Rainer Mandel and Dominic Scheider

Bifurcations of nontrivial solutions of a cubic Helmholtz system

De Gruyter | Published online: October 19, 2019

Abstract

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system

Δ u μ u = u 2 + b v 2 u  on  R 3 , Δ v ν v = v 2 + b u 2 v  on  R 3 .

It is shown that every point along any given branch of radial semitrivial solutions (u0, 0, b) or diagonal solutions (ub, ub, b) (for μ = ν) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior and the decay of solutions at infinity.

1 Introduction and main results

Systems of two coupled nonlinear Helmholtz equations arise, for instance, in models of nonlinear optics. In this paper, we analyze the physically relevant and technically easiest case of a Kerr-type nonlinearity in N = 3 space dimensions, that is, we study the system

Δ u μ u = u 2 + b v 2 u  on  R 3 , Δ v ν v = v 2 + b u 2 v  on  R 3 (H)

for given μ, ν > 0 and a constant coupling parameter b ∈ ℝ. We are mostly interested in existence results for fully nontrivial radially symmetric solutions of this system that we will obtain using bifurcation theory. Such an approach is new in the context of nonlinear Helmholtz equations or systems. In order to describe the methods used in related works we briefly discuss the available results for scalar nonlinear Helmholtz equations of the form

Δ u λ u = Q ( x ) | u | p 2 u on  R N , λ > 0. (1)

Here, the main difficulty is that solutions typically oscillate and do not belong to H1(ℝN). In the past years, Evéquoz and Weth have developed several methods allowing to find nontrivial solutions of (1) under certain conditions on Q and p, some of which we wish to mention. In [1, 2], they discuss the case of compactly supported Q and 2 < p < 2 := 2 N N 2 . The idea in [1] is to solve an exterior problem where the nonlinearity vanishes and knowledge about the far-field expansion of solutions is available. The remaining problem on a bounded domain can be solved using variational techniques. The approach in [2] uses Leray-Schauder continuation with respect to the parameter λ in order to find solutions of (1). Existence of solutions under the assumption that QL(ℝN) decays as |x| → ∞ or is periodic is proved in [3] using a dual variational approach, which yields (dual) ground state solutions and, in the case of decaying Q, infinitely many bound states. The technique relies on the Limiting Absorption Principle of Gutiérrez, see Theorem 6 in [4], which leads to the additional constraint 2 ( N + 1 ) N 1 < p < 2. Furthermore, assuming that Q is radial, the existence of a continuum of radially symmetric solutions of (1) has been shown by Montefusco, Pellacci and the first author in [5], generalizing earlier results in [1]. Their results rely on ODE techniques and only require p > 2 and a monotonicity assumption on Q.

To our knowledge, the only available result on nonlinear Helmholtz systems like (H) has been provided by the authors in [6] where, using the methods developed in [3], the existence of a nontrivial dual ground state solution is proved for the system

Δ u μ u = a ( x ) | u | p 2 + b ( x ) | v | p 2 | u | p 2 2 u on  R N , Δ v ν v = a ( x ) | v | p 2 + b ( x ) | u | p 2 | v | p 2 2 v on  R N , u , v L p ( R N )

for N ≥ 2, ℤN-periodic coefficients a, bL(ℝN) with a(x) ≥ a0 > 0, 0 ≤ b(x) ≤ p – 1 and 2 ( N + 1 ) N 1 < p < 2. Under additional easily verifiable assumptions the ground state can be shown to be fully nontrivial, i.e., both components are nontrivial. Assuming constant coefficients and working on spaces of radially symmetric functions, this variational existence result for dual ground states extends to the case p = 4, N = 3 which we dicuss in the present paper. In contrast to [6] we construct fully nontrivial radial solutions for arbitrarily large and small b ∈ ℝ that, however, need not be dual ground states.

Motivated by the decay properties of radial solutions of nonlinear Helmholtz equations in [5], e.g. Theorem 1.2 (iii), we look for solutions of (H) in the Banach space X1 where, for q ≥ 1,

X q := w C rad ( R 3 , R ) | w X q < with w X q := sup x R 3 ( 1 + | x | 2 ) q 2 | w ( x ) | . (2)

Working on these spaces, we will be able to derive compactness properties which are crucial when proving our bifurcation results. Throughout, we discuss classical, radially symmetric solutions u, vX1C2(ℝ3) of the system (H) and related equations. Let us remark here only briefly that, using elliptic regularity, all weak solutions u, v L rad 4 (ℝ3) are actually smooth and, thanks to Proposition 6 in the next section, belong to X1C2(ℝ3).

We study bifurcation of solutions (u, v, b) of the nonlinear Helmholtz system (H) from a branch of semitrivial solutions of the form

T u 0 := { ( u 0 , 0 , b ) | b R } X 1 × X 1 × R

in the Banach space X1 × X1 × ℝ. Here u0 : ℝ3 → ℝ denotes any of the uncountably many nontrivial radial solutions of the scalar Helmholtz equation

Δ u 0 μ u 0 = u 0 3 on  R 3 , (h)

which all belong to the space X1, see [5]. In contrast to the Schrödinger case, we will demonstrate that every point in 𝓣u0 is a bifurcation point for fully nontrivial solutions of (H). Our strategy is to use bifurcation from simple eigenvalues with b acting as a bifurcation parameter. The existence of isolated and algebraically simple eigenvalues will be ensured by assuming radial symmetry and by imposing suitable conditions on the asymptotic behavior of the solutions u, v. For τ, ω ∈ [0, π), we define 𝓢 ⊆ X1 × X1 × ℝ ∖ 𝓣u0 as the set of all solutions (u, v, b) ∈ X1 × X1 × ℝ ∖ 𝓣u0 of (H) satisfying the asymptotic conditions

u ( x ) u 0 ( x ) = c 1 sin ( | x | μ + τ ) | x | + O 1 | x | 2 v ( x ) = c 2 sin ( | x | ν + ω ) | x | + O 1 | x | 2 as  | x | (A)

for some c1, c2 ∈ ℝ. Propositions 4 and 5 below show that it is natural to assume such an asymptotic behavior for solutions of (H). For notational convenience, we do not denote the dependence of 𝓢 and of the asymptotic conditions (A) on the choice τ, ω ∈ [0, π). As we will show in Proposition 6, there exists a unique τ0 = τ0(u0) ∈ [0, π) such that, for τ ∈ [0, π),

Δ w μ w = 3 u 0 2 ( x ) w o n R 3 , w ( x ) = sin ( | x | μ + τ ) | x | + O 1 | x | 2 a s | x |  has no radial solution for  τ τ 0 . (N)

With that, we obtain the following

Theorem 1

Let μ, ν > 0, fix any u0X1 ∖ {0} solving the nonlinear Helmholtz equation (h) and choose τ ∈ [0, π) ∖ {τ0} according to (N). Then, for every ω ∈ [0, π), there exists a strictly increasing sequence (bk(ω))k∈ℤ such that (u0, 0, bk(ω)) ∈ 𝓢 where 𝓢 denotes the set of all solutions (u, v, b) ∈ X1 × X1 × ℝ ∖ 𝓣u0 of (H) satisfying (A). Moreover,

  1. the connected component 𝓒k of (u0, 0, bk(ω)) in 𝓢 is unbounded in X1 × X1 × ℝ; and

  2. each bifurcation point (u0, 0, bk(ω)) has a neighborhood where 𝓒k is a smooth curve in X1 × X1 × ℝ which, except for the bifurcation point, consists of fully nontrivial solutions.

The main tools in proving this statement are the Crandall-Rabinowitz Bifurcation Theorem, which will be used to show the local statement (ii) of Theorem 1, and Rabinowitz’ Global Bifurcation Theorem, which will provide (i). For a reference, see [7], Theorem 1.7 and [8], Theorem 1.3. We add some remarks the proof of which will also be given in Section 3.

Remark 2

  1. We will also see that fully nontrivial solutions of (H) satisfying the asymptotic condition (A) bifurcate from some point (u0, 0, b) ∈ 𝓣u0 if and only if b = bk(ω) for some k ∈ ℤ. Moreover, the proof will show that the values bk(ω) do not depend on the choice of τ.

  2. The map ℝ → ℝ, k π + ωbk(ω) where 0 ≤ ω < π, k ∈ ℤ is strictly increasing and onto with bk(ω) → ± ∞ as k → ± ∞. In particular, every point (u0, 0, b) ∈ 𝓣u0, b ∈ ℝ, is a bifurcation point for fully nontrivial radial solutions of (H), which is in contrast to Schrödinger systems where bifurcation points are isolated, cf. [9], Satz 2.1.6.

