant coefficients L has a Fourier symbol P ( ξ ) withthe property that { ξ ∈ R n ∶ P ( ξ ) = } is a compact manifold with nonvanishing Gaussiancurvature. Notice that this assumption makes the method of stationary phase work andprovides 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 Fouriermultiplier 1 / P ( ξ ) as a mapping between Lebesgue spaces. At least in the case P ( ξ ) = P ( ξ )(∣ ξ ∣ − λ ) ⋅ . . . ⋅ (∣ ξ ∣ − λ k ) with 0 < λ < . . . < λ k and P positive this can be establishedas in Theorem 3.3 in [7]. With these tools our fixed point approach can be adapted to findnontrivial solutions of Lu = f ( x, u ) in R n .Finally, we discuss nonlinear curl-curl equations of the form(10) ∇ × ∇ × E − λE = f ( x, E ) in R that describe the electric field E ∶ R → R of an electromagnetic wave in a nonlinear medium.This equation has been studied in the past years on bounded domains in R [3, 4] but alsoon the entire space R , which is the situation we focus on. Up to our knowledge there is onlyone result for solutions of nonlinear curl-curl equations on R without symmetry assumption.In [23] Mederski proves the existence of a weak solution of (10) by variational methods when λ is replaced by a small nonnegative potential V ( x ) that decays suitably fast to zero atinfinity, see assumption (V) in [23]. In particular, constant functions V ( x ) = λ can not betreated by this method so that our setting must be considered as entirely different from theone in [23].In the cylindrically symmetric setting the existence of solutions can be proved using variousapproaches. Here, the electrical field is assumed to be of the form(11) E ( x , x , x ) = E (√ x + x , x )√ x + x ⎛⎜⎝ − x x ⎞⎟⎠ where E ∶ [ , ∞ ) × R → R . Such functions are divergence-free so that ∇ × ∇ × E = − ∆ E implies that one actually has todeal with the elliptic 3 × − ∆ E − λE = f ( x, E ) in R , which may equally be expressed in terms of E provided the nonlinearity f ( x, E ) is compatiblewith this symmetry assumption, see page 3 in [2]. In this special case further results [2,18,26]are known but none of those applies in the case λ > f ( x, E ) = ± ∣ E ∣ p − E that we aremainly interested in.Given our earlier results for the nonlinear Helmholtz equation (3) it is not surprising thatwe obtain an existence result for (12) that is entirely analogous to the one from Theorem 1. OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 7
Since this result fills a gap in the literature, we state it in part (i) of our theorem even thoughits proof is a straightforward adaptation of the fixed point approach used in the proof ofTheorem 1. The corresponding assumption on the nonlinearity is the following.(A’) f ∶ R × R → R , ( x, E ) ↦ f (√ x + x , x , ∣ E ∣ ) E is a Carath´eodory function satisfy-ing (5) for some Q ∈ L s ( R ) ∩ L ∞ ( R ) .Assumption (A’) ensures that f is compatible with cylindrical symmetry. Indeed, for E asin (11) one can check that f ( ⋅ , E ) is of the form (11), too.In the general non-symmetric case the construction of solutions is more difficult since thecurl-curl operator satisfies a much weaker Limiting Absorption Principle as in the cylindri-cally symmetric setting, cf. Theorem 6. Moreover, well-known regularity results for ellipticproblems are not available so that we have to consider a substantially smaller class of non-linearities satisfying the following:(B) f ∶ R × R → R is a Carath´eodory function satisfying for some Q ∈ L s ( R ) ∩ L ∞ ( R ) the estimate ∣ f ( x, E )∣ ≤ Q ( x )∣ E ∣ p − ( + ∣ E ∣) ˜ p − p ( x, E ∈ R ) , ∣ f ( x, E ) − f ( x, E )∣ ≤ Q ( x )(∣ E ∣ + ∣ E ∣) p − ( + ∣ E ∣ + ∣ E ∣) ˜ p − p ∣ E − E ∣ ( x, E , E ∈ R ) (13) where 1 ≤ s ≤ p ≤ ≤ p < ∞ .Additionally, we will have to require that ∥ Q ∥ s + ∥ Q ∥ ∞ is small enough in order to obtainsolutions of (10). In the cylindrically symmetric respectively non-symmetric setting thecounterparts of the Herglotz waves (introduced in Section 2) are parametrized by functions h ∈ Z δcyl respectively h ∈ Z where Z ∶ = { h ∈ C ( S λ ; C ) ∶ h ( ξ ) = h ( − ξ ) , ⟨ h ( ξ ) , ξ ⟩ = ∀ ξ ∈ S λ } ,Z δcyl ∶ = { h ∈ Z ∶ ∥ h ∥ C < δ and Re ( h ) , Im ( h ) satisfy (11) } . Notice that both sets are nonempty. With these definitions we can formulate our main resultfor the nonlinear curl-curl equation (10).
Theorem 3. (i) Assume (A’) and λ > . Then there is δ > and a family ( E h ) h ∈ Z δcyl of mutually dif-ferent cylindrically symmetric solutions of (10) that form a W ,r ( R ; R ) -continuumand satisfy ∥ E h ∥ W ,r ( R ; R ) → as ∥ h ∥ C → for any given r ∈ ( , ∞ ) . If additionally p > s − s holds, then ∣ E h ( x )∣ ≤ C h ( + ∣ x ∣) − .(ii) Assume (B) and λ > , < q < s ( s − ) + . If ∥ Q ∥ s + ∥ Q ∥ ∞ is sufficiently small then there isa family ( E h ) h ∈ Z of mutually different weak solutions of (10) lying in H loc ( curl; R ) ∩ L q ( R ; R ) . Moreover:(a) If additionally ( p, s ) ≠ ( , ) holds, then E h ∈ L r ( R ; R ) for all r ∈ ( , q ) .(b) If additionally ˜ p < < p holds, then E h ∈ L r ( R ; R ) for all r ∈ ( q, s ( p − )( s − ) + ) . RAINER MANDEL
As an application we obtain uncountably many distinct weak solutions of the curl-curlequation (10) with saturated nonlinearities of the form f ( x, E ) = δ ∣ E ∣ Γ ( x ) E + P ( x )∣ E ∣ where inf P >
0, Γ ∈ L s ( R ; R × ) ∩ L ∞ ( R ; R × ) and δ > C will denote a generic constant that can change from line to line and r + stands for r if r > ∞ if r ≤
0. The symbol F f = ˆ f represents the Fourier transform of(the tempered distribution) f ∈ L q ( R n ) and F , F n − are the Fourier transforms in R , R n − ,respectively. For R > B R denotes the open ball of radius R around the originin R n and ⟨ ⋅ , ⋅ ⟩ is the inner product in R n extended by bilinearity to C n . L q ( R n ) , L q,w ( R n ) denote the classical respectively weak Lebesgue spaces on R n equipped with the standardnorms ∥ ⋅ ∥ q , ∥ ⋅ ∥ q,w .2. Herglotz waves and Limiting absorption principles
In this section we review some partly well-known results on Herglotz waves and LimitingAbsoprtion Principles for the linear differential operators we are interested in. A classicalHerglotz wave associated with the Helmholtz operator − ∆ − λ is defined via the formula F ( h dσ λ )( x ) ∶ = ̂ h dσ λ ( x ) ∶ = ( π ) n / ∫ S n − λ h ( ξ ) e − i ⟨ x,ξ ⟩ dσ λ ( ξ ) where h ∈ L ( S n − λ ; C ) and σ λ denotes the canonical surface measure of S n − λ = { ξ ∈ R n ∶ ∣ ξ ∣ = λ } . Herglotz waves are analytic functions that solve the linear Helmholtz equation − ∆ φ − 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 couldnot find a quantitative version of this result telling how smooth the density h needs to be inorder to ensure that ̂ h dσ λ ( x ) decays like ∣ x ∣ − n as ∣ x ∣ → ∞ in the pointwise sense. In our firstauxiliary result we provide such an estimate and its proof will be given in Appendix A. Fornotational convenience we introduce the quantity(14) m h ( x ) ∶ = e i ( n − π −√ λ ∣ x ∣) h (√ λ ˆ x ) + e − i ( n − π −√ λ ∣ x ∣) h ( − √ λ ˆ x ) so that our claim is the following. OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 9
Proposition 1.
Let n ∈ N , n ≥ and m ∶ = ⌊ n − ⌋ + . Then for all h ∈ C m ( S n − λ ; C ) theHerglotz wave ̂ h dσ λ is an analytic solution of − ∆ φ − λφ = in R n and satisfies the estimate ∣( ̂ h dσ λ )( x )∣ ≤ C ∥ h ∥ C m ( + ∣ x ∣) − n as well as lim R →∞ R ∫ B R RRRRRRRRRRRRR ̂ h dσ λ ( x ) − √ π ( √ λ ∣ x ∣ ) n − m h ( x )RRRRRRRRRRRRR dx = . In particular, we have ∥ ̂ h dσ λ ∥ r ≤ C r ∥ h ∥ C m for all r > nn − . While Herglotz waves solve the homogeneous Helmholtz equation, we also need to discussthe inhomogeneous equation. Since λ lies in the essential spectrum of − ∆ it is a nontrivialtask to solve − ∆ u − λu = f in R n . The method to find such solutions is to study the limit ofsolutions u ε ∶ = R ( λ + iε ) f ∈ H ( R n ; C ) of − ∆ u ε − ( λ + iε ) u ε = f in a suitable topology. Thecomplex-valued limit of these functions as ε → + is denoted by R ( λ + i ) f and we define R λ f ∶ = Re ( R ( λ + i )) f as its real part since we are interested in real-valued solutions. Theseoperators have the following properties: Theorem 4 (Theorem 6 [17], Theorem 2.1 [9]) . Let n ∈ N , n ≥ . The operator R λ ∶ L t ( R n ) → L q ( R n ) is a bounded linear operator provided t > n + n , q < n − n , n + ≤ t − q ≤ n ( n ≥ ) , t > n + n , q < n − n , n + ≤ t − q < n ( n = ) . (15) Moreover, for f ∈ L t ( R n ) the function R λ f ∈ W ,tloc ( R n ) is a real-valued strong solution of − ∆ u − λu = f in R n . The last statement is actually not included in the references given above, but it is a conse-quence of elliptic regularity theory for distributional solutions. We refer to Proposition A.1in [10] for a similar result. Next we discuss the asymptotic behaviour of the solutions R λ f that we will deduce from the following result. Proposition 2.
Let n ∈ N , n ≥ and assume f ∈ L p ′ ( R n ) for ( n + ) n − ≤ p ≤ n ( n − ) + , ( n, p ) ≠ ( , ∞ ) . Then: lim R →∞ R ∫ B R RRRRRRRRRRRRR R ( λ + i ) f ( x ) − √ π λ ( √ λ ∣ x ∣ ) n − e i ( n − π −√ λ ∣ x ∣) ̂ f ( − √ λ ˆ x )RRRRRRRRRRRRR dx = . Proof.
