Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 38, pp. 1–29.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI:

https://doi.org/10.58997/ejde.2023.38

PRINCIPAL EIGENVALUES FOR THE FRACTIONAL

p-LAPLACIAN WITH UNBOUNDED SIGN-CHANGING

WEIGHTS

OUMAROU ASSO, MABEL CUESTA, JONAS TÊLÉ DOUMATÈ, LIAMIDI LEADI

Abstract. Let Ω be a bounded regular domain of RN , N > 1, p ∈ (1,+∞),

and s ∈ (0, 1). We consider the eigenvalue problem

(−∆p)su+ V |u|p−2u = λm(x)|u|p−2u in Ω

u = 0 in RN \ Ω,

where the potential V and the weight m are possibly unbounded and are sign-
changing. After establishing the boundedness and regularity of weak solutions,

we prove that this problem admits principal eigenvalues under certain condi-

tions. We also show that when such eigenvalues exist, they are simple and
isolated in the spectrum of the operator.

1. Introduction

For p ∈ (1,+∞) and s ∈ (0, 1), the fractional (s, p)-Laplacian is an extension of
the s-fractional Laplacian and it is defined, for a regular function u : RN → R, as

(−∆p)
s u(x) := 2K(1− s) P.V.

(∫
RN

|u(x)− u(y)|p−2
(
u(x)− u(y)

)
|x− y|N+sp

dy
)

for all x ∈ RN with

K = p
(∫

SN−1

|〈ω, e〉|pdH N−1(ω)
)−1

, e ∈ SN−1,

where H N−1 denotes the (N−1)-dimensional Hausdorff measure of the unit sphere
SN−1 of RN . Let us recall that for all measurable function on a subset D of RN and
for all x ∈ RN , the principal value function on the integral

∫
D
Ψ(x, y)dy is denoted

by

P.V.
(∫

D

Ψ(x, y)dy
)

:= lim
ε→0

∫
D\Bε(x)

Ψ(x, y)dx,

where Bε(x) is a ball centered at x ∈ RN with radius ε > 0.

2020 Mathematics Subject Classification. 35J70, 35P30.
Key words and phrases. Fractional p-Laplacian; fractional Sobolev space; indefinite weight;

principal eigenvalues.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted July 20, 2022. Published June 19, 2023.

1



2 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

In this article, we study the conditions under which the principal eigenvalues of
the following homogeneous Dirichlet problem exist

(−∆p)
su+ V |u|p−2u = λm(x)|u|p−2u in Ω,

u = 0 in RN \ Ω,
(1.1)

where Ω is a bounded regular domain of RN , V and m are indefinite sign-changing
functions and satisfying the following conditions:

(C1) V , m ∈ Lr(Ω) with r ∈ (1, +∞) ∩ (Nsp , +∞),

(C2) m+ = max(m, 0) 6≡ 0.

Our aim is to extend some results obtained by Del-Pezzo et al. in [12] for the eigen-
value problem (1.1). These authors studied, among other issues, the existence of
eigenvalues, the positivity of the eigenfunctions associated with the first eigenvalue
of (1.1) with m ≡ 1 and V satisfying (C1). We want here to address the question
of existence of principal eigenvalue in a wide range of weights, precisely when m
and V changing sign. The presence of such weights in problem (1.1) brings us to
proceed by a considerably different approach called “eigencurve arguments” which
requires the construction of some equivalent problem.

To illustrate this eigencurve argument, let us mention the work of Fleckinger et
al. [15], where the following eigenvalue problem is considered.

−∆u+ a0(x)u = λm(x)u, in Ω, u = 0 on ∂Ω (1.2)

with Ω a bounded smooth domain, a0,m ∈ Lr(Ω), r > N
2 are indefinite and m is

unbounded. After separating the positive and negative parts of a0 and m one find
equation (1.2) as

−∆u+ a+
0 (x)u+ λm−(x)u = λm+(x)u+ a−0 (x)u. (1.3)

So, for any fixed λ, they were led to study the following eigenvalue problem of
eigenvalue parameter σ(λ),

−∆u+ (a+
0 (x) + 1)u+ λm−(x)u = σ(λ)

(
m+(x) +

a−0 (x) + 1

λ

)
u in Ω,

u = 0 on ∂Ω .

