Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 13, pp. 1–12.

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

REMARKS ON THE SECOND NEUMANN EIGENVALUE

JOSÉ C. SABINA DE LIS

Abstract. This work reviews some basic features on the second (first non-

trivial) eigenvalue λ2 to the Neumann problem

−∆pu = λ|u|p−2u x ∈ Ω

|∇u|p−2 ∂u

∂ν
= 0 x ∈ ∂Ω,

where Ω is a bounded Lipschitz domain of RN , ν is the outer unit normal, and
∆pu = div(|∇u|p−2∇u) is the p-Laplacian operator. We are mainly concerned

with the variational characterization of λ2 and place emphasis on the range

1 < p < 2, where the nonlinearity |u|p−2u becomes non smooth. We also
address the corresponding result for the p-Laplacian in graphs.

1. Introduction

The analysis of the eigenvalues of the p-Laplacian operator under different types
of boundary conditions is one of the most interesting issues in nonlinear analysis
[13, 14, 17]. Here, we focuss on the Neumann eigenvalue problem

−∆pu = λ|u|p−2u x ∈ Ω

|∇u|p−2 ∂u

∂ν
= 0 x ∈ ∂Ω,

(1.1)

where Ω ⊂ RN is a C0,1 bounded domain, ν stands for its outer unit normal on ∂Ω
and p > 1.

We recall that u ∈ W 1,p(Ω) \ {0} is said to be a weak eigenfunction to (1.1)
associated with the eigenvalue λ ∈ R if the equality∫

Ω

|∇u|p−2∇u∇v dx = λ

∫
Ω

|u|p−2uv dx, (1.2)

holds for arbitrary test functions v in W 1,p(Ω).
From the definition of eigenvalue it follows by choosing v = u in (1.2) that

eigenvalues λ must be nonnegative. Thus λ1 = 0 becomes the “first” (lowest)
eigenvalue whose eigenfunctions are constant functions. As all those eigenfunctions
are a multiple of u = 1, this amounts to say that λ1 is simple (in a proper sense).
Moreover, λ1 = 0 is an isolated eigenvalue as it is going to be checked below
(Theorem 1.1). It should be stressed that for the Dirichlet boundary condition
u = 0 on ∂Ω, proving the simplicity of the first (lowest) eigenvalue λD1 turned out to

2020 Mathematics Subject Classification. 35J70, 35J92, 35P30.

Key words and phrases. p-Laplacian operator; eigenvalues; Neumann conditions.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted August 16, 2021. Published February 20, 2022.

1



2 J. C. SABINA DE LIS EJDE-2022/13

be a quite hard question. A successful answer was given in [3], where the isolation
of λD1 was furthermore shown (a sharpened result was later proved in [15, 16]).
Moreover, the existence of a second Dirichlet eigenvalue λD2 is a consequence of
the latter assertion. In addition, it is worth remarking that obtaining a variational
characterization of λD2 is by no means an easy task. Reader is referred to [4,
11, 12] for different variational expressions of λD2 . Just to grasp an insight, the
corresponding one in [11] is

λD2 = inf
γ

max
u∈γ(I)

∫
Ω
|∇u|p dx∫

Ω
|u|p dx

, (1.3)

where γ varies in the set of all continuous curves γ : I = [0, 1]→W 1,p
0 (Ω)\{0} such

that γ(0) = φ1, γ(1) = −φ1, φ1 being a fixed normalized eigenfunction associated
with λD1 . See also [5, 6] for related results in a “nonsymmetric” version of the
Dirichlet and Neumann eigenvalue problems.

