1 Introduction
In this work we consider the non-local sinh-Poisson equation given by
(1.1)
{
(
-
Δ
)
1
2
u
=
λ
(
e
u
-
e
-
u
)
in
I
:=
(
-
1
,
1
)
,
u
≡
0
on
ℝ
∖
I
,
with λ∈ℝ+. This equation is related to mathematical models for the description of galvanic corrosion of a planar electrochemical system consisting of an electrolyte solution and an adjoining metal surface. If the electrolyte is confined in a domain Ω⊆ℝ2, the electrolytic voltage potential satisfies the non-linear boundary value problem
(1.2)
{
Δ
v
=
0
in
Ω
,
∂
v
∂
ν
=
λ
(
e
β
v
-
e
-
(
1
-
β
)
v
)
+
g
on
∂
Ω
,
where λ and β are constants depending on the constituents of the system and g models an externally imposed current. We refer to [13, 36] for the mathematical derivation of this model, which is due to Butler and Volmer. If one takes β=12 and g=0, the problem becomes equivalent to
(1.3)
{
Δ
v
=
0
in
Ω
,
∂
v
∂
ν
=
λ
(
e
v
-
e
-
v
)
on
∂
Ω
.
Problems (1.2)–(1.3), and corresponding ones in higher dimension, have been studied by several authors in recent years (see e.g. [36, 21, 10, 29, 28]). Among several generalizations, we mention that Alessandrini and Sincich [1, 2] have considered problems of the form
{
Δ
v
=
0
in
Ω
⊆
ℝ
2
,
∂
v
∂
ν
=
λ
(
e
v
-
e
-
v
)
on
Γ
1
,
∂
v
∂
ν
=
g
on
Γ
2
,
v
=
0
on
∂
Ω
∖
(
Γ
1
∪
Γ
2
)
,
where Γ1 and Γ2 are two open, disjoint portions of ∂Ω. Here Γ1 represents the corroded part of ∂Ω, which is not accessible to direct inspection, Γ2 is the portion of ∂Ω where current density can be directly measured, and the remaining part of ∂Ω is assumed to be grounded. In this work we want to consider the strictly related problem in which Ω=ℝ+2 is the upper half plane, Γ1 is a segment and Γ2=∅. Namely, we have
(1.4)
{
Δ
v
=
0
in
Ω
,
∂
v
∂
ν
=
λ
(
e
v
-
e
-
v
)
on
I
×
{
0
}
,
v
=
0
on
(
ℝ
∖
I
)
×
{
0
}
.
There is a strict connection between problems (1.4) and (1.1). Indeed, the Poisson harmonic extension of any solution to (1.1) solves (1.4). Viceversa, if v solves (1.4) and has finite Dirichlet energy, then the boundary trace u=v(⋅,0) is a solution of (1.1).
Equation (1.1) can also be considered as a one-dimensional version of the planar sinh-Poisson problem
(1.5)
{
-
Δ
u
=
λ
(
e
u
-
e
-
u
)
in
Ω
⊆
ℝ
2
,
u
=
0
on
∂
Ω
,
which arises in the statistical mechanics approach proposed by Onsager [27] and Joyce and Montgomery [20, 26] to the description of two-dimensional turbulent Euler flows with null total vorticity (we refer to [7, 8, 23, 27] for a physical discussion of this problem).
In recent years, there has been a great interest in the construction of sign-changing solutions for problems (1.3) and (1.5). When Ω is the unit disk of ℝ2, explicit families of solutions to (1.3) were exhibited by Bryan and Vogelius in [5]. As λ→0, such solutions develop an even number of sign-alternating peaks concentrating in separate points of ∂Ω. In [10], Dávila, Del Pino, Musso and Wei proved that solutions of (1.3) with an analogous behavior exist on arbitrary bounded domains with smooth boundary. In fact, for any even k∈ℕ, they constructed two independent branches of solutions developing k sign-alternating peaks on ∂Ω. In this result, k must be even to ensure the existence of sign-alternating peaks configurations on ∂Ω, whose connected components are closed curves. We also refer to [21, 25] for a-priori analysis of blowing-up solutions to (1.3).
Concerning problem (1.5), the existence of sign-changing solutions developing exactly two peaks with different sign has been obtained by Bartolucci and Pistoia in [3]. More generally, they proved that if ξ1,…,ξk∈Ω correspond to a stable critical point of a generalized version of the k-point Kirchhoff–Routh path function, then there is a solution uλ of (1.5) such that
λ
(
e
u
-
e
-
u
)
→
8
π
∑
i
=
1
k
(
-
1
)
i
δ
ξ
i
as λ→0 in the sense of measures, where δξi denotes the Dirac delta at ξi, i=1,…,k. The existence of stable critical points of the generalized Kirchhoff–Routh function for k≥3 was studied by Bartsch, Pistoia and Weth in [4]. They give existence results on any domain for k=3,4, and on axially symmetric domains for arbitrary k≥1. In the latter case, the points ξ1,…,ξk are located on the symmetry axis of Ω and the corresponding solution of (1.5) develops sign-alternating peaks at ξ1,…,ξk.
Inspired by these results, the main purpose of this paper is to discuss the existence, for small values of λ, of a branch solutions of (1.1) with an arbitrarily prescribed number of nodal regions. Specifically, for any k∈ℕ, we will construct a branch of solutions with exactly k sign-alternating peaks in the interval I=(-1,1).
Theorem 1.1.
For any k∈N, there exist λ0>0 and a family (uλ), defined for λ∈(0,λ0), of weak solutions to (1.1) with exactly k nodal regions. Moreover, for any sequence (λn)n∈N⊂(0,λ0) with λn→0 as n→+∞, there exist ξ1,…,ξk∈I with ξ1<ξ2<…<ξk such that (along a subsequence) we have:
u
λ
n
blows-up at
ξ
i
as
n
→
∞
for
i
=
1
,
…
,
k
. Namely, for any
ε
>
0
, we have
(1.6)
sup
(
ξ
i
-
ε
,
ξ
i
+
ε
)
(
-
1
)
i
-
1
u
λ
n
→
+
∞
as
n
→
+
∞
.
u
λ
n
→
2
π
∑
i
=
1
k
(
-
1
)
i
-
1
G
ξ
i
, in
C
loc
∞
(
I
∖
{
ξ
1
,
…
,
ξ
k
}
)
∩
C
loc
0
,
1
2
(
I
¯
∖
{
ξ
1
,
…
,
ξ
k
}
)
as
n
→
+
∞
, where
G
ξ
i
is the Green function for
(
-
Δ
)
1
2
with singularity at
ξ
i
(explicitly given by (
2.3
)).
The solutions provided in Theorem 1.1 can be considered as the analogue of the ones constructed in [4] for problem (1.5) but, working in dimension 1, we are able to obtain a stronger result and to show that the number of nodal regions coincides with the number of peaks, which does not seem to be known for the solutions in [4], except for the case k=2 (see [3]).
Our solutions are also strictly related to the solutions of (1.3) with an even number of peaks constructed in [10]. However, here we do not need to impose the evenness of k as the interaction between the first and the last peak is weaker than the interaction with intermediate peaks (unless k=2). It should be noted that if u solves (1.1), then -u is a solution as well. Thus, Theorem 1.1 provides two distinct branches of solutions. But, due to the lack of topology of I, we cannot expect the existence of other independent branches of solutions with separate peaks as in [10]. Nevertheless, we strongly believe it would be possible to find a different branch of solutions with k nodal regions which develops a tower of peaks at the origin. Solutions of this kind were constructed for problem (1.5) by Grossi and Pistoia in [17] (see also [30]).
Differently from the approach in [10], we will not rely on the connection between (1.1) and the extended problem (1.4). Instead, the proof of Theorem 1.1 will be based on the Lyapunov–Schmidt finite-dimensional reduction method, which has successfully been used to find solutions to (1.5) and other similar problems (see e.g. [3, 12, 15]). Here, we will apply this technique on the fractional-order Sobolev space X01/2(I), which is defined as the space of all the functions in H12(ℝ) which vanish identically outside I. A Hilbert structure on X01/2(I) is determined by the scalar product
(1.7)
〈
u
,
v
〉
:=
∫
ℝ
(
-
Δ
)
1
4
u
(
-
Δ
)
1
4
v
𝑑
x
,
u
,
v
∈
X
0
1
/
2
(
I
)
,
with the corresponding norm given by
(1.8)
∥
u
∥
:=
∥
u
∥
X
0
1
/
2
=
∥
(
-
Δ
)
1
4
u
∥
L
2
(
ℝ
)
≡
∥
(
-
Δ
)
1
4
u
∥
2
,
u
∈
X
0
1
/
2
(
I
)
.
In order to prove Theorem 1.1, we proceed as follows. For δ>0, ξ∈ℝ, let us consider the one-dimensional bubble
(1.9)
U
δ
,
ξ
(
x
)
:=
log
(
2
δ
δ
2
+
|
x
-
ξ
|
2
)
,
which solves the fractional-order Liouville equation [9, 18]
(1.10)
(
-
Δ
)
1
2
U
δ
,
ξ
=
e
U
δ
,
ξ
in
ℝ
.
Note that Uδ,ξ blows-up at ξ as δ→0+, namely limδ→0+Uδ,ξ(ξ)=+∞ and limδ→0+Uδ,ξ(x)=-∞, x∈ℝ∖{ξ}.
For any δ>0, ξ∈ℝ, let PUδ,ξ be the projection of Uδ,ξ on X01/2(I), that is
P
U
δ
,
ξ
:=
(
-
Δ
)
-
1
2
e
U
δ
,
ξ
,
where (-Δ)-12 represents the inverse of the half Laplacian (-Δ)12. Intuitively, for suitable choices of δ and ξ we will use PUδ,ξ to model each peak developed by solutions of (1.1) as λ→0. Indeed, Theorem 1.1 can be deduced by the following result:
Theorem 1.2.
For any k∈N, we can find λ0>0 such that, for any λ∈(0,λ0), there exist δi=δi(λ)∈(0,+∞), ξi=ξi(λ)∈I, i=1,…,k, and φλ∈X01/2(I)∩L∞(I) such that:
∑
i
=
1
k
(
-
1
)
i
-
1
P
U
δ
i
,
ξ
i
+
φ
λ
is a solution of (
1.1
) for any
λ
∈
(
0
,
λ
0
)
.
lim
λ
→
0
δ
i
(
λ
)
=
0
for any
i
=
1
,
…
,
k
.
There exists a small
η
0
∈
(
0
,
2
k
+
1
)
(not depending on λ ) such that -1+η0≤ξ1<…<ξk≤1-η0, and min1≤i≤k-1ξi+1-ξi≥η0 for any λ∈(0,λ0).
∥
φ
λ
∥
+
∥
φ
λ
∥
L
∞
(
I
)
→
0
as
λ
→
0
.
This work is organized as follows. In Section 2, we introduce the notation and state some preliminary results. For the reader’s convenience, we also include, in Section 2.1, an outline of the proof of Theorem 1.2. The technical aspects of the proof are discussed in Sections 3, 4, 5 and 6, as we will detail in Section 2.1. Finally, the conclusion of the proof of Theorem 1.2 and the proof of Theorem 1.1 are given in Section 7.
2 Notation and Preliminary Results
We start by recalling the definition and the main properties of the fractional Laplacian operator. For a given function u in the Schwartz space 𝒮 of rapidly decreasing functions (see e.g. [35]), we can define
(
-
Δ
)
s
u
=
ℱ
-
1
(
|
ξ
|
2
s
ℱ
(
u
)
(
ξ
)
)
,
s
∈
(
0
,
1
)
,
where ℱ and ℱ-1 denote respectively the Fourier and the inverse Fourier transform operators. In fact, this definition makes sense when |ξ|2sℱ(u)∈L2(ℝ). More generally, if u belongs to the space
L
s
(
ℝ
)
:=
{
u
∈
L
loc
1
(
ℝ
)
:
∫
ℝ
|
u
(
x
)
|
1
+
|
x
|
1
+
2
s
𝑑
x
<
+
∞
}
,
it is possible to define (-Δ)s as the tempered distribution
〈
(
-
Δ
)
s
u
,
φ
〉
=
∫
ℝ
u
(
-
Δ
)
s
φ
𝑑
x
,
φ
∈
𝒮
.
For s∈(0,1), let Hs(ℝ) denote the fractional-order Bessel potential space
H
s
(
ℝ
)
:=
{
u
∈
L
2
(
ℝ
)
:
|
ξ
|
s
ℱ
(
u
)
∈
L
2
(
ℝ
)
}
.
This space can be equivalently defined as the space of L2 functions for which the Gagliardo seminorm is finite (see e.g. [14, Section 3]). Similarly, (-Δ)s can be characterized in terms of singular integrals as
(
-
Δ
)
s
u
(
x
)
=
c
1
,
s
P. V.
∫
ℝ
u
(
x
)
-
u
(
y
)
|
x
-
y
|
1
+
2
s
𝑑
y
,
c
1
,
s
=
-
2
2
s
Γ
(
1
2
+
s
)
π
1
2
Γ
(
-
s
)
.
Throughout the paper, we will always denote I:=(-1,1)⊆ℝ and we will consider the space
X
0
1
/
2
(
I
)
=
{
u
∈
H
1
2
(
ℝ
)
:
u
≡
0
in
ℝ
∖
I
}
,
which is a Hilbert space with respect to the scalar product given in (1.7). The corresponding norm will be denoted as in (1.8).
Given p≥1 and a function f∈Lp(I), we say that u is a weak solution of the problem
(2.1)
{
(
-
Δ
)
1
2
u
=
f
in
I
,
u
=
0
in
ℝ
∖
I
,
if u∈X01/2(I) and it satisfies
∫
ℝ
(
-
Δ
)
1
4
φ
(
-
Δ
)
1
4
u
𝑑
x
=
∫
ℝ
f
φ
𝑑
x
for all
φ
∈
C
c
∞
(
I
)
.
Let also (-Δ)-12 represent the inverse of (-Δ)12. For any given p∈(1,+∞), the restriction of (-Δ)-12 to Lp(I) is defined by
(
-
Δ
)
-
1
2
:
L
p
(
I
)
→
X
0
1
/
2
(
I
)
,
f
↦
u
solution to (2.1)
.
This operator coincides with the adjoint of the inclusion operator ip′:X01/2(I)→Lp′(I), where p′=pp-1. In particular, for any p∈(1,+∞), there exists a constant C(p) such that
(2.2)
∥
(
-
Δ
)
-
1
2
f
∥
≤
C
(
p
)
∥
f
∥
L
p
(
I
)
for all
f
∈
L
p
(
I
)
.
The operator (-Δ)-12 can also be defined via an explicit representation formula. Throughout the paper, for any ξ∈I we will denote by Gξ the Green function for (-Δ)12 on I with singularity at ξ, which is given explicitly (see [6]) by the formula
(2.3)
G
ξ
(
x
)
:=
{
1
π
log
(
1
-
ξ
x
+
(
1
-
ξ
2
)
(
1
-
x
2
)
|
x
-
ξ
|
)
,
x
∈
I
,
0
,
x
∈
ℝ
∖
I
.
We shall often use the notation G(ξ,x) in place of Gξ(x). For any f∈Lp(I), we have the representation formula
(
-
Δ
)
-
1
2
f
(
x
)
=
∫
I
G
x
(
y
)
f
(
y
)
𝑑
y
,
x
∈
I
.
