2021 UNC Greensboro PDE Conference,

Electronic Journal of Differential Equations, Conference 26 (2022), pp. 151–169.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ON SOLUTIONS ARISING FROM RADIAL SPATIAL

DYNAMICS OF SOME SEMILINEAR ELLIPTIC EQUATIONS

DARÍO A. VALDEBENITO

Abstract. We consider the semilinear elliptic equation

∆u+ f(x, u) = 0,

where x ∈ RN \ {0}, N ≥ 2, and f satisfies certain smoothness and structural

assumptions. We construct solutions of the form u(r, φ) = r(2−N)/2ũ(log r, φ),
where r = |x| > 0, φ ∈ SN−1, and ũ is quasiperiodic in its first argument with

two nonresonant frequencies. These solutions are found using some recent

developments in the theory of spatial dynamics, in which the radial variable r
takes the role of time, combined with classical results from dynamical systems

and the KAM theory.

1. Introduction

We consider the semilinear elliptic equation

∆u+ f1(x, u) = 0, x ∈ RN \ {0}, (1.1)

where N is a positive integer, ∆ is the Laplace operator in x, and f1 : (RN \ {0})×
R→ R is a sufficiently smooth function satisfying f1(·, 0) ≡ 0.

The study of geometrical properties of solutions of semilinear elliptic equations
on the entire space RN has been extensive. For instance, if one considers solutions
decaying in all variables (also known as fully localized solutions), together with some
assumptions on the nonlinearity, the classical result of Gidas, Ni, and Nirenberg [20]
yields that all fully localized solutions are radially symmetric around some point
in RN . On the other extreme, if no decay conditions are imposed, then a variety
of solutions have been found, especially in the case of homogeneous problems (i.e.,
f1 = f1(u)). Just to give some examples we point to multi-bump solutions decaying
along all but finitely many rays [30], saddle shaped solutions and general multiple-
end solutions [17, 18, 28], as well as solutions having both fronts (transitions) and
bumps [44].

Equation (1.1) is defined on a punctured domain. Such equations, along with
equations on exterior domains, have been extensively studied as well. We mention
only a few problems in this field: non-radial singular solutions to the Lane-Emden

2020 Mathematics Subject Classification. 35B08, 35B15, 35J61, 37J40.
Key words and phrases. Semilinear elliptic equations; quasiperiodic solutions;

center manifold theorem; radial spatial dynamics.
©2022 This work is licensed under a CC BY 4.0 license.

Published August 25, 2022.

151



152 D. VALDEBENITO EJDE-2022/CONF/26

equation [14, 15], finite energy solutions in an exterior domain [6], equations involv-
ing supercritical exponents [13, 16], a priori estimates for solutions of superlinear
elliptic equations and systems [37], a problem involving a singular nonlinearity [23],
and the study of anisotropic singularities for a power nonlinearity [10]. Among the
(very incomplete) list of references provided, [10, 14, 23] are of special relevance to
us: their constructions are based on solving elliptic equations on spheres which are
then used to obtain solutions on the punctured space. Our approach to construct
solutions of (1.1) will be to some extent similar.

Among the wide variety of solutions of semilinear elliptic equations, one finds
quasiperiodic solutions, which will be the focus of our attention in this paper. In
previous articles [38, 40, 41], Poláčik and the author have studied the existence of
solutions to some semilinear elliptic equations on the entire space with the follow-
ing property: writing x = (x′, xN ) ∈ RN−1 × R, the solutions constructed decay
to 0 as |x′| → ∞ uniformly in xN , and are quasiperiodic (and not periodic) in xN .
Such solutions were found using a spatial dynamics approach to elliptic equations
and results from the Kolmogorov-Arnold-Moser (KAM) theory [2, 27, 34]. Previ-
ously, related ideas for finding quasiperiodic solutions of elliptic equations on an
unbounded strip have been used by Scheurle [46] and Valls [50] (see [38] for a more
detailed discussion and further related references).

The main contribution of [38] is the outlining of a general scheme to find quasiperi-
odic solutions which, in principle, could be applied in other settings, yielding differ-
ent conditions that may imply the existence of the desired quasiperiodic solutions.
For instance, in [41] a different type of KAM theorem permitted the application of
the general strategy from [38] to construct quasiperiodic solutions in such a way
that the cubic terms (in u) of the nonlinear part of the equation are not involved in
the usual nondegeneracy conditions: the nonlinearity may even be purely quadratic
in some cases. (For another perspective on this issue and a KAM-type result for
the Boussinesq equation with a quadratic nonlinearity see [48].) In [40] it is shown
that the scheme can be applied to some homogeneous semilinear equations.

A common approach to spatial dynamics found in the literature applies to cylin-
drical domains of the form Ω×R, with Ω a domain in RN−1 which is often, but not
always, assumed to be bounded. The unbounded variable xN takes the role of time,
in the sense that the partial differential equation being considered is rewritten as
an abstract equation in terms of xN . In certain settings, such as elliptic problems,
the Cauchy problem for the abstract equation is ill posed, yet in many situations it
is still possible to find solutions. A number of authors have made contributions to
the spatial dynamics approach to study partial differential equations, for instance,
[9, 19, 22, 24, 26, 31, 32, 33, 35, 36, 51]. Several of the aforementioned works develop
and make use of center manifold theory to successfully employ spatial dynamics, but
other approaches can be found in the literature: just to give an example, we point
to the work of Chen, Matano, and Vénon [10], where a strongly order-preserving
semiflow is used to construct an entire orbit connecting two distinct solutions of a
certain equation on the circle, which in turn allows the authors to obtain a singular
solution of an equation of the form ∆u = |u|q−1u in R2 \ {0}, 1 < q < 3, and its
behavior near the origin and infinity is characterized in terms of the foregoing two
solutions connected by the entire orbit.

In this article our approach to spatial dynamics considers the use of the ra-
dial variable as the time-like variable, and the “cross-sections” are now concentric



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 153

spheres. Although the idea of using the radial variable to take the role of time is
not entirely new (see, e.g., [10, 29, 43, 45]), recently it has been explored in detail
in the context of elliptic PDE by Beck et al. in [4, 5]. An interesting property of
this approach is that the functional spaces involved consist of functions defined on
spheres (or “sphere-like” bounded manifolds), so the study of the resulting equa-
tions could potentially be simpler. On the other hand, the abstract equation will in
general depend on the time-like variable, which complicates its analysis even if one
can construct a suitable invariant manifold. Under some structural assumptions
and a suitable change of variables, the abstract equation does not depend on the
time-like variable, which allows one to employ standard center manifold results.
Although there are some center manifold results which may apply to more general
settings than the one we consider here (e.g., [11, 12]), and it is likely that they
could be used to construct new solutions, we do not make use of such results here:
we expect that applying KAM theory to the equations resulting from such center
manifold reductions would incur significant difficulties.