  3. Close to the respective bifurcation point (u0, 0, bk(ω)) ∈ 𝓣u0, each continuum 𝓒k is characterized by a phase parameter ων(v2 + bu2) = ω + derived from the asymptotic behavior of v (see (10)). It seems that, in the Helmholtz case of oscillating solutions, the integer k takes the role of the nodal characterizations in the Schrödinger case, cf. Satz 2.1.6 in [9]. That phase parameter is constant on connected subsets of the continuum until it possibly runs into another family of semitrivial solutions 𝓣u1 with u1u0; unfortunately we cannot provide criteria deciding whether or not this happens. For this reason we cannot claim that 𝓒k contains an unbounded sequence of fully nontrivial solutions.

  4. For δ ≠ 0, let us assume that uδC2(ℝ3) ∩ X1 solvesΔuδμuδ = u δ 3 on3 with uδ(0) = u0(0) + δ, see Theorem 1.2 in [5]. Then w := d d δ | δ = 0 u δ satisfiesΔw - μw = 3 u 0 2 w on3, w(0) = 1. We define τ0 ∈ [0, π) as the constant appearing in the asymptotic expansion of w,

    w ( x ) = c sin ( | x | μ + τ 0 ) | x | + O 1 | x | 2 a s | x |

    for some unique c ≠ 0 and τ0 ∈ [0, π), see Proposition 6. With that in mind, the condition ττ0 is a nondegeneracy condition which by means of (N) ensures that the simplicity requirements of the above-mentioned bifurcation theorems are satisfied.

Our results are inspired by known bifurcation results for the nonlinear Schrödinger system

Δ u + λ 1 u = μ 1 u 3 + b u v 2 on  R N , Δ v + λ 2 v = μ 2 v 3 + b v u 2 on  R N , u , v H 1 ( R N ) , u > 0 , v > 0 (3)

where one assumes λ1, λ2 > 0 in contrast to (H). We focus on bifurcation results by Bartsch, Wang and Wei in [10] and Bartsch, Dancer and Wang in [11] and refer to the respective introductory sections for a general overview of methods and results for (3). In Theorem 1.1 of [10] the authors show that a continuum consisting of positive radially symmetric solutions (u, v, λ1, λ2, μ1, μ2, b) of (3) with topological dimension at least 5 bifurcates from a two-dimensional set of semipositive solutions (u, v) = (uλ1,μ1, 0) parametrized by λ1, μ1 > 0. The existence of countably many bifurcation points giving rise to sign-changing radially symmetric solutions was proved by the first author in his dissertation thesis (Satz 2.1.6 of [9]).

In Theorem 1 above, we analyze the corresponding case of bifurcation from a semitrivial family 𝓣u0 in the Helmholtz case. In contrast to the Schrödinger case, our result shows bifurcation at every point in the topology of X1 × X1 × ℝ, see Remark 2 (b). Looking more closely, we find the same structure of discrete bifurcation points as in the Schrödinger case when fixing parameters τ, ω prescribing the oscillatory behavior of solutions as |x| → ∞ as in the condition (A). In the Schrödinger case, the bifurcating solutions are characterized by their nodal structure; in the Helmholtz case, we use instead a condition on the “asymptotic phase” of the solution (disguised as an integral), which at least close to the j-th bifurcation point takes the value ω + as described in Remark 2 (c).

Similar observations can be made for bifurcation from families of diagonal solutions of the Schrödinger system (3) in the special case N = 2, 3 and λ1 = λ2 > 0 and μ1, μ2 > 0; in order to keep the presentation short, we assume in addition μ1 = μ2 = 1. Bartsch, Dancer and Wang proved in [11] the existence of countably many mutually disjoint global continua of solutions bifurcating from some diagonal solution family of the form

{ ( u b , u b , b ) : b > 1 } H rad 1 ( R N ) × H rad 1 ( R N ) × R

with a concentration of bifurcation points as b ↘ –1. Here ub := (1 + b)–1/2 u0 where u0 H rad 1 (ℝN) is a nondegenerate solution of –Δu + u = u3. Moreover, having introduced a suitable labeling of the continua, the authors showed that the k-th continuum consists of solutions where the radial profile of uv has exactly k – 1 nodes, cf. Theorem 2.3 in [11].

We provide a counterpart for the Helmholtz system (H) in our second result, Theorem 3, using the same functional analytical setup as in Theorem 1. Here we assume ν = μ. For nonzero u0 solving (h), we can then} introduce the diagonal solution family

T u 0 := ( u b , u b , b ) | b > 1 X 1 × X 1 × R with  u b := ( 1 + b ) 1 / 2 u 0 .

Given τ, ω ∈ [0, π), we denote by 𝔖 the set of all solutions (u, v, b) ∈ X1 × X1 × ℝ ∖ 𝔗u0 of the nonlinear Helmholtz system (H) with

u ( x ) + v ( x ) = 2 u b ( x ) + c 1 sin ( | x | μ + τ ) | x | + O 1 | x | 2 u ( x ) v ( x ) = c 2 sin ( | x | μ + ω ) | x | + O 1 | x | 2 as  | x | (Adiag)

for some c1, c2 ∈ ℝ. Our existence result for fully nontrivial solutions of (H) bifurcating from 𝔗u0 with asymptotics (Adiag) reads as follows.

Theorem 3

Let ν = μ > 0, fix any u0X1 ∖ {0} solving the nonlinear Helmholtz equation (h) and choose τ ∈ [0, π) ∖ {τ0} according to (N). Then, for every ω ∈ [0, π), there exists a sequence (𝔟k(ω))k∈ℕ such that (u𝔟k(ω), u𝔟k(ω), 𝔟k(ω)) ∈ 𝔖 where 𝔖 denotes the set of all solutions (u, v, b) ∈ X1 × X1 × ℝ ∖ 𝔗u0 of (H) satisfying (Adiag). Moreover,

  1. the connected componentk of (u𝔟k(ω), u𝔟k(ω), 𝔟k(ω)) in 𝔖 is unbounded in X1 × X1 × ℝ; and

  2. each bifurcation point (u𝔟k(ω), u𝔟k(ω), 𝔟k(ω)) has a neighborhood where the setk contains a smooth curve in X1 × X1 × ℝ which, except for the bifurcation point, consists of fully nontrivial, non-diagonal solutions.

Again, similar statements as in Remark 2 can be proved. In particular, one can check that every point on 𝔗u0 is a bifurcating point by a suitable choice of ω.

We point out that our methods in Theorems 1 and 3 also apply for nontrivial radial solutions of

Δ u 0 μ u 0 = u 0 3 on  R 3

and corresponding modifications in the system (H). Such solutions u0 exist in the Helmholtz case (but not in the Schrödinger case) and belong to the space X1, see Theorem 1.2 in [5].

Let us give a short outline of this paper. In Section 2, we introduce the concepts and technical results we use in the proof of Theorems 1 and 3, which are presented in Section 3 and Section 4. In the final section, we provide the proofs of the auxiliary results of Section 2.

2 Properties of the scalar problem

The main challenge in proving Theorem 1 is the thorough analysis of the linearized problem provided in this chapter. Throughout, we fix λ > 0 and discuss the linear Helmholtz equation

Δ w λ w = f on  R 3 (4)

for some fX3, where X3 is defined in (2). We will frequently identify radially symmetric functions xw(x) with their profiles; in particular, we denote by w′ := rw, w″ = r 2 w the radial derivatives. The results we establish in this section will demonstrate how to rewrite the system (H) in a way suitable for Bifurcation Theory.

2.1 Representation Formulas

First, we discuss a representation formula for solutions of the linear Helmholtz equation (4). The results resemble a more general Representation Theorem by Agmon, Theorem 4.3 in [12]. We introduce the fundamental solutions

Ψ λ , Ψ ~ λ : R 3 R , Ψ λ ( x ) := cos ( λ | x | ) 4 π | x | and Ψ ~ λ ( x ) := sin ( λ | x | ) 4 π | x | ( x 0 ) (5)

of the equation –Δwλw = 0 on ℝ3. We observe that Ψ̃λ is, up to multiplication with a constant, its unique global classical solution. We will frequently require knowledge of the mapping properties of convolutions with Ψλ resp. Ψ̃λ. Various results of such type have been found by Evéquoz and Weth in [3] and further publications, assuming fLp(ℝN), wLp(ℝN) for suitable p, p′ ∈ (1, ∞). In the spaces X3 resp. X1, which satisfy the continuous embeddings

X 1 L r a d p ( R 3 )  for  3 < p , X 3 L r a d q ( R 3 )  for  1 < q , (6)

we prove the following statements.