We will also denote by H(ξ,x) the regular part of Gξ, namely
H
(
ξ
,
x
)
:=
G
ξ
(
x
)
-
1
π
log
1
|
x
-
ξ
|
.
2.1 Outline of the Proof of Theorem 1.2: The Lyapunov–Schmidt Reduction Method
For a given k∈ℕ, we fix a small η>0 with 0<η<2k+1 and we define
𝒫
k
,
η
:=
{
𝝃
=
(
ξ
1
,
…
,
ξ
k
)
:
1
+
ξ
1
>
η
,
ξ
k
<
1
-
η
and
ξ
i
+
i
-
ξ
i
>
η
,
i
=
1
,
…
,
k
-
1
}
.
For 𝒂=(a1,…,ak)∈{-1,1}k, 𝜹=(δ1,…,δk)∈(0,1)k, and 𝝃=(ξ1,…,ξk)∈𝒫k,η, we denote
(2.4)
ω
𝒂
,
𝜹
,
𝝃
:=
∑
i
=
1
k
a
i
P
U
δ
i
,
ξ
i
.
For the proof of Theorems 1.1 and 1.2 we could fix ai=(-1)i-1, 1≤i≤k. But, since many of the estimates given throughout the paper hold true for a generic choice of the coefficients ai, we will only fix them when it is necessary.
Our goal is to find solutions for (1.1) of the form
u
=
ω
𝒂
,
𝜹
,
𝝃
+
φ
,
where φ∈X01/2(I)∩L∞(I) is small with respect to both the X01/2(I) and the L∞(I) norm. Throughout the paper we will denote fλ(u):=λ(eu-e-u). Then in terms of φ equation (1.1) reads as
(
-
Δ
)
1
2
φ
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
)
-
(
-
Δ
)
1
2
ω
𝒂
,
𝜹
,
𝝃
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
-
(
-
Δ
)
1
2
ω
𝒂
,
𝜹
,
𝝃
⏟
=
:
E
+
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
+
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
)
-
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
⏟
=
:
N
(
φ
)
,
that is
(
-
Δ
)
1
2
φ
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
=
E
+
N
(
φ
)
.
It is convenient to rewrite this equation as
(2.5)
L
φ
=
(
-
Δ
)
-
1
2
E
+
(
-
Δ
)
-
1
2
N
(
φ
)
,
where
L
:=
Id
-
(
-
Δ
)
-
1
2
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
.
For simplicity, here and in the rest of the paper, we will not specify the dependence of E,N and L on λ,𝒂,𝜹, and 𝝃.
We will prove that there exists a k-dimensional subspace K𝜹,𝝃 of X01/2(I) such that L is invertible on K𝜹,𝝃⊥. The space K𝜹,𝝃 is spanned by the functions PZ1,i:=(-Δ)-12eUδi,ξiZ1,i, i=1,…,k, where Z1,i is the unique solution which vanishes at infinity of the linearization of equation (1.10) around Uδi,ξi (see Section 4). Let π:X01/2(I)↦K𝜹,𝝃 and π⊥:X01/2(I)↦K𝜹,𝝃⊥ be the projections of X01/2(I) into K𝜹,𝝃 and K𝜹,𝝃⊥, respectively. Since X01/2(I)=K𝜹,𝝃⊕K𝜹,𝝃⊥, equation (2.5) is equivalent to the following couple of non-linear problems:
(2.6)
π
⊥
L
φ
=
π
⊥
(
-
Δ
)
-
1
2
E
+
π
⊥
(
-
Δ
)
-
1
2
N
(
φ
)
,
(2.7)
π
L
φ
=
π
(
-
Δ
)
-
1
2
E
+
π
(
-
Δ
)
-
1
2
N
(
φ
)
.
Exploiting the invertibility of π⊥L on K𝜹,𝝃, one formulates equation (2.6) in terms of a fixed point problem for φ. Such a problem can be solved if the error term E has small Lp(I) norm for some p∈(1,2), the non-linear term N(φ) decays faster than ∥φ∥, and the operator norm of (π⊥L)-1 can be controlled in terms of λ. We will show that for any choice of η∈(0,2k+1), 𝝃=(ξ1,…,ξk)∈𝒫k,η and λ small enough, these conditions are satisfied by a suitable choice of 𝜹=(δ1,…,δk) depending on λ and 𝝃. Specifically, there exists 𝜹=𝜹λ,ξ∈(0,1)k and φ=φλ,𝝃∈X01/2(I) such that (2.6) holds. In other words, there exist coefficients ci=ci(λ,ξ), i=1,…,k, which depend continuously on ξ such that
L
φ
λ
,
𝝃
=
(
-
Δ
)
-
1
2
E
+
(
-
Δ
)
-
1
2
N
(
φ
λ
,
𝝃
)
+
∑
j
=
1
k
c
j
P
Z
1
,
j
.
Then φλ,ξ solves equation (2.7) if and only if
(2.8)
c
i
(
λ
,
𝝃
)
=
0
,
i
=
1
,
…
,
k
.
The proof of Theorem 1.2 can be concluded by proving that for any small λ, there exist ξ1,…,ξk depending on λ solving the finite-dimensional system (2.8).
The rest of this paper is organized as follows. In Section 3 we choose the parameters δ1,…,δk and we provide point-wise and Lp estimates on the error terms E and N. Section 4 contains the precise definition of K𝜹,𝝃 and the analysis of the invertibility properties of π⊥L. The fix point argument which allows to solve (2.6) is explained in Section 5, while system (2.8) is studied in Section 6. Finally, we complete the proof of Theorems 1.1 and 1.2 in Section 7.
For the proof described above it is important to point out that all the estimates in Sections 3–6 will be uniform with respect to the choice of 𝝃∈𝒫k,η and of small values of λ and 𝜹. For this reason, given two quantities Θ1, Θ2 depending on λ,𝜹,𝝃 and η (and eventually other parameters), it is convenient to write Θ1=O(Θ2) to indicate |Θ1|≤CΘ2, for some constant C>0 that does not depend on 𝝃, 𝜹 and λ (but may depend on η and the other parameters, unless otherwise specified). This notation will be used several times throughout the paper.
3 Choice of the Concentration Parameters and Estimates of the Error Terms
Let ω𝒂,𝜹,𝝃 be as in (2.4). In order to perform the perturbation argument explained before, we need to be sure that ω𝒂,𝜹,𝝃 is a good approximate solution to (1.1). This means we need to estimate the error term
E
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
-
(
-
Δ
)
1
2
ω
𝒂
,
𝜹
,
𝝃
,
as defined in Section 2. As a first step, we need the following lemma.
Lemma 3.1.
For any η>0, there exists a constant Cη such that
|
P
U
δ
,
ξ
-
U
δ
,
ξ
+
log
(
2
δ
)
-
2
π
H
(
ξ
,
⋅
)
|
≤
C
η
δ
2
,
for any δ∈(0,1), ξ∈(-1+η,1-η). In particular, we have that
P
U
δ
,
ξ
=
{
U
δ
,
ξ
-
log
(
2
δ
)
+
2
π
H
(
ξ
,
ξ
)
+
O
(
|
⋅
-
ξ
|
)
+
O
(
δ
2
)
uniformly in
(
ξ
-
η
2
,
ξ
+
η
2
)
,
2
π
G
ξ
+
O
(
δ
2
)
uniformly in
I
∖
(
ξ
-
η
2
,
ξ
+
η
2
)
,
independently of the choice of ξ∈(-1+η,1-η) and δ∈(0,1).
Proof.
Let uδ,ξ:=PUδ,ξ-Uδ,ξ+log(2δ)-2πH(ξ,⋅). First, we observe that
(
-
Δ
)
1
2
u
δ
,
ξ
=
0
in
I
,
since (-Δ)12H(ξ,⋅)=0 on I and (-Δ)12PUδ,ξ=(-Δ)12Uδ,ξ in I (by the definition of PUδ,ξ). Next, we study the values of uδ,ξ in ℝ∖I. Here, since PUδ,ξ∈X01/2(I), we have PUδ,ξ=0. Then, recalling the expression of Uδ,ξ given in (1.9), and noting that 2πH(ξ,x)=2log|x-ξ| in ℝ∖I, we find that
u
δ
,
ξ
(
x
)
=
-
U
δ
,
ξ
(
x
)
+
log
(
2
δ
)
-
2
log
|
x
-
ξ
|
=
log
(
δ
2
+
|
x
-
ξ
|
2
)
-
2
log
|
x
-
ξ
|
.
Since x∈ℝ∖I and ξ∈(-1+η,1-η), we get |x-ξ|≥η. Then we can find Cη s.t. |uδ,ξ|≤Cηδ2 in ℝ∖I. By the Maximum Principle (see [33, Lemma 6]) we have the desired result. ∎
Next, we shall fix δ1,…,δk in order to make the error term E small near each of the points ξ1,…,ξk. Note that, for any 1≤i≤k, in the interval (ξi-η2,ξi+η2) we have the uniform expansion
(3.1)
E
=
λ
e
∑
j
=
1
k
a
j
P
U
δ
j
,
ξ
j
-
λ
e
-
∑
j
=
1
k
a
j
P
U
δ
j
,
ξ
j
-
∑
j
=
1
k
a
j
e
U
δ
j
,
ξ
j
=
λ
(
2
δ
i
)
a
i
e
a
i
U
δ
i
,
ξ
i
+
2
π
a
i
H
(
ξ
i
,
ξ
i
)
+
2
π
∑
j
≠
i
a
j
G
ξ
j
+
∑
i
=
1
k
O
(
δ
i
2
)
+
O
(
|
⋅
-
ξ
i
|
)
-
λ
(
2
δ
i
)
a
i
e
-
a
i
U
δ
i
,
ξ
i
-
2
π
a
i
H
(
ξ
i
,
ξ
i
)
-
2
π
∑
j
≠
i
a
j
G
ξ
j
+
∑
i
=
1
k
O
(
δ
i
2
)
+
O
(
|
⋅
-
ξ
i
|
)
-
a
i
e
U
δ
i
,
ξ
i
-
∑
j
≠
i
a
j
e
U
δ
j
,
ξ
j
=
λ
a
i
2
δ
i
e
U
δ
i
,
ξ
i
e
2
π
H
(
ξ
i
,
ξ
i
)
+
2
π
a
i
∑
j
≠
i
a
j
G
ξ
j
+
O
(
|
𝜹
|
2
)
+
O
(
⋅
-
ξ
i
|
)
+
O
(
λ
δ
i
e
-
U
δ
i
,
ξ
i
)
-
a
i
e
U
δ
i
,
ξ
i
-
∑
j
≠
i
a
j
e
U
δ
j
,
ξ
j
,
where we have used Lemma 3.1 and ai∈{-1,1}, i=1,…,k. Moreover, we have that
(3.2)
δ
i
e
-
U
δ
i
,
ξ
i
(
x
)
=
δ
i
2
+
|
x
-
ξ
i
|
2
2
=
O
(
δ
i
2
)
+
O
(
|
x
-
ξ
i
|
2
)
and, for j≠i, that
e
U
δ
j
,
ξ
j
(
x
)
=
2
δ
j
δ
j
2
+
|
x
-
ξ
j
|
2
=
2
δ
j
|
x
-
ξ
j
|
2
+
O
(
δ
j
3
)
=
2
δ
j
|
ξ
i
-
ξ
j
|
2
+
O
(
δ
j
|
x
-
ξ
i
|
)
+
O
(
δ
j
3
)
=
O
(
δ
j
)
.
For i∈{1,…,k}, let us consider the functions
(3.3)
F
i
(
𝝃
)
:=
2
π
H
(
ξ
i
,
ξ
i
)
+
2
π
a
i
∑
j
≠
i
a
j
G
ξ
j
(
ξ
i
)
,
so that estimate (3.1) rewrites as
(3.4)
E
=
a
i
e
U
δ
i
,
ξ
i
(
λ
2
δ
i
e
F
i
(
𝝃
)
+
O
(
|
𝜹
|
2
)
+
O
(
|
⋅
-
ξ
i
|
)
-
1
)
+
O
(
λ
)
+
∑
j
≠
i
O
(
δ
j
)
.
In order to make the main term of the above expansion small, we choose
(3.5)
δ
i
=
δ
i
(
λ
,
𝝃
)
:=
λ
2
e
F
i
(
𝝃
)
=
λ
2
e
2
π
H
(
ξ
i
,
ξ
i
)
+
2
π
a
i
∑
j
≠
i
a
j
G
(
ξ
i
,
ξ
j
)
,
i
=
1
,
…
,
k
.
With this choice, we get the following integral estimate on E.
Lemma 3.2.
Let 𝛅=(δ1,…,δk) be as in (3.5). For any p∈(1,∞) one has
∥
E
∥
L
p
(
I
)
=
O
(
λ
1
p
)
,
uniformly with respect to the choice of λ∈(0,1) and 𝛏=(ξ1,…,ξk)∈Pk,η.
Proof.
Thanks to (3.5), we have δi=O(λ), i=1,…,k uniformly for 𝝃∈𝒫k,η. Then (3.4) and (3.5) yield
E
(
x
)
=
a
i
e
U
δ
i
,
ξ
i
(
O
(
λ
2
)
+
O
(
|
x
-
ξ
i
|
)
)
+
O
(
λ
)
=
O
(
e
U
δ
i
,
ξ
i
|
x
-
ξ
i
|
)
+
O
(
λ
)
,
uniformly in (ξi-η2,ξi+η2), i=1,…,k, where the last equality follows by eUδi,ξi≤2δi=O(λ-1). Moreover, using Lemma 3.1, we get
E
=
λ
e
2
π
∑
j
=
1
k
a
j
G
ξ
j
+
O
(
λ
2
)
-
λ
e
-
2
π
∑
j
=
1
k
a
j
G
ξ
j
+
O
(
λ
2
)
-
∑
j
=
1
k
a
j
e
U
δ
j
,
ξ
j
=
O
(
λ
)
,
uniformly in I∖⋃i=1k(ξi-η2,ξi+η2). Using these estimates, we can assert that
∥
E
∥
L
p
(
I
)
p
=
∑
i
=
1
k
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
O
(
e
U
δ
i
,
ξ
i
|
x
-
ξ
i
|
)
+
O
(
λ
)
)
p
𝑑
x
+
∫
I
∖
⋃
i
=
1
k
(
ξ
i
-
η
2
,
ξ
i
+
η
2
)
O
(
λ
)
p
𝑑
x
=
∑
i
=
1
k
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
O
(
e
U
δ
i
,
ξ
i
|
x
-
ξ
i
|
)
)
p
𝑑
x
+
O
(
λ
)
p
.
Since p>1, with the change of variable y=|x-ξi|δi, we find
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
e
U
δ
i
,
ξ
i
|
x
-
ξ
i
|
)
p
𝑑
x
=
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
2
δ
i
|
x
-
ξ
i
|
δ
i
2
+
|
x
-
ξ
i
|
2
)
p
𝑑
x
=
δ
i
∫
-
η
2
δ
i
η
2
δ
i
(
2
y
1
+
y
2
)
p
𝑑
y
=
O
(
δ
i
)
=
O
(
λ
)
for i=1,…,k. We can so conclude that
∥
E
∥
L
p
(
I
)
p
=
O
(
λ
)
+
O
(
λ
p
)
=
O
(
λ
)
for all
p
>
1
.