The claim for n ≥ λ = R λ f ( x ) = λ R ( f ( λ − / ⋅ ))(√ λx ) . The proof in the case n = R ≥ ( R ∫ B R ∣ R ( λ + i ) f ( x )∣ dx ) / ≤ C ∥ f ∥ p ′ are valid in the case n = p . For the Stein-Tomas Theorem thisis clear. The inequality (16) is due to Ruiz and Vega [24], but we could not find an accuratereference for it that covers our range of exponents and all space dimensions n ≥
2. We providethe estimate (16) 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 thefourth order problem (7). In Theorem 3.3 in [7] the following extension of Theorem 4 tolinear differential operators of the form ∆ − β ∆ + α was proved. Theorem 5 (Theorem 3.3 [7]) . Let n ∈ N , n ≥ and assume (i) or (ii). Then there is abounded linear operator R ∶ L t ( R n ) → L q ( R n ) for t > n + n , q < n − n , n + ≤ t − q ≤ n if n ≥ , t > n + n , q < n − n , n + ≤ t − q < if n = , t > n + n , q < n − n , n + ≤ t − q ≤ if n = , such that for f ∈ L t ( R n ) the function R f belongs to W ,tloc ( R n ) and is a real-valued strongsolution of ∆ u − β ∆ u + αu = in R n . Finally we provide the tools for proving Theorem 3. As for the previous results we need afamily 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 dσ λ ( x ) ∶ = ( π ) / ∫ S λ h ( ξ ) e − i ⟨ x,ξ ⟩ dσ λ ( ξ ) (the integral to be understood componentwise) where h ∶ S λ → C is a tangential vectorfieldfield, i.e. ⟨ h ( ξ ) , ξ ⟩ = ξ ∈ S λ . These functions are real-valued whenever h ( ξ ) = h ( − ξ ) . Applying the results from Proposition 1 in each component, we deduce the followingproperties. Proposition 3.
For all h ∈ Z the function ̂ h dσ λ is an analytic solution of ∇ × ∇ × φ − λφ = in R and satisfies the pointwise estimate ∣ ̂ h dσ λ ( x )∣ ≤ ∥ h ∥ C ( + ∣ x ∣) − for all x ∈ R . Inparticular, ∥ ̂ h dσ λ ∥ r ≤ C r ∥ h ∥ C for all r > . If h ∈ Z δcyl , then ̂ h dσ λ is cylindrically symmetric. Having described the analoga of the Herglotz waves we finally discuss a Limiting AbsorptionPrinciple for the curl-curl operator. So let R ( λ + iε ) denote the resolvent of E ↦ ∇ × ∇ × E − ( λ + iε ) E which we will prove to exist in Proposition 6. As for the Helmholtz operatorone is interested in the (complex-valued) limit R ( λ + i ) G for G ∈ L ( R ; R ) . It turns outthat G decomposes into two parts behaving quite differently. So we split G into a curl-free(gradient-like) part G ∶ R → R and a divergence-free remainder G ∶ R → R of G whichare defined via G ∶ = F − (⟨ ˆ G ( ξ ) , ξ ∣ ξ ∣ ⟩ ξ ∣ ξ ∣) , G ∶ = F − ( ˆ G ( ξ ) − ⟨ ˆ G ( ξ ) , ξ ∣ ξ ∣ ⟩ ξ ∣ ξ ∣) . OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 11
This splitting corresponds to a Helmholtz decomposition of a vector field in R . Theorem 6.
Let λ > and assume that t, q ∈ ( , ∞ ) satisfy (15) . Then there is a boundedlinear operator R λ ∶ L t ( R ; R ) ∩ L q ( R ; R ) → L q ( R ; R ) such that R λ G ∈ H loc ( curl; R ) is aweak solution of ∇ × ∇ × E − λE = G provided G ∈ L t ( R ; R ) ∩ L q ( R ; R ) . Moreover, we have ∥ R λ G ∥ q ≤ C (∥ G ∥ q + ∥ G ∥ t ) ≤ C (∥ G ∥ q + ∥ G ∥ t ) and R λ G = − λ G + R λ G for R λ from Theorem 4 (applied componentwise). If G ∈ L t ( R ; R ) is cylindrically symmetric then so is R λ G and R λ G ∈ W ,qloc ( R ; R ) is a strong solution satis-fying ∥ R λ G ∥ q ≤ C ∥ G ∥ t . The proof of Theorem 6 will be given in Appendix B. With these technical preparationswe 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 approachby Guti´errez [17]. We consider the map T ( ⋅ , h ) ∶ L q ( R n ) → L q ( R n ) given by T ( u, h ) ∶ = ̂ h dσ λ + R λ ( f ( ⋅ , χ ( u ))) (17)where h ∈ X δλ as in Proposition 1 and χ is a smooth function such that ∣ χ ( z )∣ ≤ min {∣ z ∣ , } and χ ( z ) = z for ∣ z ∣ ≤ . In view of the properties of the Herglotz waves and R λ mentionedearlier, a fixed point of T ( ⋅ , h ) is a strong solution of the equation − ∆ u − λu = f ( x, χ ( u )) in R n . Since that fixed point u will belong to a small ball in L q ( R n ) , we will be able to show χ ( u ) = u so that a solution of (3) is found. The choice of the exponent q is delicate; it’s themajor technical issue in our approach. Our assumption (4) implies that the setΞ s,p ∶ = { q ∈ ( nn − , n ( n − ) + ) ∶ q < min { s ( n + )( p − )( s − ( n + )) + , ns ( p − )( s ( n + ) − n ) + }} is non-empty, so we may choose some arbitrary but fixed q ∈ Ξ s,p throughout this section. Proposition 4.
Assume (A) and λ > , h ∈ X δλ . Then the map T ( ⋅ , h ) ∶ L q ( R n ) → L q ( R n ) from (17) is well-defined and we have ∥ T ( u, h )∥ q ≤ C (∥ u ∥ αq + ∥ h ∥ C m ) , ∥ T ( u, h ) − T ( v, h )∥ q ≤ C ∥ u − v ∥ q (∥ u ∥ q + ∥ v ∥ q ) α − (18) for some α > and all u, v ∈ L q ( R n ) .Proof. Using (A), H¨older’s inequality and ∣ χ ( u )∣ ≤ min {∣ u ∣ , } we get for all u ∈ L q ( R n )∥ f ( ⋅ , χ ( u ))∥ t ≤ ∥∣ Q ∣∣ χ ( u )∣ p − ∥ t ≤ ∥ Q ∥ ˜ s ∥ χ ( u ) p − ∥ t ˜ s ˜ s − t ≤ ∥ Q ∥ ˜ s ∥ u ∥ qt − q ˜ s q < ∞ , provided t ∗ ( q, ˜ s ) ∶ = max { , ˜ sq ˜ s ( p − ) + q } ≤ t ≤ ˜ s, ˜ s ∈ [ s, ∞ ] . (19) In particular we find f ( ⋅ , χ ( u )) ∈ L t ( R n ) for all t ∈ [ t ∗ ( q, s ) , ∞ ] . For any such t we choose˜ s ∶ = tq ( q − t ( p − )) + ∈ [ s, ∞ ] (largest possible) so that t ∗ ( q, ˜ s ) ≤ t ≤ ˜ s holds. So the previous estimategives for α t ∶ = qt − q ˜ s = min { qt , p − }∥ f ( ⋅ , χ ( u ))∥ t ≤ C ∥ u ∥ α t q < ∞ for all t ∈ [ t ∗ ( q, s ) , ∞ ] . (20)Now we have to choose t ∈ [ t ∗ ( q, s ) , ∞ ] in such a way that the mapping properties of R λ fromTheorem 4 ensure R λ ( f ( ⋅ , χ ( u ))) ∈ L q ( R n ) . In view of (15) we have to require(21) nqn + q ≤ t < ( n + ) qn + + q and t < nn + . Since q ∈ Ξ s,p implies q < n ( n − ) + and hence nqn + q < nn + , we can find such t if and only if(22) t ∗ ( q, s ) < ( n + ) qn + + q and t ∗ ( q, s ) < nn + . These two inequalities hold due to q ∈ Ξ s,p . From this, Proposition 1 and Theorem 4 we get ∥ T ( u, h )∥ q ≤ ∥ R λ ( f ( ⋅ , χ ( u )))∥ q + ∥ ̂ h dσ λ ∥ q ≤ C (∥ f ( ⋅ , χ ( u ))∥ t + ∥ h ∥ C m ) (20) ≤ C (∥ u ∥ α t q + ∥ h ∥ C m ) . Since t was chosen according to (21), we have t < q and thus α t = min { p − , qt } >
1. Moreover,from (A) and H¨older’s inequality ( t = s + q + α t − q ) we get ∥ f ( ⋅ , χ ( u )) − f ( ⋅ , χ ( v ))∥ t ≤ C ∥∣ Q ∣∣ χ ( u ) − χ ( v )∣(∣ χ ( u )∣ + ∣ χ ( v )∣) p − ∥ t ≤ C ∥ Q ∥ ˜ s ∥ χ ( u ) − χ ( v )∥ q ∥(∣ χ ( u )∣ + ∣ χ ( v )∣) p − ∥ qαt − ≤ C ∥ u − v ∥ q ∥(∣ u ∣ + ∣ v ∣) α t − ∥ qαt − ≤ C ∥ u − v ∥ q (∥ u ∥ q + ∥ v ∥ q ) α t − . (23)Here p − ≥ α t − > ∥ T ( u, h ) − T ( v, h )∥ q ≤ ∥ R λ ( f ( ⋅ , χ ( u ))) − R λ ( f ( ⋅ , χ ( v )))∥ q ≤ C ∥ u − v ∥ q (∥ u ∥ q + ∥ v ∥ q ) α t − , which finishes the proof. ◻ Proof of Theorem 1:
Step 1: Existence of a solution continuum ( u h ) in L q ( R n ) : We apply Banach’s Fixed PointTheorem to T ( ⋅ , h ) on a closed small ball around zero B ρ ⊂ L q ( R n ) , q ∈ Ξ s,p ≠ ∅ and h ∈ X δ with δ > T ( ⋅ , h ) ∶ B ρ → B ρ is a contraction provided ρ, δ > h . Moreover, T is continuous with respect to the topology of L q ( R n ) × C m ( S n − λ ; C ) , which follows from Proposition 1 and Proposition 4. Hence, Banach’s OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 13