Among the challenges encountered when using a spatial dynamics approach to
construct quasiperiodic solutions, a particularly relevant one is the verification of
certain nondegeneracy conditions required to apply the KAM theory. In some
settings it is possible to formulate such conditions explicitly in terms of the functions
appearing in the original equation, but in general one often needs to restrict the
scope of the results, for instance by restricting the number of frequencies or requiring
the presence of a parameter in the equation, in order to obtain tangible hypotheses
that can be shown to apply for certain classes of equations.

The rest of this paper is organized as follows. In Section 2 we provide some
definitions and the statement of our main result, including the precise structure of
the sought-after solutions. In Section 3 we apply the spatial dynamics approach
to obtain a Hamiltonian structure for our equation, so that some previous results,
based on KAM-type theorems, can be applied to obtain the desired solutions in
Section 4.

2. Main result

In this section we introduce some terminology and provide the statement of our
main result. Afterwards, we give an outline of the proof.

Throughout the paper, C(X,Y ) denotes the class of continuous functions f :
X → Y . Given a positive integer k, Ck(X,Y ) denotes the class of functions f :
X → Y with continuous derivatives up to order k. Occasionally the spaces X and
Y will be omitted from the notation if they are clear from the context. We write
Ck(X) for Ck(X,R). We denote the unit sphere in RN by SN−1, and the space
Hk(SN−1) is the usual Sobolev space of square-integrable functions on SN−1 with
weak derivatives up to the kth order. When needed, all the aforementioned spaces
are equipped with the usual norms.

Given integers n ≥ 2, k ≥ 1, a vector ω = (ω1, . . . , ωn) ∈ Rn is said to be
nonresonant up to order k if

ω · α 6= 0 for all α ∈ Zn \ {0} such that |α| ≤ k. (2.1)

(Here |α| = |α1|+ · · ·+ |αn|, and ω · α is the usual dot product.) If (2.1) holds for
all k = 1, 2, . . . , we say that ω is nonresonant, or, equivalently, that the numbers
ω1, . . . , ωn are rationally independent.



154 D. VALDEBENITO EJDE-2022/CONF/26

A function v : (τ, φ) 7→ v(τ, φ) : R × SN−1 → R is said to be quasiperiodic in τ
if there exist an integer n ≥ 2, a nonresonant vector ω∗ = (ω∗1 , . . . , ω

∗
n) ∈ Rn, and

an injective function V defined on Tn (the n-dimensional torus) with values in the
space of real-valued functions on SN−1 such that

v(τ, φ) = V (ω∗1τ, . . . , ω
∗
nτ)(φ) (τ ∈ R, φ ∈ SN−1). (2.2)

The vector ω∗ is called a frequency vector of v.
A function u : RN \ {0} 7→ R is said to be log-radially quasiperiodic if there exist

a constant a and a quasiperiodic function v (as in (2.2)) such that

u(r, φ) = rav(log r, φ) (r > 0, φ ∈ SN−1). (2.3)

We also say that ω∗ is a frequency vector of u if ω∗ is a frequency vector of v in
the sense of the foregoing definition.

We emphasize that the nonresonance of the frequency vector is a part of our
definitions. In particular, a quasiperiodic function is not periodic and, if it has some
regularity properties, its image is dense in an n-dimensional manifold diffeomorphic
to Tn. As a consequence, a log-radially quasiperiodic function is also not periodic
in log r (even if a = 0).

We now make precise the equation we study in this article. Denoting by (r, φ) ∈
(0,∞)× SN−1 the spherical coordinates of x ∈ RN \ {0}, with r = |x|, we consider
the following elliptic equation:

∆u+ a1(φ; s)r−2u+ F (r, φ, u; s) = 0, x ∈ RN \ {0}, (2.4)

where ∆ is the Laplace operator in RN , N ≥ 2, s ≈ 0 is a parameter, and, setting

A := (N − 2)/2, (2.5)

F takes the form

F (r, φ, u; s) = r−(2+A)f(φ, rAu; s), (2.6)

for

f(φ, v; s) = a2(φ; s)v2 + v3g(φ, v; s). (2.7)

Next, we provide some assumptions on the functions involved in equations (2.4)
and (2.7). We assume that, for some δ > 0 and for some integers K, m such that

K ≥ 18, m >
N

2
, (2.8)

the functions a1, a2, and g satisfy the following hypotheses:

(A1) a1(·; s) ∈ Cm+1(SN−1) for each s ∈ (−δ, δ), and the map s ∈ (−δ, δ) 7→
a1(·; s) ∈ Cm+1(SN−1) is of class CK+1.

(A2) a2(·; s) ∈ Cm+1(SN−1) for each s ∈ (−δ, δ), the map s ∈ (−δ, δ) 7→ a2(·; s) ∈
Cm+1(SN−1) is of class CK+1; g ∈ CK+m+4(SN−1 × R × (−δ, δ)), and for
all χ > 0 the function g is bounded on SN−1× [−χ, χ]× [0, δ) together with
all its partial derivatives up to order K +m+ 4.

Denote by ∆SN−1 the spherical Laplace operator on SN−1. The next hypotheses
concern the Schrödinger operator A1(s) := −∆SN−1 − a1(φ; s), acting on L2(SN−1)
with domain H2(SN−1).

(A3) For all s ∈ [0, δ), A1(s) has exactly two eigenvalues in
(
−∞,−A2

]
. De-

noting these two eigenvalues µ1(s) < µ2(s), µ2(s) is simple, and one has
µ2(s) < −A2 for all s ∈ (0, δ) and µ2(0) = −A2.



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 155

(A4) Denoting

ϑj(s) :=
(√|µj(s)| − A√
|µj(s)|+A

)1/2
, (2.9)

j = 1, 2, the vector

ω̃(s) = (ω̃1(s), ω̃2(s)) :=
(
(
√
|µ1(s)|+A)ϑ1(s), (

√
|µ2(s)|+A)ϑ2(s)

)
is nonresonant up to order K for all s ∈ (0, δ).

Hypotheses (A3) and (A4) are assumed in our main theorem, but in some of our
results we consider more general versions of (A3) and (A4), namely:

(A3’) There is an integer n ≥ 2 such that for all s ∈ (0, δ), A1(s) has exactly
n eigenvalues in (−∞,−A2), namely, µ1(s) < µ2(s) < · · · < µn(s), all of
which are simple. In addition, if µn+1(s) is the (n + 1)-th eigenvalue of
A1(s), one has µn+1(s) > −A2 for all s ∈ [0, δ). (µn(0) = −A2 is not
required here.)

(A4’) With ϑj as in (2.9), where now j = 1, . . . , n, the vector ω̃(s) =
(
(
√
|µ1(s)|+

A)ϑ1(s), . . . , (
√
|µn(s)| + A)ϑn(s)

)
is nonresonant up to order K for all

s ∈ (0, δ), with K a positive integer satisfying

K ≥ 6(n+ 1). (2.10)

When hypotheses (A3’) and (A4’) are assumed in lieu of (A3) and (A4), the constant
K in (A1), (A2) is also assumed to satisfy (2.10). Note that if N = 2, then A = 0,

ϑ1(s) ≡ · · · ≡ ϑn(s) ≡ 1, and ω̃(s) =
(√
|µ1(s)|, . . . ,

√
|µn(s)|

)
.