∎
.
Remark 3.3.
Using the change of variable of the proof above, one can easily verify that, for any p,q≥0, the following useful estimate holds as δ→0, uniformly with respect to ξ∈ℝ:
(3.6)
∫
ξ
-
η
2
ξ
+
η
2
e
p
U
δ
,
ξ
|
x
-
ξ
|
q
𝑑
x
=
{
O
(
δ
q
-
p
+
1
)
if
2
p
-
q
>
1
,
O
(
δ
p
|
log
δ
|
)
if
2
p
-
q
=
1
,
O
(
δ
p
)
if
2
p
-
q
<
1
.
Remark 3.4.
For p=1, the argument of Lemma 3.2 gives
∥
E
∥
L
1
(
I
)
=
O
(
λ
|
log
λ
|
)
.
3.1 Estimates on the Non-Linear Error Term
In this subsection we look for estimates on the non-linear error term
N
(
φ
)
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
)
-
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
,
as defined in Section 2. The following lemma shows that N depends quadratically on φ.
Lemma 3.5.
Let 𝛅=(δ1,…,δk) be as in (3.5). For any p≥1 and s>p, there exists a constant Cp,s,η>0, depending only on p, s and η, such that
∥
N
(
φ
1
)
-
N
(
φ
2
)
∥
L
p
(
I
)
≤
C
p
,
s
,
η
λ
1
s
-
1
∥
φ
1
-
φ
2
∥
(
∥
φ
1
∥
+
∥
φ
2
∥
)
for any λ∈(0,1), 𝛏∈Pk,η and φ1,φ2∈X01/2(I) satisfying ∥φ1∥,∥φ2∥≤1.
Proof.
First of all, we observe that for any x∈I, there exist t1=t1(x),t2=t2(x)∈[0,1] such that
N
(
φ
1
)
-
N
(
φ
2
)
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
1
)
-
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
2
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
1
+
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
2
=
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
t
1
φ
1
+
(
1
-
t
1
)
φ
2
)
(
φ
1
-
φ
2
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
(
φ
1
-
φ
2
)
=
f
λ
′′
(
ω
𝒂
,
𝜹
,
𝝃
+
t
2
t
1
φ
1
+
t
2
(
1
-
t
1
)
φ
2
)
(
φ
1
-
φ
2
)
(
t
1
φ
1
+
(
1
-
t
1
)
φ
2
)
.
Denoting τ1=t1t2∈[0,1] and τ2=t2(1-t1)∈[0,1], we get that
|
N
(
φ
1
)
-
N
(
φ
2
)
|
≤
|
f
λ
′′
(
ω
𝒂
,
𝜹
,
𝝃
+
τ
1
φ
1
+
τ
2
φ
2
)
|
|
φ
1
-
φ
2
|
|
t
1
φ
1
+
(
1
-
t
1
)
φ
2
|
≤
|
f
λ
′′
(
ω
𝒂
,
𝜹
,
𝝃
+
τ
1
φ
1
+
τ
2
φ
2
)
|
|
φ
1
-
φ
2
|
(
|
φ
1
|
+
|
φ
2
|
)
.
Noting that fλ′′=fλ and that |fλ(t)|≤2λe|t|, we get
(3.7)
|
N
(
φ
1
)
-
N
(
φ
2
)
|
≤
2
λ
e
|
ω
𝒂
,
𝜹
,
𝝃
|
+
|
φ
1
|
+
|
φ
2
|
|
φ
1
-
φ
2
|
(
|
φ
1
|
+
|
φ
2
|
)
≤
2
λ
e
|
ω
𝒂
,
𝜹
,
𝝃
|
+
φ
3
|
φ
1
-
φ
2
|
φ
3
,
where φ3=|φ1|+|φ2|. Additionally, for any choice of s>p≥1, we can find s1,s2,s3>1 such that
1
s
1
+
1
s
2
+
1
s
3
+
1
s
=
1
p
.
Then Hölder’s inequality implies that
(3.8)
∥
λ
e
|
ω
𝒂
,
𝜹
,
𝝃
|
e
|
φ
3
|
|
φ
1
-
φ
2
|
φ
3
∥
L
p
(
I
)
≤
∥
λ
e
ω
𝒂
,
𝜹
,
𝝃
∥
L
s
(
I
)
∥
e
φ
3
∥
L
s
1
(
I
)
∥
φ
1
-
φ
2
∥
L
s
2
(
I
)
∥
φ
3
∥
L
s
3
(
I
)
.
Now, using Lemma 3.1, we see that λe|ω𝒂,𝜹,𝝃|=O(λ) in I∖⋃i=1k(ξi-η2,ξi+η2), and that
λ
e
|
ω
𝒂
,
𝜹
,
𝝃
|
=
λ
e
P
U
δ
i
,
ξ
i
+
O
(
1
)
=
O
(
e
U
δ
i
,
ξ
i
)
in (ξi-η2,ξi+η2) for i=1,…,k. Therefore
(3.9)
∥
λ
e
|
ω
𝒂
,
𝜹
,
𝝃
|
∥
L
s
(
I
)
s
=
∑
i
=
1
k
∫
ξ
i
-
η
2
ξ
i
+
η
2
O
(
e
s
U
δ
i
,
ξ
i
)
𝑑
x
+
O
(
λ
s
)
=
O
(
λ
1
-
s
)
+
O
(
λ
s
)
=
O
(
λ
1
-
s
)
,
where we used (3.6) and 1-s<s. Note that the quantity O(λ1-s) depends on η and s.
Using the Moser–Trudinger inequality (see [24]), we get that
(3.10)
∫
ℝ
e
s
1
|
φ
3
|
𝑑
x
≤
e
s
1
2
4
π
∥
φ
3
∥
2
∫
I
e
π
φ
3
2
∥
φ
3
∥
2
𝑑
x
≤
C
e
s
1
2
4
π
∥
φ
3
∥
2
≤
C
(
s
1
)
.
Finally, thanks to Sobolev’s inequality, we have the estimates
(3.11)
∥
φ
1
-
φ
2
∥
L
s
2
(
I
)
≤
C
(
s
2
)
∥
φ
1
-
φ
2
∥
and
∥
φ
3
∥
L
s
3
(
I
)
≤
C
(
s
3
)
∥
φ
3
∥
≤
C
(
s
3
)
(
∥
φ
1
∥
+
∥
φ
2
∥
)
.
Thus, replacing (3.8)–(3.11) into (3.7), we obtain
∥
N
(
φ
1
)
-
N
(
φ
2
)
∥
L
p
(
I
)
≤
C
λ
1
-
s
s
∥
φ
1
-
φ
2
∥
(
∥
φ
1
∥
+
∥
φ
2
∥
)
,
with C depending only on η,s,s1,s2 and s3. Since the choice of s1,s2 and s3 depends only on s and p, we get the conclusion. ∎
Remark 3.6.
Repeating the argument of the above proof, we can show that, for any s,s1>p≥1 such that 1s1+1s<1p, there exists a constant C=C(p,s,s1,η) such that
∥
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
)
-
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
∥
L
p
(
I
)
+
∥
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
∥
L
p
(
I
)
≤
C
λ
1
-
s
s
e
s
1
2
4
π
∥
φ
∥
2
∥
φ
∥
for any φ∈X01/2(I), 𝝃∈𝒫k,η and λ∈(0,1). Note that whenever s>p, it is possible to choose s1>p large enough so that 1s1+1s<1p.
4 Properties of the Linearized Operator
This section is devoted to the study of the linear operator L:X01/2(I)→X01/2(I), defined as
L
φ
=
φ
-
(
-
Δ
)
-
1
2
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
.
In particular, we are interested in exhibiting an approximate kernel of L. As a first step we describe the behavior of the term fλ′(ω𝒂,𝜹,𝝃).
Lemma 4.1.
For any i=1,…,k, we have the expansion
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
=
{
e
U
δ
i
,
ξ
i
(
1
+
O
(
|
⋅
-
ξ
i
|
)
+
O
(
λ
2
)
)
uniformly in
(
ξ
i
-
η
2
,
ξ
i
+
η
2
)
,
O
(
λ
)
uniformly in
I
∖
⋃
i
=
1
k
(
ξ
i
-
η
2
,
ξ
i
+
η
2
)
.
In particular, ∥fλ′(ω𝐚,𝛅,𝛏)∥L1(I)=O(1).
Proof.
Indeed, arguing as in (3.1), we get
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
=
λ
(
2
δ
i
)
a
i
e
a
i
U
δ
i
,
ξ
i
+
2
π
a
i
H
(
ξ
i
,
⋅
)
+
2
π
∑
j
≠
i
a
j
G
ξ
j
+
O
(
λ
2
)
+
λ
(
2
δ
i
)
a
i
e
-
a
i
U
δ
i
,
ξ
i
-
2
π
a
i
H
(
ξ
i
,
⋅
)
-
2
π
∑
j
≠
i
a
j
G
ξ
j
+
O
(
λ
2
)
=
λ
2
δ
i
e
U
δ
i
,
ξ
i
+
F
i
(
𝝃
)
+
O
(
|
⋅
-
ξ
i
|
)
+
O
(
λ
2
)
+
O
(
λ
δ
i
e
-
U
δ
i
,
ξ
i
)
=
e
U
δ
i
,
ξ
i
+
O
(
|
⋅
-
ξ
i
|
)
+
O
(
λ
2
)
+
O
(
λ
)
=
e
U
δ
i
,
ξ
i
(
1
+
O
(
|
⋅
-
ξ
i
|
)
+
O
(
λ
2
)
)
+
O
(
λ
)
=
e
U
δ
i
,
ξ
i
(
1
+
O
(
|
⋅
-
ξ
i
|
)
+
O
(
λ
2
)
)
in (ξi-η2,ξi+η2), where in the last equality we used (3.2) to estimate O(λ) as
O
(
λ
)
=
e
U
δ
i
,
ξ
i
(
x
)
O
(
λ
e
-
U
δ
i
,
ξ
i
(
x
)
)
=
e
U
δ
i
,
ξ
i
(
x
)
(
O
(
λ
2
)
+
O
(
|
x
-
ξ
|
2
)
)
.
Moreover, fλ′(ω𝒂,𝜹,𝝃)=O(λ) in I∖⋃i=1k(ξi-η2,ξi+η2), since on this set we have
ω
𝒂
,
𝜹
,
𝝃
=
2
π
∑
j
=
1
k
a
j
G
ξ
j
+
O
(
λ
2
)
=
O
(
1
)
.
∎
Next, we focus on the kernel of L. Observe that if φ∈X01/2(I) and Lφ=0, then the scaled functions
Φ
i
(
x
)
:=
φ
(
ξ
i
+
δ
i
x
)
are weak solutions to
(
-
Δ
)
1
2
Φ
i
+
δ
i
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
(
ξ
i
+
δ
i
⋅
)
)
=
0
,
i
=
1
,
…
,
k
,
in the expanding intervals (-ξi-1δi,1-ξiδi). According to Lemma 4.1, for any fixed y∈ℝ, we have
δ
i
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
(
ξ
i
+
δ
i
y
)
)
∼
δ
i
e
U
δ
i
,
ξ
i
(
ξ
i
+
δ
i
y
)
=
2
1
+
y
2
.
Then Φi should behave locally as a solution of the problem
(4.1)
(
-
Δ
)
1
2
Φ
=
2
Φ
1
+
|
⋅
|
2
in
ℝ
.
This equation was studied by many authors. In particular, Santra [32, Theorem 1.4] (see also [11]) proved that the only bounded solutions to (1.10) are linear combinations of the functions
(4.2)
Z
0
(
y
)
:=
1
-
y
2
1
+
y
2
and
Z
1
(
y
)
:=
2
y
1
+
y
2
.
Here we will need a small modification of this classification result. Let us consider the spaces
(4.3)
ℒ
:=
{
u
∈
L
loc
1
(
ℝ
)
:
|
u
|
2
(
1
+
x
2
)
-
1
∈
L
1
(
ℝ
)
}
,
ℋ
:=
{
u
∈
ℒ
:
(
-
Δ
)
1
4
u
∈
L
2
(
ℝ
)
}
.
These spaces are endowed with the norms
∥
u
∥
ℒ
2
=
∫
ℝ
u
(
x
)
2
1
+
|
x
|
2
𝑑
x
and
∥
u
∥
ℋ
2
=
∥
(
-
Δ
)
1
4
u
∥
L
2
(
ℝ
)
2
+
∥
u
∥
ℒ
2
.
It is known that one can construct an isometry between L2(ℝ) and ℒ and between H12(S1) and ℋ via the standard stereographic projection. In particular, ℋ is compactly embedded into ℒ.
Lemma 4.2.
Let Φ∈H be a solution to (4.1). Then there exist κ0,κ1∈R such that Φ=κ0Z0+κ1Z1, where Z0 and Z1 are the functions in (4.2).
Proof.
First of all, we observe that any solution φ to (4.1) is smooth. This follows by standard regularity results (see [22, Theorem 13], the appendix in [19], and [31, Corollaries 2.4 and 2.5])
Using the density of Cc∞(ℝ) in ℋ (which can be proved using the arguments of [16, Lemmas 11 and 12], since ∥(-Δ)14u∥L2(ℝ) is equivalent to the Gagliardo seminorm), we can find a sequence ψn∈Cc∞(ℝ) such that ψn→1 in ℋ (note that constant functions belong to ℋ). Then, for any n, have
∫
ℝ
(
-
Δ
)
1
4
φ
(
-
Δ
)
1
4
ψ
n
𝑑
x
=
∫
ℝ
φ
(
-
Δ
)
1
2
ψ
n
𝑑
x
=
∫
ℝ
f
φ
ψ
n
𝑑
x
,
where fφ(x):=2φ(x)1+x2. Passing to the limit as n→∞, we get
(4.4)
∫
ℝ
f
φ
𝑑
x
=
0
.
Let us now consider the functions
Γ
(
x
,
y
)
=
1
π
log
(
1
+
|
y
|
|
x
-
y
|
)
and
Φ
(
x
)
:=
∫
ℝ
Γ
(
x
,
y
)
f
φ
(
y
)
𝑑
y
.
Since φ∈ℒ⊆L12(ℝ), according to [18, Lemma 2.4], we have φ=Φ+c for some c∈ℝ. Now, observing that Γ(1x,y)=log|x|π+Γ(x,1y) for any x,y∈ℝ∖{0} with x≠1y, we have that
(4.5)
φ
(
1
x
)
=
Φ
(
1
x
)
+
c
=
log
|
x
|
π
∫
ℝ
f
φ
(
y
)
𝑑
y
⏟
=
0
by (4.4)
+
∫
ℝ
Γ
(
x
,
1
y
)
f
φ
(
y
)
𝑑
y
+
c
=
2
∫
ℝ
Γ
(
x
,
z
)
φ
(
1
z
)
1
+
z
2
𝑑
z
+
c
for a.e. x∈ℝ∖{0}. Denoting φ~(x):=φ(1x) and fφ~(x):=2φ~(x)1+x2, via a simple change of variable we can show that φ~∈ℒ and fφ~∈L1(ℝ). Since fφ~∈L1(ℝ), [18, Lemma 2.3] implies that
Φ
~
(
x
)
:=
∫
ℝ
Γ
(
x
,
z
)
f
φ
~
(
z
)
𝑑
z
is a distributional solution to (-Δ)12Φ~=fφ~ in ℝ. Moreover, using that φ~∈ℒ (and in particular fφ~∈Lloc2(ℝ)), we can repeat the first part of the proof and show that Φ~∈C∞(ℝ). By (4.5), we infer that φ~ can be extended to a smooth function on ℝ. In particular, this gives that φ∈L∞(ℝ)∩C∞(ℝ). Then we can conclude using directly the classification result in [32, Theorem 1.4]. ∎
In the following, for 𝝃∈𝒫k,η and λ>0, we shall denote Zi,j(x):=Zi(x-ξjδj), i=0,1, j=1,…,k, where δj is defined as in (3.5). Namely, we consider
(4.6)
Z
0
,
j
(
x
)
:=
δ
j
2
-
(
x
-
ξ
j
)
2
δ
j
2
+
(
x
-
ξ
j
)
2
and
Z
1
,
j
(
x
)
:=
2
δ
j
(
x
-
ξ
j
)
δ
j
2
+
(
x
-
ξ
j
)
2
,
which are solutions of the problem
(
-
Δ
)
1
2
φ
=
e
U
δ
j
,
ξ
j
φ
in
ℝ
.
