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BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET
LAPLACIAN
FALKO BAUSTIAN AND VLADIMIR BOBKOV
Abstract. We provide improved sufficient assumptions on sequences of Fučík eigenvalues
of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fučík
eigenfunctions form a Riesz basis in 𝐿2(0, 𝜋). For that purpose, we introduce a criterion for
a sequence in a Hilbert space to be a Riesz basis.
1. Introduction
We study basis properties of sequences of eigenfunctions of the Fučík eigenvalue problem
for the one-dimensional Dirichlet Laplacian
−𝑢′′(𝑥) = 𝛼𝑢+(𝑥)−𝛽𝑢−(𝑥), 𝑥 ∈(0, 𝜋),
𝑢(0) = 𝑢(𝜋)=0,(1.1)
where 𝑢+= max(𝑢, 0) and 𝑢−= max(−𝑢, 0). The Fučík spectrum is the set Σ(0, 𝜋)of pairs
(𝛼, 𝛽)∈R2for which (1.1) possesses a non-zero classical solution. Any (𝛼, 𝛽)∈Σ(0, 𝜋)
is called Fučík eigenvalue and any corresponding non-zero classical solution of (1.1) is called
Fučík eigenfunction. The Fučík eigenvalue problem (1.1) was introduced in [4] and [6] to study
elliptic equations with “jumping” nonlinearities, and afterwards it has been widely investigated
in various aspects and for different operators, see, e.g., the surveys [3], [8, Chapter 9.4], and
references therein. To the best of our knowledge, basisness of sequences of Fučík eigenfunc-
tions was considered for the first time in [2]. In that article, we provided several sufficient
assumptions on sequences of Fučík eigenvalues to obtain Riesz bases of 𝐿2(0, 𝜋)consisting of
Fučík eigenfunctions. Let us recall that a sequence is a Riesz basis in a Hilbert space if it
is the image of an orthonormal basis of that space under a linear homeomorphism, see, e.g.,
[9]. The aim of the present note is to use more general techniques to significantly improve the
results of [2].
Let us describe the structure of the Fučík spectrum Σ(0, 𝜋). It is not hard to see that
the lines {1} × Rand R× {1}are subsets of Σ(0, 𝜋), since they correspond to sign-constant
solutions of (1.1) which are constant multiples of sin 𝑥, the first eigenfunction of the Dirichlet
2010 Mathematics Subject Classification. 34L10, 34B25, 34B08, 47A70.
Key words and phrases. Fucik spectrum, Fucik eigenfunctions, Riesz basis, Paley-Wiener stability.
The main part of the research was done during a stay of F. Baustian at the Ufa Federal Research Centre.
The stay was financed by the German-Russian Interdisciplinary Science Center (G-RISC), grant no. F-2021b-
8_d. V. Bobkov was supported in the framework of implementation of the development program of Volga
Region Mathematical Center (agreement no. 075-02-2021-1393).
1
arXiv:2111.08329v1 [math.CA] 16 Nov 2021
2 F. BAUSTIAN AND V. BOBKOV
Laplacian in (0, 𝜋). The remaining part of Σ(0, 𝜋)is exhausted by the hyperbola-type curves
Γ𝑛=(𝛼, 𝛽)∈R2:𝑛
2
𝜋
√𝛼+𝑛
2
𝜋
√𝛽=𝜋
for even 𝑛∈N, and
Γ𝑛=(𝛼, 𝛽)∈R2:𝑛+ 1
2
𝜋
√𝛼+𝑛−1
2
𝜋
√𝛽=𝜋,
Γ𝑛=(𝛼, 𝛽)∈R2:𝑛−1
2
𝜋
√𝛼+𝑛+ 1
2
𝜋
√𝛽=𝜋
for odd 𝑛≥3, see, e.g., [6, Lemma 2.8]. Evidently, (𝛼, 𝛽)∈Γ𝑛for odd 𝑛≥3implies
(𝛽, 𝛼)∈
Γ𝑛. If 𝑢is a Fučík eigenfunction for some (𝛼, 𝛽), then so is 𝑡𝑢 for any 𝑡 > 0, while
−𝑡𝑢 is a Fučík eigenfunction for (𝛽, 𝛼). Hence, we neglect the curve
Γ𝑛from our investiga-
tion of the basis properties of Fučík eigenfunctions. Each sign-changing Fučík eigenfunction
consists of alternating positive and negative bumps, where positive bumps are described by
𝐶1sin(√𝛼(𝑥−𝑥1)), while negative bumps are described by 𝐶2sin(√𝛽(𝑥−𝑥2)), for proper
constants 𝐶1, 𝐶2, 𝑥1, 𝑥2∈R.