Proposition 4

For constants α, α̃ ∈ ℝ, we let 𝓡λf := (αΨλ + α̃Ψ̃λ) ∗ f. Then,

  1. the linear map L r a d 4 3 ( R 3 ) L r a d 4 ( R 3 ) , f R λ f is well-defined and continuous;

  2. the linear map X3X1, f ↦ 𝓡λf is well-defined, continuous and compact;

  3. for fX3, we have w := 𝓡λfX1C2(ℝ3) withΔwλw = αf on3; and

  4. for fX3, the profile of w := 𝓡λf and its radial derivative satisfy as r → ∞

    w ( r ) = π 2 f ^ ( λ ) α cos ( r λ ) + α ~ sin ( r λ ) r + O 1 r 2 , w ( r ) = π 2 f ^ ( λ ) α λ sin ( r λ ) + α ~ λ cos ( r λ ) r + O 1 r 2 (7)

    where f ^ ( λ ) = 2 π 0 f ( r ) sin ( r λ ) r λ r 2 d r . Further, Ψ ~ λ f = 4 π π 2 f ^ ( λ ) Ψ ~ λ .

As a consequence of Proposition 4, we state the representation formulae we require later to construct the functional analytic setting in the proof of Theorem 1. For ω ∈ (0, π), we define

R λ ω : X 3 X 1 , f Ψ λ f + cot ( ω ) Ψ ~ λ f (8)

which provide solutions of the Helmholtz equation (4) the asymptotic behavior of which is described by the phase parameter ω as follows.

Corollary 5

Let ω ∈ (0, π) and fX3. Then, for wX1, we have w = R λ ω f if and only if w is a C2 solution ofΔwλw = f on3 with, for some c ∈ ℝ,

w ( x ) = c sin ( | x | λ + ω ) | x | + O 1 | x | 2 a s | x | .

We point out that the operator R λ ω is not well-defined for ω = 0 due to the pole of the cotangent. We will comment on suitable modifications during the proofs of Theorems 1 and 3.

2.2 The Asymptotic Phase

Frequently, equations of interest will take the form (4) with f = gw for some gX2, see (2). We can then use ODE methods, more specifically the Prüfer transformation, to discuss the corresponding initial value problem for the profiles,

w 2 r w λ w = g ( r ) w on  ( 0 , ) with  w ( 0 ) = 1 , w ( 0 ) = 0. (9)

Proposition 6

Assume gX2. Then the initial value problem (9) has a unique (global) solution w: [0, ∞) → ℝ which satisfies

w ( r ) = ϱ λ ( g ) sin ( r λ + ω λ ( g ) ) r + O 1 r 2 , w ( r ) = ϱ λ ( g ) λ cos ( r λ + ω λ ( g ) ) r + O 1 r 2

as r → ∞ for some ϱλ(g) > 0 and ωλ(g) ∈ ℝ. Here, the value of ωλ(g) is given by

ω λ ( g ) = 1 λ 0 g ( r ) sin 2 ( ϕ ( r ) λ ) d r w h e r e ϕ : [ 0 , ) R s o l v e s ϕ = 1 + 1 λ g ( r ) sin 2 ( ϕ λ ) , ϕ ( 0 ) = 0. (10)

In particular, given u0X1 ∖ {0} solving (h), Proposition 6 with g := 3 u 0 2 X2 shows that the nondegeneracy condition (N) holds with τ0 ∈ [0, π) such that ωμ(3 u 0 2 ) ∈ τ0 + πℤ.

Comparing Proposition 6 with Corollary 5, we observe that Proposition 6 guarantees ϱλ(g) > 0, that is, the solution has a nonvanishing term of leading order as r → ∞. The asymptotic conditions imposed in Corollary 5 with f = gw now take the form ωλ(g) ∈ ω + πℤ. Such boundary conditions at infinity will provide operators with spectral properties suitable for building the functional analytic framework in which to prove Theorem 1. As a first auxiliary result, we prove the following continuity property.

Proposition 7

The asymptotic phase is continuous as a map ωλ : X2 → ℝ, gωλ(g).

When studying eigenvalue problems of a linearization of (H), we need to know the dependence of the asymptotic phase ωλ(b u 0 2 ) on the parameter b ∈ ℝ.

Proposition 8

Let u0X1C2(ℝ3) be some nonzero solution of (h). Then the map ℝ → ℝ, bωλ(b u 0 2 ) is continuous, strictly increasing and onto with ωλ(0) = 0.

2.3 The spectrum of the linearization

In the proof of Theorem 1, we will rewrite the nonlinear Helmholtz system (H) in the form

u = R μ τ ( u ( u 2 + b v 2 ) ) , v = R ν ω ( v ( v 2 + b u 2 ) ) , u , v X 1

for some τ, ω ∈ (0, π), which additionally imposes a certain asymptotic behavior on the solutions, see Corollary 5. In order to analyze the linearized problem, we fix some nontrivial u0X1C2(ℝ3) with –Δu0μu0 = u 0 3 on ℝ3 and study the spectra of the linear operators

R λ ω : X 1 X 1 , w R λ ω ( u 0 2 w ) = Ψ λ + cot ( ω ) Ψ ~ λ [ u 0 2 w ] , (11)

which are compact thanks to Proposition 4 (b).

Proposition 9

Let ω ∈ (0, π), λ > 0 and u0 as before. Then the spectrum of R λ ω is

σ ( R λ ω ) = { 0 } σ p ( R λ ω ) , σ p ( R λ ω ) = 1 b k ( ω , λ , u 0 2 ) | k Z

where, for k ∈ ℤ, b = bk(ω, λ, u 0 2 ) ∈ ℝ is the unique solution of ωλ(b u 0 2 ) = ω + , see Proposition 8. Moreover, all eigenvalues are algebraically simple, and the sequence (bk(ω, λ, u 0 2 ))k∈ℤ is strictly increasing and unbounded below and above.

This excludes the case ω = 0, even though the values bk(0, λ, u 0 2 ) ∈ ℝ, k ∈ ℤ, can be defined accordingly. Indeed, the first step of the proof of Proposition 9 above shows for all ω ∈ [0, π):

Remark 10

Fix ω ∈ [0, π). Then the problem

Δ w λ w = b u 0 2 w o n R 3 , w ( x ) = sin ( | x | λ + ω ) | x | + O 1 | x | 2 a s | x |

has a nontrivial radial solution wX1C2(ℝ3) if and only if b ∈ {bk(ω, λ, u 0 2 ) | k ∈ ℤ}.

3 Proof of Theorem 1

We first discuss the case ω, τ ∈ (0, π), ττ0. Afterwards, we sketch the modifications required if ω = 0 or τ = 0.

The case ω ∈ (0, π) and τ ∈ (0, π) ∖ {τ0}

  1. Step 1

    The Setting.

    We define the map

    F : X 1 × X 1 × R X 1 × X 1 ,

    F ( w , v , b ) := w R μ τ ( w 3 + 3 u 0 w 2 + 3 u 0 2 w + b ( u 0 + w ) v 2 ) v R ν ω ( v 3 + b v ( u 0 + w ) 2 )

    with the convolution operators R μ τ , R ν ω : X3X1 from (8). Observe that F is well-defined since u, v, wX1 implies uvwX3. Recalling Corollary 5 and (h), we have

    F ( w , v , b ) = 0 ( u , v , b ) := ( u 0 + w , v , b )  satisfies (H) with asymptotics ( A ) .

    So we aim to find nontrivial zeros of F. Second, we observe that F has a trivial solution family, that is F(0, 0, b) = 0 holds for every b ∈ ℝ. Third, F(⋅, b) is a compact perturbation of the identity on X1 × X1 since the operators R μ τ , R ν ω : X3X1 are compact thanks to Proposition 4 (b). Moreover, F is twice continuously Fréchet differentiable; we have for φ, ψX1 and b ∈ ℝ, denoting by D the Fréchet derivative w.r.t. the w and v components,

    D F ( 0 , 0 , b ) [ ( φ , ψ ) ] = φ ψ 3 R μ τ ( u 0 2 φ ) b R ν ω ( u 0 2 ψ ) = φ 3 R μ τ φ ψ b R ν ω ψ (12)

    with compact linear operators R μ τ , R ν ω : X1X1 as in (11). We deduce that, due to (N) and ττ0, DF(0, 0, b)[φ, ψ] = 0 implies φ = 0. So nontrivial elements of ker DF(0, 0, b) are of the form (0, ψ) where ψ satisfies ψ = b R ν ω ψ. By Proposition 9, a nontrivial solution exists if and only if b = bk(ω, ν, u 0 2 ), i.e. ων(b u 0 2 ) = + ω for some k ∈ ℤ, and that the associated eigenspaces are one-dimensional. We abbreviate bk(ω) := bk(ω, ν, u 0 2 ) and write

    ker D F ( 0 , 0 , b k ( ω ) ) = span  0 ψ k

    for some ψkX1 ∖ {0}. Thus b ∈ {bk(ω) | k ∈ ℤ} is a necessary condition for bifurcation of solutions of F(w, v, b) = 0 from (0, 0, b). We show in the following that it is also sufficient.