We let PZi,j:=(-Δ)-12(eUδj,ξjZi,j) be the projection of Zi,j on X01/2(I). Then we have the following expansions.
Lemma 4.3.
As λ→0, we have
P
Z
0
,
j
=
Z
0
,
j
+
1
+
O
(
λ
2
)
,
P
Z
1
,
j
=
Z
1
,
j
+
2
δ
j
∂
H
∂
ξ
(
ξ
j
,
⋅
)
+
O
(
λ
3
)
,
uniformly in R, for j=1,…,k. In particular, PZ0,j=O(λ2) and PZ1,j=O(λ) in R∖(ξj-η2,ξj+η2).
Proof.
First, note that for any x∈ℝ the function ξ→H(ξ,x) belongs to C1(I), with derivative
∂
H
∂
ξ
(
ξ
,
x
)
=
{
-
1
π
x
+
ξ
1
-
x
2
1
-
ξ
2
1
-
x
ξ
+
(
1
-
ξ
2
)
(
1
-
x
2
)
for
x
∈
I
,
1
π
1
ξ
-
x
for
x
∈
ℝ
∖
I
.
We claim that ∂H∂ξ(ξ,⋅) is 12-harmonic in I, for any ξ∈I. We prove that this is true in the sense of distributions. To show this, we observe that
∫
ℝ
∂
H
∂
ξ
(
ξ
,
x
)
(
-
Δ
)
1
2
φ
(
x
)
𝑑
x
=
0
for all
φ
∈
C
c
∞
(
I
)
.
Indeed, if we take ψ∈Cc∞(-1,1), we have
∫
ℝ
ψ
(
ξ
)
∫
ℝ
∂
H
∂
ξ
(
ξ
,
x
)
(
-
Δ
)
1
2
φ
(
x
)
𝑑
x
𝑑
ξ
=
∫
ℝ
(
-
Δ
)
1
2
φ
(
x
)
∫
ℝ
ψ
(
ξ
)
∂
H
∂
ξ
(
ξ
,
x
)
𝑑
ξ
𝑑
x
=
-
∫
ℝ
(
-
Δ
)
1
2
φ
(
x
)
∫
ℝ
ψ
′
(
ξ
)
H
(
ξ
,
x
)
𝑑
ξ
𝑑
x
=
-
∫
ℝ
ψ
′
(
ξ
)
∫
ℝ
(
-
Δ
)
1
2
φ
(
x
)
H
(
ξ
,
x
)
𝑑
x
𝑑
ξ
=
0
,
where the last equality follows from (-Δ)12H(ξ,x)=0. Since φ and ψ are arbitrary, we have proved the claim.
Now, the statement can be proved as in Lemma 3.1. Let us fix 1≤j≤k. Since ∂H∂ξ(ξj,⋅) is 12-harmonic in I, the definitions of PZ0,j and PZ1,j imply that also the functions
v
0
,
j
:=
P
Z
0
,
j
-
Z
0
,
j
-
1
and
v
1
,
j
:=
P
Z
1
,
j
-
Z
1
,
j
-
2
π
δ
j
∂
H
∂
ξ
(
ξ
j
,
⋅
)
are 12-harmonic in I. Additionally, for x∈ℝ∖I, we have that
v
0
,
j
(
x
)
=
-
δ
j
2
-
(
x
-
ξ
j
)
2
δ
j
2
+
(
x
-
ξ
j
)
2
-
1
=
O
(
δ
j
2
)
=
O
(
λ
2
)
,
v
1
,
j
(
x
)
=
-
2
δ
j
(
x
-
ξ
j
)
δ
j
2
+
(
x
-
ξ
j
)
2
+
2
δ
j
(
x
-
ξ
j
)
=
O
(
δ
j
3
)
=
O
(
λ
3
)
.
Thus, we conclude via the maximum principle as in the proof of Lemma 3.1. ∎
Remark 4.4.
For i,j∈{0,1} and h,l∈{1,…,k}, we have the orthogonality condition
∫
ℝ
(
-
Δ
)
1
4
P
Z
i
,
h
⋅
(
-
Δ
)
1
4
P
Z
j
,
l
𝑑
x
=
∫
ℝ
e
U
δ
h
,
ξ
h
Z
i
,
h
P
Z
j
,
l
𝑑
x
=
π
δ
i
,
j
δ
h
,
l
+
O
(
λ
)
,
where δi,j denotes the Kronecker delta symbol. Indeed, for h≠l we have PZj,l=O(λ) in ℝ∖(ξl-η2,ξl+η2) and eUδh,ξh=O(λ) in (ξl-η2,ξl+η2), while for h=l, we have
∫
ℝ
e
U
δ
l
,
ξ
l
Z
i
,
h
P
Z
j
,
l
𝑑
x
=
∫
ℝ
e
U
δ
l
,
ξ
l
Z
i
,
h
Z
j
,
l
𝑑
x
+
O
(
λ
)
=
∫
ℝ
2
Z
i
(
y
)
Z
j
(
y
)
1
+
y
2
𝑑
y
+
O
(
λ
)
=
π
δ
i
,
j
+
O
(
λ
)
.
A standard procedure consists in inverting the operator L on the orthogonal of the space generated by the functions PZi,j, i=0,1, j=1,…,k, which can be considered as an approximate kernel for L. However, Lemma 4.3 shows that PZ0,j is not close to Z0,j, as their difference approaches 1 as λ→0. For this reason we can construct a smaller approximate kernel for L using only the functions PZ1,j, j=1,…,k.
In the following, for 𝜹=(δ1,…,δk) defined as in (3.5) and for any 𝝃∈𝒫k,η, we shall denote
K
𝜹
,
𝝃
=
〈
{
P
Z
1
,
j
,
j
∈
{
1
,
…
,
k
}
}
〉
.
Let also π and π⊥ be the projections of X01/2(I) respectively into K𝜹,𝝃 and K𝜹,𝝃⊥. We now establish the invertibility of L on K𝜹,𝝃⊥.
Lemma 4.5.
There exist λ¯, C>0 such that
∥
ψ
∥
≤
C
|
log
λ
|
∥
π
⊥
L
ψ
∥
for any λ∈(0,λ¯), 𝛏∈Pk,η and ψ∈K𝛅,𝛏⊥, with 𝛅 given by (3.5).
Proof.
We argue by contradiction. Suppose that there exist sequences λn→0, 𝝃n=(ξ1,n,…,ξk,n)∈𝒫k,η, and ψn∈K𝜹n,𝝃n (where 𝜹n=(δ1,n,…,δk,n) with δi,n=δi,n(λn,ξn) given by (3.5)) such that
∥
ψ
n
∥
=
1
and
|
log
λ
n
|
∥
h
n
∥
→
0
,
where
h
n
:=
π
⊥
L
ψ
n
.
Throughout this proof we will write fn:=fλn, ωn:=ω𝒂,𝜹n,𝝃n and Ui,n=Uδi,n,ξi,n. For any i=1,…,k, we also let Z0,i,n and Z1,i,n denote the functions in (4.6) with ξi=ξi,n and δi=δi,n.
By the definition of π⊥ there exists ζn∈K𝜹n,𝝃n such that Lψn=hn+ζn. This means that
(4.7)
∫
ℝ
(
-
Δ
)
1
4
ψ
n
(
-
Δ
)
1
4
v
𝑑
x
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
v
𝑑
x
+
∫
ℝ
(
-
Δ
)
1
4
h
n
(
-
Δ
)
1
4
v
𝑑
x
+
∫
ℝ
(
-
Δ
)
1
4
ζ
n
(
-
Δ
)
1
4
v
𝑑
x
for any v∈X01/2(I). Note that taking v=ψn∈K𝜹n,𝝃n⊥, one finds
∥
ψ
n
∥
2
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
2
𝑑
x
+
∫
ℝ
(
-
Δ
)
1
4
ψ
n
(
-
Δ
)
1
4
h
n
𝑑
x
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
2
𝑑
x
+
O
(
∥
h
n
∥
)
,
from which we get
(4.8)
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
2
𝑑
x
→
1
,
as n→∞. Since fn′(ωn) is bounded in L1(I) by Lemma 4.1, Hölder’s inequality also gives
(4.9)
∫
ℝ
f
n
′
(
ω
n
)
|
ψ
n
|
𝑑
x
≤
(
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
2
𝑑
x
)
1
2
(
∫
I
f
n
′
(
ω
n
)
𝑑
x
)
1
2
=
O
(
1
)
.
Keeping in mind the relations above, we split the rest of the proof into several steps.
Step 1.
Since ζn∈K𝛅n,𝛏n, we can write ζn=∑i=1kci,nPZ1,i,n.
We have ci,n=O(∥ζn∥) for i=1,…,k. In particular, ∥ζn∥L∞(R)=O(∥ζn∥).
Indeed, setting c¯n=max{|ci,n|:0≤i≤k} and using Remark 4.4, we find that
∥
ζ
n
∥
2
=
∑
i
=
1
k
∑
j
=
1
k
c
i
,
n
c
j
,
n
∫
ℝ
(
-
Δ
)
1
4
P
Z
1
,
i
,
n
(
-
Δ
)
1
4
P
Z
1
,
j
,
n
𝑑
x
=
π
∑
i
=
1
k
c
i
,
n
2
+
O
(
λ
n
c
¯
n
2
)
≥
π
c
¯
n
2
+
O
(
λ
n
c
¯
n
2
)
.
Since λn→0, this implies that c¯n=O(∥ζn∥).
Step 2.
For i=1,…,k, and s=1,2, we have that
(4.10)
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
|
f
n
′
(
ω
n
)
-
e
U
i
,
n
|
|
ψ
n
|
s
𝑑
x
=
O
(
λ
n
)
.
Moreover, we have
(4.11)
∫
ℝ
e
U
i
,
n
|
ψ
n
|
s
𝑑
x
=
O
(
1
)
𝑎𝑛𝑑
∫
ℝ
|
f
n
′
(
ω
n
)
-
e
U
i
,
n
|
|
ψ
n
|
s
|
P
Z
j
,
i
,
n
|
𝑑
x
=
O
(
λ
n
)
,
j
=
0
,
1
.
Indeed, in view of Lemma 4.1, we have
(4.12)
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
|
f
n
′
(
ω
n
)
-
e
U
i
,
n
|
|
ψ
n
|
s
𝑑
x
=
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
O
(
|
x
-
ξ
i
,
n
|
)
𝑑
x
+
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
O
(
λ
n
2
)
𝑑
x
.
By Hölder’s inequality, estimate (3.6) and Sobolev’s inequality, we get
(4.13)
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
x
-
ξ
i
,
n
|
|
ψ
n
|
s
𝑑
x
≤
(
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
2
U
i
,
n
|
x
-
ξ
i
,
n
|
2
𝑑
x
)
1
2
∥
ψ
n
s
∥
L
2
(
I
)
=
O
(
λ
n
)
.
Furthermore, using again Lemma 4.1 and (4.8)–(4.9), we find that
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
𝑑
x
=
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
f
n
′
(
ω
n
)
|
ψ
n
|
s
𝑑
x
+
O
(
λ
n
)
+
O
(
λ
n
2
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
𝑑
x
)
=
O
(
1
)
+
O
(
λ
n
2
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
𝑑
x
)
,
which implies that
(4.14)
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
|
ψ
n
|
s
𝑑
x
=
O
(
1
)
.
Then we get (4.10) by substituting (4.13) and (4.14) in (4.12). The first estimate in (4.11) follows by (4.14) and the bound eUi,n=O(λn) in ℝ∖(ξi,n-η2,ξi,n+η2). Similarly, the second estimate in (4.11) is a consequence of (4.10) and the bounds PZ0,i,n=O(λn2), PZ1,i,n=O(λn) in ℝ∖(ξi,n-η2,ξi,n+η2).
Step 3.
We have ∥ζn∥=o(|logλn|-1) as n→∞.
Taking v=ζn in (4.7) and recalling that ζn∈K𝜹n,𝝃n, ψn,hn∈K𝜹n,𝝃n⊥ we find that
(4.15)
0
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
ζ
n
𝑑
x
+
∥
ζ
n
∥
2
=
∑
i
=
1
k
c
i
,
n
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
P
Z
1
,
i
,
n
𝑑
x
+
∥
ζ
n
∥
2
.
Now, for i=1,…,k, Step 2 and Lemma 4.3 give
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
P
Z
1
,
i
,
n
𝑑
x
=
∫
ℝ
e
U
i
,
n
P
Z
1
,
i
,
n
ψ
n
𝑑
x
+
O
(
λ
n
)
=
∫
ℝ
ψ
n
(
-
Δ
)
1
2
P
Z
1
,
i
,
n
𝑑
x
⏟
=
0
by
ψ
n
∈
K
𝜹
n
,
𝝃
n
⊥
+
O
(
λ
n
)
.
Then, using also Step 1, we can rewrite (4.15) as
∥
ζ
n
∥
=
O
(
λ
n
)
=
o
(
|
log
λ
n
|
-
1
)
.
Step 4.
For i=1,…,k, we have that
∫
ℝ
e
U
i
,
n
ψ
n
d
x
=
o
(
|
log
λ
n
|
-
1
)
𝑎𝑛𝑑
∫
ℝ
e
U
i
,
n
U
i
,
n
ψ
n
d
x
→
0
.
First of all, taking v=PZ0,i,n in (4.7), and using Steps 2–3 and Lemma 4.3, we find that
∫
ℝ
(
-
Δ
)
1
4
P
Z
0
,
i
,
n
⋅
(
-
Δ
)
1
4
ψ
n
𝑑
x
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
P
Z
0
,
i
,
n
𝑑
x
+
O
(
∥
P
Z
0
,
i
,
n
∥
∥
h
n
∥
)
+
O
(
∥
P
Z
0
,
i
,
n
∥
∥
ζ
n
∥
)
=
∫
ℝ
e
U
i
,
n
ψ
n
P
Z
0
,
i
,
n
d
x
+
o
(
|
log
λ
n
|
-
1
)
=
∫
ℝ
e
U
i
,
n
ψ
n
Z
0
,
i
,
n
d
x
+
∫
ℝ
e
U
i
,
n
ψ
n
d
x
+
o
(
|
log
λ
n
|
-
1
)
.