We want to uniquely specify a Fučík eigenfunction for each point of Σ(0, 𝜋). In slight
contrast to [2], we normalize Fučík eigenfunctions in such a way that they are “close” to the
functions
𝜙𝑘(𝑥) = 2
𝜋sin(𝑘𝑥), 𝑘 ∈N,
which form a complete orthonormal system in 𝐿2(0, 𝜋). This choice will be helpful in the
proof of our main result, Theorem 1.3, below.
Definition 1.1. Let 𝑛≥2and (𝛼, 𝛽 )∈Γ𝑛. The normalized Fučík eigenfunction 𝑔𝑛
𝛼,𝛽 is the
𝐶2-solution of the boundary value problem (1.1) with (𝑔𝑛
𝛼,𝛽)′(0) >0and which is normalized
by
‖𝑔𝑛
𝛼,𝛽‖∞= sup
𝑥∈[0,𝜋]|𝑔𝑛
𝛼,𝛽(𝑥)|=2
𝜋.
For 𝑛= 1, we set 𝑔1
𝛼,𝛽 =𝜙1for every (𝛼, 𝛽)∈({1} × R)∪(R× {1}).
Piecewise definitions of the Fučík eigenfunctions 𝑓𝑛
𝛼,𝛽 =𝜋/2𝑔𝑛
𝛼,𝛽 can be found in the
equations (1.2) and (1.3) in [2]. In accordance to [2], we study the basisness of sequences of
Fučík eigenfunctions described by the following definition.
Definition 1.2. We define the Fučík system 𝐺𝛼,𝛽 ={𝑔𝑛
𝛼(𝑛),𝛽(𝑛)}as a sequence of normal-
ized Fučík eigenfunctions with mappings 𝛼, 𝛽 :N→Rsatisfying 𝛼(1) = 𝛽(1) = 1 and
(𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛for every 𝑛≥2.
We can now formulate our main result on the basisness of Fučík systems which presents a
non-trivial generalization of [2, Theorems 1.4 and 1.9].
Theorem 1.3. Let 𝐺𝛼,𝛽 be a Fučík system. Let 𝑁be a subset of the even natural numbers
and 𝑁*=N∖𝑁. Assume that
𝑛∈𝑁*1−⟨𝑔𝑛
𝛼,𝛽, 𝜙𝑛⟩2
‖𝑔𝑛
𝛼,𝛽‖2+𝐸2sup
𝑛∈𝑁4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<1,(1.2)
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 3
with sup𝑛∈𝑁4 max(𝛼(𝑛), 𝛽(𝑛))/𝑛2∈[4,9). Here, 𝐸: [4,9) →Ris a strictly increasing
function defined as
𝐸(𝛾) = 2√2
𝜋
𝛾2
√𝛾−1
(√𝛾−2) sin 𝜋
√𝛾
(𝛾−1)(2√𝛾−1) +((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)
3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2)
+4
√3𝜋
𝛾2
√𝛾−1
(√𝛾−2) sin −3𝜋
√𝛾
(9 −𝛾)(2√𝛾−3)(4√𝛾−3) +2
𝜋
𝛾2
√𝛾−1
(√𝛾−2)
(16 −𝛾)(3√𝛾−4)(5√𝛾−4)
+6
5
2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
∞
𝑘=5
1
(𝑘2−𝛾)((𝑘−1)√𝛾−𝑘)((𝑘+ 1)√𝛾−𝑘).(1.3)
Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
The proof of this theorem is given in Section 3and it is based on a general basisness criterion
provided in Section 2. We visualize special cases of domains on the (𝛼, 𝛽)-plane described in
Theorem 1.3 in Figures 1and 2below.