As mentioned above, λ1 = 0 is isolated and so the Neumann problem (1.1)
admits a second eigenvalue λ2. More interestingly, I came across reference [7] when
searching for sharp lower estimates for λ2 in a general domain Ω. I was amazed by
the existence of the elegant characterization of this eigenvalue, freely used there.
The expression is considerably much simpler than its Dirichlet counterpart (1.3),
and seems indeed closer to the familiar ‘Rayleigh quotient’ for −∆. Namely,

λ2 = inf

∫
Ω
|∇u|p dx∫

Ω
|u|p dx

, (1.4)

the infimum being extended over all those nonvanishing functions u ∈ W 1,p(Ω)
satisfying the “null average” type condition (see Section 2),∫

Ω

|u|p−2u dx = 0. (1.5)

In fact, by setting v = 1 as a test function in (1.2) it follows that the eigenfunctions
u ∈W 1,p(Ω) associated with all possible eigenvalues λ 6= 0 must satisfy (1.5).

On the other hand, it is more or less straightforward for the specialist to check
(1.4) in the case p ≥ 2. On the contrary, the proof for the complementary range
1 < p < 2 is far from obvious and this brief note is just devoted to this goal. Since
I have not been able to find a proof, I decided to publish one of my own. It should
be mentioned that the omission of the case 1 < p < 2 is striking in some references
(see [14, Chapter IV, §2]).

It should be also remarked that C∗ = λ
−1/p
2 can be regarded as the optimum

constant C in Poincaré’s inequality

inf
t∈R
‖u− t‖Lp(Ω) ≤ C‖|∇u|‖Lp(Ω) , (1.6)

where u ∈ W 1,p(Ω). In spite of reference [18] containing general versions of this
inequality, a connection between (1.6) and λ2 is not reported there.

Our main result reads as follows.

Theorem 1.1. Let Ω ⊂ RN be a bounded C0,1 domain and set

λ̂ = inf
u∈M0\{0}

∫
Ω
|∇u|p dx∫

Ω
|u|p dx

, (1.7)



EJDE-2022/13 SECOND NEUMANN EIGENVALUE 3

where

M0 = {u ∈W 1,p(Ω) :

∫
Ω

|u|p−2u dx = 0}.

Then the following properties hold

(i) The infimum in (1.7) is achieved at some u ∈M0 and thus λ̂ > 0.

(ii) Every eigenvalue λ 6= 0 to the Neumann problem (1.1) satisfies λ ≥ λ̂. In
particular, λ = 0 is an isolated eigenvalue.

(iii) λ̂ is an eigenvalue and therefore

λ2 = λ̂. (1.8)

Note that (i) and (ii) are well-known and that (iii) can be proved in a standard
way when p ≥ 2 (see Remark 3.2). Therefore we focuss on showing (1.8) when p
falls in the ‘singular’ range 1 < p < 2.

Moving now to a further scenario, the version of the p-Laplacian operator −∆p

for graphs G has been recently studied in [2], where its second eigenvalue λ2 has
been introduced and analyzed in detail (see also [8, 9, 10]. The latter specifically
concerns a variety of features on the spectrum of −∆). Such eigenvalue plays an
important rôle in the so-called ‘clustering problem’ for undirected graphs [8]. Our
next result extends [2, Theorem 1] to the range p > 1. In fact, the proof contained
there is only valid in the case p ≥ 2, and the same remark applies to [8, Theorem
3.2] (see also [9]). The reader is referred to Section 4 for the necessary background
material concerning graph theory.

Theorem 1.2. Let G = (V, E), V = {v1, . . . , vn}, be a connected graph with weights
matrix A = (ωij)1≤i≤n, ωij = ωji, ωij ≥ 0, ωii = 0, A 6= 0. Consider the eigenvalue
problem

−∆p(f) = λνφp(f), (1.9)

where f = (fi) ∈ Rn, ν = (νi) ∈ Rn+, φp(f) = (|pi|p−2pi) and −∆p is the p-
Laplacian in G. Then the following properties hold

(i) λ1 = 0 is the first eigenvalue which is isolated, simple and has span{1} as
the set of associated eigenfunctions.

(ii) The second eigenvalue λ2 is expressed as

λ2 = inf
f∈M0\{0}

1

2

∑
i,j ωij |fi − fj |p∑

i νi|fi|p
, (1.10)

where M0 = {f :
∑
i νi|fi|p−2fi = 0}.