  2. Step 2

    Local Bifurcation.

    We apply the Crandall-Rabinowitz Theorem at the point (0, 0, bk(ω)). As F(⋅, b) is a compact perturbation of the identity on X1 × X1, the Riesz-Schauder Theorem implies that DF(0, 0, bk(ω)) is a Fredholm operator of index zero with one-dimensional kernel spanned by (0, ψk), see above. To verify the transversality condition, we first compute

    b D F ( 0 , 0 , b k ( ω ) ) [ ( 0 , ψ k ) ] = ( 12 ) 0 R ν ω ψ k = 1 b k ( ω ) 0 ψ k .

    Then, assuming there is vX1 with vbk(ω) R ν ω v = ψk, we conclude

    v ker ( I b k ( ω ) R ν ω ) 2 ker ( I b k ( ω ) R ν ω ) ,

    which contradicts the algebraic simplicity of the eigenvalue bk(ω)–1 of R ν ω proved in Proposition 9. Thus b DF(0, 0, bk(ω)) [(0, ψk)] ∉ ran DF(0, 0, bk(ω)), and the Crandall-Rabinowitz Theorem provides the smooth curve of solutions of F(w, v, b) = 0 as in (ii). Further, possibly shrinking the neighborhood where the local result holds, we may w.l.o.g. assume fully nontrivial solutions (u0 + w, v) of (H) since the direction of bifurcation is given by (0, ψk).

  3. Step 3

    Global Bifurcation.

    We have already seen that F(⋅, b), b ∈ ℝ, is a compact perturbation of the identity on X1 × X1. Thus the application of Rabinowitz’ Global Bifurcation Theorem only requires to verify that the index of F(⋅, b) in (0, 0) changes sign at each value b = bk(ω), k ∈ ℤ. By the identity (12), for b ∉ {bk(ω) | k ∈ ℤ},

    ind X 1 × X 1 ( F ( , b ) , ( 0 , 0 ) ) = ind X 1 × X 1 ( D F ( 0 , 0 , b ) , ( 0 , 0 ) ) = ( 12 ) ind X 1 ( I 3 R μ τ , 0 ) ind X 1 ( I b R ν ω , 0 ) ,

    and hence indX1×X1 (F(⋅, b), (0, 0)) changes sign at b = bk(ω) if and only if so does indX1 (Ib R ν ω , 0). The latter change of index occurs since bk(ω) is an isolated eigenvalue of algebraic multiplicity 1 of R ν ω , see Proposition 9.

The Global Bifurcation Theorem by Rabinowitz asserts that (u0, 0, bk(ω)) ∈ 𝓢 and that the associated connected component 𝓒k of 𝓢 is unbounded or returns to 𝓣u0 at some point (u0, 0, b). We prove that, in any case, the component is unbounded.

The asymptotic phase satisfies ων(bk(ω) u 0 2 ) = ω + by definition of bk(ω), see Step 1, and ων(v2 + bu2) ∈ ω + πℤ for all (u, v, b) ∈ 𝓒k with v ≠ 0. This is due to (A) and Proposition 6. So if all elements (u, v, b) ∈ 𝓒k ∖ 𝓣u0 satisfy v ≠ 0, then as a consequence of the continuity of ων (see Proposition 7) and of the fact that 𝓒k is connected, we infer that ων(v2 + bu2) = ω + for all (u, v, b) ∈ 𝓒k. Let us now assume that 𝓒k returns to the trivial family in some point (u0, 0, b) ∈ 𝓣u0, bbk(ω). Then ων(b u 0 2 ) ≠ ω + by strict monotonicity (see Proposition 8), hence (u, v, b) ↦ ων(v2 + bu2) is not constant on 𝓒k. Thus, there exists a semitrivial element (u1, 0, b1) ∈ 𝓒k ∖ 𝓣u0, u1u0. Since 𝓒k is maximal connected, it contains the unbounded semitrivial family 𝓣u1 = {(u1, 0, b) | b ∈ ℝ}.

The case ω = 0 and τ ∈ (0, π) ∖ {τ0}

  1. Step 1

    The Setting.

    We recall that, in case ω = 0, the map F resp. R ν ω is not well-defined due to the pole of the cotangent. To write down a suitable replacement, we use the Hahn-Banach Theorem to define functionals α(ν), β(ν) X 1 with the following property: For wX1 with

    w ( x ) = α w sin ( | x | ν ) 4 π | x | + β w cos ( | x | ν ) 4 π | x | + O 1 | x | 2  as  | x | (13)

    for some αw, βw ∈ ℝ, we have α(ν)(w) := αw and β(ν)(w) := βw. We then define for σ = ± 1

    G σ : X 1 × X 1 × R X 1 × X 1 , G σ ( w , v , b ) := w R μ τ ( w 3 + 3 u 0 w 2 + 3 u 0 2 w + b ( u 0 + w ) v 2 ) v Ψ ν [ v ( v 2 + b ( w + u 0 ) 2 ) ] ( α ( ν ) ( v ) + σ β ( ν ) ( v ) ) Ψ ~ ν .

    Using Proposition 4, we see that Gσ(w, v, b) = 0 if and only if (u0 + w, v, b) solves the nonlinear Helmholtz system (H) with asymptotics (A), ω = 0. Indeed, by (13), one sees that Gσ(w, v, b) = 0 implies β(ν)(v) = 0. Further, recalling the property (N), (φ, ψ) ∈ ker DGσ(0, 0, b) if and only if

    φ 0 , Δ ψ ν ψ = b u 0 2 ψ , ψ ( x ) = β ( ν ) ( ψ ) = 0 c sin ( | x | ν ) | x | + O 1 | x | 2

    for some c ≠ 0. As before, Propositions 6 and 8 allow to conclude that solutions of (H), (A) for ω = 0 can bifurcate from (u0, 0, b) ∈ 𝓣u0 only if b = bk(0) for some k ∈ ℤ, and that there exist ψkX1 ∖ {0} with

    ker D G σ ( 0 , 0 , b k ( 0 ) ) = span  0 ψ k .

  2. Step 2

    Local Bifurcation.

    The proof of transversality as required in the Crandall-Rabinowitz Theorem has to be adapted. Assuming for contradiction that there are φ, ψX1 with DGσ(0, 0, bk(0))[(φ, ψ)] = b DGσ(0, 0, bk(0))[(0, ψk)], a short calculation gives φ = 0 (due to (N)) and

    ψ = b k ( 0 ) Ψ ν [ u 0 2 ψ ] + ( α ( ν ) ( ψ ) + σ β ( ν ) ( ψ ) ) Ψ ~ ν Ψ ν [ u 0 2 ψ k ] .

    Applying the functional α(ν) to this identity, we infer β(ν)(ψ) = 0. Moreover, Proposition 4 (c) gives ψ 2 r ψ ν ψ = b k ( 0 ) u 0 2 ψ u 0 2 ψ k . Further, by Step 1, β(ν)(ψk) = 0 as well as ψ k 2 r ψ k ν ψ k = b k ( 0 ) u 0 2 ψ k . Using these differential equations, one finds

    ( r 2 ( ψ k ψ ψ ψ k ) ) = r 2 u 0 2 ( r ) ψ k 2 for  r > 0.

    Integrating by parts and exploiting the asymptotic behavior of ψ resp. ψk and their derivatives, see Proposition 4 (d) and equation (13), this finally implies

    0 R r 2 u 0 2 ( r ) ψ k 2 ( r ) d r = R 2 ψ k ( R ) ψ ( R ) ψ ( R ) ψ k ( R ) = O 1 R ,

    which is a contradiction as R → ∞; hence transversality holds.

  3. Step 3

    Global Bifurcation.