Notice that, thanks to the orthonormality of {𝜙𝑛}, the terms in the first sum in (1.2) satisfy
0≤1−⟨𝑔𝑛
𝛼,𝛽, 𝜙𝑛⟩2
‖𝑔𝑛
𝛼,𝛽‖2=‖𝑔𝑛
𝛼,𝛽 −𝜙𝑛‖2−(‖𝑔𝑛
𝛼,𝛽‖2− ⟨𝑔𝑛
𝛼,𝛽, 𝜙𝑛⟩)2
‖𝑔𝑛
𝛼,𝛽‖2≤ ‖𝑔𝑛
𝛼,𝛽 −𝜙𝑛‖2,(1.4)
and we have the following explicit bounds:
‖𝑔𝑛
𝛼,𝛽 −𝜙𝑛‖2≤
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
8(3 + 𝜋2)
9
(max(√𝛼, √𝛽)−𝑛)2
𝑛2for even 𝑛,
8𝑛2(𝑛2+ 1)
(𝑛−1)4
(√𝛼−𝑛)2
𝑛2for odd 𝑛≥3with 𝛼≥𝑛2,
10𝑛2(𝑛2+ 1)
(𝑛+ 1)4
(√𝛽−𝑛)2
𝑛2for odd 𝑛≥3with 𝛽 > 𝑛2,
(1.5)
see the estimates (3.2), (3.4), (3.5), (3.6) in [2, Section 3]. In view of (1.4), if we chose 𝑁=∅,
then Theorem 1.3 is an improvement of [2, Theorem 1.4].
Let us summarize a few properties of the function 𝐸defined in Theorem 1.3, see the end
of Section 3for discussion.
Lemma 1.4. The function 𝐸has the following properties:
(i) 𝐸is continuous in [4,9).
(ii) Each summand in the definition (1.3)of 𝐸is strictly increasing in [4,9).
(iii) We have 𝐸(4) = 0 and 𝐸(6.49278 . . .)=1.
(iv) The infinite sum in the definition (1.3)of 𝐸in (4,9) can be expressed as follows:
6
5
2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
∞
𝑘=5
1
(𝑘2−𝛾)((𝑘−1)√𝛾−𝑘)((𝑘+ 1)√𝛾−𝑘)
=6
5
2
𝜋
√𝛾
√𝛾−1
∞
𝑘=5 1
𝑘2−𝛾−1
𝑘2−𝛾
(√𝛾−1)2
4 F. BAUSTIAN AND V. BOBKOV
=6
5
1
𝜋(√𝛾−1) 𝜋(√𝛾−1) cot 𝜋√𝛾
√𝛾−1−𝜋cot(𝜋√𝛾)−(√𝛾−2)
−6
5
2
𝜋
𝛾2(√𝛾−2)
√𝛾−1
4
𝑘=1
1
(𝑘2−𝛾)((𝑘−1)√𝛾−𝑘)((𝑘+ 1)√𝛾−𝑘).
The interval [4,9) appears naturally in the proof of Theorem 1.3. In fact, Lemma 1.4 (iii)
indicates that the highest possible value of sup𝑛∈𝑁4 max(𝛼(𝑛), 𝛽(𝑛))/𝑛2to satisfy the
assumption (1.2) is even smaller than 9.
We obtain the following practical corollary of Theorem 1.3 by applying the upper bounds
(1.5) for the case that 𝑁is the set of all even natural numbers, see Figure 1.
Corollary 1.5. Let 𝐺𝛼,𝛽 be a Fučík system, and 𝜀 > 0. Assume that
sup
𝑛∈Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<6.49278 . . .
and
max(𝛼(𝑛), 𝛽(𝑛)) ≤𝑛+√𝑐𝑛𝑛(1−𝜀)/22for all odd 𝑛≥3,
where
0≤𝑐𝑛<
1−𝐸2sup
𝑛∈Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2
45 1−1
21+𝜀𝜁(1 + 𝜀)−1
with the Riemann zeta function 𝜁. Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
(a) (b)
Figure 1. The assumptions of Corollary 1.5 are satisfied for (𝛼(𝑛), 𝛽(𝑛)) be-
longing to bold lines inside the shaded regions. We have 𝜀= 0.5for both panels
and sup𝑛∈Neven 4 max(𝛼(𝑛),𝛽(𝑛))
𝑛2= 5,6in panel (A), (B), respectively.
If we assume that the first sum of (1.2) in Theorem 1.3 is vanishing, which corresponds to
𝑐𝑛= 0 for all odd 𝑛≥3in the previous corollary, we obtain the following result.