(iii) The maximum eigenvalue λ∗ is provided by the expression

λ∗ = sup
f∈M0\{0}

1

2

∑
i,j ωij |fi − fj |p∑

i νi|fi|p
. (1.11)

Remark 1.3. In this work we denote by −∆p what is commonly defined as the
p-Laplacian operator in graphs, usually designated as ∆p. We have proceeded in
this way to preserve the analogies with partial differential equations (see Section
4).

This work is organized as follows. Section 2 analyzes the differentiability prop-
erties of the variance functional in Lp(X,µ), where (X,µ) is a measurable space.



4 J. C. SABINA DE LIS EJDE-2022/13

The results attained seem to be new and are instrumental in the subsequent sec-
tions. Section 3 is devoted to proving Theorem 1.1. There, we are dealing with the
slightly more general version of (1.1),

−∆pu = λm(x)|u|p−2u x ∈ Ω

|∇u|p−2 ∂u

∂ν
= 0 x ∈ ∂Ω,

(1.12)

where m ∈ Lr(Ω) is a weight function such that m(x) > 0 a.e. in Ω, while exponent
r ≥ 1 is suitably chosen.

The tools developed in Section 2 turn also out to be useful for studying the
nonlinear diffusion problem

−∆pu = λm(x)|u|q−2u x ∈ Ω

|∇u|p−2 ∂u

∂ν
= 0 x ∈ ∂Ω,

(1.13)

where 1 ≤ q < p∗ with p∗ = pN
N−p if p < N , p∗ = ∞ otherwise, m ∈ Lr(Ω)+ and r

varies in a convenient range. It should be pointed out that the Dirichlet counterpart
of (1.13) was discussed in full detail in [13] where m ∈ L∞(Ω)+. In Section 4 we
analyze the main existence issues concerning (1.13). Finally Section 5 contains the
proof of Theorem 1.2.

2. Variance functional in Lp(X,µ)

Our forthcoming results will be obtained in the general framework of a measur-
able space (X,A, µ) where µ is a finite measure defined in some fixed σ-algebra A
in X. As usual, Lp(X,µ) stands for the space of measurable functions f such that∫
X
|f |p dµ < ∞. We are proceeding in this way for the sake of completeness. We

begin by reviewing some elementary features which permit us refreshing the notion
of variance.

Lemma 2.1. Let (X,A) be a measurable space endowed with a finite measure µ.
Then, for every u ∈ Lp(X,µ) there exists a unique ũ ∈ R such that∫

X

|u− ũ|p−2(u− ũ) dµ = 0.

Moreover, ũ is variationally characterized by the expression∫
X

|u− ũ|p dµ = inf
t∈R

∫
X

|u− t|p dµ. (2.1)

Furthermore, ũ defines a continuous functional of u ∈ Lp(X,µ).

Definition 2.2. For a function u ∈ Lp(X,µ) its variance is defined as

Vp(u) =

∫
X

|u− ũ|p dµ,

while ũ is said to be the average of u relative to Lp(X,µ).

Remark 2.3. This is of course the standard definition when p = 2. In fact,

ũ =
1

|X|

∫
X

u dµ, |X| :=
∫
X

dµ = µ(X),

if p = 2 and ũ coincides with the standard average of u in X.



EJDE-2022/13 SECOND NEUMANN EIGENVALUE 5

Proof of Lemma 2.1. The function g(t) =
∫
X
|u− t|p−2(u− t) dµ is continuous and

decreasing while limt→±∞ g(t) = ∓∞. Hence, the existence and uniqueness asser-
tions follow.