Fixed Point Theorem for continuous uniform contractions yields a continuum of (uniquelydetermined) fixed points u h ∈ B ρ of T ( ⋅ , h ) for h ∈ X δ provided δ > Step 2: The continuum property in L r ( R n ) for r ∈ ( q, ∞ ] : As a fixed point of T ( ⋅ , h ) thefunction u h solves − ∆ u − λu = f ( x, χ ( u )) in the strong sense on R n , see Theorem 4. Since f ( ⋅ , χ ( u h )) is bounded, we even have u h ∈ W ,rloc ( R n ) for all r ∈ [ , ∞ ) . Fixing now ˜ q ≥ q suchthat ˜ q > n we get from Theorem 8.17 in [15] (Moser iteration) ∥ u h − u h ∥ ∞ ≤ ∥ u h − u h ∥ q + ∥ f ( ⋅ , χ ( u h )) − f ( ⋅ , χ ( u h ))∥ ˜ q ≤ ∥ u h − u h ∥ q + ∥ Q ∥ ∞ ∥(∣ χ ( u h )∣ p − + ∣ χ ( u h )∣ p − )( χ ( u h ) − χ ( u h ))∥ ˜ q ≤ ∥ u h − u h ∥ q + p − + ˜ q − qq ∥ Q ∥ ∞ ∥∣ u h − u h ∣ q / ˜ q ∥ ˜ q ≤ C (∥ u h − u h ∥ q + ∥ u h − u h ∥ q / ˜ qq ) so that ( u h ) is also a continuum in L ∞ ( R n ) and hence in L r ( R n ) for all r ∈ ( q, ∞ ] . Step 3: The continuum property in L r ( R n ) for r ∈ ( nn − , q ) : From u h ∈ L q ( R n ) we deduce f ( ⋅ , χ ( u h )) ∈ L t ∗ ( q,s ) ( R n ) ∩ L ∞ ( R n ) for t ∗ ( q, s ) defined in (19). We set˜ q ∶ = max { r, ( n + ) t ∗ ( q, s ) n + − t ∗ ( q, s ) } so that Theorem 4 gives u h ∈ L ˜ q ( R n ) since the tuple of exponents ( t ∗ ( q, s ) , ˜ q ) satisfies theinequalities (15). Notice that q < s ( n + )( p − )( s −( n + )) + (because of q ∈ Ξ s,p ) implies ˜ q < q . Moreover, wehave ∥ u h − u h ∥ ˜ q ≤ ∥ F (( h − h ) dσ λ )∥ ˜ q + ∥ R λ ( f ( ⋅ , χ ( u h )) − f ( ⋅ , χ ( u h )))∥ ˜ q ≤ C ∥ h − h ∥ C m + C ∥ f ( ⋅ , χ ( u h )) − f ( ⋅ , χ ( u h )))∥ t ∗ ( q,s ) (23) ≤ C (∥ h − h ∥ C m + ∥ u h − u h ∥ q (∥ u h ∥ q + ∥ u h ∥ q ) p − ) ≤ C (∥ h − h ∥ C m + ∥ u h − u h ∥ q ) . Taking now ˜ q as the new q and repeating the above arguments we get after finitely manysteps u h ∈ L r ( R n ) as well as ∥ u h − u h ∥ r ≤ C (∥ h − h ∥ C m + ∥ u h − u h ∥ q ) . Hence, ( u h ) is a continuum in L r ( R n ) for all r ∈ ( nn − , q ) . Step 4: The continuum property in W ,r ( R n ) for r ∈ ( nn − , ∞ ) and (3) : From step 2 and step3 we get − ∆ u h + u h = ( + λ ) u h + f ( ⋅ , χ ( u h )) ∈ L r ( R n ) because of ∣ f ( x, χ ( u h ))∣ ≤ ∥ Q ∥ ∞ ∣ u h ∣ ∈ L r ( R n ) . Bessel potential estimates imply u h ∈ W ,r ( R n ) and as above one finds ∥ u h − u h ∥ W ,r ( R n ) ≤ C ∥ u h − u h ∥ r so that the continuum property isproved. In particular, ∥ h ∥ C m → ∥ u h ∥ ∞ = ∥ u h − u ∥ ∞ → χ ( u h ) = u h is a W ,r ( R n ) -solution of (3) for all h ∈ X δλ provided δ > Step 5: Asymptotics of u h : From the previous steps we get u h ∈ L r ( R n ) for all r ∈ ( nn − , ∞ ] and this implies ∣ f ( ⋅ , u h )∣ ≤ Q ∣ χ ( u h )∣ p − ∈ L t ( R n ) for t > ns n + s ( n − )( p − ) . In particular we have f ( ⋅ , u h ) ∈ L ( n + ) n + ( R n ) because p (4) > s ( n + n − ) − n ( n + )( n − ) s > s ( n + n − ) − n ( n + )( n − ) s . Hence, Proposition 2 yieldslim R →∞ R ∫ B R RRRRRRRRRRRRR R λ ( f ( ⋅ , u h ))( x ) − √ π λ ( √ λ ∣ x ∣ ) n − Re ( e i ( n − π −√ λ ∣ x ∣) ̂ f ( ⋅ , u h )( − √ λ ˆ x ))RRRRRRRRRRRRR dx = . Using h ( − ξ ) = h ( ξ ) and Proposition 1 we moreover getlim R →∞ R ∫ B R RRRRRRRRRRRRR ̂ h dσ λ ( x ) − √ π ( √ λ ∣ x ∣ ) n − Re ( e i ( n − π −√ λ ∣ x ∣) h (√ λ ˆ x ))RRRRRRRRRRRRR dx = . So u h = T ( u h , h ) = ̂ h dσ λ + R λ ( f ( ⋅ , u h )) and the above asymptotics imply ( ?? ). Finally, in thecase p > s ( n − )− n ( n − ) s we have V ∶ = Q ∣ u ∣ p − ∈ L t ( R n ) for some t < nn + because of Q ∈ L s ( R n ) and u ∈ L r ( R n ) for all r > nn − . Hence, Lemma 2.9 in [10] yields the pointwise bounds if n ≥ n = Step 6: Distinguishing u h , u h : From Step 2 we deduce that h ≠ h implies u h ≠ u h .Indeed, assuming u h = u h we get T ( u h , h ) = T ( u h , h ) and thus F (( h − h ) dσ λ ) =
0. Weshow that this implies h = h . To see this we use the scaling property F ( h dσ λ )( x ) = λ n − F ( h (√ λ ⋅ ) dσ )(√ λx ) . From Corollary 4.6 in [1] we infer0 = lim R →∞ R ∫ B R ∣ F (( h − h ) dσ λ )∣ dx = λ n − lim R →∞ R ∫ B R ∣ F (( h − h )(√ λ ⋅ ) dσ )(√ λx )∣ dx = λ n − ⋅ lim R →∞ √ λR ∫ B √ λR ∣ F (( h − h )(√ λ ⋅ ) dσ )( x )∣ dx = ( π ) n − λ n − ∫ S n − ∣( h − h )(√ λx )∣ dσ ( x ) , which implies h = h . ◻ Remark 2. (a) Let us describe how Theorem 1 provides nonsymmetric solutions of sym-metric nonlinear Helmholtz equations as mentioned in Remark 1(b). We assume f ( γx, z ) = f ( x, z ) for almost all x ∈ R n and all z ∈ R , γ ∈ Γ where Γ ⊂ O ( n ) , Γ ≠ { id } OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 15 is a subgroup. Since Γ ≠ { id } we can find h ∈ X δ and γ ∈ Γ satisfying h ≠ h ○ γ andour claim is that this implies u h ≠ u h ○ γ . Indeed, otherwise we would have ̂ h dσ λ + R λ ( f ( ⋅ , χ ( u h ))) = T ( u h , h ) = u h = u h ○ γ = T ( u h , h ) ○ γ = ̂ h dσ λ ○ γ + R λ ( f ( ⋅ , χ ( u h ))) ○ γ = ̂ h ○ γ dσ λ + R λ ( f ( ⋅ , χ ( u h )) ○ γ ) = ̂ h ○ γ dσ λ + R λ ( f ( ⋅ , χ ( u h ○ γ ))) = ̂ h ○ γ dσ λ + R λ ( f ( ⋅ , χ ( u h ))) . From the second to the third line we used that R λ is a convolution operator with aradially symmetric and hence Γ -symmetric kernel and from the third to the fourthline we used that f is Γ -invariant. So we conclude ̂ h dσ λ = ̂ h ○ γ dσ λ , which implies h = h ○ γ as in Step 6 above, a contradiction. Hence, u h ≠ u h ○ γ so that u h is not Γ -symmetric.(b) In Remark 1(c) we claimed that Theorem 1 provides radial solutions assuming theweaker condition (6) instead of (4) . This is due to an improved version of the resol-vent estimates from (15) where in both lines n + ≤ t − q can be replaced by n − n < t − q .This was demonstrated in Remark 3.1 in [7]. Let us explain how these improved re-solvent estimates allow to obtain radial solutions for a larger range of exponents. Theonly radially symmetric Herglotz waves ̂ h dσ λ are given by real-valued and constantdensities h . So for h ∈ R we get that T ( ⋅ , h ) maps L qrad ( R n ) into itself provided f ( x, u ) = f (∣ x ∣ , u ) is radially symmetric. Here, the exponent q may be chosen from Ξ rads,p ∶ = { q ∈ ( nn − , n ( n − ) + ) ∶ q < min { n s ( p − )(( n − ) s − n ) + , ns ( p − )( s ( n + ) − n ) + }} , which now is nonempty due to (6) . Replacing in Proposition 4 the first inequalityin (22) by t ∗ ( q, s ) < n q n +( n − ) 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 dualvariational methods originally developed by Ev´equoz and Weth [10]. To demonstratethis we consider the special case f ( x, z ) = ∣ z ∣ p − z with ( n + ) n − < p < nn − . The dualfunctional J h ∶ L p ′ ( R n ) → R is then given by J h ( v ) ∶ = p ′ ∫ R n ∣ v ∣ p ′ − ∫ R n v ⋅ F ( h dσ λ ) −
12 p . v . ∫ R n ∣ ˆ v ( ξ )∣ ∣ ξ ∣ − λ dξ and a local minimizer of J h lying in the interior of a small ball may be shown to existfor h ∈ X δ for δ > sufficiently small using Ekeland’s variational principle. Sinceevery critical point v h of J h provides a fixed point of T ( ⋅ , h ) vai u h ∶ = ∣ v h ∣ p ′ − v h , seeSection 4 in [10], we rediscover the solutions found in Theorem 1. Proof of Theorem 2
In this section we discuss how the above approach needs to be modified in order to getsolutions of the fourth order problem (7).We first consider the case (i) in (8). From Theorem 5 we know that there is a resolvent-type operator R associated with ∆ − β ∆ + α which is linear and bounded between the same(and even more) pairs of Lebesgue spaces as R λ . So all estimates in Proposition 4 involving R λ equally hold for R . As a replacement for the Herglotz wave of the Helmholtz operatorwe take again a Herglotz wave ̂ h dσ λ where now h ∈ Y δ = X δλ and λ was defined in (9) independence of α, β . This definition of λ was made in such a way that ̂ h dσ λ satisfies thehomogeneous equation ∆ φ − β ∆ φ + αφ = λ − βλ + α =
0. As a consequence, alsothe Herglotz-wave part of the map T ( u, h ) ∶ = ̂ h dσ 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 L q ( R n ) for q ∈ Ξ s,p exactly as in the proof of Theorem 1. The qualitative properties can also beproved the same way, see also Section 5 in [7] where u h ∈ W ,r ( R n ) for all r ∈ ( nn − , ∞ ) as wellas its pointwise decay rate was proved in the special case f ( x, z ) = Γ ( x )∣ z ∣ p − z, Γ ∈ L ∞ ( R n ) .In the case (ii) the proof is essentially the same. The only difference is that the fixed pointoperator now reads T ( u, h ) ∶ = ̂ h dσ λ + ̂ h dσ λ + R ( f ( ⋅ , χ ( u ))) , for h = ( h , h ) ∈ Y δ = X δλ × X δλ . Besides that all arguments are identical and we concludeas above. ◻ Proof of Theorem 3