    We apply Rabinowitz’ Global Bifurcation Theorem from [13], Theorem II.3.3, which as above yields unbounded connected components 𝓒k𝓢 once we show that the index

    ind X 1 ( I K b , 0 ) where  K b := b Ψ ν [ u 0 2 ] + ( α ( ν ) + σ β ( ν ) ) Ψ ~ ν

    changes sign at b = bk(0), k ∈ ℤ. More precisely, we analyze bifurcation at bk(0) ≥ 0 using the map G+ and at bk(0) < 0 using G. In the following, we present the main ideas how to verify that 1 is an algebraically simple eigenvalue of Kbk(0) and that the corresponding perturbed eigenvalue λb ≈ 1 of Kb for bbk(0) crosses 1 as b crosses bk(0). For the existence, algebraic simplicity and continuous dependence of λb on b we refer to Kielhöfer’s book [13], p. 203.

Algebraic Simplicity

We adapt the proof of Proposition 9 to the case ω = 0. Assuming ker (IKbk(0)) = span {w} and v ∈ ker (IKbk(0))2 ∖ ker (IKbk(0)), we have w.l.o.g.

w = K b k ( 0 ) w and v = K b k ( 0 ) ( v + w ) . (14)

Then, Proposition 4 (c) implies that the profiles satisfy

w 2 r w ν w = b k ( 0 ) u 0 2 w , v 2 r v ν v = b k ( 0 ) u 0 2 ( v + w ) on  R 3 . (15)

We let q(r) := r2 (w(r)v′(r) – v(r) w′(r)) for r ≥ 0. Using (15), we find

q ( r ) = r 2 b k ( 0 ) u 0 2 ( r ) w 2 ( r ) ( r > 0 ) , q ( 0 ) = 0

hence q is nondecreasing if bk(0) ≤ 0 and nonincreasing if bk(0) ≥ 0. On the other hand, applying α(ν) to (14), we infer β(ν)(w) = 0 and β(ν)(v) = –σα(ν)(w). Then the asymptotic expansions of v, w due to equation (14) and Proposition 4 (d) imply as r → ∞

q ( r ) = σ α ( ν ) ( w ) 2 ( 4 π ) 2 ν + O 1 r .

Since α(ν)(w) ≠ 0 by Proposition 6, and since we choose σ = +1 for bk(0) ≥ 0 and σ = –1 for bk(0) < 0, this contradicts the monotonicity of q. Hence ker (IKbk(0)) = ker (IKbk(0))2.

Perturbation of the eigenvalue

Throughout the following lines, we consider a perturbed value bbk(0), bbk(0) and the corresponding eigenpair with Kb wb = λb wb. The latter implies

( λ b 1 ) α ( ν ) ( w b ) = σ β ( ν ) ( w b ) (16)

and hence λb ≠ 1 due to β(ν)(wb) ≠ 0, see Proposition 8. We recall that

ω ν ( b k ( 0 ) u 0 2 ) π Z and α ( ν ) ( w b ) β ( ν ) ( w b ) = cot ( ω ν ( b λ b 1 u 0 2 ) ) ( b b k ( 0 ) , b b k ( 0 ) ) (17)

where the second identity can be deduced comparing the expansions in equation (13) and in Corollary 5 resp. Proposition 6.

We now discuss the values bk(0) ≥ 0, i.e. σ = +1. In case b > bk(0) we show that λb > 1. Assuming λb < 1, we infer from (16) that sgn α(ν)(wb) ≠ sgn β(ν)(wb) and thus ω ν ( b λ b 1 u 0 2 ) π 2 , 0 + π Z due to (17). But since b λ b 1 > bk(0), the monotonicity stated in Proposition 8 implies ω ν ( b λ b 1 u 0 2 ) ω ν ( b k ( 0 ) u 0 2 ) + 0 , π 2 0 , π 2 + π Z , a contradiction. In the same way, for b < bk(0), we can show that λb < 1. Following the same strategy, we see for bk(0) < 0, i.e. σ = –1, that b > bk(0) implies λb < 1 and b < bk(0) implies λb > 1.

We have thus proved that, as b crosses bk(0), the perturbed eigenvalue λb crosses λbk(0) = 1 and hence the sign of the Leray-Schauder index indX1×X1 (Gσ(⋅, b), (0, 0) ) changes at b = bk(0) for all k ∈ ℤ and for σ ∈ {±1} chosen as above.

The case τ = 0

This is covered by redefining the first components of F resp. Gσ,

( w , v , b ) w Ψ μ [ w 3 + 3 u 0 w 2 + 3 u 0 2 w + b ( u 0 + w ) v 2 ] α ( μ ) ( w ) + β ( μ ) ( w ) Ψ ~ μ

instead of ( w , v , b ) w R μ τ ( w 3 + 3 u 0 w 2 + 3 u 0 2 w + b ( u 0 + w ) v 2 ) similar to the modification of the second component in the case ω = 0. The argumentation works as before, which is why we omit the details.□

Proof of Remark 2

  1. The Steps 1 of the proof above in fact show that solutions of (H), (A) bifurcate from (u0, 0, b) ∈ 𝓣u0 only if b = bk(ω) for k ∈ ℤ; Steps 2 show that this is also sufficient. Moreover, since bk(ω) is the unique solution of ων(b u 0 2 ) = ω + , its value does not change when choosing another asymptotic parameter τ in (A).

  2. By Proposition 8, the map q : ℝ → ℝ, q(b) := ων(b u 0 2 ) is strictly increasing and onto. Having chosen bk(ω) = q–1(ω + ) for ω ∈ [0, π), k ∈ ℤ, we infer strict monotonicity and surjectivity of the map ℝ → ℝ, ω + bk(ω).

  3. In Steps 2 we have seen that in a neighborhood of the bifurcation point (u0, 0, bk(ω)), the continuum 𝓒k contains only fully nontrivial solutions apart from (u0, 0, bk(ω)) itself. In Step 3, we infer for all (u, v, b) ∈ 𝓒k from this neighborhood that the asymptotic phase of v satisfies ων(v2 + bu2) = ω + k π. More generally, ων(v2 + bu2) = ω + k π holds on every connected subset of 𝓒k containing (u0, 0, bk(ω)) but no other semitrivial solution with v = 0.

  4. By Proposition 6, the (formally derived) initial value problem has a unique radial solution with c = ϱμ(3 u 0 2 ) ≠ 0 and τ0 = ωμ(3 u 0 2 ) ∈ [0, π).□

4 Proof of Theorem 3

We now prove the occurence of bifurcations from the diagonal solution family 𝔗u0. To this end we first rewrite the system (H) in an equivalent but more convenient way. Looking for solutions (u, v, b) ∈ X1 × X1 × ℝ ∖ 𝔗u0, we introduce the functions w1, w2X1 via

u =: u b + w 1 w 2 , v =: u b + w 1 + w 2 .

A few computations then yield that bifurcation at the point (ub, ub, b) occurs if and only if we have bifurcation from the trivial solution of the nonlinear Helmholtz system

Δ w 1 μ w 1 = ( 1 + b ) ( ( w 1 + u b ) 3 u b 3 ) + ( 3 b ) ( w 1 + u b ) w 2 2 on  R 3 , Δ w 2 μ w 2 = ( 1 + b ) w 2 3 + ( 3 b ) ( w 1 + u b ) 2 w 2 on  R 3 , (18)

and the asymptotic conditions (Adiag) are equivalent to

w 1 ( x ) = c 1 sin ( | x | μ + τ ) | x | + O 1 | x | 2 , w 2 ( x ) = c 2 sin ( | x | μ + ω ) | x | + O 1 | x | 2 (19)

as |x| → ∞ for some c1, c2 ∈ ℝ. As in the proof of Theorem 1, the functional analytical setting in the special cases ω = 0 or τ = 0 is different from the general one since a substitute for the operators R μ τ , R μ ω has to be found, see the definition of Gσ in the proof of Theorem 1. In order to keep the presentation short we only discuss the case τ, ω ∈ (0, π) and refer to the proof of Theorem 1 for the modifications in the remaining cases. So we introduce the map F : X1 × X1 × (–1, ∞) → X1 × X1 via

F ( w 1 , w 2 , b ) := w 1 w 2 R μ τ ( ( 1 + b ) ( ( w 1 + u b ) 3 u b 3 ) + ( 3 b ) ( w 1 + u b ) w 2 2 ) R μ ω ( ( 1 + b ) w 2 3 + ( 3 b ) ( w 1 + u b ) 2 w 2 ) .