For s ∈ [0, δ) and j = 1, . . . , n, we denote by ϕj(·; s) the eigenfunction of A1(s)
associated with µj(s), normalized in the L2-norm. This determines each ϕj uniquely
up to a sign. Making a choice of sign for each j, the map s ∈ [0, δ) 7→ ϕj ∈
H2(SN−1) is well defined and of class CK+1 [25]. Note that the exact choice of sign
is inconsequential for our purposes.

Our last hypothesis concerns the coefficient a2 and the eigenfunction ϕ2 when
s = 0:

(A5) One has ∫
SN−1

a2(φ; 0)ϕ3
2(φ; 0)dφ 6= 0.

Hypotheses (A1), (A2), (A3’), and (A4’) with m > N/2 and K ≥ 6(n + 1) are
assumed throughout the paper. In our main theorem and its proof (Section 4), we
take n = 2 and assume also that (A3) and (A5) hold.

Remark 2.1. (i) Since the eigenvalues of A1(s) are isolated in σ(A1(s)), hy-
potheses (A3) and (A3’) imply that there is γ > −A2 such that (−A2, γ)∩
σ(A1(s)) = ∅ for all s ∈ [0, δ). Note that the operator A1(s) acts on func-
tions defined on SN−1, so under our assumptions its spectrum consists only
of eigenvalues.

(ii) Hypothesis (A1) implies that the eigenvalues µ1(s), µ2(s) in (A3) (or µ1(s),
. . . , µn(s) in (A3’)) are functions of s of class CK+1 (see [25]). The sim-
plicity of a finite set of eigenvalues of the Schrödinger operator A1(s) is
a generic property (in a suitable sense) of the potential a1, see [1]. Note,
however, that the case of a1 being constant in φ must be excluded, since
the second eigenvalue of −∆SN−1 is a multiple eigenvalue.



156 D. VALDEBENITO EJDE-2022/CONF/26

(iii) Note that if f is sufficiently smooth, then (2.7) is just a Taylor expansion of
f around v = 0. The specific dependence on r in (2.6) is the most significant
restriction we impose on F , and it is necessary for the applicability of
standard center manifold results.

(iv) Condition (A4) holds automatically as long as δ > 0 is sufficiently small:
if µ2(s) is sufficiently close to −A2, then one has 0 < Kω̃2(s) < ω̃1(s) for
all s ∈ (0, δ), and (A4) can be easily verified using this fact. For (A4’),
being a finite-order nonresonance condition, one can combine ideas from
[1] with the scheme used in [39] to obtain that (A4’) holds generically with
respect to the potential a1. Condition (A5) is obviously satisfied for “most”
functions a2(·; 0).

(v) Our hypotheses are for the most part analogous to some hypotheses in
[38, 41]. This will allow us to use certain technical results from [38]. Hy-
pothesis (A5) is specific to our approach to verify a certain nondegeneracy
condition, in which we use Arnold’s condition. There are other conditions
used in KAM theory, such as Kolmogorov’s or Bruno’s conditions. In the
presence of parameters other conditions can be used, see, e.g., [8, 47]. In
principle any condition in a KAM-type theorem which permits the per-
turbed Hamiltonian to have only finite differentiability should suffice for
our purposes.

We can now state our main theorem.

Theorem 2.2. Suppose that hypotheses (A1)–(A5) with K, m as in (2.8) are
satisfied. Then the following statements are valid, possibly after making δ > 0
smaller, for each s ∈ (0, δ). There exists a solution u = u(r, φ) of equation (2.4)
such that u is log-radially quasiperiodic. In fact, there is an uncountable family of
such solutions, their frequency vectors forming an uncountable subset of R2.

Remark 2.3. (i) For technical reasons (the verification of a nondegeneracy
relation), in this theorem we need the parameter s > 0 to be sufficiently
small and the number of frequencies to be restricted to n = 2. Below,
we include a theorem – see Theorem 4.1 – where, assuming (A1), (A2),
(A3’), and (A4’), we give a different sufficient condition for the existence
of log-radially quasiperiodic solutions of (2.4) with any given number of
frequencies and for a fixed value of s. Unlike (A5), that condition is rather
implicit, and in general we are unable to formulate it as a specific condition
on a1, a2.

(ii) We have taken a1 and f (cf. (2.4) and (2.7), respectively) depending on
φ ∈ SN−1 for the sake of simplicity, but one could actually consider other
“spherical-like” coordinate systems. For instance, if M is a sufficiently
smooth manifold enclosing a star-shaped domain with respect to the origin,
then one could consider a1, f , and the sought-after solutions as functions
depending on r > 0, φ ∈ M , and our statements can be easily modified
to apply in this new setting. The simplicity of the eigenvalues of A1(s)
and hypothesis (A4) should also be generic in a suitable sense, again by
arguments from [1].

(iii) The specific dependence of (2.4) in r allows us to apply standard center
manifold results, see, e.g., [24, 51]. Such results are well suited to our
approach because the resulting reduced equation inherits the Hamiltonian



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 157

structure of the original equation. There are center manifold theorems
for equations where the linear part of the equation is allowed to be non-
autonomous, e.g., [11, 12, 45]. Such theorems may apply to a broader set of
equations, and it is an interesting question, which we do not address in this
article, whether such a reduction could be used to construct new solutions
of equations of the form (2.4).

The proof of Theorem 2.2 follows a general scheme from [38, 41]. We express
(2.4) in an abstract form and, after a reparametrization – where the logarithm
of the radial variable takes the role of time, we apply a center manifold theorem.
The resulting equation (the “reduced equation”) is endowed with a Hamiltonian
structure. After some transformations, the resulting Hamiltonian system is put in
a form appropriate for some results from [41] to be applied, yielding quasiperiodic
solutions of the abstract equation. These solutions correspond, in turn, to log-
radially quasiperiodic solutions of (2.4).

3. Hamiltonian setting

To a significant extent, this section uses results from [38, 41], with changes
to account for the setting of the present article. We first write equation (2.4)
in abstract form, then apply a center manifold reduction, and endow the resulting
equation with a Hamiltonian structure, which will be transformed to a form suitable
for an application of a KAM-type theorem. Throughout this section we assume that
hypotheses (A1), (A2), (A3’), and (A4’) hold with m > N/2 and K ≥ 6(n+ 1).

To write (2.4) in abstract form, with log r taking the role of time, we start by
recalling that

∆u = urr +
N − 1

r
ur +

1

r2
∆SN−1u,

where ∆SN−1 is the spherical Laplace operator on SN−1, the unit sphere in RN . Let
F be as in (2.6), and consider the Nemytskii operator F : (0,∞)×Hm+2(SN−1)×
(−δ, δ)→ Hm+1(SN−1) given by

F(r, u; s)(φ) = F (r, φ, u(φ); s) (φ ∈ SN−1).