Besides, by the definition of PZ0,i,n, we have
∫
ℝ
(
-
Δ
)
1
4
P
Z
0
,
i
,
n
⋅
(
-
Δ
)
1
4
ψ
n
𝑑
x
=
∫
ℝ
e
U
i
,
n
ψ
n
Z
0
,
i
,
n
𝑑
x
.
If we combine the two estimates above, we find that
∫
ℝ
e
U
i
,
n
ψ
n
d
x
=
o
(
|
log
λ
n
|
-
1
)
.
Since ∥Ui,n∥L∞(I)=O(|logλn|) (in fact 0≤|x-ξi|≤2 implies 2δi,nδi,n2+4≤eUi,n≤2δi,n in I) and ψn=0 in ℝ∖I, we get the conclusion.
Step 5.
For i=1,…,k, the function Ψi,n:=ψn(ξi,n+δi,n⋅) satisfies Ψn→0 in L, where L is defined in (4.3).
First of all, we observe that
∥
Ψ
i
,
n
∥
=
∥
ψ
n
∥
=
1
and
2
∫
ℝ
|
Ψ
i
,
n
|
2
1
+
|
x
|
2
𝑑
x
=
∫
ℝ
e
U
i
,
n
ψ
n
2
𝑑
x
≤
C
,
by Step 2. Then Ψi,n is uniformly bounded in the space ℋ (see (4.3)), which is compactly embedded in ℒ. Thus, there exists Ψ∞∈ℋ such that, up to subsequences, we have Ψi,n⇀Ψ∞ weakly in ℋ and Ψi,n→Ψ∞ in ℒ as n→+∞. The weak convergence in ℋ implies that
∫
ℝ
(
-
Δ
)
1
4
Ψ
i
,
n
⋅
(
-
Δ
)
1
4
w
𝑑
x
→
∫
ℝ
(
-
Δ
)
1
4
Ψ
∞
⋅
(
-
Δ
)
1
4
w
𝑑
x
for any w∈Cc∞(ℝ). Besides, using (4.7) with v=vn:=w(⋅-ξi,nδi,n), we get
∫
ℝ
(
-
Δ
)
1
4
Ψ
i
,
n
⋅
(
-
Δ
)
1
4
w
𝑑
x
=
∫
ℝ
(
-
Δ
)
1
4
ψ
n
⋅
(
-
Δ
)
1
4
v
n
𝑑
x
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
v
n
𝑑
x
+
∫
ℝ
(
-
Δ
)
1
4
h
~
n
(
-
Δ
)
1
4
v
n
𝑑
x
,
where h~n=hn+ζn. Since h~n→0 (by Step 3), we get that
∫
ℝ
(
-
Δ
)
1
4
h
~
n
(
-
Δ
)
1
4
v
n
𝑑
x
≤
∥
h
~
n
∥
∥
v
n
∥
=
∥
h
~
n
∥
∥
w
∥
→
0
.
Moreover, noting that vn is supported in (ξi,n-Rδi,n,ξi,n+δi,nR) for some R>0, we have
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
v
n
𝑑
x
=
∫
ξ
i
,
n
-
R
δ
i
,
n
ξ
i
,
n
+
R
δ
i
,
n
e
U
i
,
n
(
1
+
O
(
|
x
-
ξ
i
,
n
|
)
+
O
(
λ
n
)
)
ψ
n
v
n
=
∫
-
R
R
2
1
+
y
2
(
1
+
O
(
λ
n
(
|
y
|
+
1
)
)
)
Ψ
i
,
n
w
𝑑
y
→
∫
ℝ
2
1
+
y
2
Ψ
∞
w
𝑑
y
,
where the convergence in the last line follows by the convergence of Ψi,n in ℒ. Then it follows that Ψ∞ is a solution in ℋ to the problem
(
-
Δ
)
1
2
Ψ
∞
=
2
1
+
x
2
Ψ
∞
in
ℝ
.
Then, by Lemma 4.2, there exist κ0,κ1∈ℝ such that Ψ∞=κ0Z0+κ1Z1. But using again the convergence in ℒ and recalling that ψn∈K𝜹i,n,𝝃i,n⊥, we have
0
=
∫
ℝ
(
-
Δ
)
1
4
ψ
n
(
-
Δ
)
1
4
P
Z
1
,
i
,
n
𝑑
x
=
∫
ℝ
ψ
n
e
U
i
,
n
Z
1
,
i
,
n
𝑑
x
=
∫
ℝ
2
Ψ
i
,
n
Z
1
1
+
y
2
𝑑
y
→
κ
0
∫
ℝ
2
Z
0
Z
1
1
+
y
2
𝑑
y
+
κ
1
∫
ℝ
2
Z
1
2
1
+
y
2
𝑑
y
=
π
κ
1
,
Hence, κ1=0. Similarly, thanks to Step 4, we know that
0
=
lim
n
→
∞
∫
ℝ
ψ
n
e
U
δ
i
,
n
,
ξ
i
,
n
(
U
δ
i
,
n
,
ξ
i
,
n
+
log
δ
i
,
n
)
𝑑
x
=
lim
n
→
∞
∫
ℝ
2
Ψ
i
,
n
1
+
y
2
log
(
2
1
+
y
2
)
𝑑
y
=
κ
0
∫
ℝ
2
Z
0
1
+
y
2
log
(
2
1
+
y
2
)
𝑑
y
=
π
κ
0
,
which implies κ0=0 and Ψ∞≡0.
Step 6.
Conclusion of the proof.
We know that
1
+
o
(
1
)
=
∫
ℝ
f
n
′
(
ω
n
)
ψ
n
2
𝑑
x
=
∑
i
=
1
k
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
ψ
n
2
𝑑
x
+
O
(
λ
n
)
by (4.8), Lemma 4.1 and Step 2. But, using Step 5, one gets
∫
ξ
i
,
n
-
η
2
ξ
i
,
n
+
η
2
e
U
i
,
n
ψ
n
2
𝑑
x
≤
∫
ℝ
e
U
i
,
n
ψ
n
2
𝑑
x
=
∫
ℝ
2
Ψ
i
,
n
2
1
+
y
2
𝑑
y
→
0
for any i=1,…,k. This gives a contradiction. ∎
The a priori estimates of Lemma 4.5 imply the following invertibility property.
Corollary 4.6.
For λ∈(0,λ¯) and 𝛏∈Pk,η, the operator A:=π⊥L:K𝛅,𝛏⊥→K𝛅,𝛏⊥ is invertible and
∥
A
-
1
∥
ℒ
(
K
𝜹
,
𝝃
⊥
)
=
O
(
|
log
λ
|
)
,
where ∥F∥L(K𝛅,𝛏⊥):=sup{h∈K𝛅,𝛏⊥:∥h∥≤1}∥Fh∥.
Proof.
By Lemma 4.5, for any φ∈K𝜹,𝝃⊥, we have ∥φ∥≤C|logλ|∥Aφ∥. In particular, A is injective. Since K𝜹,𝝃⊥ is a Hilbert space, and since A is a Fredholm operator of index 0 (indeed it decomposes as the identity of K𝜹,𝝃⊥ plus a compact operator), we can assert that A is invertible. Moreover, we have
∥
A
-
1
∥
ℒ
(
K
𝜹
,
𝝃
⊥
)
=
sup
{
h
∈
K
𝜹
,
𝝃
⊥
:
∥
h
∥
≤
1
}
∥
A
-
1
h
∥
≤
C
|
log
λ
|
.
∎
5 Fix Point Argument
As we have outlined in Section 2, equation (1.1) can be reduced to the couple of non-linear problems (2.6) and (2.7). With the notation of the previous section, let us consider the operator A=π⊥L:K𝜹,𝝃⊥→K𝜹,𝝃⊥. Thanks to Corollary 4.6, we can rewrite equation (2.6) as
φ
=
A
-
1
(
π
⊥
(
-
Δ
)
-
1
2
E
+
π
⊥
(
-
Δ
)
-
1
2
N
(
φ
)
)
.
We now prove that this equation admits a solution for any small λ and any 𝝃∈𝒫k,η.
Lemma 5.1.
Let p∈(1,2) be fixed. Then there exist κ=κ(p,η)>0 and λ0=λ0(p,η)>0 such that, for any 𝛏∈Pk,η and λ∈(0,λ0), the operator
T
(
φ
)
:=
A
-
1
(
π
⊥
(
-
Δ
)
-
1
2
E
+
π
⊥
(
-
Δ
)
-
1
2
N
(
φ
)
)
has a fixed point on
(5.1)
B
:=
{
φ
∈
K
𝜹
,
𝝃
⊥
:
∥
φ
∥
≤
κ
|
log
λ
|
λ
1
p
}
.
Proof.
By Corollary 4.6, Lemma 3.2, estimate (2.2), and the fact that the projection π⊥ reduces the norm, we can find constants CA and CE such that
(5.2)
∥
A
-
1
∥
ℒ
(
K
𝜹
,
𝝃
⊥
)
≤
C
A
|
log
λ
|
and
∥
π
⊥
(
-
Δ
)
1
2
E
∥
≤
C
E
λ
1
p
for any 𝝃∈𝒫k,η and any small λ. Note that CA and CE do not depend neither on 𝝃 nor on λ. Similarly, for any s>p, Lemma 3.5 and estimate (2.2) imply the existence of a constant CN, depending only on s,p,η such that
(5.3)
∥
π
⊥
(
-
Δ
)
1
2
(
N
(
φ
1
)
-
N
(
φ
2
)
)
∥
≤
C
N
λ
1
s
-
1
|
log
λ
|
∥
φ
1
-
φ
2
∥
(
∥
φ
1
∥
+
∥
φ
2
∥
)
,
for any 𝝃∈𝒫k,η, λ small enough and any φ1,φ2∈X01/2(I) with ∥φi∥≤1 for i=1,2.
Let us set κ:=2CACE. We shall prove that T is a contraction on B. First, taking λ small enough so that κλ1p|logλ|≤1, we get ∥φ∥≤1 for any φ∈B. Hence, (5.2) and (5.3) give
∥
T
(
φ
)
∥
≤
∥
A
-
1
∥
ℒ
(
K
𝜹
,
𝝃
⊥
)
∥
π
⊥
(
-
Δ
)
-
1
2
E
∥
+
∥
A
-
1
∥
ℒ
(
K
𝜹
,
𝝃
⊥
)
∥
π
⊥
(
-
Δ
)
-
1
2
N
(
φ
)
∥
≤
C
A
C
E
λ
1
p
|
log
λ
|
+
C
A
C
N
λ
1
s
-
1
∥
φ
∥
2
≤
C
A
C
E
λ
1
p
|
log
λ
|
(
1
+
4
C
A
2
C
E
C
N
λ
1
p
+
1
s
-
1
|
log
λ
|
)
,
where last inequality follows from the definition of B in (5.1) and our choice of κ. Since p∈(1,2), it is enough to take s∈(p,pp-1) and λ such that 4CA2CECN|logλ|2λ1p+1s-1≤1 to get T(φ)∈B for all φ∈B.
Arguing as above, we now prove that ∥Tφ1-Tφ2∥≤12∥φ1-φ2∥ for all φ1,φ2∈B. Indeed, thanks to (5.3), it is sufficient to choose λ small enough such that
4
C
A
2
C
E
C
N
|
log
λ
|
2
λ
1
p
+
1
s
-
1
≤
1
2
to get
∥
T
(
φ
1
)
-
T
(
φ
2
)
∥
=
∥
A
-
1
π
⊥
(
-
Δ
)
-
1
2
(
N
(
φ
1
)
-
N
(
φ
2
)
)
∥
≤
C
A
|
log
λ
|
C
N
λ
1
s
-
1
∥
φ
1
-
φ
2
∥
(
∥
φ
1
∥
+
∥
φ
2
∥
)
≤
4
C
A
2
C
E
C
N
|
log
λ
|
2
λ
1
p
+
1
s
-
1
⏟
≤
1
2
∥
φ
1
-
φ
2
∥
.
Thus we have proved that T is a contraction on the ball B, so it has a unique fix point in B. ∎
For 𝝃∈𝒫k,η and λ small enough, let φλ,ξ be the fix point for the operator T constructed in Lemma 5.1. By definition, φλ,ξ satisfies (2.6). Then, since K𝜹,𝝃 is spanned by PZ1,1,…,PZ1,k, as a consequence of Lemma 5.1, we get the following proposition:
Proposition 5.2.
Fix p∈(1,2) and let λ0 and κ be as in Lemma 5.1. Then, for any λ∈(0,λ0) and any 𝛏∈Pk,η, there exists a unique function φλ,𝛏∈Kδ,𝛏⊥ such that ∥φλ,𝛏∥≤κλ1p|logλ| and such that we can find k coefficients ci=ci(λ,𝛏), i=1,…,k, such that
(5.4)
(
-
Δ
)
1
2
φ
λ
,
𝝃
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
λ
,
𝝃
=
E
+
N
(
φ
λ
,
𝝃
)
+
∑
i
=
1
k
c
i
e
U
δ
i
,
ξ
i
Z
1
,
i
.
Moreover, by the definition of E, N and L, we also have
(5.5)
(
-
Δ
)
1
2
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
=
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
+
∑
i
=
1
k
c
i
e
U
δ
i
,
ξ
i
Z
1
,
i
.
Remark 5.3.
By testing equation (5.4) against PZ1,i, i=1,…,k, we get that
c
i
(
λ
,
𝝃
)
=
-
∑
j
=
1
k
b
i
j
∫
ℝ
(
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
φ
λ
,
𝝃
+
E
+
N
(
φ
λ
,
𝝃
)
)
P
Z
1
,
j
𝑑
x
,
where the bij=bij(λ,ξ) are the coefficients of the inverse of the matrix (bij)1≤i,j≤k with
b
i
j
=
∫
ℝ
e
U
δ
i
,
ξ
i
Z
1
,
i
P
Z
1
,
j
𝑑
x
.
The matrix (bij)1≤i,j≤k is symmetric and invertible by Remark 4.4.
We conclude this section by proving the regularity of φλ,𝝃 with respect to 𝝃. From now on, with some abuse of notation we will use the notation λ0 to refer different constants possibly smaller than the one given by Lemma 5.1 and Proposition 5.2.
Lemma 5.4.
For any λ∈(0,λ0), the map 𝛏→φλ,𝛏 is a C1 map from Pk,η into X01/2(I).
Proof.
For the study of the regularity of φλ,𝝃 it is important to recall that π, π⊥, L and T depend on λ and 𝝃. For this reason, throughout this proof these operators will be denoted respectively by πλ,𝝃,πλ,𝝃⊥, Lλ,𝝃 and Tλ,𝝃. For a fixed λ∈(0,λ0), let us consider the C1 map Gλ:𝒫k,η×X01/2(I)→X01/2(I) defined by
G
λ
(
ξ
,
φ
)
=
φ
+
π
λ
,
𝝃
⊥
[
ω
𝒂
,
𝜹
,
𝝃
-
(
-
Δ
)
-
1
2
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
π
λ
,
𝝃
⊥
φ
)
]
.