On the other hand, f(t) =
∫
X
|u− t|p dµ is a convex coercive function such that

f ′ = −pg. This implies (2.1).
As for the continuity assertion assume that un → u in Lp(X,µ). The inequalities

|X||ũ|p ≤ 2p−1

∫
X

{|u− ũ|p + |u|p} dµ ≤ 2p
∫
X

|u|p dµ

imply

|ũ| ≤ 2

|X|1/p
‖u‖p, (2.2)

with ‖u‖p = ‖u‖Lp(X,µ). Thus ũn is bounded and a convergent subsequence ũn′ →
ũ′ can be extracted from ũn. Since |un′ − ũn′ |p−2(un′ − ũn′)→ |u− ũ′|p−2(u− ũ′)
in Lp

′
(X,µ), it follows that∫

X

|u− ũ′|p−2(u− ũ′) dµ = 0.

This means that ũ′ = ũ and the continuity is shown. �

The main result of this section is the next one. It states the differentiability of
the variance functional Vp and seems new at the best of our knowledge. In this
regard, it should be remarked that Vp is defined in a variational way and so this is
not a straightforward issue.

Theorem 2.4. Under the assumptions of Lemma 2.1, the variance functional Vp
is Fréchet differentiable in Lp(X,µ) for all p > 1. Moreover, its differential DVp(u)
at u is represented as

〈DVp(u), v〉 = p

∫
X

|u− ũ|p−2(u− ũ)v dµ, v ∈ Lp(X,µ).

The proof of Theorem 2.4 relies on the next crucial lemma.

Lemma 2.5. Let u ∈ Lp(X,µ) be fixed. Then the Gâteaux derivative dVp(u, v) of
the Variance Vp at u in the direction v ∈ Lp(X,µ) exists and is given by

dVp(u, v) = p

∫
X

|u− ũ|p−2(u− ũ)v dµ. (2.3)

Proof. Fix u, v ∈ Lp(X,µ). For t ∈ R define

ut = u+ tv, ũt = ũt = ũ+ tv,

together with
V (t) = Vp(ut) = Vp(u+ tv).

By setting

ft(σ) =

∫
X

|u+ tv − σ|p dµ =

∫
X

|ut − σ|p dµ,

it holds that

V (t) = ft(ũt) = inf
σ∈R

ft(σ), V (0) = f0(ũ) = inf
σ∈R

f0(σ).

On the other hand,

ft(ũt)− f0(ũt) ≤ V (t)− V (0) ≤ ft(ũ)− f0(ũ). (2.4)



6 J. C. SABINA DE LIS EJDE-2022/13

The first term in inequality (2.4) can be written as

ft(ũt)− f0(ũt) = pt

∫
X

{∫ 1

0

|u+ tsv − ũt|P−2(u+ tsv − ũt) ds
}
v dµ.

By using (2.2) and assuming that |t| ≤ 1, the integrand in the last term can be
estimated as ∣∣∣ ∫ 1

0

|u+ tsv − ũt|p−2(u+ tsv − ũt) ds
∣∣∣

≤
∫ 1

0

|u+ tsv − ũt|p−1 ds

≤ C{|u|p−1 + |v|p−1 + ‖u‖p−1
p + ‖v‖p−1

p },
where C > 0 is a constant only depending on p and |X|. Since∫ 1

0

|u+ tsv − ũt|p−2(u+ tsv − ũt) ds→ |u− ũ|p−2(u− ũ),

a.e. in X as t→ 0, the Lebesgue dominated convergence theorem implies that

lim
t→0

1

t

(
ft(ũt)− f0(ũt)

)
= p

∫
X

|u− ũ|p−2(u− ũ)v dµ.

An identical argument shows that

lim
t→0

1

t

(
ft(ũ)− f0(ũ)

)
= p

∫
X

|u− ũ|p−2(u− ũ)v dµ,

and the desired result follows from dividing the three terms in (2.4) by t 6= 0 and
passing to the limit as t→ 0. �

Proof of Theorem 2.4. The Gâteaux derivative dVp(u, v) is linear continuous in v ∈
Lp(X,µ) for u fixed. On the other hand, mapping u→ dVp(u, ·) regarded as taking

values in (Lp(X,µ))∗ = Lp
′
(X,µ) is continuous. This entails that Vp is Fréchet

differentiable at u (see for instance [1, Chapter 1]). �

We next single out two special cases where Theorem 2.4 is applied. In the first
one, X = Ω is a bounded set of Rn, endowed with the measure dµ = m(x)dx where
m ∈ L1(Ω), m(x) > 0 a.e. in Ω.