This map is well defined and of class CK+1 in u. This fact can be proven using
that m > N/2 (so Hm(SN−1) is a Banach algebra) and arguments from [49] or [38,
Theorem A.1(b)].

For t > 0 and s ∈ (−δ, δ), let

u1(t; s)(φ) = u(t, φ; s),

u2(t; s)(φ) =
∂u

∂r
(t, φ; s) (φ ∈ SN−1).

Here we use t > 0 to emphasize that the radial variable r now takes the role of
time. Equation (2.4) can thus be written in the form

d

dt

(
u1
u2

)
=

[
0 1

−t−2∆SN−1 − t−2a1(φ; s) −(N − 1)t−1

](
u1
u2

)
−
(

0
F(t, u1; s)

)
,

(3.1)
for t > 0.



158 D. VALDEBENITO EJDE-2022/CONF/26

Following [4, Section 2], we consider the reparametrization τ = log t, and the
functions

ũ1(τ) = eAτu1(eτ ),

ũ2(τ) = e(1+A)τu2(eτ ),

f̃(ũ1)(φ) = e(2+A)τF(eτ , u1(eτ ))(φ) = f(φ, ũ1),

(3.2)

defined for τ ∈ R. The last equality in the third line of (3.2) is obtained using (2.6).

Here A = (N − 2)/2, as in (2.5), and f̃ : Hm+2(SN−1)× (−δ, δ)→ Hm+1(SN−1) is

the Nemytskii operator associated to f . From the regularity of F it follows that f̃
is of class CK . Note that ũ1, ũ2, and f̃ all depend on the parameter s, but for the
sake of notational simplicity we will drop that dependence from the notation when
not needed (this will also apply to a1 and other functions involving s). Note also

that f̃ does not explicitly depend on τ .
Substituting (3.2) into (3.1), and expressing the system in terms of τ , we obtain

d

dτ

(
ũ1
ũ2

)
=

[
A 1

−∆SN−1 − a1(φ) −A

](
ũ1
ũ2

)
−
(

0

f̃(φ, ũ1)

)
, τ ∈ R. (3.3)

Denote

ũ = (ũ1, ũ2),

A1(s) = −∆SN−1 − a1(·),

A(s) =

[
A 1

A1(s) −A

]
,

R(ũ1, ũ2; s) =

(
0

−f̃(·, ũ1)

)
,

so (3.3) becomes
d

dτ
ũ = A(s)ũ+R(ũ; s), τ ∈ R. (3.4)

Here, for each s ∈ (−δ, δ), A(s) is considered as an operator on the space X :=
Hm+1(SN−1)×Hm(SN−1) and domainD(A(s)) = Z := Hm+2(SN−1)×Hm+1(SN−1),
and R as a CK+1-map from Z × (−δ, δ) to Z. The concept of a solution of (3.4)
on an interval I is as in [24, 51]: it is a function in C1(I, X) ∩ C(I, Z) satisfying
(3.4).

Given s ∈ [0, δ), to find the spectrum of A(s) we consider the eigenvalue problem

A(s)(ũ1, ũ2)T = ν(s)(ũ1, ũ2)T ,

where the sought-after eigenvalues ν(s) depend on s. Using the definition of A(s),
this equation can be expanded as follows:

Aũ1 + ũ2 = ν(s)ũ1

A1(s)ũ1 −Aũ2 = ν(s)ũ2.

Eliminate ũ2 from the system to find

A1(s)ũ1 = (ν(s)2 −A2)ũ1;

i.e., ν(s) is an eigenvalue of A(s) if and only if ν(s)2−A2 is an eigenvalue of A1(s).
Denoting the eigenvalues of A1(s) as µ`(s), ` = 1, 2, . . . in an increasing manner we
find

ν±` (s) = ±
√
µ`(s) +A2.



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 159

Using (A3’), we find that ν±` (s) ∈ iR (the imaginary axis) for ` = 1, . . . , n and

s ∈ [0, δ), while there is some positive constant c such that ν±` (s) ∈ R \ (−c, c) for
all s ∈ [0, δ) and ` ≥ n+ 1. Note also that, for each s ∈ [0, δ), the eigenvalues lying
on the imaginary axis are all simple.

For s ∈ [0, δ), let ϕj(·; s), j = 1, . . . , n, be the eigenfunction of A1(s) corre-
sponding to µj as introduced in Section 2 – in particular, owing to (A3’), the maps
s 7→ ϕj(·; s), j = 1, . . . , n are well defined. By elliptic regularity, (A1) implies that
ϕj(·; s) ∈ Hm+2(SN−1), for j = 1, . . . , n and s ∈ [0, δ). Moreover, by [25] and the
regularity of µj with respect to s (cf. Remark 2.1(ii)), the maps s 7→ ϕj(·; s) are of
class CK+1 as Hm+2(SN−1)-valued functions of s.

We define the space

Xc(s) :=
{

(h, h̃)T : h, h̃ ∈ span{ϕ1(·; s), . . . , ϕn(·; s)}
}
⊂ Z,

the orthogonal projection operator Π(s) : L2(SN−1)→ span{ϕ1(·; s), . . . , ϕn(·; s)},
and let Pc(s) : X → Xc(s) be given by Pc(s)(v1, v2) = (Π(s)v1,Π(s)v2). This
operator is the spectral projection for the operator A(s) associated with the spectral
set {ν±` (s) : ` = 1, . . . , n}, cf. [38, Section 3.2], and it is well defined, since the rest of
the spectrum of A(s) is at a positive distance (independent of s) from the imaginary
axis. Due to (A1), the map s 7→ Pc(s) is of class CK+1 from s ∈ [0, δ) to the class of
linear bounded operators on X; moreover, the smoothness of the maps s 7→ ϕj(·; s)
implies that s 7→ Pc(s) is of class CK+1 as a map on [0, δ) with values on the class
of linear bounded operators from X to Z.

Also we define Ph(s) = IX − Pc(s), IX being the identity map on X, and, for
j = 1, . . . , n,

ψj(·; s) = (ϕj(·; s), 0)T , ζj(·; s) = (0, ϕj(·; s))T . (3.5)

A basis of Xc(s) is given by

B(s) := {ψ1(·; s), . . . , ψn(·; s), ζ1(·; s), . . . , ζn(·; s)}.

For z ∈ Xc(s), we denote by {z}B the coordinates of z with respect to the basis
B(s). Denote further

ψ(s) := (ψ1(·; s), . . . , ψn(·; s)),
ζ(s) := (ζ1(·; s), . . . , ζn(·; s)).