It is clear that λ > 0 is an eigenvalue of (1.2) if and only if σ(λ) = λ. For this
purpose, they studied the properties of continuity, concavity and monotonicity of
the curve λ 7→ σ(λ) and they proved that, under certain conditions, the existence
of λ > 0 satisfies σ(λ) = λ. For more details see [15].

Our construction of the equivalent problem is different from the one made in [15]
and it is closer to the one used by Binding and Huang [3]. These authors considered,
for bounded potential V and bounded weight m, the principal eigencurve µ1(λ),
that is, µ1(λ) is the principal eigenvalue of

−∆pu+ (V (x)− λm(x))|u|p−2u = µ1(λ)|u|p−2u in Ω, u = 0 on ∂Ω

and deduced the existence of λ ∈ R such that µ1(λ) = 0 under some conditions
on V and m. This technique has generated several results which have enriched the
scientific literature (see for example [2, 3, 9, 19, 21]). For instance, recently [8]
made use of such arguments when solving the above problem for a potential V and
a weight function m that may change sign and may be unbounded. They looked
and established additional conditions on V and m that guarantee the existence



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 3

of principal eigenvalues. In this work, our main results extend those of [8] and
references therein to the fractional p-Laplacian.

This article is organized as follows. We start by recalling some basic proper-
ties of essential the fractional Sobolev spaces in Section 2. In Section 3 we prove
the boundedness and regularity of the weak solutions. Section 4 is devoted to the
existence of principal eigenvalues. In Section 5, we show that when principal eigen-
values exist, they are isolated in the spectrum and we give a lower bound of the
measure of the nodal domains for changing sign eigenfunctions. Finally in Section
6 we prove some sort of continuity of the principal eigenvalues when varying s.
We collect in appendix the proof of a discrete version of some well known identity
as well as a regularity result for more general equations involving the fractional
p-Laplacian with unbounded terms.

2. Preliminaries

The Lebesgue measure of a Lebesgue measurable set Z ⊂ RN is denoted by |Z|.

2.1. Basic results about fractional Sobolev spaces. Let p ∈ [1,+∞), s ∈
(0, 1) and let Ω ⊂ RN be an open set.

• The (s, p)-fractional Sobolev space, denoted by W s,p(Ω), is defined by

W s,p(Ω) :=
{
u ∈ Lp(Ω) :

∫
Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy < +∞

}
.

The space W s,p(Ω) is a separable Banach space endowed with the norm∥∥u∥∥
W s,p(Ω)

:=
(∫

Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy +

∫
Ω

|u|pdx
)1/p

.

W s,p(Ω) is reflexive if p > 1.
• For any function u of W s,p(Ω) we denote the Gagliardo semi-norm by[

u
]
Ws,p(Ω)

:=
(∫

Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

.

• The space W̃ s,p(Ω) is defined as the space of all u ∈ W s,p(Ω) such that

ũ ∈W s,p(RN ), where ũ is the extension by zero of u, outside of Ω. W̃ s,p(Ω)
is a Banach space endowed with the norm

‖u‖
W̃ s,p(Ω)

:= ‖ũ‖W s,p(RN )

and it is a reflexive space if p > 1.

Let us quote some properties of these spaces that will be used later. Here we
will denote by C(N, p) any positive constant depending only on N and p.

Proposition 2.1 ([10]). Let Ω be a bounded open set of RN .

(1) There exists C(N, p) such that, for any u ∈ W̃ s,p(Ω), it holds

‖u‖pLp(Ω) ≤ C(N, p)(diam(Ω))sp(1− s)
[
u
]p
W s,p(RN )

. (2.1)

Thus, the Gagliardo semi-norm
[
·
]
W s,p(RN )

is a norm in W̃ s,p(Ω) equivalent

to the previous norm ‖ · ‖
W̃ s,p(Ω)

(c.f. [10, Lemma 2.5]).