Theorem 3 will be proved via the Contraction Mapping Theorem on a small ball in R (part (i)) or on R (part (ii)). The reason for this is that the Limiting Absorption Principlein the latter case is much weaker and forces us to consider nonlinear curl-curl equations withnonlinearities that grow sublinearly at infinity. Notice that the growth of the nonlinearitywith respect to E cannot be ignored by using a truncation as in our results proved above. Infact, an equivalent of local elliptic regularity theory and in particular ( L r , L ∞ ) -estimates forthe 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 ofTheorem 1 and Theorem 2. So let f satisfy (A’). For q ∈ Ξ s,p defined in (4) (for n =
3) weconsider the map T ( ⋅ , h ) ∶ L qcyl ( R ; R ) → L qcyl ( R ; R ) where T ( E, h ) ∶ = ̂ h dσ λ + R λ ( f ( ⋅ , χ (∣ E ∣) E /∣ E ∣)) . (24)Here, L qcyl ( R ; R ) denotes the Banach space of cylindrically symmetric functions lying in L q ( R ; R ) and the function χ ∈ C ∞ ( R ) is chosen as before, i.e., it satisfies ∣ χ ( z )∣ ≤ min {∣ z ∣ , } as well as χ ( z ) = z provided ∣ z ∣ ≤ . The map T ( ⋅ , h ) is well-defined for h ∈ Z δcyl , δ > ̂ h dσ λ are smooth cylindrically symmetric solutions of ∇ × ∇ × E − λE = OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 17 the same estimates as their scalar counterparts used in the proof of Theorem 1. Likewise,Theorem 6 implies that R λ restricted to the space of cylindrically symmetric functions hasthe same L p − L q mapping properties as R λ . Moreover, not only f but also the function ( x, E ) ↦ f ( x, χ (∣ E ∣) E /∣ E ∣) satisfies (A’) because z ↦ χ ( z )/ z is smooth. So the operator T defined in (24) also satisfies the estimates from Proposition 4 and one obtains a unique fixedpoint E h of T ( ⋅ , h ) on a small ball in L qcyl ( R ; R ) via the Contraction Mapping Theorem. Asa cylindrically symmetric and hence divergence-free solution of (10) the vector field E h evensolves the elliptic system (12) so that elliptic regularity theory implies χ ( E h ) = E h providedthe ball in L qcyl ( R ; R ) and h ∈ Z δcyl , δ > ∣ E h ( x )∣ ≤ C h ( + ∣ x ∣) − n is proved as in step 5 of the proof of Theorem 1 under the assumption p > s ( n − )− n ( n − ) s .From now on we prove part (ii), so let f satisfy assumption (B). We fix an exponent q suchthat 3 < q < s ( s − ) + , which is possible due to s ∈ [ , ] . For h ∈ Z we set T ( E, h ) ∶ = ̂ h dσ λ + R λ ( f ( ⋅ , E )) . (25)We first verify that T ( ⋅ , h ) ∶ L q ( R ; R ) → L q ( R ; R ) is well-defined and Lipschitz continuous.In the proof of these estimates we use the number α p, ˜ p ∶ = sup z ∈ R ∣ z ∣ p − ( + ∣ z ∣) ˜ p − p = ( p − ) p − ( − ˜ p ) − ˜ p ( p − ˜ p ) p − ˜ p where 0 ∶ = p ≤ ≤ p. Proposition 5.
Assume (B) and λ > , h ∈ Z . Then the map T ( ⋅ , h ) ∶ L q ( R ; R ) → L q ( R ; R ) from (25) is well-defined with ∥ T ( E , h ) − T ( E , h )∥ q ≤ Cα p, ˜ p (∥ Q ∥ s + ∥ Q ∥ ∞ )∥ E − E ∥ q . (26) where C only depends on q and s .Proof. By Proposition 3 the functions ̂ h dσ λ belong to L q ( R ; R ) for all h ∈ Z . So thedefinition of T from (25) 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 ) ∈ L q ( R ; R ) ∩ L t ( R ; R ) for some t ∈ ( , ∞ ) satisfying (15). To verify this we set ˜ s ∶ = max { s, } for s ∈ [ , ] as inassumption (B) and choose t ∶ = ˜ sq ˜ s + q . This implies 1 < t < , ≤ ˜ s ≤ ( t, q ) indeedsatisfies (15). So we infer from assumption (B) ∥ f ( ⋅ , E )∥ q ≤ ∥ Q ∣ E ∣ p − ( + ∣ E ∣) ˜ p − p ∥ q ≤ α p, ˜ p ∥ Q ∣ E ∣∥ q ≤ α p, ˜ p ∥ Q ∥ ∞ ∥ E ∥ q < ∞ , ∥ f ( ⋅ , E )∥ t ≤ ∥ Q ∣ E ∣ p − ( + ∣ 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 ) − f ( ⋅ , E )∥ q ≤ α p, ˜ p ∥ Q ∥ ∞ ∥ E − E ∥ q , ∥ f ( ⋅ , E ) − f ( ⋅ , E )∥ t ≤ α p, ˜ p ∥ Q ∥ ˜ s ∥ E − E ∥ q ≤ α p, ˜ p (∥ Q ∥ s + ∥ Q ∥ ∞ )∥ E − E ∥ q . Combining these estimates with Theorem 6 one gets (26) from the definition of T ( ⋅ , h ) . ◻ Proof of Theorem 3 (ii):
From the previous proposition we get that T ( ⋅ , h ) maps L q ( R ; R ) into itself and it is a contraction provided α p, ˜ p (∥ Q ∥ s + ∥ Q ∥ ∞ ) is small enough, whichis guaranteed by the assumptions of the Theorem 3. So for any given h ∈ Z the Contrac-tion Mapping Theorem and Theorem 6 provide a unique weak solution E h ∈ H loc ( curl; R ) ∩ L q ( R ; R ) of (3). It remains to discuss the integrability properties of E h . In this discussionwe will w.l.o.g. assume that assumption (B) holds with ˜ p ∈ ( , ] because otherwise we maysimply increase ˜ p . Proof of (ii)(a):
Under the additional assumption ( p, s ) ≠ ( , ) we want to show E h ∈ L r ( R ; R ) for all r ∈ ( , q ) . To achieve this iteratively we use E h ∈ L q ( R ; R ) and hence, byassumption (B), f ( ⋅ , E h ) ∈ L t ( R ; R ) ∩ L ˜ q ( R ; R ) for all t, ˜ q ∈ [ t ∗ ( q, s ) , q ˜ p − ] . This follows as in (19), where also t ∗ ( q, s ) is defined. In order to prove E h ∈ L ˜ q ( R ; R ) with˜ q < q we use E h = T ( E h , h ) = ̂ h dσ λ + R λ ( f ( ⋅ , E h )) . In view of Proposition 3 and the mappingproperties of R λ from Theorem 6 we obtain E h ∈ L ˜ q ( R ; R ) with ˜ q < q provided the pair ( t, ˜ q ) satisfies 1 < t, ˜ q < ∞ as well as the inequalities from (15) in the three-dimensional case n =
3. In other words we require(27) 1 < t < , < ˜ q < q, ≤ t − q ≤ , t ∗ ( q, s ) ≤ t, ˜ q ≤ q ˜ p − . So we set ˜ q ∶ = max { r, t − t , t ∗ ( q, s )} = max { r, t − t } ≥ r >
3. (Notice that our choice for t willensure t ∗ ( q, s ) < t < < < r .) Plugging in the definition of t ∗ ( q, s ) from (19) we obtain aftersome calculations that the inequalities (27) hold ifmax { , r + r , qsq + s ( p − ) } < t < min { , qq + , q ˜ p − } . Here non-strict inequalities in (27) were sharpened to strict inequalities for notational con-venience. Such a choice for t is possible because of 1 ≤ s ≤ ≤ p, ( p, s ) ≠ ( , ) and3 < r < q << s ( s − ) + < s ( p − )( s − ) + . For instance we may choose t = t q ∶ = ⋅ ( max { , r + r , qsq + s ( p − ) } + min { , qq + , q ˜ p − }) and Theorem 6 implies E h ∈ L ˜ q ( R ; R ) . In the case ˜ q = r we are done. Otherwise, we mayrepeat this argument replacing q by ˜ q so that r ∈ ( , ˜ q ) . The corresponding iteration yields E h ∈ L r ( R ; R ) after finitely many steps. Proof of (ii)(b):
Using ˜ p < < p we now prove E h ∈ L r ( R ; R ) for all r ∈ ( q, s ( p − )( s − ) + ) . In viewof (B) and E h ∈ L q ( R ; R ) we now have to choose t, ˜ q ∈ [ t ∗ ( q, s ) , q ˜ p − ] with ˜ q > q such thatthe pair ( t, ˜ q ) satisfies 1 < t, ˜ q < ∞ as well as1 < t < , q < ˜ q < ∞ , ≤ t − q ≤ , t ∗ ( q, s ) ≤ t, ˜ q ≤ q ˜ p − . OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 19
So we set ˜ q ∶ = min { t − t , q ˜ p − } and due to 1 < ˜ p < t such thatmax { , q + q , qsq + s ( p − ) } < t < min { , qq + ( ˜ p − ) , q ˜ p − } . Such a choice is possible thanks to q < s ( p − )( s − ) + and 1 ≤ s ≤ , < ˜ p <
2. For instance we maychoose t = t q ∶ = ⋅ ( max { , q + q , qsq + s ( p − ) } + min { , qq + ( ˜ p − ) , q ˜ p − }) . Then Theorem 6 implies E h ∈ L ˜ q ( R ; R ) and we have ˜ q > q . We may repeat this argumentas long as ˜ q < s ( p − )( s − ) + and thereby obtain u ∈ L r ( R ; R ) for all r ∈ ( q, s ( p − )( s − ) + ) , which finishesthe proof. ◻ Appendix A: Proof of Proposition 1
In this section we give the proof of Proposition 1 which we repeat for convenience.
Proposition.