(3.6)

Proposition 3.1. Using the above notation the following statement is valid, pos-
sibly after making δ > 0 smaller. There exist a map σ : (ξ, η; s) ∈ R2n × [0, δ) 7→
σ(ξ, η; s) ∈ Z of class CK+1 and a neighborhood N of 0 in Z such that for each
s ∈ [0, δ) one has

σ(ξ, η; s) ∈ Ph(s)Z ((ξ, η) ∈ R2n), (3.7)

σ(0, 0; s) = 0, D(ξ,η)σ(0, 0; s) = 0, (3.8)

and the manifold

Wc(s) = {ξ · ψ(s) + η · ζ(s) + σ(ξ, η; s) : (ξ, η) = (ξ1, . . . , ξn, η1, . . . , ηn) ∈ R2n} ⊂ Z

has the following properties:

(a) If ũ(τ) is a solution of (3.4) on I = R and ũ(τ) ∈ N for all τ ∈ R,
then ũ(τ) ∈ Wc(s) for all τ ∈ R; that is, Wc(s) contains the orbit of each
solution of (3.4) which stays in N for all τ ∈ R.



160 D. VALDEBENITO EJDE-2022/CONF/26

(b) If z : R→ Xc(s) is a solution of the equation

dz

dτ
= A(s)

∣∣
Xc(s)

z + Pc(s)R(z + σ({z}B; s); s) (3.9)

on some interval I, and ũ(τ) := z(τ) + σ({z(τ)}B; s) ∈ N for all τ ∈ I,
then ũ : I → Z is a solution of (3.4) on I.

Moreover, σ satisfies the following relation:
(c) If 2 ≤ ` ≤ K is an integer, then σ({ũ}B; s) = O(‖ũ‖`) as ũ→ 0 whenever

s ∈ [0, δ) is such that R(ũ; s) = O(‖ũ‖`) as ũ→ 0.

From now on, the function σ is called the reduction function, Wc(s) is the cen-
ter manifold, and equation (3.9) is the reduced equation. In the sequel it will be
convenient to write σ = (σ1, σ2), where σ1 ∈ Hm+2(SN−1), σ2 ∈ Hm+1(SN−1).

The proof of Proposition 3.1 can be found in [41]. For the most part, the conclu-
sions of Proposition 3.1 are standard conclusions of center manifold theorems found
in the literature [24, 51], but some additional work is needed to obtain the desired
regularity in s, since the parameter s appears in the linear term (albeit only in the
bounded part of the linear term). Note that in [41] the space Z was taken to be
Hm+2(RN )×Hm+1(RN ), but the specifics of the space (other than the fact that it
is a Hilbert space) are not relevant in the proofs. Similarly, the regularity assump-

tions for the nonlinear term R rely on the regularity of the Nemytskii operator f̃ ,
discussed above, so all these results apply in the present setting.

Remark 3.2. (i) In our case statement (c) of Proposition 3.1 applies with
` = 2, so σ({ũ}B; s) = O(‖ũ‖2) as ũ → 0, or, equivalently, σ(ξ, η; s) =
O(|(ξ, η)|2) as (ξ, η)→ (0, 0) uniformly in s.

(ii) The components σ1 and σ2 of σ take values in the orthogonal complement
(with respect to the L2-inner product) of span{ϕ1(·; s), . . . , ϕn(·; s)}. In
addition, span{ϕ1(·; s), . . . , ϕn(·; s)} and its orthogonal complement are in-
variant under the operator A1(s). These facts will be used below.

To endow the reduced equation corresponding to (3.4) with a Hamiltonian struc-
ture, we first study the (formal) Hamiltonian structure of (3.4), since this structure
is inherited (in a precise sense) by the reduced equation [32].

Let F (φ, u; s) :=
∫ u
0
f(φ,w; s)dw for s ∈ [0, δ), φ ∈ SN−1 (f is as in (2.7)), and,

for (ũ1, ũ2) ∈ Z,

H(ũ1, ũ2; s) :=

∫
SN−1

(
Aũ1ũ2 +

1

2
ũ22−

1

2
|∇ũ1|2 +

1

2
a1(φ; s)ũ21 + F (ũ1)

)
dφ, (3.10)

where ∇ stands for the spherical gradient.
Equation (3.3) has a formal Hamiltonian structure with respect to the functional

H and the canonical symplectic structure on L2(SN−1)×L2(SN−1). Its restriction
to the center manifold yields the Hamiltonian of the reduced equation. More pre-
cisely, let

Φ(ξ, η; s) = H
(
ξ · ϕ(s) + σ1(ξ, η; s), η · ϕ(s) + σ2(ξ, η; s); s

)
, (3.11)

where ϕ(s) = (ϕ1(s), . . . , ϕn(s)), ξ ·ϕ(s) = ξ1ϕ1(s)+ · · ·+ξnϕn(s) and similarly for
η ·ϕ(s). Then Φ is a map from R2n×[0, δ) to R, and (3.9) is the Hamiltonian system
with respect to the Hamiltonian Φ and a certain symplectic structure defined in a
neighborhood of (0, 0) ∈ R2n. This can be proved using general statements in [32],
but in [38, 41] we can find results which contain additional information regarding



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 161

the dependence of Φ on (ξ, η) and s. The computations performed in those papers
apply here as well, thus, aside from stating the relevant equations to account for
the differences in our current setting, we will omit most of the proofs, which are
quite technical and do not require any meaningful changes to be valid in the present
setting. The following result, also used in [38, 41] will be relevant later on.

Lemma 3.3. The quadratic and cubic terms (in (ξ, η)) of Φ are independent of the
reduction function σ.

Proof. Noting that σ = (σ1, σ2) is of order O(|(ξ, η)|2) as (ξ, η)→ (0, 0) (cf. Remark
3.2(i)), we see that the quadratic terms (in (ξ, η)) of Φ do not involve the function
σ. In order to study the terms of degree 3, we note first that∫

SN−1

(
− 1

2
|∇ũ1|2 +

1

2
a1(φ; s)ũ21

)
dφ =

1

2

∫
SN−1

(∆SN−1 ũ1 + a1ũ1)ũ1 dφ.

Recalling that ∆SN−1ϕj + a1ϕj = −A1(s)ϕj = −µjϕj , we notice that the cubic
terms resulting from taking ũ1 = ξ ·ϕ(s) + σ1(ξ, η; s) and ũ2 = η ·ϕ(s) + σ2(ξ, η; s)
as in (3.11) are terms that either do not involve σ, or of the form∫

SN−1

( n∑
j=1

(ajξjϕj + bjηjϕj)
)
G(ξ, η) dφ, (3.12)

where aj , bj are some constants depending on s, independent of (ξ, η) and φ, and
G is equal to either σ`(ξ, η), ` = 1, 2, or −A1(s)σ1(ξ, η). In either case, G and
ϕj are orthogonal by Remark 3.2(ii). We thus conclude that the integral in (3.12)
vanishes, whence (3.11) does not contain any nonzero cubic terms involving σ. �

The Hamiltonian system with functional Φ can be successively transformed by
performing three coordinate changes:

a Darboux transformation, normal form transformation, and
action-angle variables.