4 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

(2) Let 0 < s ≤ s′ < 1. Then there exists a positive constant C(N, p) such that[
u
]p
W s,p(RN )

≤
[
u
]p
W s′,p(RN )

+ C(N, p)
( 1

sp
− 1

s′p

)
‖u‖p

Lp(RN )

for any u ∈W s′,p(RN ) (cf. [10, Lemma 2.3]).

Proposition 2.2 ([7, 13, 17]). Let Ω ⊂ RN be a bounded open set with Lipschitz
boundary. Then

(1) C∞0 (Ω) is dense in W̃ s,p(Ω) (c.f. [17, Theorem 1.4.2.2]).

(2) If u ∈ W̃ s,p(Ω) and f is a Lipschitz function then f(u) ∈ W̃ 1,p(Ω).
(3) Let 0 < s ≤ s′ < 1. Then there exists a positive constant C(N, p) such that

(1− s)[u]pW s,p(Ω) ≤ 2(1−s)p diam(Ω)(s′−s)p(1− s′)[u]p
W s′,p(Ω)

for any u ∈W s′,p(Ω) (c.f. [7, Lemma 2]; [13, Lemmas 4.3 and 4.4]).
(4) For any u ∈W 1,p(Ω),

lim
s→1−

(1− s)[u]pW s,p(Ω =

∫
Ω

|∇u|p dx.

(c.f. [7, Corollary 2 ]).

2.2. Embeddings. Let the fractional critical exponent of Sobolev be defined by

p∗
s

=

{
Np
N−sp if sp < N,

+∞ if sp > N.

The following results are versions of the classical Sobolev injection theorem in the
case of fractional Sobolev spaces (c.f. [14, pages 218 and 219]).

Theorem 2.3. [14] Let Ω be an open set with a Lipschitz boundary. We have the
following continuous injections:

(1) If sp < N, W s,p(Ω) ↪→ Lq(Ω) for all q ∈ [p, p∗s].
(2) If sp = N, W s,p(Ω) ↪→ Lq(Ω) for all q ∈ [p, +∞).

(3) If sp > N , W s,p(Ω) ↪→ C0, α(Ω̄) with α ∈
(

0, s− N
p

]
.

Furthermore we have the following compact injections when Ω is an open bounded
domain of RN with a Lipschitz boundary:

4. If sp 6 N , then W s,p(Ω) ↪→c L
q(Ω) for all q ∈ [1, p∗s).

5. If sp > N , then W s,p(Ω) ↪→c C
0, α(Ω̄) with α ∈ (0, s− N

p ).

6. W s,p(Ω) ↪→c L
pq′(Ω) with max{1, Nsp} < q < +∞ and 1

q + 1
q′ = 1.

Throughout this work we will assume that Ω is a bounded domain of RN with a
Lipschitz boundary.

3. Weak solutions of the eigenvalue problem and regularity results

For simplicity, from now on we will denote by u, instead of ũ, the extension by

0 of any function u ∈ W̃ s,p(Ω).

Definition 3.1. (1) We will say that a function u ∈ W̃ s,p(Ω) is a weak solution
of (1.1) if

H(u, v) +

∫
Ω

V (x)|u|p−2uvdx = λ

∫
Ω

m(x)|u|p−2uv dx (3.1)



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 5

for all v ∈ W̃ s,p(Ω), where

H(u, v) := K(1− s)
∫
RN

∫
RN

|u(x)− u(y)|p−2
(
u(x)− u(y)

)
|x− y|N+sp

(
v(x)− v(y)

)
dx dy.

(3.2)

It should be noted that for all u ∈ W̃ s,p(Ω), we have

H(u, u) = K(1− s)
[
u
]p
Ws,p(RN )

.