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 5
Corollary 1.6. Let 𝐺𝛼,𝛽 be a Fučík system such that 𝑔𝑛
𝛼,𝛽 =𝜙𝑛for any odd 𝑛. Assume that
sup
𝑛∈Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2<6.49278 . . . (1.6)
Then 𝐺𝛼,𝛽 is a Riesz basis in 𝐿2(0, 𝜋).
Figure 2. The assumption (1.6) is satisfied for (𝛼(𝑛), 𝛽(𝑛)) belonging to bold
lines inside the shaded region.
We remark that Corollaries 1.5 and 1.6 are significant improvements of [2, Theorem 1.9]
since each point (𝛼(𝑛), 𝛽(𝑛)) ∈Γ𝑛for even 𝑛≥2is free to belong to the whole angular sector
in between the line
𝛽=sup
𝑛∈Neven 4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2−1−2
𝛼
and its reflection with respect to the main diagonal 𝛼=𝛽, and the angle of that sector is
allowed to be larger than the one provided by [2, Theorem 1.9]. We refer to Figure 2for
the domain on the (𝛼, 𝛽)-plane given by Corollary 1.6. Moreover, Corollary 1.5 improves [2,
Theorem 1.9] in the sense that 𝑔𝑛
𝛼,𝛽 for odd 𝑛≥3might differ from 𝜙𝑛, see Figure 1.
2. Basisness criterion
In this section, we formulate a useful generalization of the separation of variables approach
of [5] in a real Hilbert space 𝑋. The provided criterion will be applied to the space 𝐿2(0, 𝜋)
to prove our main result, Theorem 1.3, in the subsequent section.
Theorem 2.1. Let 𝑀∈N. Let 𝑁*, 𝑁𝑚⊂N,1≤𝑚≤𝑀, be pairwise disjoint sets which
form a decomposition of the natural numbers, i.e.,
𝑁*∪
𝑀
𝑚=1
𝑁𝑚=N.
6 F. BAUSTIAN AND V. BOBKOV
Let {𝜑𝑛}be a complete orthonormal sequence in 𝑋and {𝑓𝑛} ⊂ 𝑋be a sequence that can be
represented as
𝑓𝑛=𝜑𝑛+∞
𝑘=1
𝐶𝑚
𝑛,𝑘𝑇𝑚
𝑘𝜑𝑛for every 𝑛∈𝑁𝑚,1≤𝑚≤𝑀, (2.1)
and satisfies
Λ*:=
𝑛∈𝑁*1−⟨𝑓𝑛, 𝜑𝑛⟩2
‖𝑓𝑛‖21
2
<∞.
In the representation formula (2.1),{𝑇𝑚
𝑘}is a family of bounded linear mappings from 𝑋to
itself with bounds ‖𝑇𝑚
𝑘‖*≤𝑡𝑚
𝑘on the operator norm and {𝐶𝑚
𝑛,𝑘}is a family of constants with
uniform bounds |𝐶𝑚
𝑛,𝑘| ≤ 𝑐𝑚
𝑘that satisfy
Λ𝑚:= ∞
𝑘=1
𝑐𝑚
𝑘𝑡𝑚
𝑘<∞.(2.2)
Then {𝑓𝑛}is a basis in 𝑋provided that
Λ2
*+
𝑀
𝑚=1
Λ2
𝑚<1.(2.3)
If, in addition, the subsequence {𝑓𝑛}𝑛∈𝑁*is bounded, then {𝑓𝑛}is a Riesz basis in 𝑋.