(3.13)

By the first change of coordinates, we achieve that the transformed system is
Hamiltonian with respect to the standard symplectic form on R2n (and the trans-
formed Hamiltonian functional). The existence of such a local transformation is
guaranteed by the Darboux theorem, but we need some more precise statements
found in [38], which provide additional details on the dependence of the transforma-
tion on the parameter s and on the coordinates (ξ, η). In particular, the Darboux
transformation can be chosen as the sum of the identity map (on R2n) and terms
of order O(|(ξ, η)|3), with the cubic terms having coefficients of class CK in s. This
implies that the Darboux transformation does not change the quadratic or cubic
terms of Φ, but it may alter terms of degree 4 and higher.

In the new coordinates (still denoted (ξ, η)) resulting from the aforementioned
Darboux transformation, the Hamiltonian takes the following form for (ξ, η) ≈
(0, 0):

Φ(ξ, η; s) =
1

2

n∑
j=1

(
−µj(s)ξ2j + 2Aξjηj + η2j

)
+

1

3

∫
SN−1

a2(φ; s)(ξ · ϕ(φ; s))3 dφ+ Φ4(ξ, η; s) + Φ′(ξ, η; s).

(3.14)



162 D. VALDEBENITO EJDE-2022/CONF/26

Here Φ4 is a homogeneous polynomial in (ξ, η) of degree 4 whose coefficients are
of class CK in s ∈ [0, δ) (in particular, their CK-norm is bounded), and Φ′ is a
function of class CK in all its arguments and of order O(|(ξ, η)|5) as (ξ, η)→ (0, 0).
Note that, thanks to Lemma 3.3 and our choice of Darboux transformation, the
quadratic and cubic terms of Φ are explicitly known, as the reduction function σ
and the terms introduced by the Darboux transformation are present only in terms
of degree 4 and higher. Also, all the changes of variables we consider below will be
canonical changes, that is, the (canonical) symplectic structure will be preserved.

For j = 1, . . . , n, denote ωj =
√
|µj | (ωj depends on s, but for the sake of

notational clarity we omit the dependence), and consider the change of coordinates

ξj = (ωj)
−1/2ξ′j , ηj = (ωj)

1/2η′j ,

so the Hamiltonian Φ becomes

Φ(ξ′, η′) =
1

2

n∑
j=1

(
ωjξ
′2
j + 2Aξ′jη′j + ωjη

′2
j

)
+

1

3

∫
SN−1

a2(φ)(ξ · ϕ(φ))3 dφ+ Φ4(ξ′, η′) + Φ′(ξ′, η′).

(3.15)

Here Φ(ξ′, η′) stands for Φ(ξ(ξ′), η(η′)) (same for Φ4 and Φ′). For the time being
we postpone expanding ξ · ϕ in terms of ξ′.

Next, we diagonalize the quadratic terms of Φ. If N = 2, then A = 0, and
nothing needs to be done. IfN ≥ 3, define ϑj as in (2.9) and consider the (canonical)
transformation

ξ̃j =

√
ϑj√
2

(ξ′j − η′j), η̃j =
1√

2
√
ϑj

(ξ′j + η′j).

In the new coordinates,

Φ(ξ̃, η̃)

=

n∑
j=1

(ωj +A)ϑj

( ξ̃2j + η̃2j
2

)
+

1

3

∫
SN−1

a2(φ)
[ n∑
j=1

1√
2
√
ωj

( 1√
ϑj
ξ̃j +

√
ϑj η̃j

)
ϕj(φ)

]3
dφ+ h.o.t.,

(3.16)

where h.o.t. stands for terms of order O(|(ξ̃, η̃)|4) as |(ξ̃, η̃)| → 0, and the term in

brackets is the expansion of (ξ · ϕ(φ)) from (3.15), now written in terms of (ξ̃, η̃).
The Hamiltonian Φ in (3.16) (or in (3.15) if N = 2) has thus been written in

a suitable form so that the second transformation in (3.13) can be performed: for
s > 0, the Hamiltonian Φ(·, ·; s) is transformed to its normal form up to order
2kB + 1, where kB := [K/2] − 1, [K/2] being the integer part of K/2. More
precisely, near (0, 0) there is a canonical coordinate transformation such that in
the new coordinates (ξ̄, η̄) the Hamiltonian can be written as follows. Let (ξ̄, η̄) =
(ξ̄1, . . . , ξ̄n, η̄1, . . . , η̄n),

Ij =
1

2
(ξ̄2j + η̄2j ) (j = 1, . . . , n), (3.17)

and I = (I1, . . . , In). Then

Φ(ξ̄, η̄; s) = ω̃(s) · I + Φ0(I; s) + Φ1(ξ̄, η̄; s), (3.18)



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 163

where

ω̃(s) = (ω̃1(s), . . . , ω̃n(s)) :=
(
(ω1(s) +A)ϑ1(s), . . . , (ωn(s) +A)ϑn(s)

)
, (3.19)

Φ0 is a polynomial in I of degree at most kB , and Φ1 a CK function of order
O(|(ξ̄, η̄)|2kB+2) as (ξ̄, η̄) → (0, 0). (Note that if N = 2, then ω̃j = ωj for j =
1, . . . , n.) The polynomial Φ0 is of the form

Φ0(I; s) =
1

2
I ·M(s)I + P̂ (I; s), (3.20)

where, for s ∈ (0, δ), M(s) is an n × n matrix and P̂ (I; s) a polynomial in I (of
degree at most kB) with no constant, linear, or quadratic terms. The entries of

M(s) and the coefficients of P̂ (·; s) are of class CK in s.
For the final transformation in (3.13), we introduce the action-angle variables

I = (I1, . . . , In) ∈ Rn, θ = (θ1, . . . , θn) ∈ Tn by

(ξ̄j , η̄j) =
√

2Ij(cos θj , sin θj).

The change of coordinates from (ξ̄j , η̄j) to (θ, I) is defined in regions where Ij =
(ξ̄2j + η̄2j )/2 > 0 for all j ∈ {1, . . . , n}, and it is well known that this transformation
is canonical. In these coordinates, Φ looks as follows:

Φ(θ, I; s) = ω̃(s) · I + Φ0(I; s) + Φ1(θ, I; s). (3.21)

(Φ(θ, I; s) actually stands for the function Φ(ξ̄(θ, I), η̄(θ, I); s), and similarly for Φ0,
Φ1.) Thus, the Hamiltonian Φ is the sum of an integrable Hamiltonian (the first
two terms on the right hand side of (3.21)) and a “perturbation” (the last term in
(3.21)). This is a form suitable for an application of a KAM-type theorem.