Then F(0, 0, b) = 0 for b > –1, F(⋅, b) is a compact perturbation of the identity on X1 × X1 and it remains to find bifurcation points for F(w1, w2, b) = 0. First we identify candidates for bifurcation points, i.e. those b ∈ (–1, ∞) where ker DF(0, 0, b) is nontrivial. Using

D F ( 0 , 0 , b ) [ ( ϕ 1 , ϕ 2 ) ] = ϕ 1 ϕ 2 R μ τ ( 3 ( 1 + b ) u b 2 ϕ 1 ) R μ ω ( ( 3 b ) u b 2 ϕ 2 ) = ϕ 1 ϕ 2 3 R μ τ ϕ 1 3 b 1 + b R μ ω ϕ 2 ,

we get that nontrivial kernels occur exactly if 3 b 1 + b = bk(ω) for some k ∈ ℤ, cf. Steps 1 in the previous proof. For the analogous result in the Schrödinger case, see Lemma 3.1 [11]. So

ker D F ( 0 , 0 , b ) = span  { 0 ψ k } provided  b = 3 b k ( ω ) 1 + b k ( ω ) > 1

for some ψkX1 ∖ {0}. Using the algebraic simplicity of ψk proved in Proposition 9 we infer exactly as in the proof of Theorem 1 that the transversality condition holds and that the Leray-Schauder index changes at the bifurcation point. So, choosing b k ( ω ) := 3 b k ( ω ) 1 + b k ( ω ) for all k ∈ ℤ with bk(ω) > –1, the Crandall-Rabinowitz Theorem and Rabinowitz’ Global Bifurcation Theorem yield statements (ii) and (i) of the Theorem, respectively. We remark that, to be consistent with the labeling in the Theorem, we might have to shift the index in such way that b0(ω) ≤ –1 < b1(ω).

Unboundedness of the components can also be deduced as before. Indeed, assuming that ℭk is bounded, it returns to 𝔗u0 at some point (ub, ub, b) ≠ (u𝔟k(ω), u𝔟k(ω), 𝔟k(ω)) by Rabinowitz’ Theorem. We then infer that the phase ωμ((1 + b) w 2 2 + (3 – b)(w1 + ub)2) cannot be constant along ℭk. Due to Proposition 6 applied to w2 in (18), this requires the existence of some element (u, v, b) ∈ ℭk with w 2 = 1 2 ( v u ) = 0 , and hence the associated unbounded diagonal family belongs to ℭk.□

5 Proofs of the Results in Section 2

Before proving Proposition 4, we state two auxiliary results. The first one provides a formula for the Fourier transform of radially symmetric functions, see e.g. [14], p. 430.

Lemma 11

For fX3 and x ∈ ℝ3 ∖ {0}, we have

f ^ ( x ) = 2 π 0 f ( r ) sin ( | x | r ) | x | r r 2 d r .

We denote by Φ λ ( x ) := e i λ | x | 4 π | x | = ( 5 ) Ψ λ ( x ) + i Ψ ~ λ ( x ) for λ > 0, x ≠ 0 a (complex) fundamental solution of the Helmholtz equation –Δϕλϕ = 0 on ℝ3. A short calculation using spherical coordinates provides the following pointwise formula for convolutions with kernel Φλ.

Lemma 12

For fX3 and x ∈ ℝ3 ∖ {0}, we have

( Φ λ f ) ( x ) = e i λ | x | | x | 0 | x | sin ( λ r ) λ r f ( r ) r 2 d r + sin ( λ | x | ) | x | | x | e i λ r λ r f ( r ) r 2 d r = π 2 f ^ ( λ ) e i λ | x | | x | + | x | f ( r ) e i λ r sin ( λ | x | ) e i λ | x | sin ( λ r ) λ r | x | r 2 d r .

5.1 Proof of Proposition 4

We now prove, one by one, the assertions of Proposition 4 for convolutions with Φλ in place of αΨλ + α̃Ψ̃λ. The latter (real-valued) case can be deduced from the former via Φλ = Ψλ + i Ψ̃λ. Unless stated otherwise, we extend norms defined on spaces of real-valued functions to complex-valued functions g : ℝ3 → ℂ by considering the respective norm of |g| : ℝ3 → ℝ. (a) is a consequence of Theorem 2.1 in [3]. The solution properties stated in (c) can be verified by direct computation.

  1. Step 1

    Proof of (b), first part. Continuity.

    Due to the continuous embedding X3 L r a d 4 3 (ℝ3), see (6), the convolution is well-defined for fX3. Using Young’s convolution inequality, we get

    | ( Φ λ f ) ( x ) | 1 B 1 ( 0 ) Φ λ f L ( R 3 ) + 1 R 3 B 1 ( 0 ) Φ λ f L ( R 3 ) 1 B 1 ( 0 ) Φ λ L 1 ( R 3 ) f L ( R 3 ) + 1 R 3 B 1 ( 0 ) Φ λ L 4 ( R 3 ) f L 4 3 ( R 3 ) f L ( R 3 ) B 1 ( 0 ) d y 4 π | y | + f L 4 3 ( R 3 ) R 3 B 1 ( 0 ) d y ( 4 π | y | ) 4 1 4 ( 6 ) C 1 f X 3

    for some C1 ≥ 0. Next, by means of Lemma 12, we estimate for x ∈ ℝ3 ∖ {0}

    | | x | ( Φ λ f ) ( x ) | = e i λ | x | 0 | x | sin ( λ r ) λ r f ( r ) r 2 d r + sin ( λ | x | ) | x | e i λ r λ r f ( r ) r 2 d r 0 | x | 1 λ r f X 3 ( 1 + r 2 ) 3 2 r 2 d r + | x | 1 λ r f X 3 ( 1 + r 2 ) 3 2 r 2 d r = C 2 f X 3

    with some C2 ≥ 0. Combining both estimates, we have ∥ΦλfX1 ≤ [C1 + C2] ⋅ ∥fX3.

  2. Step 2

    Proof of (d). Asymptotics of w and w.

    Given fX3, we let w := Φλf. Then for r = |x| > 0, Lemma 12 implies

    w ( r ) π 2 f ^ ( λ ) e i λ r r = r f ( s ) e i λ s sin ( λ r ) e i λ r sin ( λ s ) λ s r s 2 d s r f X 3 ( 1 + s 2 ) 3 2 2 λ r s d s = f X 3 2 λ r 2 . (20)

    To understand the asymptotic behavior of the radial derivative w′, a short calculation shows that the auxiliary function δ ( r ) := r w ( r ) π 2 f ^ ( λ ) e i λ r satisfies

    δ ( r ) = O 1 r , δ ( r ) = λ δ ( r ) r f ( r ) = O 1 r as  r .

    Then for r > 0, we find τr ∈ (0, 1) with δ ( r + 1 ) = δ ( r ) + δ ( r ) + 1 2 δ ( r + τ r ) , whence also δ ( r ) = O 1 r . This shows the asserted properties of w′ since

    r w ( r ) = i λ π 2 f ^ ( λ ) e i λ r w ( r ) + O 1 r as  r .

    As a consequence of equation (20), we derive the formula stated for Ψ̃λf. Due to (c), Ψ̃λf is a radial solution of the homogeneous Helmholtz equation –Δwλw = 0 on ℝ3 and hence a scalar multiple of Ψ̃λ itself. The asymptotics in (20) justify the asserted constant.

  3. Step 3

    Proof of (b), second part. Compactness.

    We consider a bounded sequence (fn)n in the space X3 and aim to prove convergence of a subsequence of (un)n, un := Φλfn, in the space X1. First, due to the continuous embeddings into reflexive Lp spaces stated in (6), we can pass to a subsequence with

    f n k f  weakly in  L 4 ( R 3 ) L 4 3 ( R 3 ) , u n k u  weakly in  L 4 ( R 3 )

    for some f L rad 4 ( R 3 ) L rad 4 3 ( R 3 ) , u L rad 4 ( R 3 ) . Then the regularity properties in Proposition A.1 in [3] and the Rellich-Kondrachov Embedding Theorem 6.3 in [15] allow to extract a subsequence with unku strongly in C loc 1 (ℝ3), in particular for any R > 0

    1 B R ( 0 ) u n k 1 B R ( 0 ) u n l X 1 0 as  k , l . (21)

    On the other hand, on the unbounded set ℝ3BR(0), convergence in X1 follows essentially from the asymptotic expansion in equation (20) where w.l.o.g. f ^ n k ( λ ) f ^ ( λ ) as k → ∞. Then,

    1 R 3 B R ( 0 ) u n k 1 R 3 B R ( 0 ) u n l X 1 ( 20 ) sup | x | R π 2 | f ^ n k ( λ ) f ^ n l ( λ ) | ( 1 + | x | 2 ) 1 2 | x | + 2 ( 1 + | x | 2 ) 1 2 λ | x | 2 f n k f n l X 3 π | f ^ n k ( λ ) f ^ n l ( λ ) | + 1 R 4 2 λ sup n N f n X 3 . (22)

    Thus given ε > 0, we can choose R(ε) > 0 large enough and k(ε) ∈ ℕ such that (21) and (22) imply ∥unkunlX1 < ε for all k, lk(ε). Hence (unk)k∈ℕ is a Cauchy sequence in X1.□