Let n ∈ N , n ≥ and m ∶ = ⌊ n − ⌋ + . Then for all h ∈ C m ( S n − λ ; C ) the Her-glotz wave ̂ h dσ λ is an analytic solution of − ∆ φ − λφ = in R n and satisfies the estimate ∣( ̂ h dσ λ )( x )∣ ≤ C ∥ h ∥ C m ( + ∣ x ∣) − n as well as lim R →∞ R ∫ B R RRRRRRRRRRRRR ̂ h dσ λ ( x ) − √ π ( √ λ ∣ x ∣ ) n − m h ( x )RRRRRRRRRRRRR dx = . In particular, we have ∥ ̂ h dσ λ ∥ r ≤ C r ∥ h ∥ C m for all r > nn − . The asymptotics of the functions ̂ h dσ λ are proved using the method of stationary phase,but typically it is assumed that h is smooth, see for instance Proposition 4,5,6 in ChapterVIII § n ≥ m . For thisreason, we decided to present a proof here. Proof of Proposition 1:
We consider the Herglotz wave ̂ h dσ λ ( x ) = ( π ) n / ∫ S n − λ h ( ξ ) e − i ⟨ x,ξ ⟩ dσ λ ( ξ ) . To investigate its pointwise behaviour as ∣ x ∣ → ∞ let Q = Q x ∈ O ( n ) satisfy Q T x = ∣ x ∣ e n , so ̂ h dσ λ ( x ) = ( π ) n / ∫ S n − λ h ( Qξ ) e − i ∣ x ∣ ξ n dσ λ ( ξ ) . Now we choose η , . . . η n ∈ C ∞ ( R n ) such that η + . . . + η n = S n − λ and, for k = , . . . , n ,supp ( η k − ) ⊂ { ξ = ( ξ , . . . , ξ n ) ∈ R n ∶ ξ k > + √ λδ } , supp ( η k ) ⊂ { ξ = ( ξ , . . . , ξ n ) ∈ R n ∶ ξ k < − √ λδ } , where δ ∈ ( , √ n ) is fixed. Notice that such a partition of unity exists since the open sets onthe right hand side cover the sphere S n − λ . We define h j ( ξ ) ∶ = ( π ) − n / h ( Qξ ) η j ( ξ ) so that wehave to investigate the integrals I j ∶ = ∫ S n − λ h j ( ξ ) e − i ∣ x ∣ ξ n dσ λ ( ξ ) ( j = , . . . , n ) . We first estimate the oscillatory integrals I , . . . , I n − where the resonant poles ± e n are cutout by the choice of η , . . . , η n − . To estimate I j we use the local parametrization given byIf j = k − ∶ ψ j ( ξ ′ ) ∶ = √ λ ( ξ , . . . , ξ k − , + √ − ∣ ξ ′ ∣ , ξ k , . . . , ξ n − ) ∈ S n − λ If j = k ∶ ψ j ( ξ ′ ) ∶ = √ λ ( ξ , . . . , ξ k − , − √ − ∣ ξ ′ ∣ , ξ k , . . . , ξ n − ) ∈ S n − λ for ξ ′ ∶ = ( ξ , . . . , ξ n − ) belonging to the unit ball B ⊂ R n − . The function H j ( ξ ′ ) ∶ = λ n − h j ( ψ j ( ξ ′ ))( − ∣ ξ ′ ∣ ) − / then satisfies supp ( H j ) ⊂ B so that integration by parts yields for all ∣ x ∣ ≥ j = , . . . , n − ∣ I j ∣ = ∣ ∫ B H j ( ξ ′ ) e − i √ λ ∣ x ∣ ξ n − dξ ′ ∣ = ∣ ∫ R n − H j ( ξ ′ ) ∂ m ∂ ( ξ n − ) m ( e − i √ λ ∣ x ∣ ξ n − ) dξ ′ ∣ ⋅ ∣√ λx ∣ − m = ∣ ∫ R n − ( ∂ m ∂ ( ξ n − ) m H j ( ξ ′ )) e − i √ λ ∣ x ∣ ξ n − dξ ′ ∣ ⋅ ∣√ λx ∣ − m = ∫ B ∣ ∇ m H j ( ξ ′ )∣ dξ ′ ⋅ ∣√ λx ∣ − m ≤ C ∥ h ∥ C m ⋅ ∣ x ∣ − m Since the estimate for ∣ x ∣ ≤ m = ⌊ n − ⌋ + ≥ n , we conclude(28) ∣ I j ∣ ≤ C ∥ h ∥ C m ( + ∣ x ∣) − n − α for all x ∈ R n , j = , . . . , n − , α ∈ ( , ) . Next we analyze the integrals I n − , I n − . With ψ j , H j as above we define H ∗ j ( η ) ∶ = H j ( η √ − ∣ η ∣ ) ⋅ ( − ∣ η ∣ )( − ∣ η ∣ ) n − ( j = n − , n ) . Again the supports of H j , H ∗ j are contained in the interior of the unit ball B so that neitherfunction is singular. Performing twice a change of coordinates we get I n − = ∫ R n − H n − ( ξ ′ ) e − i √ λ ∣ x ∣√ −∣ ξ ′ ∣ dξ ′ = e − i √ λ ∣ x ∣ ∫ R n − H ∗ n − ( η ) e i √ λ ∣ x ∣∣ η ∣ dη,I n = ∫ R n − H n ( ξ ′ ) e + i √ λ ∣ x ∣√ −∣ ξ ′ ∣ dξ ′ = e + i √ λ ∣ x ∣ ∫ R n − H ∗ n ( η ) e − i √ λ ∣ x ∣∣ η ∣ dη. OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 21
The integrals may be estimated using Proposition 11 in [22] for s ∶ = m − α . Notice that α ∈ ( , ) ensures s ≥ n + − α > n − so that the estimate from this proposition is valid. Using H ∗ n − ( ) = n − H n − ( ) = ( λ ) n − h n − ( + √ λe n ) = √ π ( λπ ) n − h ( + √ λ ˆ x ) ,H ∗ n ( ) = n − H n ( ) = ( λ ) n − h n ( − √ λe n ) = √ π ( λπ ) n − h ( − √ λ ˆ x ) as well as ∥ H j ∥ H m ( R n − ) ≤ C ∥ h ∥ C m for j ∈ { n − , n } we deduce from the first inequality inProposition 11 [22] RRRRRRRRRRRRR I n − − e i ( n − π −√ λ ∣ x ∣) √ π ( √ λ ∣ x ∣ ) n − h (√ λ ˆ x )RRRRRRRRRRRRR ≤ C ∥ h ∥ C m ∣ x ∣ − n − α , RRRRRRRRRRRRR I n − e − i ( n − π −√ λ ∣ x ∣) √ π ( √ λ ∣ x ∣ ) n − h ( − √ λ ˆ x )RRRRRRRRRRRRR ≤ C ∥ h ∥ C m ∣ x ∣ − n − α . (29)Combining (28),(29) and ̂ h dσ λ = I + . . . + I n we find for ∣ x ∣ ≥ RRRRRRRRRRRRR ̂ h dσ λ ( x ) − √ π ( √ λ ∣ x ∣ ) n − m h ( x )RRRRRRRRRRRRR ≤ C ∥ h ∥ C m ∣ x ∣ − n − α , where m h was introduced in (14). In view of the estimate ∣ ̂ h dσ λ ( x )∣ ≤ C ∥ h ∥ ∞ for ∣ x ∣ ≤ ∣ ̂ h dσ λ ( x )∣ ≤ C ∥ h ∥ C m ( + ∣ x ∣) − n as well as 1 R ∫ B R RRRRRRRRRRRRR ̂ h dσ λ ( x ) − √ π ( √ λ ∣ x ∣ ) n − m h ( x )RRRRRRRRRRRRR dx ≤ C ∥ h ∥ ∞ ⋅ R ∫ B ( + ∣ x ∣ − n ) dx + C ∥ h ∥ C m ⋅ R ∫ B R ∖ B ∣ x ∣ − n − α dx ≤ C ( R + R α )∥ h ∥ C m → R → ∞ . This finishes the proof of Proposition 1. ◻ Appendix B: Proof of Theorem 6
In this section we prove the Limiting Absorption Principle from Theorem 6. We recall thestatement for the convenience of the reader.
Theorem.
Let λ > and assume that t, q ∈ ( , ∞ ) satisfy (15) . Then there is a boundedlinear operator R λ ∶ L t ( R ; R ) ∩ L q ( R ; R ) → L q ( R ; R ) such that R λ G ∈ H loc ( curl; R ) is aweak solution of ∇ × ∇ × E − λE = G provided G ∈ L t ( R ; R ) ∩ L q ( R ; R ) . Moreover, we have ∥ R λ G ∥ q ≤ C (∥ G ∥ q + ∥ G ∥ t ) ≤ C (∥ G ∥ q + ∥ G ∥ t ) and R λ G = − λ G + R λ G for R λ from Theorem 4 (applied componentwise). If G ∈ L t ( R ; R ) is cylindrically symmetric then so is R λ G and R λ G ∈ W ,qloc ( R ; R ) is a strong solution satis-fying ∥ R λ G ∥ q ≤ C ∥ G ∥ t . To prove this, we first show that the domain of the selfadjoint realization of the curl-curloperator LE ∶ = ∇ × ∇ × E is given by D ∶ = { E ∈ L ( R ; R ) ∶ LE ∈ L ( R ; R )} ∶ = { E ∈ L ( R ; R ) ∶ ξ ↦ ∣ ξ ∣ ˆ E ( ξ ) − ⟨ ˆ E ( ξ ) , ξ ⟩ ξ ∈ L ( R ; R )} and that its spectrum is [ , ∞ ) . Using the Helmholtz decomposition(30) ˆ G ( ξ ) ∶ = ⟨ ˆ G ( ξ ) , ξ ∣ ξ ∣ ⟩ ξ ∣ ξ ∣ , ˆ G ( ξ ) ∶ = ˆ G ( ξ ) − ˆ G ( ξ ) of a vector field G ∈ L ( R ; R ) we get the following. Proposition 6.