4. Proof of Theorem 2.2

Once the Hamiltonian Φ of the reduced equation corresponding to the abstract
equation (3.4) has been rewritten in the form (3.21), one can use results from [41]
to obtain the existence of quasiperiodic solutions of (3.4). Using a theorem from
[4], those solutions correspond to log-radially quasiperiodic solutions of our original
equation (2.4).

We first consider the more general case of log-radially quasiperiodic solutions
of (2.4) with n frequencies. In order to do so, we need the following additional
hypothesis on Φ, the transformed Hamiltonian of the reduced equation as in (3.21):

(A6) Consider the (n+ 1)× (n+ 1) matrix

M(s) :=

[
D2Φ0(0; s) ω̃(s)
ω̃T (s) 0

]
. (4.1)

Then at least one of the matrices D2Φ0(0; s) and M(s) is nonsingular.

Theorem 4.1. Assume that hypotheses (A1), (A2), (A3’), (A4’) are satisfied, and
that (A6) holds for some fixed s ∈ (0, δ). Then there exists a solution u = u(r, φ)
of equation (2.4) such that u is log-radially quasiperiodic with a (nonresonant) fre-
quency vector in Rn. Moreover, there is an uncountable family of such log-radially
quasiperiodic solutions, their frequency vectors forming an uncountable subset of
Rn.



164 D. VALDEBENITO EJDE-2022/CONF/26

The proof of this theorem consists of two parts. The first step is obtaining a
pair of quasiperiodic functions (ũ1, ũ2) which satisfy (3.4), that is, ũ1 and ũ2 are
such that the maps (τ, φ) 7→ ũj(τ)(φ), j = 1, 2, are quasiperiodic in the sense of
the definition in Section 2 (cf. equation (2.2)). This step is analogous to a theorem
in [41]. Once such a pair is obtained, we need to establish that there is a solution
u of (2.4) corresponding to the pair (ũ1, ũ2). In order to do so, we make use of the
following result contained in [4, Theorem 3.6]:

Theorem 4.2. Suppose 0 < T < ∞. If (u1, u2) is a solution of (3.1) on (0, T )
(for a fixed value of s), then there exists a weak solution u of (2.4) on B(0, T )\{0}
such that u(t, ·) = u1(t), ∂u

∂r (t, ·) = u2(t) for each t ∈ (0, T ).

The definition of weak solution used in [4] is as follows. Given b > a > 0, let
Ω = {x ∈ RN : a < |x| < b}. Then u is a weak solution of (2.4) on Ω if u ∈ H1(Ω),
a1(φ; s)r−2u+ F (r, φ, u; s) ∈ L2(Ω), and∫ b

a

∫
SN−1

∇u · ∇v dφdr =

∫ b

a

∫
SN−1

(
a1(φ; s)r−2u+ F (r, φ, u; s)

)
v dφdr

holds for all v ∈ H1
0 (Ω). (Here∇ is the usual gradient.) In the case Ω = B(0, b)\{0},

we say u is a weak solution on Ω if u is a weak solution on B(0, b)\B(0, b′) for each
b′ ∈ (0, b). By standard regularity arguments, if u is a weak solution of (2.4) on a
domain away from the origin, then u is a classical solution as well.

Remark 4.3. The aforementioned theorem in [4] applies to a wider class of geo-
metrical settings (see Hypothesis 3.1 in [4]), in which case the abstract formulation
of a semilinear elliptic equation is more involved (cf. [4, Equation (14)]). In general
it is to be expected that the linear part of the abstract formulation will not be
autonomous, and that this will not be remedied by a change of variables such as
(3.2). This would preclude the application of classical center manifold reductions
as found in, say, [24, 51], where it is essential to have the linear part of the equation
to be autonomous.

Proof of Theorem 4.1. One can follow the proof of [41, Theorem 4.4] to construct
quasiperiodic solutions (ũ1, ũ2) of (3.4), parametrized by their frequency vectors,
which are nonresonant and form an uncountable subset of Rn. Each pair (ũ1, ũ2)
corresponds to a solution (u1, u2) of (3.1) via (3.2) and the reparametrization t = eτ ,
which implies that u1 and u2 are log-radially quasiperiodic (with a = −A and
a = −A−1, respectively). Using Theorem 4.2, there is a corresponding log-radially
quasiperiodic solution u of (2.4) on domains of the form {x ∈ RN : 0 < |x| < T}
for any T > 0, which satisfies u(r, ·) = u1(r), ∂u

∂r (r, ·) = u2(r). This allows us to

define u on RN \ {0}. �

Although we do not reproduce the proof of [41, Theorem 4.4] here, for the reader’s
convenience we provide a brief sketch. The Hamiltonian (3.21) can be seen as a
near-integrable Hamiltonian, in the sense that if I is sufficiently small, then Φ
is the sum of an analytic integrable Hamiltonian (namely, the first two terms in
(3.21)) and a perturbation term Φ1, which is of order O(|I|kB+1) (kB is the constant
considered in the paragraph after (3.16)), so this term is small if the domain for I
is sufficiently small. This is the standard setting for KAM-type results. In order to
apply a KAM-type theorem, one usually requires a Diophantine condition and some
nondegeneracy condition, the former condition being relatively easy to verify once



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 165

it is shown that the latter holds. Hypothesis (A6) provides two options to verify a
nondegeneracy condition: if D2Φ0 is nonsingular (often referred to as Kolmogorov’s
condition), then a theorem by Pöschel [42] can be applied (as in [38]) to yield the
existence of the desired quasiperiodic solutions for (3.4); ifM is nonsingular (known
as Arnold’s condition), then a result from [7] allows one to apply the result in [42] to
an auxiliary Hamiltonian, which, after a suitable rescaling, yields again the desired
quasiperiodic solutions for (3.4).

Note that the Hamiltonian Φ in (3.21) takes the same form as the Hamiltonian
in [41, Equation (3.25)]. Throughout the proof of [41, Theorem 4.4] the original
elliptic equation and the abstract equation play no role whatsoever, which allows
one to use the same arguments to obtain a solution of (3.4).

We can now prove Theorem 2.2. We henceforth fix n = 2 (the number of
frequencies), and assume (A1)–(A5) hold. The proof relies on a careful study of
the normal form procedure, similar to [41]. Recall that the Birkhoff normal form
algorithm consists of successive transformations eliminating inessential terms of
a given degree, which introduces new terms of higher degree, but leaving lower
order terms unchanged. In our setting, the first transformation eliminates all cubic
terms, introducing new terms of degree 4 and higher. The second transformation
eliminates nonresonant terms of degree 4 and leaves the remaining resonant terms
of degree 4 unchanged (see, e.g., [21] for a detailed discussion of the Birkhoff normal
form algorithm). Careful computations allow us to study the asymptotic behavior
of detM(s) as s → 0; more precisely, we determine which term of degree 4 (after
the first transformation) grows at the fastest rate as s→ 0.