5.2 Proof of Proposition 6

Let gX2. Then the profile w : [0, ∞) → ℝ is a (global) solution of the initial value problem (9) if and only if y : [0, ∞) → ℝ, y(r) = rw(r) solves

y λ y = g ( r ) y on  ( 0 , ) , y ( 0 ) = 0 , y ( 0 ) = 1. (23)

Moreover, wX1 if y is bounded. Global existence and uniqueness of such yC2([0, ∞)) are consequences of the Picard-Lindelöf Theorem and of Gronwall’s Lemma since gL1([0, ∞)). We apply the Prüfer transformation. Since y ≢ 0, uniqueness implies that y(r)2 + y′(r)2 > 0 for all r ≥ 0. We thus parametrize using polar coordinates in the phase space

y ( r ) = ϱ ( r ) sin ( ϕ ( r ) λ ) , y ( r ) = ϱ ( r ) λ cos ( ϕ ( r ) λ ) ( r 0 ) (24)

with functions ϱ: [0, ∞) → (0, ∞) and ϕ : [0, ∞) → ℝ. A short calculation shows that we thus obtain a solution of (23) if and only if ϱ and ϕ satisfy the first-order system

( log ϱ ) = g ( r ) 2 λ sin ( 2 ϕ λ ) on  ( 0 , ) , ϕ = 1 + g ( r ) λ sin 2 ( ϕ λ ) on  ( 0 , ) , ϱ ( 0 ) = 1 λ , ϕ ( 0 ) = 0. (25)

Equivalently, for r ≥ 0,

ϱ ( r ) = 1 λ exp 0 r g ( t ) 2 λ sin ( 2 ϕ ( t ) λ ) d t , ϕ ( r ) = r + 0 r g ( t ) λ sin 2 ( ϕ ( t ) λ ) d t . (26)

We will frequently refer to the estimate

r 0 1 λ exp 0 r g ( t ) 2 λ sin ( 2 ϕ ( t ) λ ) d t 1 λ exp π 4 λ g X 2 =: C g . (27)

Indeed, (26) and (27) immediately yield boundedness of y and moreover convergence of the integrals in

ω λ ( g ) := 0 g ( t ) λ sin 2 ( ϕ ( t ) λ ) d t and ϱ λ ( g ) := 1 λ exp 0 g ( t ) 2 λ sin ( 2 ϕ ( t ) λ ) d t .

Thus ϱλ(g) > 0, and we verify the asserted asymptotic behavior of y as r → ∞:

y ( r ) ϱ λ ( g ) sin ( r λ + ω λ ( g ) ) = ( 24 ) ϱ ( r ) sin ( ϕ ( r ) λ ) ϱ λ ( g ) sin ( r λ + ω λ ( g ) ) ϱ ( r ) ϱ λ ( g ) sin ( ϕ ( r ) λ ) + ϱ λ ( g ) sin ( ϕ ( r ) λ ) sin ( r λ + ω λ ( g ) ) ( 27 ) C g ϱ ( r ) ϱ λ ( g ) 1 + sin ( ϕ ( r ) λ ) sin ( r λ + ω λ ( g ) )

where we estimate both terms as follows, using |g(r)| ≤ ∥gX2 ⋅ (1 + r2)–1 for r > 0,

ϱ ( r ) ϱ λ ( g ) 1 + sin ( ϕ ( r ) λ ) sin ( r λ + ω λ ( g ) ) ( 26 ) exp r g ( t ) 2 λ sin ( 2 ϕ ( t ) λ ) d t 1 + ϕ ( r ) λ r λ ω λ ( g ) ( 26 ) , ( 27 ) C g r g ( t ) 2 λ sin ( 2 ϕ ( t ) λ ) d t + r g ( t ) λ sin 2 ( ϕ ( t ) λ ) d t C g 2 + 1 g X 2 λ 1 r .

Thus y ( r ) ϱ λ ( g ) sin ( r λ + ω λ ( g ) ) = O 1 r , and similarly y ( r ) ϱ λ ( g ) λ cos ( r λ + ω λ ( g ) ) = O 1 r as r → ∞. Since y(r) = rw(r), all assertions are proved.□

5.3 Proof of Proposition 7

We consider gn, g0 in X2 with gng0 in X2 and aim to show that ωλ(gn) → ωλ(g0). By ϕnC1((0, ∞)) ∩ C([0, ∞)) we denote the unique solution of

ϕ n = 1 + g n ( r ) λ sin 2 ( ϕ n λ ) , ϕ n ( 0 ) = 0.

Then we have pointwise convergence, ϕn(r) → ϕ0(r) for all r ≥ 0. Indeed, let us fix any R > 0 and estimate for 0 ≤ rR and n ∈ ℕ

ϕ n ( r ) ϕ 0 ( r ) = 0 r g n ( t ) λ sin 2 ( ϕ n ( t ) λ ) g 0 ( t ) λ sin 2 ( ϕ 0 ( t ) λ ) d t 1 λ 0 r g n ( t ) g 0 ( t ) d t + 1 λ 0 r g 0 ( t ) sin 2 ( ϕ n ( t ) λ ) sin 2 ( ϕ 0 ( t ) λ ) d t 1 λ 0 g n g 0 X 2 d t 1 + t 2 + 2 g 0 λ 0 r ϕ n ( t ) ϕ 0 ( t ) d t π 2 λ g n g 0 X 2 + 2 g 0 λ 0 r ϕ n ( t ) ϕ 0 ( t ) d t .

Thus, by Gronwall’s Lemma, we have for 0 ≤ rR

ϕ n ( r ) ϕ 0 ( r ) π 2 λ g n g 0 X 2 e 2 g 0 λ r π 2 λ g n g 0 X 2 e 2 g 0 λ R .

Since gng0 in X2, we conclude ϕnϕ0 locally uniformly on [0, ∞), in particular pointwise. Now we can deduce the convergence of the asymptotic phase,

ω λ ( g n ) = 1 λ 0 g n ( r ) sin 2 ( ϕ n ( r ) λ ) d r 1 λ 0 g 0 ( r ) sin 2 ( ϕ 0 ( r ) λ ) d r = ω λ ( g 0 ) ,

which follows by dominated convergence since supn∈ℕgnX2 < ∞.□

5.4 Proof of Proposition 8

Let us first recall that, given the assumptions of Proposition 8, equation (10) implies for b ∈ ℝ

ω λ ( b u 0 2 ) = b λ 0 u 0 2 ( r ) sin 2 ( ϕ b ( r ) λ ) d r

where ϕb satisfies ϕ b = 1 + b λ u 0 2 ( r ) sin 2 ( ϕ b λ ) on ( 0 , ) , ϕ b ( 0 ) = 0. We immediately see that ωλ(0) = 0 and sgn ωλ(b u 0 2 ) = sgn(b) for all b ∈ ℝ ∖ {0}. Further, continuity of bωλ(b u 0 2 ) is a consequence of Proposition 7. The assertions are proved once we show that bωλ(b u 0 2 ) is strictly increasing with ωλ(b u 0 2 ) → ±∞ as b → ±∞.

  1. Step 1

    Strict monotonicity. We let b1 < b2, define

    χ ( r ) := sin 2 ( ϕ b 2 ( r ) λ ) sin 2 ( ϕ b 1 ( r ) λ ) ϕ b 2 ( r ) λ ϕ b 1 ( r ) λ if  ϕ b 2 ( r ) ϕ b 1 ( r ) , 2 sin ( ϕ b 1 ( r ) λ ) cos ( ϕ b 1 ( r ) λ ) else

    and observe that χ is bounded with 0 ≤ |χ(r)| ≤ 2 and continuous. ψ := ϕb2ϕb1 satisfies

    ψ = b 2 b 1 λ u 0 2 ( r ) sin 2 ( ϕ b 2 ( r ) λ ) + b 1 λ u 0 2 ( r ) χ ( r ) ψ , ψ ( 0 ) = 0.

    The unique solution is given by the Variation of Constants formula. We have

    ω λ ( b 2 u 0 2 ) ω λ ( b 1 u 0 2 ) = λ lim r ψ ( r ) = 0 b 2 b 1 λ u 0 2 ( ϱ ) sin 2 ( ϕ b 2 ( ϱ ) λ ) e ϱ b 1 λ u 0 2 ( t ) χ ( t ) d t d ϱ > 0

    since the integrand is nonnegative and not identically zero.

  2. Step 2

    Asymptotic behavior as b → ∞.