Proof. Denote
𝑓𝑛=𝜌𝑛𝑓𝑛, where 𝜌𝑛= 1 for 𝑛∈N∖𝑁*, and the values of 𝜌𝑛for 𝑛∈𝑁*will
be specified later. Let {𝑎𝑛}𝑛∈
𝑁be an arbitrary finite sequence of constants with a finite index
set
𝑁⊂N. Setting
𝑁*=𝑁*∩
𝑁and
𝑁𝑚=𝑁𝑚∩˜
𝑁for every 1≤𝑚≤𝑀, we obtain
⃦
⃦
𝑛∈
𝑁
𝑎𝑛(
𝑓𝑛−𝜑𝑛)⃦
⃦≤
𝑀
𝑚=1 ⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛(𝑓𝑛−𝜑𝑛)⃦
⃦+⃦
⃦
𝑛∈
𝑁*
𝑎𝑛(𝜌𝑛𝑓𝑛−𝜑𝑛)⃦
⃦.(2.4)
For the first sum on the right-hand side of (2.4), we apply the representation (2.1) and get
𝑀
𝑚=1 ⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛(𝑓𝑛−𝜑𝑛)⃦
⃦=
𝑀
𝑚=1 ⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛
∞
𝑘=1
𝐶𝑚
𝑛,𝑘𝑇𝑚
𝑘𝜑𝑛⃦
⃦=
𝑀
𝑚=1 ⃦
⃦∞
𝑘=1
𝑇𝑚
𝑘
𝑛∈
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛⃦
⃦
≤
𝑀
𝑚=1
∞
𝑘=1 ⃦
⃦𝑇𝑚
𝑘
𝑛∈
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛⃦
⃦≤
𝑀
𝑚=1
∞
𝑘=1
𝑡𝑚
𝑘⃦
⃦
𝑛∈
𝑁𝑚
𝐶𝑚
𝑛,𝑘𝑎𝑛𝜑𝑛⃦
⃦
≤
𝑀
𝑚=1
∞
𝑘=1
𝑡𝑚
𝑘𝑐𝑚
𝑘⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛𝜑𝑛⃦
⃦=
𝑀
𝑚=1
Λ𝑚⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛𝜑𝑛⃦
⃦,
while for the second sum we obtain
⃦
⃦
𝑛∈
𝑁*
𝑎𝑛(𝜌𝑛𝑓𝑛−𝜑𝑛)⃦
⃦≤
𝑛∈
𝑁*
‖𝜌𝑛𝑓𝑛−𝜑𝑛‖21
2
𝑛∈
𝑁*
|𝑎𝑛|21
2.
Let us choose 𝜌𝑛to be a minimizer of the distance ‖𝜌𝑓𝑛−𝜑𝑛‖2with respect to 𝜌. Since
‖𝜌𝑓𝑛−𝜑𝑛‖2=𝜌2‖𝑓𝑛‖2−2𝜌⟨𝑓𝑛, 𝜑𝑛⟩+ 1,
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 7
we readily see that
‖𝜌𝑛𝑓𝑛−𝜑𝑛‖2= min
𝜌∈R‖𝜌𝑓𝑛−𝜑𝑛‖2= 1 −⟨𝑓𝑛, 𝜑𝑛⟩2
‖𝑓𝑛‖2=‖𝑓𝑛−𝜑𝑛‖2−(‖𝑓𝑛‖2− ⟨𝑓𝑛, 𝜑𝑛⟩)2
‖𝑓𝑛‖2
with 𝜌𝑛=⟨𝑓𝑛, 𝜑𝑛⟩/‖𝑓𝑛‖2. Evidently, we have |𝜌𝑛| ≤ 1. We remark that in case of 𝜌𝑛= 0,
we get Λ*≥1which violates the assumption (2.3). Applying now the Cauchy inequality, we
deduce from (2.4) that
⃦
⃦
𝑛∈
𝑁
𝑎𝑛(
𝑓𝑛−𝜑𝑛)⃦
⃦≤
𝑀
𝑚=1
Λ𝑚⃦
⃦
𝑛∈
𝑁𝑚
𝑎𝑛𝜑𝑛⃦
⃦+ Λ*
𝑛∈
𝑁*
|𝑎𝑛|21
2
≤𝑀
𝑚=1
Λ2
𝑚+ Λ2
*1
2⃦
⃦
𝑛∈
𝑁
𝑎𝑛𝜑𝑛⃦
⃦.
We conclude from the assumption (2.3) that the sequence {
𝑓𝑛}is Paley-Wiener near to the
complete orthonormal sequence {𝜑𝑛}and, thus, it is a Riesz basis in 𝑋, see, e.g., [9, Chapter 1,
Theorem 10]. Clearly, {𝑓𝑛}={𝜌−1
𝑛
𝑓𝑛}is a basis in 𝑋. Assume that the subsequence {𝑓𝑛}𝑛∈𝑁*
is bounded. Then there exists 0< 𝑐 < 1such that |𝜌𝑛| ≥ 𝑐for all 𝑛∈
𝑁*. This is evident for
finite 𝑁*since 𝜌𝑛̸= 0. In the case of infinte 𝑁*, if we suppose that 𝜌𝑛goes to zero up to a
subsequence, then the sum
Λ*=
𝑛∈𝑁*1−⟨𝑓𝑛, 𝜑𝑛⟩2
‖𝑓𝑛‖21
2
=
𝑛∈𝑁*1−𝜌2
𝑛‖𝑓𝑛‖21
2
does not converge. Recalling 𝜌𝑛= 1 for every 𝑛∈N∖𝑁*, we obtain 1≤ |𝜌−1
𝑛| ≤ 𝑐−1for all
𝑛∈Nwhich implies that {𝑓𝑛}is a Riesz basis in 𝑋, see, e.g., [9, Chapter 1, Theorem 9].