Corollary 2.6. Let Vp be the variance functional defined in Lp(Ω,mdx). Then, at
any u ∈ Lp(Ω,mdx) we have

〈DVp(u), v〉 = p

∫
Ω

|u− ũ|p−2(u− ũ)vmdx, v ∈ Lp(Ω,mdx).

In the second example X is a finite set V = {v1, . . . , vn} where the measure is

µ =

n∑
i=1

νiδ(x− vi),

while δ is the Dirac’s delta and νi > 0, 1 ≤ i ≤ n. Functions f : V → R are
identified to vectors in Rn by means of the expression f = (fi) with fi = f(i).
Then

Vp(f) =
∑
i

|fi − f̃ |pνi = inf
t∈R

∑
i

|fi − t|pνi,

where t = f̃ is the unique number so that
∑
i |fi − t|p−2(fi − t)νi = 0.



EJDE-2022/13 SECOND NEUMANN EIGENVALUE 7

Corollary 2.7. Function Vp is differentiable at any f ∈ Rn and

〈DVp(f), g〉 = p
∑
i

|fi − f̃ |p−2(fi − f̃)giνi, g ∈ Rn. (2.5)

3. Proof of Theorem 1.1

In this section we prove a slightly more general version of Theorem 1.1 that can
be stated as follows.

Theorem 3.1. Assume that Ω ⊂ Rn is class C0,1 bounded domain, m ∈ Lr(Ω),
m(x) > 0 a.e. in Ω, where

r


>
(
p∗

p

)′
= Np if 1 < p < N,

> 1 if p = N,

= 1 if p > N.

We define

λ̂(m) = inf
u∈M0\{0}

∫
Ω
|∇u|p dx∫

Ω
|u|pmdx

, (3.1)

with M0 = {u ∈ W 1,p(Ω) :
∫

Ω
|u|p−2umdx = 0}. Then eigenvalue problem (1.12)

satisfies the assertions (i)–(iii) in Theorem 1.1 with λ2 replaced by λ2(m), the
second eigenvalue to (1.12).

Proof. To show Theorem 3.1 for problem (1.12) we first notice that

λ̂(m) = inf
u∈M0\{0}

∫
Ω
|∇u|p dx∫

Ω
|u|pmdx

= inf
u∈M1

J (u),

where J (u) =
∫

Ω
|∇u|p dx, and

M0 =
{
u ∈W 1,p(Ω) :

∫
Ω

|u|p−2u mdx = 0
}
, M1 =M0 ∩

{∫
Ω

|u|pmdx = 1
}
.

Functional J is coercive, i.e. J (u) → ∞ as ‖u‖W 1,p(Ω) → ∞, and weakly lower

semicontinuous, while the election of r entails thatM1 is weakly closed in W 1,p(Ω).
Thus, a well-known result in Calculus of Variations ([19, Chapter I]) ensures us the
existence of a global minimizer u1 ∈M1. Hence

0 < λ̂(m) = J (u1) = inf
u∈M0\{0}

∫
Ω
|∇u|p dx∫

Ω
|u|p mdx

.

Since every possible eigenfunction u associated with a nonzero eigenvalue λ to (1.12)

lies in M0 then λ ≥ λ̂(m).