    By the uniqueness statement of the Picard-Lindelöf Theorem, u0 ≢ 0 requires u0(0) ≠ 0. We can thus choose r0 > 0 with

    1 2 u 0 2 ( 0 ) < u 0 2 ( r ) < 3 2 u 0 2 ( 0 ) for all  r [ 0 , r 0 ] . (28)

    To keep notation short, we let in this paragraph ξ := 1 2 λ u 0 2 ( 0 ) . We have for b > 0

    ϕ b = 1 + b λ u 0 2 ( r ) sin 2 ( ϕ b λ ) ( 28 ) 1 + b ξ sin 2 ( ϕ b λ )  on  [ 0 , r 0 ] , ϕ b ( 0 ) = 0.

    We now study the modified initial value problem

    ψ b = 1 + b ξ sin 2 ( ψ b λ )  on  [ 0 , r 0 ] , ψ b ( 0 ) = 0.

    For 0 ≤ rr0 with r π 2 + π Z , its unique solution is given by the expression

    ψ b ( r ) = 1 λ n π + arctan tan r λ 1 + b ξ 1 + b ξ for  1 + b ξ λ r n π < π 2

    where n ∈ ℕ0. We deduce immediately ψb(r0) → ∞ as b → ∞. Since by construction ϕbψb on [0, r0], this implies ϕb(r0) → ∞ as b → ∞. Substituting t := λ ϕ b ( r ) , we estimate

    ω λ ( b u 0 2 ) = b λ 0 u 0 2 ( r ) sin 2 ( ϕ b ( r ) λ ) d r ( 28 ) b ξ λ 0 r 0 sin 2 ( ϕ b ( r ) λ ) d r = b ξ 0 λ ϕ b ( r 0 ) sin 2 ( t ) d t ϕ b ( ϕ b 1 ( λ 1 2 t ) ) ( 28 ) b ξ 1 + 3 b ξ 0 λ ϕ b ( r 0 ) sin 2 ( t ) d t

    and hence ϕb(r0) → ∞ implies ωλ(b u 0 2 ) → ∞ as b → ∞.

  3. Step 3

    Asymptotic behavior as b → –∞.

    For b < –1, we introduce

    r b := max r > 0 | ϕ b ( r ) λ = arcsin ( | b | 1 4 ) ,

    which is well-defined due to 1 | b | λ u 0 X 1 2 1 + r 2 ϕ b 1 and ϕb(0) = 0. In particular, we have

    ϕ b ( r b ) λ = arcsin ( | b | 1 4 ) and ϕ b ( r ) λ > arcsin ( | b | 1 4 ) for all  r > r b . (29)

    We prove below that rb → ∞ as b → –∞. Then for rrb, equation (29) and ϕ b ≤ 1 imply

    ϕ b ( r ) λ ϕ b ( r b ) λ + ( r r b ) λ = r λ + arcsin ( | b | 1 4 ) r b λ .

    Then the asymptotic phase satisfies

    ω λ ( b u 0 2 ) = λ lim r ( ϕ b ( r ) r ) arcsin ( | b | 1 4 ) r b λ as  b .

    It remains to prove that rb → ∞ as b → –∞. We assume for contradiction that we find a subsequence (bk)k∈ℕ and > 0 with bk ↘ –∞, rbk as k → ∞. Then, since ϕ b k ≤ 1 and due to equation (29), we have for sufficiently large k ∈ ℕ

    arcsin ( | b k | 1 4 ) ϕ b k ( r ) λ arcsin ( | b k | 1 4 ) + 1 < π 2 for  r b k r r b k + 1 λ . (30)

    We conclude sin sin ( ϕ b k ( r ) λ ) | b k | 1 4 and hence as k → ∞

    ϕ b k r b k + 1 λ = ϕ b k ( r b k ) + 0 1 λ ϕ b k ( r b k + t ) d t = ( 29 ) 1 λ arcsin ( | b k | 1 4 ) + 0 1 λ 1 | b k | λ u 0 2 ( r b k + t ) sin 2 ( ϕ b k ( r b k + t ) λ ) d t ( 30 ) 2 λ + 1 λ | b k | λ 0 1 λ u 0 2 ( r b k + t ) d t

    since u 0 2 > 0 almost everywhere and rbk. On the other hand, for every k ∈ ℕ, the differential equation ϕ = 1 + b k λ u 0 2 ( r ) sin 2 ( ϕ λ ) states that ϕbk(r) = 0 implies ϕ b k (r) = 1. Thus ϕbk cannot attain negative values, which contradicts the limit calculated before.□

5.5 Proof of Proposition 9

For ω ∈ (0, π) and λ > 0, we compute the spectrum of

R λ ω : X 1 X 1 , w R λ ω [ u 0 2 w ] = Ψ λ + cot ( ω ) Ψ ~ λ [ u 0 2 w ] .

Compactness of R λ ω is a consequence of Proposition 4 (b). Then immediately σ ( R λ ω ) = { 0 } σ p ( R λ ω ) with discrete eigenvalues of finite multiplicity.

  1. Step 1

    Eigenvalues.

    We find the eigenfunctions of R λ ω , that is, we look for such η ∈ ℝ, η ≠ 0 and nontrivial wX1 that R λ ω w = ηw. Corollary 5 implies that this is equivalent to η ∈ ℝ, η ≠ 0 and nontrivial wX1C2(ℝ3),

    Δ w λ w = 1 η u 0 2 ( x ) w on  R 3 with  w ( x ) = c sin ( | x | λ + ω ) | x | + O 1 | x | 2  as  | x |

    for some c ∈ ℝ. By Proposition 6, such an eigenfunction exists if and only if ω λ 1 η u 0 2 = ω + k π for some k ∈ ℤ; in this case, c ≠ 0 and every eigenspace is one-dimensional since the radially symmetric solution w is unique up to multiplication by a constant. Since we have seen in Proposition 8 that ℝ → ℝ, bωλ(b u 0 2 ) is strictly increasing and onto, we can define b k ( ω , λ , u 0 2 ) u 0 2 ) v i a ω λ ( b k ( ω , λ , u 0 2 ) u 0 2 ) = ω + k π for all k ∈ ℤ, and conclude

    σ p ( R λ ω ) = 1 b k ( ω , λ , u 0 2 ) | k Z .

  2. Step 2

    Simplicity.

    It remains to show that the eigenvalues are algebraically simple. We consider an eigenvalue η := 1 b k ( ω , λ , u 0 2 ) of R λ ω with eigenspace ker ( R λ ω ηIX1) = span {w}. We have to prove that

    ker R λ ω η I X 1 2 = ker R λ ω η I X 1 .

    So let now v ker R λ ω η I X 1 2 . We assume for contradiction that v ker R λ ω η I X 1 . By assumption on v, we have R λ ω v η v ker R λ ω η I X 1 { 0 } , and since η ≠ 0 we may assume w.l.o.g. R λ ω v η v = η w = R λ ω w . Then by Proposition 4 v, wC2(ℝ3) as well as

    w 2 r w λ w = 1 η u 0 2 ( r ) w , v 2 r v λ v = 1 η u 0 2 ( r ) ( v + w ) on  ( 0 , ) . (31)

    Furthermore, Proposition 4 (d) implies

    w ( r ) = c 1 sin ( r λ + ω ) r + O 1 r 2 , w ( r ) = c 1 λ cos ( r λ + ω ) r + O 1 r 2 , v ( r ) = c 2 sin ( r λ + ω ) r + O 1 r 2 , v ( r ) = c 2 λ cos ( r λ + ω ) r + O 1 r 2 (32)

    for some c1, c2 ∈ ℝ. Let us define q(r) = r2(w′(r)v(r) – v′(r) w(r)) for r ≥ 0. Then, using (31), we find q ( r ) = 1 η r 2 u 0 2 ( r ) w 2 ( r ) for r ≥ 0. Hence q is monotone on [0, ∞) with q(0) = 0. On the other hand, the asymptotic expansions in (32) imply q ( r ) = O 1 r as r → ∞. We conclude q(r) = 0 for r ≥ 0. Since all zeros of w are simple, one can deduce that v(r) = cw(r) for all r ≥ 0 and some c ∈ ℝ, and thus v ∈ ker( R λ ω ηIX1), a contradiction. □

Acknowledgements

The authors would like to express their gratitude to the reviewer for thorough and in-depth revision and various helpful remarks and suggestions. They also gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 ”Wave phenomena: analysis and numerics”. We acknowledge support by the KIT-Publication Fund of the Karlsruhe Institute of Technology.

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Received: 2019-01-21
Accepted: 2019-05-21
Published Online: 2019-10-19

© 2019 R. Mandel and D. Scheider, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.