Note that
∂
G
λ
∂
φ
(
𝝃
,
φ
)
[
v
]
=
v
-
π
λ
,
𝝃
⊥
(
-
Δ
)
-
1
2
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
π
λ
,
𝝃
⊥
φ
)
π
λ
,
𝝃
⊥
v
for any v∈X01/2(I). In particular, ∂Gλ∂φ(𝝃,φ) is a Fredholm operator of index 0 and thus, it is invertible if and only if it is injective. By definition, we have that Gλ(ξ,φ)=0 if and only if φ∈K𝜹,𝝃⊥ is a fix point for Tλ,𝝃. In particular, Gλ(𝝃,φλ,𝝃)=0. Moreover,
∂
G
λ
∂
φ
(
𝝃
,
φ
λ
,
𝝃
)
[
v
]
:=
v
-
π
λ
,
𝝃
⊥
(
-
Δ
)
-
1
2
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
π
λ
,
𝝃
⊥
v
=
π
λ
,
𝝃
v
+
π
λ
,
𝝃
⊥
[
π
λ
,
𝝃
⊥
v
-
(
-
Δ
)
-
1
2
(
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
π
λ
,
𝝃
⊥
v
)
]
=
π
λ
,
𝝃
v
+
π
λ
,
𝝃
⊥
L
λ
,
𝝃
(
π
λ
,
𝝃
⊥
v
)
-
π
λ
,
𝝃
⊥
[
(
-
Δ
)
-
1
2
(
(
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
)
π
λ
,
𝝃
⊥
v
)
]
.
For any p∈(1,2) and s>p such that 1p+1s>1, Remark 3.6 gives
∥
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
∥
L
p
(
I
)
=
O
(
λ
1
-
s
s
∥
φ
λ
,
𝝃
∥
)
=
O
(
λ
1
p
+
1
s
-
1
|
log
λ
|
)
.
Hence, using Sobolev’s inequality, we can find α>0 such that
∥
π
λ
,
𝝃
⊥
[
(
-
Δ
)
-
1
2
(
(
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
-
f
λ
′
(
ω
𝒂
,
𝜹
,
𝝃
)
)
π
λ
,
𝝃
⊥
v
)
]
∥
=
O
(
λ
α
∥
v
∥
)
.
Then we have
∥
∂
G
λ
∂
φ
(
𝝃
,
φ
λ
,
𝝃
)
[
v
]
∥
≥
∥
π
λ
,
𝝃
v
+
π
λ
,
𝝃
⊥
L
λ
,
𝝃
(
π
λ
,
𝝃
⊥
v
)
∥
+
O
(
λ
α
∥
v
∥
)
≥
1
2
∥
π
λ
,
𝝃
v
∥
+
1
2
∥
π
λ
,
𝝃
⊥
L
λ
,
𝝃
(
π
λ
,
𝝃
⊥
v
)
∥
+
O
(
λ
α
∥
v
∥
)
≥
1
2
∥
π
λ
,
𝝃
v
∥
+
c
2
|
log
λ
|
-
1
∥
π
λ
,
𝝃
⊥
v
∥
+
O
(
λ
α
∥
v
∥
)
≥
c
(
∥
π
λ
,
𝝃
v
∥
+
∥
π
λ
,
𝝃
⊥
v
∥
)
+
O
(
λ
α
∥
v
∥
)
≥
(
c
+
O
(
λ
α
)
)
∥
v
∥
.
This implies that ∂Gλ∂φ(𝝃,φλ,𝝃)[v] is invertible. Then the implicit function theorem gives that φλ,𝝃 is of class C1. ∎
6 Choice of the Concentration Points
Let φλ,𝝃 be as in Proposition 5.2. It is clear that if we find 𝝃=(ξ1,…,ξk) (depending on λ) such that
(6.1)
c
i
(
λ
,
𝝃
)
=
0
for all
i
=
1
,
…
,
k
,
then the function uλ:=ω𝒂,𝜹,𝝃+φλ,𝝃 is solution for our initial problem (1.1). In this section, we will prove that (6.1) is satisfied when 𝝃 is a critical point of the reduced energy functional
(6.2)
𝔉
λ
(
𝝃
)
:=
J
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
,
where
J
λ
(
u
)
:=
1
2
∥
u
∥
2
-
∫
I
g
λ
(
u
)
𝑑
x
with
g
λ
(
t
)
:=
∫
0
t
f
λ
(
s
)
𝑑
s
.
In order to prove this, we will need the following preliminary estimate.
Lemma 6.1.
Let F1,…,Fk be as in (3.3). As λ→0, we have
(6.3)
∥
ω
𝒂
,
𝜹
,
𝝃
∥
2
=
-
4
π
k
log
λ
-
2
π
∑
i
=
1
k
F
i
(
𝝃
)
+
O
(
λ
|
log
λ
|
)
,
uniformly with respect to 𝛏∈Pk,η.
Proof.
By the definition of ω𝒂,𝜹,𝝃, in order to prove (6.3), it is sufficient to show that
(6.4)
∥
P
U
δ
i
,
ξ
i
∥
2
=
-
4
π
log
λ
-
2
π
F
i
(
𝝃
)
-
4
π
2
a
i
∑
j
≠
i
a
j
G
ξ
j
(
ξ
i
)
+
O
(
λ
|
log
λ
|
)
,
for
i
=
1
,
…
,
k
,
and
(6.5)
〈
P
U
δ
i
,
ξ
i
,
P
U
δ
j
,
ξ
j
〉
=
4
π
2
G
ξ
j
(
ξ
i
)
+
O
(
λ
|
log
λ
|
)
for
i
,
j
=
1
,
…
,
k
with
i
≠
j
.
Let us prove (6.4) first. For i=1,…,k, since eUδi,ξi=O(λ) in ℝ∖(ξi-η2,ξi+η2), PUδi,ξi=0 in ℝ∖I and, by Lemma 3.1, ∥PUδi,ξi∥L∞(I)=O(|logλ|), we have
∥
P
U
δ
i
,
ξ
i
∥
2
=
∫
ℝ
P
U
δ
i
,
ξ
i
e
U
δ
i
,
ξ
i
d
x
=
∫
ξ
i
-
η
2
ξ
i
+
η
2
P
U
δ
i
,
ξ
i
e
U
δ
i
,
ξ
i
d
x
+
O
(
λ
|
log
λ
|
)
.
Moreover, thanks to the estimates
∫
ξ
i
-
η
2
ξ
i
+
η
2
e
U
δ
i
,
ξ
i
d
x
=
2
π
+
O
(
λ
)
and
∫
ξ
i
-
η
2
ξ
i
+
η
2
U
δ
i
,
ξ
i
e
U
δ
i
,
ξ
i
d
x
=
-
2
π
log
(
2
δ
i
)
+
O
(
λ
|
log
λ
|
)
,
the expansion of PUδi,ξi from Lemma 3.1 yields
∫
ξ
i
-
η
2
ξ
i
+
η
2
P
U
δ
i
,
ξ
i
e
U
δ
i
,
ξ
i
𝑑
x
=
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
-
log
(
2
δ
i
)
+
U
δ
i
,
ξ
i
+
2
π
H
(
ξ
i
,
x
)
)
e
U
δ
i
,
ξ
i
𝑑
x
+
O
(
λ
2
)
=
-
4
π
log
(
2
δ
i
)
+
4
π
2
H
(
ξ
i
,
ξ
i
)
+
O
(
λ
|
log
λ
|
)
=
-
4
π
log
(
2
δ
i
)
+
2
π
F
i
(
𝝃
)
-
4
π
2
∑
j
≠
i
G
ξ
j
(
ξ
i
)
+
O
(
λ
|
log
λ
|
)
.
Recalling that δi is chosen as in (3.5), we have log(2δi)=logλ+Fi(𝝃), and we obtain (6.4).
With similar arguments, for i≠j we get
∫
ℝ
(
-
Δ
)
1
4
P
U
δ
i
,
ξ
i
(
-
Δ
)
1
4
P
U
δ
j
,
ξ
j
𝑑
x
=
∫
ℝ
e
U
δ
i
,
ξ
i
P
U
δ
j
,
ξ
j
𝑑
x
=
∫
ξ
i
-
η
2
ξ
i
+
η
2
e
U
δ
i
,
ξ
i
P
U
δ
j
,
ξ
j
d
x
+
O
(
λ
|
log
λ
|
)
=
2
π
∫
ξ
i
-
η
2
ξ
i
+
η
2
e
U
δ
i
,
ξ
i
G
ξ
j
d
x
+
O
(
λ
|
log
λ
|
)
=
4
π
2
G
ξ
j
(
ξ
i
)
+
O
(
λ
|
log
λ
|
)
,
so that (6.5) holds. ∎
Proposition 6.2.
For λ∈(0,λ0) and 𝛏∈Pk,η, the following conditions are equivalent:
c
i
(
λ
,
𝝃
)
=
0
, for
i
=
1
,
…
,
k
.
Proof.
By the definition of Jλ, we have that
∇
J
λ
(
u
)
=
u
-
(
-
Δ
)
-
1
2
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
for any u∈X01/2(I). Then, recalling that φλ satisfies (5.4)–(5.5), we get
∇
J
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
=
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
-
(
-
Δ
)
-
1
2
f
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
=
∑
j
=
1
k
c
j
P
Z
1
,
j
.
For i=1,…,k, by the chain rule, we find
∂
𝔉
λ
∂
ξ
i
(
𝝃
)
=
〈
∇
J
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
,
d
d
ξ
i
ω
𝒂
,
𝜹
,
𝝃
+
d
d
ξ
i
φ
λ
,
𝝃
〉
=
∑
j
=
1
k
β
i
j
c
j
,
where
β
i
j
=
〈
P
Z
1
,
j
,
d
d
ξ
i
ω
𝒂
,
𝜹
,
𝝃
〉
+
〈
P
Z
1
,
j
,
d
d
ξ
i
φ
λ
,
𝝃
〉
.
Then it suffices to show that the matrix (βij) is invertible. Indeed, this gives
∇
𝔉
λ
(
𝝃
)
=
0
⇔
c
i
(
λ
,
𝝃
)
=
0
,
i
=
1
,
…
,
k
.
Let us then estimate the coefficients βij. First, for i,h=1,…,k, we observe that
d
d
ξ
i
e
U
δ
h
,
ξ
h
=
δ
i
,
h
δ
i
e
U
δ
i
,
ξ
i
Z
1
,
i
-
1
δ
h
e
U
δ
h
,
ξ
h
Z
0
,
h
∂
δ
h
∂
ξ
i
=
δ
i
,
h
δ
i
e
U
δ
i
,
ξ
i
Z
1
,
i
-
e
U
δ
h
,
ξ
h
Z
0
,
h
∂
F
h
∂
ξ
i
(
𝝃
)
,
where δi,h denotes the Kronecker delta and we have used that 𝜹=(δ1,…,δk) is given by (3.5). Consequently,
d
d
ξ
i
ω
𝒂
,
𝜹
,
𝝃
=
∑
h
=
1
k
a
h
d
d
ξ
i
P
U
δ
h
,
ξ
h
=
a
i
δ
i
P
Z
1
,
i
-
∑
h
=
1
k
a
h
P
Z
0
,
h
∂
F
h
∂
ξ
i
(
𝝃
)
.
Then we infer
(6.6)
〈
P
Z
1
,
j
,
d
ω
𝒂
,
𝜹
,
𝝃
d
ξ
i
〉
=
a
i
δ
i
〈
P
Z
1
,
j
,
P
Z
1
,
i
〉
-
∑
h
=
1
k
a
h
∂
F
h
∂
ξ
i
(
𝝃
)
〈
P
Z
1
,
j
,
P
Z
0
,
h
〉
=
π
a
i
δ
i
δ
i
,
j
+
O
(
1
)
.
Now, for i,j=1,…,k, observe that
φ
λ
,
𝝃
∈
K
𝜹
,
𝝃
⊥
⟹
〈
P
Z
1
,
j
,
φ
λ
,
𝝃
〉
=
0
⟹
〈
d
d
ξ
i
P
Z
1
,
j
,
φ
λ
,
𝝃
〉
+
〈
P
Z
1
,
j
,
d
d
ξ
i
φ
λ
,
𝝃
〉
=
0
.
Note further that we have the identity
d
d
ξ
i
e
U
δ
j
,
ξ
j
Z
1
,
j
=
δ
i
,
j
(
1
δ
i
e
U
δ
i
,
ξ
i
Z
1
,
i
2
-
e
2
U
δ
i
,
ξ
i
Z
0
,
i
)
-
2
e
U
δ
j
,
ξ
j
Z
1
,
j
Z
0
,
j
∂
F
j
(
𝝃
)
∂
ξ
i
=
O
(
1
δ
i
e
3
U
δ
i
,
ξ
i
|
x
-
ξ
i
|
2
)
+
O
(
e
2
U
δ
i
,
ξ
i
)
+
O
(
e
2
U
δ
j
,
ξ
j
)
,
where we have used that |Z0,i|,|Z0,j|≤1.
Then, since ddξiPZ1,j=(-Δ)-12ddξieUδj,ξjZ1,j, by (2.2) and (3.6) we get that
∥
d
d
ξ
i
P
Z
1
,
j
∥
=
O
(
∥
d
d
ξ
i
e
U
δ
j
,
ξ
j
Z
1
,
j
∥
L
2
(
I
)
)
=
O
(
λ
-
3
2
)
.
In particular, recalling that for p∈(1,2) we have ∥φλ,𝝃∥=O(λ1p|logλ|), we get
(6.7)
〈
P
Z
1
,
j
,
d
d
ξ
i
φ
λ
,
𝝃
〉
=
-
〈
d
d
ξ
i
P
Z
1
,
j
,
φ
λ
,
𝝃
〉
=
O
(
λ
-
3
2
∥
φ
λ
,
𝝃
∥
)
=
O
(
λ
1
p
-
3
2
|
log
λ
|
)
=
o
(
λ
-
1
)
,
uniformly for 𝝃∈𝒫k,η. For λ small enough, using (3.5), (6.6) and (6.7), we conclude that the matrix (βij) is dominant diagonal and thus invertible. This concludes the proof. ∎
The following lemma describes the asymptotic behavior of 𝔉λ as λ→0.
Lemma 6.3.
We have
𝔉
λ
(
𝝃
)
=
-
2
π
k
log
λ
-
2
π
k
-
π
∑
i
=
1
k
F
i
(
𝝃
)
+
o
(
1
)
,
where o(1)→0 as λ→0, uniformly for 𝛏∈Pk,η.
Proof.
According to Lemma 6.1, we have ∥ω𝒂,𝜹,𝝃∥2=O(|logλ|), so that
∥
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
∥
2
=
∥
ω
𝒂
,
𝜹
,
𝝃
∥
2
+
O
(
∥
ω
𝒂
,
𝜹
,
𝝃
∥
∥
φ
λ
,
𝝃
∥
)
+
O
(
∥
φ
λ
,
𝝃
∥
2
)
=
∥
ω
𝒂
,
𝜹
,
𝝃
∥
2
+
O
(
λ
1
p
|
log
λ
|
3
2
)
.
Noting that gλ=fλ′-2λ, by Remark 3.6, for any p∈(1,2) and s>p such that 1s+1p>1, one has
∥
g
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
-
g
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
∥
L
1
(
I
)
≤
C
λ
1
-
s
s
∥
φ
λ
,
𝝃
∥
=
O
(
λ
1
p
+
1
s
-
1
|
log
λ
|
)
=
o
(
1
)
as λ→0. Thus
(6.8)
J
λ
(
ω
𝒂
,
𝜹
,
𝝃
+
φ
λ
,
𝝃
)
=
J
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
+
o
(
1
)
=
1
2
∥
ω
𝒂
,
𝜹
,
𝝃
∥
2
-
∫
I
g
λ
(
ω
𝒂
,
𝜹
,
𝝃
)
𝑑
x
+
o
(
1
)
.