Thus the main feature remains to be proved. Namely that λ̂(m) is actually an
eigenvalue. The next proof, relying on the properties of the variance functional Vp
presented in Section 2, can be applied to cover both cases p ≥ 2 and 1 < p < 2.
First observe that

M0 = {u− ũ : u ∈W 1,p(Ω)},
where the notation of Section 2 has been used. Hence,

λ̂(m) = inf
v∈M0\{0}

∫
Ω
|∇v|p dx∫

Ω
|v|p mdx

= inf
u/∈span{1}

∫
Ω
|∇u|p dx∫

Ω
|u− ũ|p mdx

,



8 J. C. SABINA DE LIS EJDE-2022/13

since M0 \ {0} = {u − ũ : u ∈ W 1,p(Ω), u /∈ span{1}}. Thus, an alternative

expression for λ̂(m) reads as follows

λ̂(m) = inf
u/∈span{1}

∫
Ω
|∇u|p dx
Vp(u)

.

Next we observe that the quotient can be differentiated regardless the value of
p > 1. Accordingly, by using the expression for the Gâteaux derivative of Vp stated
in Lemma 2.5 (Corollary 2.6), evaluated at a minimizer u1 and in the direction
v ∈W 1,p(Ω) we easily arrive to

p

∫
Ω

|∇u1|p−2∇u1∇v dx = pλ̂(m)〈DVp(u), v〉,

equivalently,∫
Ω

|∇u1|p−2∇u1∇v dx = λ̂(m)

∫
Ω

|u1 − ũ1|p−2(u1 − ũ1)vmdx.

Since v ∈ W 1,p(Ω) is arbitrary, this means that λ̂(m) is an eigenvalue with associ-
ated eigenfunction u1 − ũ1. �

Remark 3.2. An alternative proof of the fact that λ̂(m) is an eigenvalue can be
given in the case p ≥ 2. In this case, Lagrange’s multiplier rule can be employed
in a standard way. Indeed, the constraint defining M0 involves the functional
I(v) =

∫
Ω
|v|p−2v which is C1 provided p ≥ 2.

4. A further subcritical problem

The results of Section 2 can be still employed to study the following nonlinear
problems

−∆pu = λm(x)|u|q−2u x ∈ Ω

|∇u|p−2 ∂u

∂ν
= 0 x ∈ ∂Ω,

(4.1)

where
1 ≤ q < p∗, (4.2)

with p∗ = pN
N−p if p < N , p∗ =∞ otherwise. In addition, m ∈ Lr(Ω) is positive a.e.

in Ω and

r


>
(
p∗

q

)′
= Np

N(p−q)+pq if 1 < p < N,

> 1 if p = N,

= 1 if p > N.

(4.3)

Theorem 4.1. Assume that the exponents q and r satisfy (4.2) and (4.3), respec-
tively. Then problem (4.1) admits a nontrivial (nonconstant) solution u ∈W 1,p(Ω)
if and only if λ > 0. These nontrivial solutions satisfy the average condition∫

Ω

|u|q−2umdx = 0. (4.4)

Proof. Solutions u ∈W 1,p(Ω) are understood in weak sense, that is, equality∫
Ω

|∇u|p−2∇u∇v dx = λ

∫
Ω

|u|q−2uvmdx, (4.5)

holds for all v ∈W 1,p(Ω). Observe that the latter integrand lays in L1(Ω) because
of conditions (4.3) satisfied by m. Thus, it is readily deduced that nonconstant



EJDE-2022/13 SECOND NEUMANN EIGENVALUE 9

nontrivial solutions are only possible when λ > 0 while in addition, every solution
must satisfy (4.4) (use v = 1 as a test function).

On the other hand, solving (4.1) for a specific λ = λ0 > 0 amounts to solving it
for every λ > 0. In fact, if u0 is a non trivial solution for λ = λ0 then

uλ =
( λ
λ0

) 1
p−q

u0,

defines a solution to (4.1) corresponding to any λ > 0. Thus it is enough with
finding out a solution at some fixed value of λ0 > 0.