In the case 𝑁1=N, Theorem 2.1 simplifies to Theorem D from [5] and for 𝑁*=Nwe get
the result of Theorem V-2.21 and Corollary V-2.22 i) from [7] which were discussed in [2].
Remark 2.2. It can be seen from the proof of Theorem 2.1 that if we weaken the definition
of Λ*to
Λ*:=
𝑛∈𝑁*
‖𝑓𝑛−𝜑𝑛‖21
2
≤Λ*,
then we can formulate the following result under the assumptions of Theorem 2.1: the sequence
{𝑓𝑛}is a Riesz basis in 𝑋provided that
Λ2
*+
𝑀
𝑚=1
Λ2
𝑚<1.
The boundedness of the subsequence {𝑓𝑛}𝑛∈𝑁*is not required under this modified assumption.
3. Proof of Theorem 1.3
We prove Theorem 1.3 by applying the general basisness criterion introduced in the previous
section. To determine the bounds on the family of constants {𝐶𝑚
𝑛,𝑘}in Theorem 2.1 we will
make use of the Fourier coefficients of Fučík eigenfunctions corresponding to Fučík eigenvalues
8 F. BAUSTIAN AND V. BOBKOV
on the first nontrivial curve Γ2. Namely, we provide estimates for the Fourier coefficients of
the odd Fourier expansion of the function
𝑔2
𝛾,𝛾/(√𝛾−1)2=∞
𝑘=1
𝐴𝑘(𝛾)𝜙𝑘(𝑥)
for 𝛾 > 4which are given by
𝐴𝑘(𝛾) = 𝜋
0
𝑔2
𝛾,𝛾/(√𝛾−1)2(𝑥)𝜙𝑘(𝑥) d𝑥=2
𝜋
𝛾2
√𝛾−1
(2 −√𝛾) sin 𝑘𝜋
√𝛾
(𝑘2−𝛾)(𝑘2(√𝛾−1)2−𝛾),
and of the function
𝑔2
𝛿/(√𝛿−1)2,𝛿 =∞
𝑘=1
𝐴𝑘(𝛿)𝜙𝑘(𝑥)
for 𝛿 > 4which are given by
𝐴𝑘(𝛿) = 𝜋
0
𝑔2
𝛿/(√𝛿−1)2,𝛿 (𝑥)𝜙𝑘(𝑥) d𝑥= (−1)𝑘𝐴𝑘(𝛿).
In the case 𝛾=𝛿= 4, we have 𝐴2= 1 and 𝐴𝑘= 0 for any other 𝑘∈N.
Obviously, we have
|𝐴1(𝛾)|=𝐵1(𝛾) := 2
𝜋
𝛾2
√𝛾−1
(√𝛾−2) sin 𝜋
√𝛾
(𝛾−1)(2√𝛾−1) (3.1)
and it was shown in [2, Section 5] that
|𝐴2(𝛾)−1| ≤ 𝐵2(𝛾) := ((3 + 𝜋2)𝛾+ (9 −2𝜋2)√𝛾−6)(√𝛾−2)
3(√𝛾−1)(√𝛾+ 2)(3√𝛾−2) .(3.2)
For 𝛾∈[4,9), we clearly have
|𝐴3(𝛾)|=𝐵3(𝛾) := 2
𝜋
𝛾2
√𝛾−1
(√𝛾−2) −sin 3𝜋
√𝛾
(9 −𝛾)(2√𝛾−3)(4√𝛾−3) (3.3)
and for 𝑘≥4we use the simple estimate
|𝐴𝑘(𝛾)| ≤ 𝐵𝑘(𝛾) := 2
𝜋
𝛾2
√𝛾−1
(√𝛾−2)
(𝑘2−𝛾)((𝑘−1)√𝛾−𝑘)((𝑘+ 1)√𝛾−𝑘).(3.4)
Evidently, the same bounds hold for
𝐴𝑘. Numerical calculations with the exact coefficients
show that the used estimates in (3.2) and (3.4) do not influence the results in a significant
way.