Using again that gλ=fλ′-2λ together with Lemma 4.1 and (3.6), we find
∫
I
g
λ
(
u
)
d
x
=
∑
i
=
1
k
∫
ξ
i
-
η
2
ξ
i
+
η
2
e
U
δ
i
,
𝝃
i
d
x
+
O
(
λ
|
log
λ
|
)
=
2
π
k
+
O
(
λ
|
log
λ
|
)
.
Then the conclusion follows by Lemma 6.1 and (6.8). ∎
The previous lemma shows that, up to constant terms that do not depend on 𝝃, the functional 𝔉λ converges uniformly to a multiple of the function
(6.9)
𝔉
(
𝝃
)
:=
1
2
π
∑
i
=
1
k
F
i
(
𝝃
)
=
∑
i
=
1
k
H
(
ξ
i
,
ξ
j
)
+
∑
i
,
j
=
1
,
i
≠
j
k
a
i
a
j
G
(
ξ
i
,
ξ
j
)
.
In the next subsection, we shall study the properties of 𝔉 and exploit them to show that 𝔉λ has a critical point (a local minimum) in 𝒫k,η, provided η is fixed small enough and the coefficients ai have alternating sign.
6.1 Existence of a Critical Point
Let us now assume ai=-ai+1 for all i∈{1,…,k-1}. We refer to the appendix for some considerations concerning different possible choices of the coefficients ai. With this assumption, the function 𝔉 defined in (6.9) becomes
𝔉
(
𝝃
)
=
∑
i
=
1
k
H
(
ξ
i
,
ξ
i
)
+
∑
i
,
j
=
1
,
i
≠
j
k
(
-
1
)
i
+
j
G
(
ξ
i
,
ξ
j
)
=
∑
i
=
1
k
1
π
log
(
2
(
1
-
ξ
i
2
)
)
+
∑
i
,
j
=
1
,
i
≠
j
k
(
-
1
)
i
+
j
1
π
log
(
1
-
ξ
i
ξ
j
+
(
1
-
ξ
i
2
)
(
1
-
ξ
j
2
)
|
ξ
i
-
ξ
j
|
)
.
The goal of this subsection is to show that the set of maximum points for 𝔉 on the set
(6.10)
𝒫
k
:=
{
𝝃
=
(
ξ
1
,
…
,
ξ
k
)
:
-
1
<
ξ
i
<
ξ
i
+
1
<
1
,
i
=
1
,
…
,
k
-
1
}
is a non-empty compact subset of 𝒫k, independently of the value of k∈ℕ. Combining this with Lemma 6.3, we will prove that the functional 𝔉λ defined in (6.2) has a critical point in 𝒫k (in fact in 𝒫k,η, if η is small enough). The proof of this result is inspired by [4, proof of Theorem 3.3]. We will provide some details here, since having the explicit expression for the Green function of our operator simplifies considerably many steps of the proof. For example, we easily get the following properties.
Lemma 6.4.
The following properties hold:
H
(
ξ
,
ξ
)
→
-
∞
as
ξ
→
∂
I
=
{
-
1
,
1
}
.
Let
ε
>
0
there exists a constant
c
(
ϵ
)
such that
|
H
(
ξ
,
ξ
)
|
≤
c
(
ε
)
if
dist
(
ξ
,
∂
I
)
≥
ε
.
Let
ε
>
0
be small enough, there exist a constant
c
(
ϵ
)
such that
|
G
(
x
,
y
)
|
≤
c
(
ε
)
if
|
x
-
y
|
≥
ε
.
For any
x
,
ξ
∈
I
, x≠ξ, we have
d
d
x
G
(
ξ
,
x
)
=
-
1
π
1
-
ξ
2
(
x
-
ξ
)
1
-
x
2
.
Given any three points
x
<
y
<
z
∈
I
, we have
G
(
x
,
z
)
-
G
(
x
,
y
)
≤
0
.
The following lemma provides upper bounds on 𝔉.
Lemma 6.5.
For any 𝛏=(ξ1,…,ξk)∈Pk, we have
∑
i
,
j
=
1
,
i
≠
j
k
(
-
1
)
i
+
j
G
(
ξ
i
,
ξ
j
)
≤
0
,
Proof.
For any 1≤i≤k-1, we set
G
i
(
𝝃
)
:=
∑
j
=
i
+
1
k
(
-
1
)
i
+
j
G
(
ξ
i
,
ξ
j
)
,
so that
∑
i
,
j
=
1
,
i
≠
j
k
(
-
1
)
i
+
j
G
(
ξ
i
,
ξ
j
)
=
2
∑
i
=
1
k
-
1
G
i
(
𝝃
)
.
Then it is sufficient to observe that Gi(𝝃)≤0 for any 1≤i≤k-1. Indeed, if k-i is even, we have that
G
i
(
𝝃
)
=
∑
j
=
1
k
-
i
(
-
1
)
j
G
(
ξ
i
,
ξ
i
+
j
)
=
∑
j
=
2
,
j
even
k
-
i
G
(
ξ
i
,
ξ
i
+
j
)
-
G
(
ξ
i
,
ξ
i
+
j
-
1
)
≤
0
,
where the last inequality follows by v of Lemma 6.4. If instead k-i is odd, then we have
G
i
(
𝝃
)
=
∑
j
=
1
k
-
i
(
-
1
)
j
G
(
ξ
i
,
ξ
i
+
j
)
=
-
G
(
ξ
i
,
ξ
k
)
+
∑
j
=
2
,
j
even
k
-
i
-
1
G
(
ξ
i
,
ξ
i
+
j
)
-
G
(
ξ
i
,
ξ
i
+
j
-
1
)
≤
0
,
where we used again property v of Lemma 6.4 together with the inequality G(ξi,ξk)≥0. ∎
Proposition 6.6.
For any k∈N, we have F(𝛏)→-∞ when dist(𝛏,∂Pk)→0. In particular, F has a maximum point in Pk. Moreover, the set MF of global maxima for F in Pk is compact.
Proof.
It suffices to show that, for any sequence 𝝃n=(ξ1n,…,ξkn)∈𝒫k with dist(𝝃n,∂𝒫k)→0 as n→+∞, up to extracting a subsequence, one has 𝔉(𝝃n)→-∞ as n→+∞. If there exists i∈{1,…,k} such that ξin→∂I, then i of Lemma 6.4 implies that H(ξin,ξin)→-∞ and, thanks to Lemma 6.5,
𝔉
(
𝝃
n
)
≤
∑
j
=
1
k
H
(
ξ
j
n
,
ξ
j
n
)
≤
H
(
ξ
i
n
,
ξ
i
n
)
+
k
-
1
π
log
2
→
-
∞
as n→+∞. Thus, up to a subsequence, we may assume that there exists an ε>0 such that |ξin|≤1-ε, 1≤i≤k. Then d(𝝃n,∂𝒫k)→0 implies ξin-ξi+1n→0 for some 1≤i≤k-1. Let i0 be the maximal index i∈{1,…,k-1} such that this property holds. Then, up to extracting another subsequence, we may assume ε≤ξi+1-ξi≤2 for any i0<i≤k-1. Note that ii and iii of Lemma 6.4 give
𝔉
(
𝝃
n
)
=
-
1
π
∑
i
≠
j
(
-
1
)
i
+
j
log
|
ξ
i
n
-
ξ
j
n
|
+
O
(
1
)
=
-
2
π
∑
i
=
1
k
-
1
∑
j
=
1
k
-
i
(
-
1
)
j
log
|
ξ
i
n
-
ξ
i
+
j
n
|
+
O
(
1
)
=
-
2
π
∑
i
=
1
i
0
∑
j
=
1
k
-
i
(
-
1
)
j
log
|
ξ
i
n
-
ξ
i
+
j
n
|
⏟
=
:
ζ
i
n
+
O
(
1
)
.
If k-i is even, then we have
e
ζ
i
n
=
∏
j
=
1
k
-
i
2
|
ξ
i
n
-
ξ
i
+
2
j
n
|
|
ξ
i
n
-
ξ
i
+
2
j
-
1
n
|
≥
1
,
while if k-i is odd, we have that
e
ζ
i
n
=
1
|
ξ
k
n
-
ξ
i
n
|
∏
j
=
1
k
-
i
-
1
2
|
ξ
i
n
-
ξ
i
+
2
j
n
|
|
ξ
i
n
-
ξ
i
+
2
j
-
1
n
|
≥
1
2
.
Then all the sequences (ζin)n∈ℕ, 1≤i≤k-1 are bounded from below and we obtain
𝔉
(
𝝃
n
)
≤
-
2
π
ζ
i
0
n
+
O
(
1
)
=
-
2
π
∑
j
=
1
k
-
i
0
(
-
1
)
j
log
|
ξ
i
0
n
-
ξ
i
0
+
j
n
|
+
O
(
1
)
=
2
π
log
|
ξ
i
0
n
-
ξ
i
0
+
1
n
|
+
O
(
1
)
→
-
∞
as n→∞. ∎
Corollary 6.7.
Let Fλ be as in (6.2). Then there exists η0∈(0,2k+1) such that Fλ has a critical point 𝛏(λ)∈Pk,η0 for any small λ.
Proof.
By Proposition 6.6, we can fix η0 such that all the maxima of 𝔉 in 𝒫k belong to 𝒫k,η0. In particular, since 𝒫k,η0 is open, we have
(6.11)
max
∂
𝒫
k
,
η
0
𝔉
<
max
𝒫
¯
k
,
η
0
𝔉
.
According to Lemma 6.3, we have that
S
λ
:=
1
π
2
(
2
π
k
log
λ
+
2
π
k
-
𝔉
λ
)
→
𝔉
,
uniformly in 𝒫¯k,η0 (in fact, in 𝒫k,η for any fixed η<η0) as λ→0. Then, by (6.11), we must have
max
∂
𝒫
k
,
η
0
S
λ
<
max
𝒫
¯
k
,
η
0
S
λ
,
which implies that Sλ has a maximum point 𝝃(λ) in 𝒫k,η0. In particular, 𝝃(λ) is a critical point for Sλ and 𝔉λ. ∎
Remark 6.8.
By construction, we also have that dist(𝝃(λ),M𝔉)→0. In particular, for any sequence λn→0, we have 𝝃(λn)→𝝃¯ up to extracting a subsequence, where 𝝃¯ is a maximum point for 𝔉.
7 Proof of the Main Theorems
We now collect the results of the previous sections to complete the proof of our main results.
Proof of Theorem 1.2.
Let ω𝒂,𝜹,𝝃 be as in (2.4), with 𝜹=(δ1,…,δk) as in (3.5). For a given p∈(1,2), let λ0 and φλ,𝝃 be as in Proposition 5.2 and let 𝔉λ be as in (6.2). By Corollary 6.7, there exists η0 small enough such that 𝔉λ has a critical point in 𝒫k,η0. By Propositions 6.2 and 5.2, setting φλ:=φλ,𝝃(λ) and 𝜹(λ):=(δ1(λ,𝝃(λ)),…,δk(λ,𝝃(λ))), we get that uλ:=ω𝒂,𝜹(λ),𝝃(λ)+φλ is a solution of equation (1.1), as claimed in Theorem 1.2. Proposition 5.2 also gives ∥φλ∥=O(λ1p|logλ|)→0 as λ→0. It remains to prove that ∥φλ∥L∞(I)→0. Let us recall that φλ∈X01/2(I) is a weak solution to
(
-
Δ
)
1
2
φ
λ
=
f
λ
′
(
ω
𝒂
,
𝜹
(
λ
)
,
𝝃
(
λ
)
)
φ
λ
+
E
+
N
(
φ
λ
)
in I. Thanks to Lemma 4.1, we have
∫
I
(
f
λ
′
(
ω
𝒂
,
𝜹
(
λ
)
,
𝝃
(
λ
)
)
|
φ
λ
|
)
p
𝑑
x
=
∑
i
=
1
k
∫
ξ
i
-
η
2
ξ
i
+
η
2
O
(
e
p
U
δ
i
,
ξ
i
)
|
φ
λ
|
p
𝑑
x
+
o
(
1
)
as λ→0. But by Hölder’s inequality (with any q>1) and (3.6), we get that
∫
ξ
i
-
η
2
ξ
i
+
η
2
(
e
U
δ
i
,
ξ
i
|
φ
λ
|
)
p
𝑑
x
≤
(
∫
ξ
i
-
η
0
2
ξ
+
η
0
2
e
p
q
U
δ
i
,
ξ
i
𝑑
x
)
1
q
(
∫
I
|
φ
λ
|
p
q
q
-
1
𝑑
x
)
q
-
1
q
=
(
O
(
λ
1
-
q
p
)
)
1
q
O
(
∥
φ
λ
∥
p
)
=
O
(
λ
1
q
+
1
-
p
|
log
λ
|
p
)
.
Since p<2, we can take q>1 such that 1q+1-p>0, so that ∥fλ′(ω𝒂,𝜹(λ),𝝃(λ))φλ∥Lp(I)→0 as λ→0. In addition, Lemma 3.2 and Lemma 3.5 give ∥E∥Lp(I)→0 and ∥N(φλ)∥Lp(I)→0 as λ→0. Thus (-Δ)12φλ→0 in Lp(I) and elliptic estimates (see [22, Theorem 13]) imply ∥φλ∥L∞(I)→0, as desired. ∎
We now turn to the proof of Theorem 1.1. From now on, we let uλ be the solution constructed in Theorem 1.2. Note that uλ has the form
u
λ
:=
∑
i
=
1
k
(
-
1
)
i
-
1
P
U
δ
i
(
λ
)
,
ξ
i
(
λ
)
+
φ
λ
,
with 𝝃(λ)=(ξ1(λ),…,ξk(λ))∈𝒫k,η0 for some small η0, 𝜹(λ)=(δ1(λ),…,δk(λ)) such that δi=O(λ) as λ→0, and φλ∈X01/2(I) satisfying ∥φλ∥+∥φλ∥L∞(I)→0. Up to extracting a subsequence, we may also assume that
ξ
i
(
λ
)
→
ξ
i
with
-
1
<
ξ
1
<
…
<
ξ
k
<
1
.
Lemma 7.1.
The following properties hold:
u
λ
blows-up with alternating sign at
ξ
1
,
…
,
ξ
k
as
λ
→
0
, that is (
1.6
) holds for any small
ε
>
0
.
u
λ
∈
C
∞
(
I
)
and
u
λ
→
u
0
:=
∑
i
=
1
k
(
-
1
)
i
-
1
G
ξ
1
in
C
loc
∞
(
I
∖
{
ξ
1
,
…
,
ξ
k
}
)
as
λ
→
0
.
For any
ε
>
0
, and
α
∈
(
0
,
1
2
)
, we have
∥
u
λ
-
u
0
d
∥
C
0
,
α
(
(
-
1
,
ξ
1
-
ε
)
)
+
∥
u
λ
-
u
0
d
∥
C
0
,
α
(
(
ξ
k
+
ε
,
1
)
)
→
0
as
λ
→
0
, where
d
(
x
)
:=
1
-
|
x
|
is the distance of
x
from
∂
I
.
Proof.