We now mimic the proof of Theorem 1.1 and minimize J (u) =
∫

Ω
|∇u|p on

M1,q = M0,q ∩
{
u ∈ W 1,p(Ω) :

∫
Ω
|u|q mdx = 1

}
, M0,q being the functions of

W 1,p(Ω) satisfying ∫
Ω

|u|q−2umdx = 0.

By similar reasons as in Section 3, J achieves a minimum at some u1 ∈ M1,q and
so

0 < µ1 := J (u1) = inf
M1,q

J

= inf
u∈M0,q\{0}

∫
Ω
|∇u|p dx( ∫

Ω
|u|q mdx

)p/q = inf
u/∈span{1}

∫
Ω
|∇u|p dx
Vq(u)p/q

.
(4.6)

In the above expression we used that M0,q \ {0} = {u − ũq : u ∈ W 1,p(Ω), u /∈
span{1}} where ũq means the average of u relative to Lq(Ω,mdx) (Section 2). By
taking the directional derivative of the latter quotient at u = u1 in the direction
v ∈W 1,p(Ω), and then equaling to zero we obtain

〈−∆pu1, v〉 = Vq(u1)
p
q−1µ1

〈1

q
DVq(u1), v

〉
,

for all v ∈ W 1,p(Ω). By using Lemma 2.5 this implies that u0 = u1 − ũ1 solves
(4.1) for the special value:

λ0 = Vq(u1)
p
q−1µ1.

As already pointed out this permits us obtaining a solution for any positive value
λ > 0. �

Remark 4.2. As in the case of Poincaré’s inequality (1.6), µ
− 1

q

1 defines the opti-
mum constant C in the inequality:

inf
t∈R
‖u− t‖Lq(Ω,mdx) ≤ C‖|∇u|‖Lp(Ω), (4.7)

associated with the embedding W 1,p(Ω) ⊂ Lq(Ω,mdx). See [18, Lemma V.2.3–2]
for the case m = 1.

5. p-Laplacian on graphs

An order n graph G is defined through a couple (V, E) where V = {v1, . . . , vn}
is a set with n ≥ 2 elements, the vertices of the graph, together with a family
E of two–elements subsets e = {u, v} of V. Members of E define the edges of G
and it is said that u is adjacent (or ‘connected’) to v when {u, v} ∈ E. It is often
convenient to associate a weight ω > 0 to every edge e = {u, v} ∈ E, which could be
understood as the ‘connection intensity’ between the vertices u and v. Observe that
no order is prescribed in the edges. A possible way of simultaneously defining both



10 J. C. SABINA DE LIS EJDE-2022/13

the edges and their weights consists in introducing the weights matrix A = (ωij)
of G. Such a matrix A is always chosen nonnegative and symmetric, ωij = ωji,
ωij ≥ 0, while ωij > 0 both means that {vi, vj} ∈ E and that has weight ωij as
an edge of E. Observe that, from the definition of E, ωii = 0 for every 1 ≤ i ≤ n.
Matrix A is termed as the ‘adjacency’ matrix when the weights ωij ∈ {0, 1}.

A subset {vi0 , vi1 , . . . , vim} of m + 1 vertices so that {vik−1
, vik} ∈ E for k =

1, . . . ,m is defined to be a path of length m connecting vi0 to vim . Accordingly, a
graph G is said to be connected if every couple of distinct vertices x, y ∈ V can be
joined through a path.

Let us next define the p-Laplacian operator −∆p on G. As pointed out at the
end of Section 2, the set H of real functions f : V → R can be identified with
Rn (f = (fi), fi = f(i)). The p-Laplacian in G is defined as a mapping from H
into itself according the next definition. The eigenvalue problem for −∆p is also
introduced there. Warning: to keep the similarities with the partial differential
equations setting, a minus sign preceding the operator is employed.