(2) We will say that a real number λ is an eigenvalue of (1.1) if there exist u 6≡ 0
satisfying (3.1). In this case, we say that u is an eigenfunction associated
with λ.

(3) Moreover, if the eigenfunction u has a constant sign on Ω, then λ is called
a principal eigenvalue of the problem (1.1).

(4) Finally, the eigenvalue λ is said to be simple if any two eigenfunctions u
and v associated with λ are such that u = cv for some real constant c.

Definition 3.2. For each u ∈ W̃ s,p(Ω), let the energy associated with the problem
(1.1) be

EV (u) := H(u, u) +

∫
Ω

V (x)|u|p dx = K(1− s)
[
u
]p
Ws,p(RN )

+

∫
Ω

V (x)|u|p dx. (3.3)

It is clear that EV is of class C 1 on W̃ s,p(Ω) with

〈E′V (u), v〉 = pH(u, v) + p

∫
Ω

V (x)|u|p−2uv dx ∀(u, v) ∈ W̃ s,p(Ω)× W̃ s,p(Ω).

Let us now state the main result of this section. Let us consider the homogeneous
problem

(−∆p)
su+ V ′|u|p−2u = 0 in Ω,

u = 0 in RN \ Ω,
(3.4)

where V ′ satisfies condition (C1).

Theorem 3.3. If u ∈ W̃ s,p(Ω) is a weak solution of (3.4), then u ∈ L∞(Ω)∩C(Ω).
Furthermore, there exists a positive constant C = C(s, p,N,Ω, ‖V ′‖Lr(Ω)) such that

‖u‖L∞(Ω) ≤ C‖u‖Lr′p(Ω). (3.5)

The proof of this theorem will follow from Lemma 3.4 below, based on the De
Giorgi-Stampacchia iteration technique (see for instance [11, 16, 22], where the case
V ′ ≡ 1 has been considered).

Lemma 3.4. Assume that sp ≤ N . Let u be a weak solution of (3.4) admitting a
positive part u+ 6≡ 0. Let us define the sequence (wk)k by

wk :=
(
u−

(
1− 1

2k

))+

.

Then there exists a positive constant σ = σ(s, p,N,Ω, ‖V ′‖Lr(Ω)) such that, if

‖u+‖
Lr
′p(Ω)

< σ, then u ≤ 1 a.e.

Proof. Let us denote Wk = ‖wk‖pLr′p(Ω)
. The conclusion of the lemma will follow

from the following results that we prove below:

(1) limk→+∞Wk = ‖(u− 1)+‖p
Lr
′p(Ω)

.



6 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

(2) limk→+∞Wk = 0.

Notice that, by definition, wk ∈W s,p(Ω) and wk = 0 a.e. in Ωc.

1. Trivially the sequence (wk)k is decreasing so, for all k ∈ N we have |wk|r
′p ≤

|w0|r
′p = |u+|r′p ∈ L1(Ω). Moreover the sequence (|wk|r

′p)k converges to ((u −
1)+)r

′p almost everywhere in Ω. Hence Wk → ‖(u− 1)+‖p
Lr
′p(Ω)

by the Lebesgue’s

dominated convergence theorem.
2. Let us first prove two claims.
Claim 1. For all k ∈ N,

‖u|p−1wk+1‖Lr′ (Ω) ≤ 2(p−1)(k+1)Wk. (3.6)

Indeed, first observe that if wk+1(x) > 0, that is, if u(x) > 2k+1−1
2k+1 , then

wk(x) = wk+1(x) +
1

2k+1
≥ 1

2k+1
, wk(x) ≥ u(x)

2k+1 − 1

and ∫
Ω

|u|r
′(p−1)wr

′

k+1dx =

∫
{wk+1>0}

|u|r
′(p−1)wr

′

k+1dx

≤
∫
{wk+1>0}

(2k+1 − 1)r
′(p−1)w

r′(p−1)
k wr

′

k (x)dx

≤ (2k+1 − 1)r
′(p−1)‖wk‖r

′p

Lr
′p(Ω)

≤ 2r
′(p−1)(k+1)W r′

k .