The curl-curl operator L ∶ D → L ( R ; R ) is selfadjoint with spectrum σ ( L ) = [ , ∞ ) . For µ ∈ C ∖ [ , ∞ ) the resolvent is given by ( L − µ ) − G = − µ G + R ( µ ) G where R ( µ ) is (in each component) the operator defined at the beginning of Section 2.Proof. The curl-curl-operator L is symmetric when defined on the Schwartz functions in R .So we have to show that all E ∈ L ( R ; R ) in the domain of its adjoint actually belong to D . So assume that E ∈ L ( R ; R ) satisfies for all F ∈ D the inequality ∣ ∫ R ⟨ E, LF ⟩∣ ≤ C ∥ F ∥ . Using Plancherel’s identity on both sides this can be rewritten as ∣ ∫ R ⟨∣ ξ ∣ ˆ E ( ξ ) − ⟨ ˆ E ( ξ ) , ξ ⟩ ξ, ˆ F ( ξ )⟩ dξ ∣ = ∣ ∫ R ⟨ ˆ E ( ξ ) , ∣ ξ ∣ ˆ F ( ξ ) − ⟨ ˆ F ( ξ ) , ξ ⟩ ξ ⟩ dξ ∣ ≤ C ∥ ˆ F ∥ . Since this holds for all F ∈ D , which is dense in L ( R ; R ) , we infer that ξ ↦ ∣ ξ ∣ ˆ E ( ξ ) − ⟨ ˆ E ( ξ ) , ξ ⟩ ξ is square-integrable, which precisely means E ∈ D .To prove the second claim let µ ∈ C ∖ [ , ∞ ) and G ∈ L ( R ; R ) . Then ∇ × ∇ × E − µE = G is equivalent to ∣ ξ ∣ ˆ E ( ξ ) − ⟨ ξ, ˆ E ( ξ )⟩ ξ − µ ˆ E ( ξ ) = ˆ G ( ξ ) . Decomposing E into E , E as in (30) and multiplying the above equation with ξ /∣ ξ ∣ ∈ R wefind ˆ E ( ξ ) = − µ ⟨ ˆ G ( ξ ) , ξ ∣ ξ ∣⟩ ξ ∣ ξ ∣ = − µ ˆ G ( ξ ) . OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 23
This implies ˆ E ( ξ )(∣ ξ ∣ − µ ) = ˆ G ( ξ ) + µ ˆ E ( ξ ) = ˆ G ( ξ ) − ⟨ ˆ G ( ξ ) , ξ ∣ ξ ∣ ⟩ ξ ∣ ξ ∣ = ˆ G ( ξ ) . So we have E = E + E = − µ G + F − ( ˆ G ( ⋅ )∣ ⋅ ∣ − µ ) = − µ G + R ( µ ) G . Since the right hand side defines a bounded linear operator provided µ ∈ C ∖ [ , ∞ ) , thisproves that C ∖ [ , ∞ ) belongs to the resolvent set of the curl-curl operator.By the closedness of the spectrum it therefore suffices to show that for all µ ∈ ( , ∞ ) thereis a Weyl sequence of the curl-curl operator. Indeed, as in the case of Laplacian one mayconsider the sequence F n ( x ) ∶ = c n χ ( x / n ) F ( x ) where F ( x ) ∶ = ⎛⎜⎝ cos (√ µx ) ⎞⎟⎠ where χ ∈ C ∞ ( R ) is identically one on the cuboid W ∶ = [ − , ] ⊂ R and zero outside 2 W .The factor c n > ∥ F n ∥ =
1. Using ∫ a − a ∫ a − a ∫ a − a cos ( n √ µx ) dx dx dx = a + o ( ) ( n → ∞ ) , ∫ a − a ∫ a − a ∫ a − a sin ( n √ µx ) dx dx dx = a + o ( ) ( n → ∞ ) for a = c n = ∥ χ ( ⋅ / n ) F ∥ = n / ∥ χF ( n ⋅ )∥ ≤ n / ∥ F ( n ⋅ )∥ L ( W ) ≤ n / ( + o ( )) . Using this as well as the above asymptotics for a = ∥ LF n − µF n ∥ = ∥ − ∆ F n − µF n ∥ = c n ⋅ ∥ χ ( x / n )( − ∆ F − µF ) − n ∇ χ ( x / n ) ⋅ ∇ F ( x ) − n ∆ χ ( x / n ) F ( x )∥ ≤ Cn − / ⋅ ( + n / ∥ ∇ χ ⋅ ∇ F ( n ⋅ )∥ + n − / ∥( ∆ χ ) F ( n ⋅ )∥ ) ≤ C ( n − ∥∣ ∇ F ( n ⋅ )∣∥ L ( W ) + n − ∥∣ F ( n ⋅ )∣∥ L ( W ) ) ≤ C (√ µn − + n − )(√ ⋅ + o ( )) = O ( n − ) as n → ∞ so that a Weyl sequence at the level µ is found. ◻ Proof of Theorem 6:
In order to construct R λ consider a Schwartz function G ∈ S ( R ; R ) so that E ε ∶ = ( L − λ − iε ) − G is well-defined and Proposition 6 implies E ε = − λ + iε G + R ( λ + iε ) G ∈ D + i D . By the Limiting Absorption Principle for the Helmholtz operator from Theorem 4 we get ( L − λ − i ) − G ∶ = lim ε → + E ε = − λ G + R ( λ + i ) G . Here, both limits exist in L q ( R ; R ) . Indeed, the Riesz transform maps L r ( R ) into itselffor all r ∈ ( , ∞ ) and thus ∥ G ∥ r ≤ C ∥ G ∥ r for r ∈ { t, q } , hence ∥ G ∥ t ≤ C ∥ G ∥ t < ∞ and ∥ G ∥ q ≤ C ∥ G ∥ q < ∞ . Taking the real part of this we obtain that R λ G ∶ = − λ G + R λ G defines a bounded linear operator from L t ( R ; R ) ∩ L q ( R ; R ) to L q ( R ; R ) with ∥ R λ G ∥ q ≤ λ ∥ G ∥ q + ∥ R λ G ∥ q ≤ C (∥ G ∥ q + ∥ G ∥ t ) . Next we prove that R λ G is a weak solution lying in H loc ( curl; R ) . We will use that E ε ∈ L q ( R ; R ) implies E ε ∈ L q ′ loc ( R ; R ) due to q > > q ′ . Testing the equation for E ε ∈ D + i D with E ε φ ∈ H ( curl; C ) we get(31) ∫ R ⟨ ∇ × E ε , ∇ × ( E ε φ )⟩ − λ ∣ E ε ∣ φ = ∫ R ⟨ G, E ε ⟩ φ for all φ ∈ C ∞ ( R ) . So for any given compact set K ⊂ R we may choose a nonnegative test function φ such that φ ∣ K ≡
1, set K ′ ∶ = supp ( φ ) . Then we get ∫ R ⟨ G, E ε ⟩ φ ≤ ∥ G ∥ L q ′ ( K ′ ; R ) ∥ E ε ∥ L q ( K ′ ; C ) ∥ φ ∥ ∞ as well as ∫ R ⟨ ∇ × E ε , ∇ × ( E ε φ )⟩ − λ ∣ E ε ∣ φ = ∫ R ∣ ∇ × ( E ε φ )∣ − ∣ ∇ φ × E ε ∣ − λ ∣ E ε ∣ φ ≥ ∫ R ∣ ∇ × ( E ε φ )∣ − (∣ ∇ φ ∣ + λφ )∣ E ε ∣ ≥ ∫ R ∣ ∇ × ( E ε φ )∣ − ∥∣ ∇ φ ∣ + λφ ∥ L qq − ( K ′ ; R ) ∥ E ε ∥ L q ( K ′ ; R ) . Here we used ∣ a × b ∣ ≤ ∣ a ∣∣ b ∣ for a, b ∈ C and q > nn − >
2. Combining the previous twoinequalities with (31) we get ∫ K ∣ ∇ × E ε ∣ ≤ ∫ R ∣ ∇ × ( E ε φ )∣ ≤ C (∥ E ε ∥ L q ( K ′ ; C ) + ∥ G ∥ L q ′ ( K ′ ; R ) ∥ E ε ∥ L q ( K ′ ; C ) ) ≤ C (∥ E ε ∥ L q ( R ; C ) + ∥ G ∥ L q ′ ( K ′ ; R ) ∥ E ε ∥ L q ( R ; C ) ) ≤ C < ∞ OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 25 because the functions E ε are equibounded in L q ( R ; C ) . The latter fact comes from the proofof Guti´errez’ Limiting Absorption Principle, see Theorem 6 in [17]. Since K was arbitrary,we conclude that ( E ε ) is bounded in H loc ( curl; C ) and therefore the weak limit of its realpart R λ G also belongs to that space.Finally we show that R λ leaves the space of cylindrically symmetric solutions invariant. If F ∈ S ( R ; R ) is cylindrically symmetric, then there is F ∶ [ , ∞ ) × R → R such that F ( x , x , x ) = F (√ x + x , x ) ⎛⎜⎝ − x x ⎞⎟⎠ for all ( x , x , x ) ∈ R . It suffices to show that there is ˜ F ∶ [ , ∞ ) × R → C such that ˜ F ( r, − z ) = − ˜ F ( r, z ) for all r ≥ , z ∈ R and(32) ˆ F ( ξ , ξ , ξ ) = ˜ F (√ ξ + ξ , ξ ) ⎛⎜⎝ − ξ ξ ⎞⎟⎠ for all ( ξ , ξ , ξ ) ∈ R . Clearly the third component of ˆ F vanishes identically. Using the symmetry of F and thedefinition of the Fourier transform we moreover obtain after some calculations that for all θ ∈ [ , π ) we have ⟨ ˆ F ( ξ , ξ , ξ ) , ⎛⎜⎝ cos ( θ ) − sin ( θ ) ( θ ) cos ( θ )
00 0 0 ⎞⎟⎠ ⎛⎜⎝ ξ ξ ⎞⎟⎠⟩ = ρ θ (√ ξ + ξ , ξ ) for some function ρ θ ∶ [ , ∞ ) × R → C , i.e., for every given θ ∈ [ , π ) the left hand side isinvariant under all rotations with respect to the ( ξ , ξ ) -variable. Using this for θ = ⟨ ˆ F ( ξ , ξ , ξ ) , ⎛⎜⎝ ξ ξ ⎞⎟⎠⟩ = ρ (√ ξ + ξ , ξ ) = ⟨ ˆ F ( , √ ξ + ξ , ξ ) , ⎛⎜⎝ √ ξ + ξ ⎞⎟⎠⟩ = ( π ) / ∫ R F (√ x + x , x ) ⟨⎛⎜⎝ − x x ⎞⎟⎠ , ⎛⎜⎝ √ ξ + ξ ⎞⎟⎠⟩ e − i ( x √ ξ + ξ + x ξ ) dx = ( π ) / ∫ R ( ∫ R F (√ x + x , x ) x dx ) √ ξ + ξ e − i ( x √ ξ + ξ + x ξ ) d ( x , x ) = because x ↦ F (√ x + x , x ) x is odd for all fixed ( x , x ) ∈ R . This showsˆ F ( ξ , ξ , ξ ) = ρ π / (√ ξ + ξ , ξ ) ξ + ξ ⎛⎜⎝ − ξ ξ ⎞⎟⎠ for all ( ξ , ξ , ξ ) ∈ R so that ˆ F can be written in the form (32). Finally, since F is real-valued we get ˆ F ( − ξ ) = ˆ F ( ξ ) and hence ˜ F ( r, − z ) = − ˜ F ( r, z ) for all r ≥ , z ∈ R . This finishes the proof. ◻ Appendix C: The Ruiz-Vega resolvent estimates
In this section we review the resolvent estimates by Ruiz and Vega that are essentiallycontained in Theorem 3.1 in [24]. Since this theorem does not exactly provide the estimateswe need, we decided to reformulate their results in the way we apply them in the proof ofProposition 2. We even show a bit more.
Theorem 7.