Definition 5.1. Let G = (V, E) be a graph with weights matrix A = (ωij). The
p-Laplacian operator −∆p : H → H is defined as

−∆p(f)(i) =
∑
j

ωij |fi − fj |p−2(fi − fj), 1 ≤ i ≤ n. (5.1)

It is said that λ ∈ R is an eigenvalue to −∆p with associated eigenvector f ∈ H\{0}
provided that

−∆p(f) = λνφp(f), (5.2)

where ν = (νi) ∈ Rn+ is a given weight function, φp(f)(i) = |fi|p−2fi, 1 ≤ i ≤ n.

Remark 5.2. When p = 2, −∆p becomes the linear operator (the Laplacian in G):

−∆2f = (D −A)f,

A being the weights matrix of G and D = diag(d1, . . . , dn) where di =
∑
j ωij . In

this case the spectrum of −∆2 corresponding to the weight ν = 1 consists of the
eigenvalues of D − A. In some cases the interest is focussed on the normalized
eigenvalues of −∆2 (respectively, −∆p). These are the eigenvalues corresponding
to the choice ν = (di) as a weight function in (5.2) (see [10, 8]).

Proof of Theorem 1.2. We first review some preliminary well known features [2, 8].
By using the Euclidean scalar product 〈·, ·〉2 of Rn it is found that

〈−∆p(f), f〉2 =
∑
i

∑
j

ωij |fi − fj |p−2(fi − fj)fi

= −
∑
i

∑
j

ωij |fi − fj |p−2(fi − fj)fj ,

which implies

〈−∆p(f), f〉2 =
1

2

∑
i,j

ωij |fi − fj |p =: D(f). (5.3)

As a consequence, all possible eigenvalues λ of −∆p must be nonnegative since the
weight function ν is positive. Another implication of (5.3) is the fact that λ = 0 is
a simple eigenvalue since its set of associated eigenfunctions is span{1}, 1 standing
for the function f = 1. In this regard, the connectedness of G is employed to show
that for every associated eigenfunction f it holds fi = fj for every couple of indices



EJDE-2022/13 SECOND NEUMANN EIGENVALUE 11

1 ≤ i, j ≤ n. In fact, the constant value f = fi is ‘propagated’ through the path
connecting vi to vj .

Next observe that

〈−∆p(f), 1〉2 =
∑
i

(−∆p(f))i =
∑
i

∑
j

ωij |fi − fj |p−2(fi − fj) = 0.

That is why any eigenfunction f associated with an eigenvalue λ 6= 0 must satisfy
the condition equivalent to (1.5), i. e.,∑

i

|fi|p−2fiνi = 0. (5.4)

Set now M1 =
{
f ∈ Rn : f satisfies (5.4) and

∑
i |fi|pνi = 1

}
. By compactness,

function D achieves its minimum at some f1 ∈M1. In addition,

0 < λ̂ := D(f1) = min
f /∈span{1}

D(f)

Vp(f)
,

since {
f ∈ Rn :

∑
|fi|p−2fiνi = 0

}
\ {0} =

{
f − f̃ : f ∈ Rn \ span{1}

}
.

Next observe that

∇D(f) = p(−∆p(f)).

Hence, it readily follows by differentiating D(u)
Vp(u) and employing (2.5) that

−∆p(f1 − f̃1) = λ̂νφp(f1 − f̃1).

This means that λ̂ defines the second eigenvalue λ2. Observe that Corollary 2.7
has permitted us handling both the case p ≥ 2 and the singular one 1 < p < 2.

By arguing in the same way one obtains that

λ∗ := max
f /∈span{1}

D(f)

Vp(f)
,

constitutes the maximum eigenvalue of −∆p in G. �

Acknowledgments. This research was partially supported by CECE (Generalitat
Valenciana) under project AICO/2021/223.

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José C. Sabina de Lis

Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, C. As-
trof́ısico Francisco Sánchez s/n, 38203 – La Laguna, Spain

Email address: josabina@ull.edu.es


	1. Introduction
	2. Variance functional in Lp(X,)
	3. Proof of Theorem ??
	4. A further subcritical problem
	5. p-Laplacian on graphs
	Acknowledgments

	References