Proof of Theorem 2.2. If hypothesis (A6) holds for the Hamiltonian Φ in (3.21) for
each s ∈ (0, δ), then the result is a direct consequence of Theorem 4.1. Therefore
we will show that in our setting hypotheses (A3) and (A5) imply (A6) for each
s ∈ (0, δ), where δ > 0 is sufficiently small. In order to do this, we return to the
Hamiltonian Φ as found in (3.16) (or (3.15) if N = 2). We recall that at this point
the Hamiltonian has been written in a standard form suitable for the application
of a Birkhoff normal form algorithm, as outlined in, e.g., [3, 21]. As before, for the
sake of clarity we will drop the dependence in s from the notation whenever it does
not play a relevant role.

We first assume N ≥ 3. The cubic terms (in (ξ̃, η̃) = (ξ̃1, ξ̃2, η̃1, η̃2)) of Φ can be
written as

Φ3(ξ̃, η̃) =

2∑
j,k,`=1

Θ(j, k, `)ξ̃j ξ̃k ξ̃` + Φr3(ξ̃, η̃),

where

Θ(j, k, `) =
1

3(ωjωkω`)1/2
1

2
√

2(ϑjϑkϑ`)1/2

∫
SN−1

a2ϕjϕkϕ`dφ, (4.2)

and

Φr3(ξ̃, η̃) =

2∑
j,k,`=1

Θ(j, k, `)
(

3ϑ`ξ̃j ξ̃kη̃` + 3ϑkϑ`ξ̃j η̃kη̃` + ϑjϑkϑ`η̃j η̃kη̃`

)
,

i.e., Φr3 comprises all cubic terms of the Hamiltonian Φ involving at least one factor
η̃1 or η̃2. As before, ϕj stands for the normalized eigenfunction of −∆SN−1 − a1



166 D. VALDEBENITO EJDE-2022/CONF/26

associated to µj , as in Section 2, ωj(s) =
√
|µj(s)|, and

ϑj(s) =
(ωj(s)−A
ωj(s) +A

)1/2
is as in (2.9), j = 1, 2.

Making δ > 0 smaller if necessary, we have that, by (A3) and our assumption
N ≥ 3, ω1 and ω2 satisfy ω1(s) > cδ > ω2(s) ≥ A > 0 for all s ∈ [0, δ), where
cδ > A is a constant depending on δ, but independent of s. We conclude that there
is a constant c > 0 such that ϑ1(s) ≥ c > 0 holds for all s ∈ [0, δ), while ϑ2(s)→ 0+
as s → 0, this limit coming from the assumption µ2(0) = −A2 in (A3). Since the
maps s ∈ [0, δ) 7→ a2(·; s) ∈ Cm+1(SN−1) and s ∈ [0, δ) 7→ ϕj(·; s) ∈ L∞(SN−1) are
continuous, the integral in (4.2) is bounded by a constant independent of s. The
foregoing statements imply that

Θ(j, k, `; s) = O
(
ϑ
−(j+k+`−3)/2
2

)
(j, k, ` ∈ {1, 2})

as s→ 0. In particular,

Θ(2, 2, 2; s) = O(ϑ
−3/2
2 ),

Θ(j, k, `; s) = O(ϑ−12 ) if (j, k, `) 6= (2, 2, 2).

From the asymptotic behavior of Θ(j, k, `; s) we also conclude that all the coeffi-
cients in Φr3 are of order O(ϑ−12 ) as s → 0. The coefficients of the terms of degree

4 in (ξ̃, η̃) can be shown to be of order O(ϑ−22 ) as s→ 0 by a similar argument.
One can now apply the Birkhoff normal form algorithm to eliminate all terms

of degree 3 (in (ξ̃, η̃)), which introduces new terms of degree 4 (and higher). As
discussed above, the next transformation eliminates some terms of degree 4, while
the remaining terms are unchanged. After the change of variables (3.17), one can
study the asymptotic behavior of the nonresonant terms of degree 2 in I = (I1, I2)
as in [41, Lemma 5.4] to obtain

Φ0(I; s) =
C

ω
3/2
2 (s)(ω2(s) +A)ϑ42(s)

(∫
SN−1

a2(φ; s)ϕ3
2(φ; s)dφ

)2
I22 +

+ Φ̃(I; s) + h.o.t.,

where C is a positive constant independent of s, and Φ̃, comprising all remaining

quadratic terms (in I), has coefficients of order O(ϑ
−7/2
2 ) as s → 0, while h.o.t.

stands for terms of degree 3 and higher in I. Recalling that N ≥ 3, so ω2(s) ≥
A > 0, we can prove that the matrix M(s), defined in (A6), is nonsingular for
all s ∈ (0, δ) by showing that its determinant is of order O(ϑ−42 ) as s → 0, hence,
detM(s)→∞ as s→ 0 (see [41, Lemma 5.5] for details), and therefore hypothesis
(A6) holds for all s ∈ (0, δ), making δ > 0 smaller if necessary. We can thus apply
Theorem 4.1 for each s ∈ (0, δ), which gives the desired log-radially quasiperiodic
solutions for (2.4), concluding the proof in the case N ≥ 3.

The case N = 2 can be treated similarly. Instead of the Hamiltonian Φ as in
(3.16), we start from the Hamiltonian (3.15). Since now A = 0, the Hamiltonian is
already diagonalized. The cubic terms are

Φ3(ξ′, η′) =
1

3

2∑
j,k,`=1

Θ(j, k, `)ξ′jξ
′
kξ
′
`,



EJDE-2018/CONF/26 RADIAL SPATIAL DYNAMICS 167

where

Θ(j, k, `) =
1

3(ωjωkω`)1/2

∫
SN−1

a2ϕjϕkϕ`dφ,

and there are no other cubic terms in Φ. One can reproduce the foregoing argument,
with ω1 > cδ > 0 and ω2 → 0 taking the role of ϑ1 and ϑ2, respectively, to derive
the asymptotic behavior of the remaining terms of degree 2 (in I) and conclude
that Φ0 takes the form

Φ0(I; s) =
C

ω4
2(s)

(∫
SN−1

a2(φ; s)ϕ3
2(φ; s)dφ

)2
I22 + Φ̃(I; s) + h.o.t.,

where C is a positive constant; the function Φ̃, comprising all remaining quadratic

terms in I, has coefficients of order O(ω
−7/2
2 ) as s→ 0; and h.o.t. stands for terms

of degree 3 and higher in I. The rest of the argument is the same as in the previous
case, the only change being that we use ω2 → 0 rather than ϑ2 → 0. This concludes
the proof in the case N = 2, and the theorem has thus been proved. �

Acknowledgments. The author would like to thank Peter Poláčik for his helpful
remarks during the preparation of this article.

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Daŕıo A. Valdebenito
Department of Mathematics, Ave Maria University, 5050 Ave Maria Blvd, Ave Maria,

FL 34142, USA

Email address: dario.valdebenito@avemaria.edu


	1. Introduction
	2. Main result
	3. Hamiltonian setting
	4. Proof of Theorem 2.2
	Acknowledgments

	References