Lemma 3.1. Let 𝛾∈[4,9) and 𝑘∈N. Then 𝐵𝑘is strictly increasing.
Proof. For simplicity, we introduce the change of variables 𝑥=√𝛾∈[2,3). The first derivative
of 𝐵𝑘(𝑥2)with 𝑘∈N∖ {1,3}is a rational function with a positive denominator and we can
easily check that the numerator is positive, as well. Hence, 𝐵𝑘(𝛾)with 𝑘∈N∖{1,3}is strictly
increasing for 𝛾∈[4,9). The first derivative of 𝐵1(𝑥2)takes the form
2𝑥2(𝑥−1) cos 𝜋
𝑥𝑥(2𝑥4−4𝑥3−𝑥2+ 15𝑥−8) tan 𝜋
𝑥−𝜋(2𝑥4−5𝑥3+ 5𝑥−2)
𝜋(𝑥−1)2(𝑥2−1)2(2𝑥−1)2.
BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE DIRICHLET LAPLACIAN 9
Noting that 𝑥(2𝑥4−4𝑥3−𝑥2+ 15𝑥−8) >0for 𝑥∈[2,3), we can use the simple lower bound
tan 𝜋
𝑥≥√3to show that the expression in square brackets is positive. Since all other terms
in the derivative are also positive, we conclude that 𝐵1(𝛾)is strictly increasing for 𝛾∈[4,9).
Finally, the numerator of the first derivative of 𝐵3(𝑥2)is given by
−2𝑥2𝑥(10𝑥5+ 90𝑥4−765𝑥3+ 1872𝑥2−1863𝑥+ 648) sin 3𝜋
𝑥
+ 3𝜋(8𝑥6−42𝑥5+ 7𝑥4+ 315𝑥3−693𝑥2+ 567𝑥−162) cos 3𝜋
𝑥,(3.5)
whereas the denominator is a positive polynomial. We have sin 3𝜋
𝑥<0and cos 3𝜋
𝑥<0for
𝑥∈[2,3), and taking into account that
𝑥(10𝑥5+ 90𝑥4−765𝑥3+ 1872𝑥2−1863𝑥+ 648) <0,
3𝜋(8𝑥6−42𝑥5+ 7𝑥4+ 315𝑥3−693𝑥2+ 567𝑥−162) >0,
we employ the estimates
sin 3𝜋
𝑥<−3𝜋
𝑥−𝜋+1
63𝜋
𝑥−𝜋3
and cos 3𝜋
𝑥>−1.
As a result, the expression (3.5) is estimated from below by a polynomial which is positive for
𝑥∈[2,3). Thus, 𝐵3(𝛾)is strictly increasing for 𝛾∈[4,9).
Now we are ready to prove our main result.
Proof of Theorem 1.3.We apply Theorem 2.1, where we consider 𝑋=𝐿2(0, 𝜋), the sequence
{𝑓𝑛}is the Fučík system, which is bounded by definition, and the complete orthonormal set
{𝜑𝑛}is given by {𝜙𝑛}. We set 𝑀= 1 and 𝑁1=𝑁and choose 𝑁*=N∖𝑁as assumed in
Theorem 1.3. We define the linear operators 𝑇1
𝑘:𝐿2(0, 𝜋)→𝐿2(0, 𝜋)as
𝑇1
𝑘𝑔(𝑥) = 𝑔*𝑘𝑥
2,
where
𝑔*(𝑥)=(−1)𝜅𝑔(𝑥−𝜋𝜅)for 𝜋𝜅 ≤𝑥≤𝜋(𝜅+ 1), 𝜅 ∈N∪ {0},
is the 2𝜋-antiperiodic extension for arbitrary functions 𝑔∈𝐿2(0, 𝜋). In particular, we have
𝑇1
𝑘sin(𝑛𝑥) = sin 𝑘𝑛𝑥
2for every even 𝑛. It was proven in [2, Appendix B] that ‖𝑇1
𝑘‖*= 1 for
even 𝑘and ‖𝑇1
𝑘‖*=1+1/𝑘 for odd 𝑘.