In order to get the first property, it is sufficient to observe that Lemma 3.1 implies
P
U
δ
i
(
λ
)
,
ξ
i
(
λ
)
(
ξ
i
(
λ
)
)
→
+
∞
and
P
U
δ
j
(
λ
)
,
ξ
j
(
λ
)
(
ξ
i
(
λ
)
)
→
G
ξ
j
(
ξ
i
)
(
ξ
i
)
for
j
≠
i
.
Since ∥φλ∥L∞(I)→0 as λ→0, this gives the conclusion.
Similarly, the second property follows by the boundedness of uλ in Lloc∞(I∖{ξ1,…,ξk}) and elliptic estimates for (-Δ)12 (see e.g. [31]).
It remains to prove the third property. We focus first on the case x∈[ξk+ϵ,1) and we let ψ be a smooth cut-off function such that ψ≡0 on (-∞,ξk+ε2) and ψ≡1 on [ξk+ε,∞). By construction, we have uλψ≡0 in ℝ∖(ξk+ε2,1). Moreover, for x∈(ξk+ε2,1) we have
(7.1)
(
-
Δ
)
1
2
(
u
λ
ψ
)
=
ψ
(
-
Δ
)
1
2
u
λ
+
u
λ
(
-
Δ
)
1
2
ψ
+
∫
ℝ
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
.
Note that the last integral is well defined since uλ∈C0,12(ℝ) and ψ∈C∞(ℝ). Moreover, ψ∈C∞(ℝ)∩L∞(ℝ) implies (-Δ)12ψ∈C∞(ℝ) (see for example [34, Proposition 2.1.4]). Then, since uλ is uniformly bounded in Lloc∞(ℝ∖{ξ1,…,ξk}), and uλ solves (1.1), we have that
∥
ψ
(
-
Δ
)
1
2
u
λ
∥
L
∞
(
(
ξ
k
+
ε
2
,
1
)
)
+
∥
u
λ
(
-
Δ
)
1
2
ψ
∥
L
∞
(
(
ξ
k
+
ε
2
,
1
)
)
≤
C
for some C>0, depending only on ε. Now, if x∈[ξk+2ε,1), then
|
∫
ℝ
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
|
=
|
∫
-
∞
ξ
k
+
ε
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
1
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
|
≤
1
ε
2
(
∥
u
λ
∥
L
∞
(
ξ
k
+
2
ε
,
1
)
+
∥
u
λ
∥
L
1
(
ℝ
)
)
≤
C
.
If instead x∈(ξk+ε2,ξk+2ε), then
∫
ℝ
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
=
∫
x
-
ε
3
x
+
ε
3
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
+
O
(
∥
ψ
∥
L
∞
(
ℝ
)
(
∥
u
λ
∥
L
1
(
ℝ
)
+
∥
u
λ
∥
L
∞
(
(
ξ
k
+
ε
2
,
1
)
)
)
)
,
with
|
∫
x
-
ε
3
x
+
ε
3
(
u
λ
(
x
)
-
u
λ
(
y
)
)
(
ψ
(
x
)
-
ψ
(
y
)
)
|
x
-
y
|
2
𝑑
y
|
≤
2
ε
3
∥
u
λ
′
∥
L
∞
(
ξ
k
+
ε
6
,
ξ
k
+
7
3
ε
)
∥
ψ
′
∥
L
∞
(
ℝ
)
.
We have so proved that the right-hand side of (7.1) is bounded in L∞((ξk+ε2,1)). Then the regularity results of Ros-Oton and Serra ([31, Theorem 1.2]) implies that
∥
u
λ
d
∥
C
0
,
β
(
(
ξ
k
+
ε
,
1
)
)
≤
∥
u
λ
ψ
d
∥
C
0
,
β
(
(
ξ
k
+
ε
2
,
1
)
)
≤
C
for any β<12. In particular, we have that ∥uλ-u0d∥C0,α→0 for any α<β<12. With similar arguments, we prove an analogue convergence result in (-1,ξ1-ε). ∎
The main step in the proof on Theorem 1.1 consists in showing that the limit profile u0 has exactly k-1 zeroes in I∖(ξ1,…,ξk). In fact, we shall prove that u0 is strictly monotone in each of the intervals (ξi,ξi+1). In the following it is useful to denote ξ0=-1 and ξk+1=1.
Proposition 7.2.
For any given k∈N and 𝛏=(ξ1,…,ξk) with -1=ξ0<ξ1<…<ξk<ξk+1=1, consider the function
u
0
=
∑
i
=
1
k
(
-
1
)
i
-
1
G
ξ
i
.
Then there exists a constant c=c(k,𝛏) such that, for any j=0,…,k, we have
(
-
1
)
j
u
0
′
(
x
)
1
-
x
2
≥
c
>
0
for
x
∈
(
ξ
j
,
ξ
j
+
1
)
.
Proof.
Throughout the proof we denote α(x):=1-x2, x∈I.
Step 1.
There exists c1=c1(ξ1,…,ξk) such that Gξi′α≥c1 in (-1,ξi) and Gξi′α≤-c1 in (ξi,1).
Fix i∈{1,…,k}. According to iv of Lemma 6.4, we have that
G
ξ
i
′
(
x
)
=
-
1
π
1
-
ξ
i
2
(
x
-
ξ
i
)
α
(
x
)
.
In particular,
|
G
ξ
i
′
(
x
)
|
α
(
x
)
=
1
-
ξ
i
2
π
|
x
-
ξ
i
|
≥
1
-
ξ
i
2
2
π
≥
1
-
max
1
≤
j
≤
k
|
ξ
j
|
2
2
π
,
where we used |x-ξi|≤2. Since Gξi′>0 in (ξ0,ξi) and Gξi′<0 in (ξi,ξk+1), the inequality above gives the conclusion.
Step 2.
Assume k≥2 and for any 1≤i≤k-1 set gi:=Gξi-Gξi+1. There exists a constant c2>0 such that gi′α≥c2 in (ξ0,ξi)∪(ξi+1,ξk+1) and gi′α≤-c2 in (ξi,ξi+1) for any 1≤i≤k.
By Step 1, we know that Gξi′α≤-c1 and Gξi+1′α≥c1 in (ξi,ξi+1). This immediately gives gi′α≤-2c1 in (ξi,ξi+1). Let us now assume x<ξi or x>ξi+1. As in Step 1, we have the explicit expression
g
i
′
(
x
)
≥
G
ξ
i
′
(
x
)
-
G
ξ
i
+
1
′
(
x
)
=
1
π
1
-
ξ
i
+
1
2
(
x
-
ξ
i
+
1
)
α
(
x
)
-
1
π
1
-
ξ
i
2
(
x
-
ξ
i
)
α
(
x
)
=
f
x
(
ξ
i
+
1
)
-
f
x
(
ξ
i
)
π
α
(
x
)
,
where fx(t):=1-t2x-t. If x<ξi or x>ξi+1, using that fx∈C1((-1,1)∖{x}), we get that
f
x
(
ξ
i
+
1
)
-
f
x
(
ξ
i
)
=
f
x
′
(
ξ
¯
)
(
ξ
i
+
1
-
ξ
i
)
,
where ξ¯ is a point between ξi and ξi+1. In particular,
f
x
′
(
ξ
¯
)
=
1
-
x
ξ
¯
(
x
-
ξ
¯
)
2
1
-
ξ
¯
2
≥
1
-
|
ξ
¯
|
(
x
-
ξ
¯
)
2
1
-
ξ
¯
2
≥
1
-
max
{
|
ξ
i
|
,
|
ξ
i
+
1
|
}
(
x
-
ξ
¯
)
2
≥
1
-
M
(
𝝃
)
4
,
where M(𝝃):=max1≤j≤k|ξj|∈(0,1). We can so conclude that
g
i
′
(
x
)
α
(
x
)
≥
(
1
-
M
(
𝝃
)
)
(
ξ
i
+
1
-
ξ
i
)
4
π
≥
(
1
-
M
(
𝝃
)
)
σ
(
𝝃
)
4
π
,
where σ(𝝃):=min1≤j≤kξj+1-ξj>0. The right-hand side is a constant depending only on k and 𝝃.
Step 3.
Conclusion of the proof.
If k=1 or k=2, the conclusion follows directly from Steps 1 and 2.
Assume k≥3, k odd. For 1≤i≤k-1 let gi be as in Step 2. We can write
(7.2)
u
0
=
∑
i
=
1
,
i
odd
k
-
2
g
i
+
G
ξ
k
,
(7.3)
u
0
=
G
ξ
1
-
∑
i
=
2
,
i
even
k
-
1
g
i
.
Note that if 1≤i≤k-2 is odd, and if 0≤j≤k-1 is even, the interval (ξj,ξj+1) is contained in (-1,ξi)∪(ξi+1,1) and in (-1,ξk). In particular, Steps 1 and 2 guarantee the product of α with any of the functions appearing in (7.2) is increasing in (ξj,ξj+1). In fact, we get
u
0
′
α
≥
k
-
1
2
c
2
+
c
1
for any
j
even
.
Similarly, when 2F≤i≤k-1 is even and 1≤j≤k is odd, then (ξj,ξj+1) is contained in (-1,ξi)∪(ξi+1,1) and in (ξ1,1). Therefore, (7.3) together with Steps 1 and 2 yields
-
u
0
′
α
≥
c
1
+
k
-
1
2
c
2
in
(
ξ
j
,
ξ
j
+
1
)
,
j
odd
.
Finally, assume k even and k≥4. Then we can decompose
(7.4)
u
0
=
∑
i
=
1
,
i
odd
k
-
1
g
i
,
(7.5)
u
0
=
G
ξ
1
-
∑
i
=
2
,
i
even
k
-
2
g
i
-
G
ξ
k
.
As before, for 0≤j≤k even, we have (ξj,ξj+1)⊆(-1,ξi)∪(ξi+1,1) and gi′α≥c2 in (ξj,ξj+1) (by Step 2), for any odd 1≤i≤k-1. Then (7.4) yields
u
0
′
α
≥
k
2
c
2
in
(
ξ
j
,
ξ
j
+
1
)
for any
j
even
.
If instead j is odd, one has (ξj,ξj+1)⊆(-1,ξi)∪(ξi+1,1) and gi′α≥c in (ξj,ξj+1) for any 2≤i≤k-2 even. Moreover, since (ξj,ξj+1)⊆(ξ1,1)∩(-1,ξk), we also get Gξ1′α≤-c1 and Gξk′α≥c1. Then thanks to (7.5) we find that
-
u
0
′
α
≥
2
c
1
+
k
-
2
2
c
2
in
(
ξ
j
,
ξ
j
+
1
)
for any
j
odd
.
∎
We can now complete the proof of Theorem 1.1.
Proof of Theorem 1.1.
Let uλ be the solution constructed in Theorem 1.2. In view of Lemma 7.1, we only need to prove that, for λ small enough, uλ has exactly k nodal regions in I or, equivalently, exactly k-1 zeroes in I. Let us fix ε>0 small enough so that
(7.6)
(
-
1
)
i
-
1
u
0
(
ξ
i
+
ε
)
>
0
,
(
-
1
)
i
-
1
u
0
(
ξ
i
-
ε
)
>
0
and
(
ξ
i
-
2
ε
,
ξ
i
+
2
ε
)
⊆
I
∖
⋃
j
≠
i
(
ξ
j
-
ε
,
ξ
j
+
ε
)
for i=1,…,k. Let us split I:=Iε1∪Iε2∪Iε3, where
I
ε
1
:=
(
-
1
,
ξ
1
-
ε
]
∪
[
ξ
k
+
ε
,
1
)
,
I
ε
2
:=
⋃
i
=
1
k
(
ξ
i
-
ε
,
ξ
i
+
ε
)
,
I
ε
3
=
I
∖
(
I
ε
1
∪
I
ε
2
)
.
First, we observe that uλ has no zeroes in Iε1. Using Proposition 7.2, in [ξk+ε,1), we can write
(
-
1
)
k
-
1
u
0
(
x
)
=
∫
x
1
(
-
1
)
k
u
0
′
(
t
)
𝑑
t
≥
∫
x
1
c
1
-
t
2
𝑑
t
≥
c
2
∫
x
1
1
1
-
t
𝑑
t
=
c
2
(
1
-
x
)
=
2
d
(
x
)
,
where d(x)=1-|x|=dist(x,∂I). Similarly, for x∈(-1,ξ1-ε], we can write
u
0
(
x
)
=
∫
-
1
x
u
0
′
(
t
)
𝑑
t
≥
∫
x
1
c
1
-
t
2
𝑑
t
≥
c
2
∫
x
1
1
1
+
t
𝑑
t
=
c
2
(
1
+
x
)
=
2
d
(
x
)
.
Thanks to Lemma 7.1, we get |uλ(x)|≥d(x) in Iε1, provided λ is sufficiently small. This shows that uλ has no zeroes in Iε1.
Next, we observe that uλ has no zeroes in Iε2. Let us fix 1≤i≤k. Lemma 3.1 gives that
u
λ
=
(
-
1
)
i
-
1
P
U
δ
i
(
λ
)
,
ξ
i
(
λ
)
+
2
π
∑
j
≠
i
(
-
1
)
j
-
1
G
(
ξ
j
(
λ
)
,
ξ
i
)
+
O
(
|
⋅
-
ξ
i
|
)
=
(
-
1
)
i
-
1
log
(
1
δ
i
(
λ
)
2
+
|
x
-
ξ
i
(
λ
)
|
2
)
+
2
π
(
-
1
)
i
-
1
H
(
ξ
i
,
ξ
i
)
+
2
π
∑
j
≠
i
(
-
1
)
j
-
1
G
(
ξ
j
,
ξ
i
)
+
O
(
ε
)
in (ξi-ε,ξi+ε) if λ is small enough. Moreover, we may assume that |ξi(λ)-ξi|≤ε and δi(λ)≤ε. In particular, we have that |x-ξi(λ)|2+δi(λ)2≤5ε2 for any x∈(ξi-ε,ξi+ε). Then we get
|
u
λ
|
≥
log
1
5
ε
2
-
O
(
1
)
.
Thus, we have |uλ|≥1 in (ξi-ε,ξi+ε) if ε is fixed small enough.
Finally, let us consider the interval Iε3. Note that Iε3 has exactly k-1 connected components, namely we have
I
ε
3
=
⊔
i
=
1
k
-
1
J
i
,
ε
,
where
J
i
,
ε
=
[
ξ
i
+
ε
,
ξ
i
+
1
-
ε
]
.
By (7.6) and Proposition 7.2, we know that for any 1≤i≤k-1, if ε is small enough, we have
(
-
1
)
i
u
0
(
ξ
i
+
ε
)
<
0
,
(
-
1
)
i
u
0
(
ξ
i
+
1
-
ε
)
>
0
and
(
-
1
)
i
u
0
′
≥
c
in
J
i
,
ε
.
Since uλ→u0 in C1(I¯ε3) by Lemma 7.1, this implies that
(
-
1
)
i
u
λ
(
ξ
i
+
ε
)
<
0
,
(
-
1
)
i
u
λ
(
ξ
i
+
1
-
ε
)
>
0
and
(
-
1
)
i
u
λ
′
≥
c
in
J
i
,
ε
.
Then uλ has exactly one zero in Ji,ε for any 1≤i≤k-1. We can so conclude that uλ has exactly k-1 zeroes in Iε3 (and thus in I), as claimed. ∎
,
Gabriele Mancini