Let n ∈ N , n ≥ and assume f ∈ L q ( R n ) for n + ≤ q − ≤ min { , n } with ( n, q ) ≠ ( , ) . Then there is a C > such that for all ε ≠ we have sup R ≥ ( R ∫ B R ∣ R ( λ + iε ) f ( x )∣ dx ) / ≤ C ∥ f ∥ q . If moreover n + ≤ q − ≤ n , ( q, n ) ≠ ( , ) holds, then sup R ≥ ( R ∫ B R ∣ ∇R ( λ + iε ) f ( x )∣ dx ) / ≤ C ∥ f ∥ q . We emphasize that this result implies that the inequality (16) from the proof of Proposi-tion 2 holds for q ∶ = p ′ because ( n + ) n − ≤ p ≤ n ( n − ) + , ( n, p ) ≠ ( , ∞ ) is equivalent to n + ≤ q − ≤ min { , n } , ( n, q ) ≠ ( , ) . It seems that our statement dealing with the two-dimensionalcase n = n ≥ n = , q =
1. The proof by Guti´errezis not carried out in detail but it is referred to the paper of Ruiz and Vega (Theorem 3.1in [24]) where a closely related but different result is proved. So we believe that an updatedand self-contained version of these resolvent estimates might be useful even though our proofbelow mainly reformulates the arguments of Ruiz and Vega.
Proof of Theorem 7:
It suffices to prove the estimates for Schwartz functions f ∈ S ( R n ) and, via rescaling, for λ =
1. Then we have R ( + iε ) f = F − ( ∣ ξ ∣ − − iε ˆ f ( ξ )) . We introduce the splitting R ( + iε ) f = v ε + w ε where v ε ∶ = F − ( φ ( ξ )∣ ξ ∣ − − iε ˆ f ( ξ )) , w ε ∶ = F − ( − φ ( ξ )∣ ξ ∣ − − iε ˆ f ( ξ )) OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 27 and φ ∈ C ∞ ( R n ) is a test function satisfying supp ( φ ) ⊂ B + δ ∖ B − δ , δ ∶ = − √ + n as well as φ ≡ w ε = G ε ∗ f where ∣ G ε ( z )∣ ≤ C { min {∣ z ∣ − n , ∣ z ∣ − − n } , n ≥ {∣ log ( z )∣ , ∣ z ∣ − } , n = ∣ ∇ G ε ( z )∣ ≤ C min {∣ z ∣ − n , ∣ z ∣ − − n } for some C > ε , see page 8-9 in [7] for related estimates. These boundsimply G ε ∈ L q q − ( R n ) whenever 0 ≤ q − ≤ min { , n } with q − < n and the correspondingnorms are uniformly bounded from above with respect to ε . For such q we getsup R ≥ ( R ∫ B R ∣ w ε ( x )∣ dx ) / ≤ ∥ w ε ∥ = ∥ G ε ∗ f ∥ ≤ ∥ G ε ∥ q q − ∥ f ∥ q ≤ C ∥ f ∥ q . It remains to prove this estimate for n ≥ q − = n . In this case we use G ε ∈ L q q − ,w ( R n ) with uniformly bounded norms so that Young’s convolution inequality for classical and weakLebesgue spaces (see Theorem 1.4.25 in [16] for the latter) yieldssup R ≥ ( R ∫ B R ∣ w ε ( x )∣ dx ) / ≤ ∥ G ε ∗ f ∥ ≤ ∥ G ε ∥ q q − ,w ∥ f ∥ q ≤ C ∥ f ∥ q . (Notice that we also have G ε ∈ L q q − ,w ( R n ) in the case q − = n and n =
4, but Theo-rem 1.4.25 [16] does not apply since each of the exponents 2 , q q − , q has to be different from1 or ∞ .) The same way, if 0 ≤ q − < n then we have ∣ ∇ G ε ∣ ∈ L q q − ( R n ) and q − = n implies ∣ ∇ G ε ∣ ∈ L q q − ,w ( R n ) with uniformly bounded norms. In the former case we getsup R ≥ ( R ∫ B R ∣ ∇ w ε ( x )∣ dx ) / ≤ ∥∣ ∇ w ε ∣∥ = ∥∣ ∇ G ε ∣ ∗ f ∥ ≤ ∥∣ ∇ G ε ∣∥ q q − ∥ f ∥ q ≤ C ∥ f ∥ q . and in the latter case we have under the additional assumption ( q, n ) ≠ ( , ) (for the samereason as above)sup R ≥ ( R ∫ B R ∣ ∇ w ε ( x )∣ dx ) / ≤ ∥∣ ∇ G ε ∣ ∗ f ∥ ≤ ∥∣ ∇ G ε ∣∥ q q − ,w ∥ f ∥ q ≤ C ∥ f ∥ q . So it remains to show that the estimate(33) sup R ≥ ( R ∫ B R ∣ v ε ( x )∣ + ∣ ∇ v ε ( x )∣ dx ) / ≤ C ∥ f ∥ q holds whenever q − ≥ n + .To prove (33) we split v ε into 2 n different pieces using a suitable partition of unity. In viewof supp ( ˆ v ε ) ⊂ B + δ ∖ B − δ we consider the covering { O , + , O , − , . . . , O n, + , O n, − } of B + δ ∖ B − δ given by the open sets O j, ± ∶ = { ξ ∈ B + δ ∖ B − δ ∶ ± ξ j > √ n ∣ ξ ∣} Let { η , + , η , − , . . . , η n, + , η n, − } be an associated partition of unity so that v = n ∑ j = ( v εj, + + v εj, − ) where v εj, ± ∶ = F − ( η j, ± ( ξ ) φ ( ξ )∣ ξ ∣ − − iε ˆ f ( ξ )) The reason for this is that we want to make use of the following inequalities: ξ ∈ supp ( ˆ v εj, ± ) Ô⇒ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∣ ξ j ∣ ≤ ∣ ξ ∣ ≤ + δ, ± ξ j > √ n ∣ ξ ∣ > max { − δ √ n , √ n ∣ ξ ′ j ∣} , ∣ ξ ′ j ∣ ≤ √∣ ξ ′ j ∣ +∣ ξ j ∣ √ + n ≤ + δ √ + n = − δ . (34)Here we used the notation ξ = ( ξ , . . . , ξ n ) ∈ R n and ξ ′ j ∶ = ( ξ , . . . , ξ j − , ξ j + , . . . , ξ n ) . Sincethe estimates for v j, ± are the same for all j ∈ { , . . . , n } up to textual modifications we onlyconsider j =
1. Moreover, the estimates for v , + , v , − only differ at one point (which we willmention), so that we only carry out the estimates for v ε , + . To simplify the notation we write v ε , η, ξ ′ instead of v ε , + , η , + , ξ ′ = ( ξ , . . . , ξ n ) .Using B R ⊂ [ − R, R ] × R n − we get1 R ∫ B R ∣ v ε ( x )∣ + ∣ ∇ v ε ( x )∣ dx ≤ R ∫ R ( ∫ R n − ∣ v ε ( x , x ′ )∣ + ∣ ∇ v ε ( x , x ′ )∣ dx ′ ) dx = R ∫ R ∥( A ε ∗ f )( x , ⋅ )∥ L ( R n − ) dx (35)where A ε ∶ = F − ( η ( ξ ) φ ( ξ )( +∣ ξ ∣ ) / ∣ ξ ∣ − − iε ) . We first provide some estimates related to this function.We have Ψ εt ( ξ ′ ) ∶ = e − i √ −∣ ξ ′ ∣ t F − ξ ( ˆ A ε ( ξ , ξ ′ ))( t ) = √ π ∫ R η ( ξ , ξ ′ ) φ ( ξ , ξ ′ )√ + ∣ ξ ∣ + ∣ ξ ′ ∣ ξ + ∣ ξ ′ ∣ − − iε e i ( ξ −√ −∣ ξ ′ ∣ ) t dξ = ∫ R ψ ε ( ξ , ξ ′ ) ⋅ e iξ t ξ dξ (36)where ψ ε ( ξ , ξ ′ ) ∶ = η ( ξ + √ − ∣ ξ ′ ∣ , ξ ′ ) φ ( ξ + √ − ∣ ξ ′ ∣ , ξ ′ )√ + ξ √ − ∣ ξ ′ ∣ + ∣ ξ ∣ √ π ( ξ + √ − ∣ ξ ′ ∣ − iε / ξ ) . In (36) we used the coordinate transformation ξ ↦ ξ + √ − ∣ ξ ′ ∣ , which is well-definedbecause of ∣ ξ ′ ∣ ≤ − δ by (34). Moreover, ξ + √ − ∣ ξ ′ ∣ > ξ ≥ − δ √ n > ξ ∈ supp ( ψ ε ) , ε ∈ R . (In the estimates for v , − the coordinate transformation to be used is ξ ↦ ξ − √ − ∣ ξ ′ ∣ andthe above estimate has to be replaced by ξ − √ − ∣ ξ ′ ∣ < ξ < − − δ √ n < OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 29 that ψ ε is smooth for every fixed ε ∈ R . Based on these properties of ψ ε we now provide someestimates for Ψ εt .From (34) we infer η ( s, ξ ′ ) = ∣ ξ ′ ∣ ≥ − δ , s ∈ R so that(37) supp ( Ψ εt ) ⊂ B − δ for all t ∈ R , ε ≠ . Moreover, we have for all m ∈ N and ξ ′ ∈ R n − , ∣ ξ ′ ∣ ≤ − δ ∣ ∇ m Ψ εt ( ξ ′ )∣ (36) = ∣ e i √ −∣ ξ ′ ∣ t F ( ∇ mξ ′ ( ψ ε ( ⋅ , ξ ′ )) ⋅ p . v . ( ⋅ )) ( t )∣ ≤ ∥ F ( ∇ mξ ′ ( ψ ε ( ⋅ , ξ ′ ))) ∗ F ( p . v . ( ⋅ ))∥ L ∞ ( R ) ≤ ∥ F ( ∇ mξ ′ ( ψ ε ( ⋅ , ξ ′ )))∥ L ( R ) ∥ iπ sign ( ⋅ )∥ L ∞ ( R ) ≤ π ∥ F ( ∇ mξ ′ ( ψ ε ( ⋅ , ξ ′ ))) ⋅ ( + ∣ ⋅ ∣ )( + ∣ ⋅ ∣ ) − ∥ L ( R ) ≤ π ∥ F ( ∇ mξ ′ ( − ∂ ξ ξ + )( ψ ε ( ⋅ , ξ ′ )))∥ L ∞ ( R ) ∥( + ∣ ⋅ ∣ ) − ∥ L ( R ) ≤ C ∥ ∇ mξ ′ ( − ∂ ξ ξ + )( ψ ε ( ⋅ , ξ ′ ))∥ L ( R ) ≤ C m . (38)From (37) and (38) we conclude that for all m ∈ N there is a C m > ∣ ∇ m Ψ εt ( ξ ′ )∣ ≤ C m ( + ∣ ξ ′ ∣) − m for all ξ ′ ∈ R n − , t ∈ R , ε ≠ . In view of (35) we now use (39) in order to estimate the term ∥( A ε ∗ f )( x , ⋅ )∥ L ( R n − ) = ∫ R n − ( A ε ∗ f )( x , y ′ )( ¯ A ε ∗ ¯ f )( x , y ′ ) dy ′ = ∫ R ∫ R ( ∫ R n − f ( z , y ′ ) S εx ,z , ˜ z ( f ( ˜ z , ⋅ ))( y ′ ) dy ′ ) dz d ˜ z for any given x , z , ˜ z ∈ R , ε ≠ S εx ,z , ˜ z ∶ S ( R n − ) → S ( R n − ) is given by S εx ,z , ˜ z g ∶ = ¯ A ε ( x − ˜ z , ⋅ ) ∗ A ε ( x − z , ⋅ ) ∗ ¯ g. The ( L , L ) -bound for S εx ,z , ˜ z results from ∥ S εx ,z , ˜ z g ∥ L ( R n − ) = ∥ F n − ( ¯ A ε ( x − ˜ z , ⋅ ) ∗ A ε ( x − z , ⋅ )) ⋅ ¯ˆ g ∥ L ( R n − ) ≤ ∥ F n − ( ¯ A ε ( x − ˜ z , ⋅ ) ∗ A ε ( x − z , ⋅ )) ∥ L ∞ ( R n − ) ∥ g ∥ L ( R n − ) ≤ sup t ∈ R ∥ F n − ( ¯ A ε ( t, ⋅ ))∥ L ∞ ( R n − ) ∥ g ∥ L ( R n − ) ≤ sup t ∈ R ,ξ ′ ∈ R n − ∣ F − ( ˆ A ε ( ⋅ , ξ ′ ))( t )∣ ⋅ ∥ g ∥ L ( R n − ) = sup t ∈ R ,ξ ′ ∈ R n − ∣ Ψ εt ( ξ ′ )∣ ⋅ ∥ g ∥ L ( R n − ) (39) ≤ C ∥ g ∥ L ( R n − ) . (40)The ( L , L ∞ ) -bound is obtained via the method of stationary phase. ∥ S εx ,z , ˜ z g ∥ ∞ ≤ ∥ ¯ A ε ( x − ˜ z , ⋅ ) ∗ A ε ( x − z , ⋅ )∥ ∞ ∥ g ∥ = ∥ F − n − ( F − ( ˆ A ε ( ⋅ , ξ ′ ))( x − ˜ z ) ⋅ F − ( ˆ A ε ( ⋅ , ξ ′ ))( x − z ))∥ ∞ ∥ g ∥ = ∥ F − n − ( e − i √ −∣ ξ ′ ∣ ( x − ˜ z ) Ψ εx − ˜ z ( ξ ′ ) ⋅ e i √ −∣ ξ ′ ∣ ( x − z ) Ψ εx − z ( ξ ′ ))∥ ∞ ∥ g ∥ = ∥ F − n − ( e i √ −∣ ξ ′ ∣ ( ˜ z − z ) Ψ εx − ˜ z ( ξ ′ ) Ψ εx − z ( ξ ′ ))∥ ∞ ∥ g ∥ = ( π ) n − sup y ′ ∈ R n − ∣ ∫ R n − e i (⟨ y ′ ,ξ ′ ⟩+√ −∣ ξ ′ ∣ ( z − ˜ z )) Ψ εx − ˜ z ( ξ ′ ) Ψ εx − z ( ξ ′ ) dξ ′ ∣ ∥ g ∥ . In the last integral the smooth phase function Φ ( ξ ′ ) ∶ = ⟨ y ′ , ξ ′ ⟩ + √ − ∣ ξ ′ ∣ ( z − ˜ z ) is stationaryprecisely at ξ ′ = sign ( z − ˜ z )√∣ y ′ ∣ +( z − ˜ z ) y ′ and the Hessian in that point D Φ ( ξ ′ ) = ˜ z − z √ − ∣ ξ ′ ∣ ( Id + − ∣ ξ ′ ∣ ξ ′ ( ξ ′ ) T ) ∈ R ( n − )×( n − ) possesses the eigenvalues 1 , . . . , , −∣ ξ ′ ∣ , which are all uniformly bounded away from zero andinfinity on the support of ξ ′ ↦ Ψ εx − ˜ z ( ξ ′ ) Ψ εx − z ( ξ ′ ) . Moreover, by (39) all derivatives of thisfunction are square integrable with L -norms that are uniformly bounded with respect to x , ˜ z , z , ε . Hence, the Morse Lemma and the method of stationary phase yield(41) ∥ S εx ,z , ˜ z g ∥ L ∞ ( R n − ) ≤ C ( + ∣ z − ˜ z ∣) − n ∥ g ∥ L ( R n − ) . Interpolating the ( L , L ) -estimate (40) and the ( L , L ∞ ) -estimate (41) we get from theRiesz-Thorin Theorem ∥ S x ,z , ˜ z g ∥ L p ′ ( R n − ) ≤ C ( + ∣ z − ˜ z ∣) ( − n )( p − ) ∥ g ∥ L p ( R n − ) for all p ∈ [ , ] . (42)With this estimate we are finally ready to conclude.So assume q − ≥ n + . Then ( + ∣ ⋅ ∣) ( − n )( q − ) lies in L q ( q − ) ,w ( R n ) so that Young’s inequalityfor weak Lebesgue spaces implies1 R ∫ B R ∣ v ε ( x )∣ + ∣ ∇ v ε ( x )∣ dx (35) ≤ sup x ∈ R ∥( A ε ∗ f )( x , ⋅ )∥ L ( R n − ) = sup x ∈ R ∫ R ∫ R ( ∫ R n − f ( z , y ′ ) S εx ,z , ˜ z ( f ( ˜ z , ⋅ ))( y ′ ) dy ′ ) dz d ˜ z ≤ sup x ∈ R ∫ R ∫ R ∥ f ( z , ⋅ )∥ L q ( R n − ) ∥ S εx ,z , ˜ z ( f ( ˜ z , ⋅ ))∥ L q ′ ( R n − ) dz d ˜ z ≤ C ∫ R ∫ R ( + ∣ z − ˜ z ∣) ( − n )( p − ) ∥ f ( z , ⋅ )∥ L q ( R n − ) ∥ f ( ˜ z , ⋅ )∥ L q ( R n − ) dz d ˜ z ≤ C ∥ f ∥ q . This finally shows (33) and the proof is finished. ◻ OLUTIONS FOR NONLINEAR HELMHOLTZ AND CURL-CURL EQUATIONS 31
Acknowledgements
The author gratefully acknowledges financial support by the Deutsche Forschungsgemein-schaft (DFG) through CRC 1173 ”Wave phenomena: analysis and numerics”.
References [1] Shmuel Agmon. A representation theorem for solutions of the Helmholtz equation and resolvent estimatesfor the Laplacian. In
Analysis, et cetera , pages 39–76. Academic Press, Boston, MA, 1990.[2] Thomas Bartsch, Tom´aˇs Dohnal, Michael Plum, and Wolfgang Reichel. Ground states of a nonlinearcurl-curl problem in cylindrically symmetric media.
NoDEA Nonlinear Differential Equations Appl. ,23(5):Art. 52, 34, 2016.[3] Thomas Bartsch and Jaros law Mederski. Nonlinear time-harmonic Maxwell equations in an anisotropicbounded medium.
J. Funct. Anal. , 272(10):4304–4333, 2017.[4] Thomas Bartsch and Jaros law Mederski. Nonlinear time-harmonic Maxwell equations in domains.
J.Fixed Point Theory Appl. , 19(1):959–986, 2017.[5] Denis Bonheure, Jean-Baptiste Cast´eras, Ederson Moreira dos Santos, and Robson Nascimento. OrbitallyStable Standing Waves of a Mixed Dispersion Nonlinear Schr¨odinger Equation.
SIAM J. Math. Anal. ,50(5):5027–5071, 2018.[6] Denis Bonheure, Jean-Baptiste Cast´eras, Tianxiang Gou, and Louis Jeanjean. Normalized solutions tothe mixed dispersion nonlinear Schr¨odinger equation in the mass critical and supercritical regime, 2018.[7] Denis Bonheure, Jean-Baptiste Cast´eras, and Rainer Mandel. On a fourth order nonlinear Helmholtzequation, 2018.[8] Denis Bonheure and Robson Nascimento. Waveguide solutions for a nonlinear Schr¨odinger equationwith mixed dispersion. In
Contributions to nonlinear elliptic equations and systems , volume 86 of
Progr.Nonlinear Differential Equations Appl. , pages 31–53. Birkh¨auser/Springer, Cham, 2015.[9] G. Ev´equoz. Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equationin the plane.
Analysis (Berlin) , 37(2):55–68, 2017.[10] G. Ev´equoz and T. Weth. Dual variational methods and nonvanishing for the nonlinear Helmholtzequation.
Adv. Math. , 280:690–728, 2015.[11] Gilles Ev´equoz. A dual approach in Orlicz spaces for the nonlinear Helmholtz equation.
Z. Angew. Math.Phys. , 66(6):2995–3015, 2015.[12] Gilles Ev´equoz and Tobias Weth. Real solutions to the nonlinear Helmholtz equation with local nonlin-earity.
Arch. Ration. Mech. Anal. , 211(2):359–388, 2014.[13] Gilles Ev´equoz and Tobias Weth. Branch continuation inside the essential spectrum for the nonlinearSchr¨odinger equation.
J. Fixed Point Theory Appl. , 19(1):475–502, 2017.[14] Gilles Ev´equoz and Tolga Yesil. Dual ground state solutions for the critical nonlinear Helmholtz equation,2017.[15] David Gilbarg and Neil S. Trudinger.
Elliptic partial differential equations of second order . Classics inMathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.[16] Loukas Grafakos.
Classical Fourier analysis , volume 249 of
Graduate Texts in Mathematics . Springer,New York, third edition, 2014.[17] S. Guti´errez. Non trivial L q solutions to the Ginzburg-Landau equation. Math. Ann. , 328(1-2):1–25,2004.[18] Andreas Hirsch and Wolfgang Reichel. Existence of cylindrically symmetric ground states to a nonlinearcurl-curl equation with non-constant coefficients.
Z. Anal. Anwend. , 36(4):419–435, 2017.[19] A. D. Ionescu and D. Jerison. On the absence of positive eigenvalues of Schr¨odinger operators with roughpotentials.
Geom. Funct. Anal. , 13(5):1029–1081, 2003.[20] Herbert Koch and Daniel Tataru. Carleman estimates and absence of embedded eigenvalues.
Comm.Math. Phys. , 267(2):419–449, 2006. [21] R. Mandel, E. Montefusco, and B. Pellacci. Oscillating solutions for nonlinear Helmholtz equations.
Zeitschrift f¨ur angewandte Mathematik und Physik , 68(6):121, 10 2017.[22] Rainer Mandel. The limiting absorption principle for periodic differential operators and applications tononlinear Helmholtz equations, 2017.[23] Jaros law Mederski. Ground states of time-harmonic semilinear Maxwell equations in R with vanishingpermittivity. Arch. Ration. Mech. Anal. , 218(2):825–861, 2015.[24] Alberto Ruiz and Luis Vega. On local regularity of Schr¨odinger equations.
Internat. Math. Res. Notices ,(1):13–27, 1993.[25] E. Stein.
Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , volume 43of
Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1993.[26] Xiaoyu Zeng. Cylindrically symmetric ground state solutions for curl-curl equations with critical expo-nent.
Z. Angew. Math. Phys. , 68(6):Art. 135, 12, 2017.
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