Let 𝑛∈𝑁be fixed and recall that 𝑛is even. To begin with, we assume that 𝛼(𝑛)> 𝑛2.
The Fučík eigenfunction 𝑔𝑛
𝛼,𝛽 has the dilated structure
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛾𝑛,𝛾𝑛/(√𝛾𝑛−1)2𝑛𝑥
2with 𝛾𝑛=4𝛼(𝑛)
𝑛2
and, thus, has the odd Fourier expansion
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛾𝑛,𝛾𝑛/(√𝛾𝑛−1)2𝑛𝑥
2=∞
𝑘=1
𝐴𝑘(𝛾𝑛)𝜙𝑘𝑛𝑥
2=∞
𝑘=1
𝐴𝑘(𝛾𝑛)𝑇1
𝑘𝜙𝑛(𝑥).
From this, we directly see that the representation (2.1) of 𝑔𝑛
𝛼,𝛽 in terms of {𝜙𝑛}holds with
the constants 𝐶1
𝑛,𝑘 =𝐴𝑘(𝛾𝑛)for 𝑘̸= 2 and 𝐶1
𝑛,2= 1 −𝐴2(𝛾𝑛). The bounds for the constants
10 F. BAUSTIAN AND V. BOBKOV
|𝐶1
𝑛,𝑘|are given by the functions 𝐵𝑘(𝛾𝑛)defined in (3.1), (3.2), (3.3), and (3.4), which are
strictly increasing in the interval [4,9) by Lemma 3.1. For the case 𝛽(𝑛)> 𝑛2, the Fučík
eigenfunction has the form
𝑔𝑛
𝛼,𝛽(𝑥) = 𝑔2
𝛿𝑛/(√𝛿𝑛−1)2,𝛿𝑛𝑛𝑥
2with 𝛿𝑛=4𝛽(𝑛)
𝑛2,
and by analogous arguments we get the bounds |𝐶1
𝑛,𝑘| ≤ 𝐵𝑘(𝛿𝑛). If 𝛼(𝑛) = 𝑛2, and hence
𝛽(𝑛) = 𝑛2, then we set 𝐶1
𝑛,𝑘 = 0 for every 𝑘∈N.
In view of the monotonicity, we have
|𝐶1
𝑛,𝑘| ≤ 𝐵𝑘sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛).
Therefore, we can provide the following upper estimate on the constant Λ1defined in (2.2):
Λ1≤√2𝐵1sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)+𝐵2sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)
+4
3𝐵3sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)+𝐵4sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)
+6
5
∞
𝑘=5
𝐵𝑘sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)
=𝐸sup
𝑛∈𝑁
max(𝛾𝑛, 𝛿𝑛)=𝐸sup
𝑛∈𝑁4 max(𝛼(𝑛), 𝛽(𝑛))
𝑛2,
with the function 𝐸introduced in Theorem 1.3, and 𝐸is strictly increasing in [4,9). Noticing
that we have
Λ*=
𝑛∈𝑁*1−⟨𝑔𝑛
𝛼,𝛽, 𝜙𝑛⟩2
‖𝑔𝑛
𝛼,𝛽‖21
2
,
the assumption (1.2) yields the assumption Λ2
*+ Λ2
1<1in Theorem 2.1. This completes the
proof of Theorem 1.3.
We conclude this note by discussing Lemma 1.4. The monotonicity statement (ii) directly
follows from Lemma 3.1, and to obtain the alternative representation (iv), we make use of the
identity
∞
𝑘=1
1
𝑘2−𝑎2=1
2𝑎2−𝜋cot(𝜋𝑎)
2𝑎, 𝑎 ̸∈ N,
see, e.g., [1, (6.3.13)]. The representation (iv) shows that the function 𝐸is continuous in
[4,9). The combination of the continuity and monotonicity of 𝐸allows us to compute values
of 𝐸with an arbitrary precision. In particular, we have 𝐸(6.49278 . . .)=1.
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(F. Baustian)
Institute of Mathematics, University of Rostock,
Ulmenstraße 69, 18057 Rostock, Germany
Email address:falko.baustian@uni-rostock.de
(V. Bobkov)
Institute of Mathematics, Ufa Federal Research Centre, RAS,
Chernyshevsky str. 112, 450008 Ufa, Russia
Email address:bobkov@matem.anrb.ru