(3.7)

Claim 2. There exist D > 1 and β > 0 such that for all k ∈ N,

Wk+1 ≤ DkW 1+β
k .

To prove this claim, let us quote the following (trivial) inequality:

∀(a, b) ∈ R2, |a+ − b+|p ≤ |a− b|p−2(a− b)(a+ − b+). (3.8)

By taking

a = u(x)−
(

1− 1

2k+1

)
, b = u(y)−

(
1− 1

2k+1

)
in (3.8) for all (x, y) ∈ RN , we obtain[
wk+1

]p
W s,p(RN )

≤
∫
RN

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
(wk+1(x)−wk+1(y)) dx dy.

Besides, by taking wk+1 in the weak formulation of (3.4), we obtain from the
previous inequality

K(1− s)
[
wk+1

]p
W s,p(RN )

≤
∫

Ω

|V ′(x)‖u|p−1wk+1dx,

and therefore, using Claim 1,

K(1− s)
[
wk+1

]p
W s,p(RN )

≤ ‖V ′‖
Lr(Ω)

[ ∫
Ω

(
|u|p−1wk+1

)r′
dx
]1/r′

≤ C2(p−1)(k+1)Wk

(3.9)



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 7

for some positive constant C depending on ‖V ′‖Lr(Ω), s, p,N, and Ω. On the other

hand, using Hölder’s inequality with the exponents q := N
r′(N−sp) if q < Ns (or any

q > 1 if N = ps), by Sobolev’s embedding we have

Wk+1 = ‖wk+1‖p
Lr
′p(Ω)

≤ ‖wk+1‖p
Lr
′pq(Ω)

|{wk+1 > 0}|
q−1
r′q

≤ C
[
wk+1

]p
W s,p(RN )

|{wk+1 > 0}|
q−1
r′q

(3.10)

for some 0 < C = C(N, p, s,Ω). Moreover, since wk = wk+1 + 1
2k+1 , then

|{wk+1 > 0}| ≤ |{wk > 2−k−1}| ≤ 2r
′p(k+1)W r′

k (3.11)

and hence, using (3.9), (3.10) and (3.11)

Wk+1 ≤ C2(p−1)(k+1)Wk × |{wk+1 > 0}|
q−1
r′q

≤ C2(p−1)(k+1)Wk ×
(
2r
′p(k+1)W r′

k

) q−1
r′q

≤ C(2p−1 × 2
p(q−1)
q )k+1W

1+ q−1
q

k

≤ DkW 1+β
k ,

with D =
{

[1 + C] 2p−1 × 2
p(q−1)
q
}2

> 1 and β = q−1
q > 0. Claim 2 is proved.

Now we complete the proof of 2. Let σ = D
− q2

p(q−1)2 , denote ρ = ‖u+‖p
Lr
′p(Ω)

and assume that ρ1/p < σ. Choose η ∈ (ρ
q−1
q , D−

q
q−1 ). It should be noted that

η ∈]0, 1[, ρ
q−1
q ≤ η, and Dη

q−1
q ≤ 1. Let us prove by induction that for all k ∈ N

Wk ≤ ρηk. (3.12)

By definition

W0 := ‖w0‖p
Lr
′p(Ω)

= ‖u+‖p
Lr
′p(Ω)

= ρ ≤ ρη0.

Assume that (3.12) holds at order k and let us show that it holds at order k + 1.
By Claim 2,

Wk+1 ≤ DkW
1+

q−1
q

k ≤ Dk(ρηk)
1+

q−1
q

= ρ
(
η
q−1
q )k

ρ
q−1
q
ηk ≤ ρηk+1.

Thus by passing to the limit in (3.12), we finally obtain that Wk → 0. �

Proof of Theorem 3.3. Take v = σu
2‖u+‖

Lr
′p(Ω)

, where σ = σ(s, p,N,Ω, ‖V ′‖Lr(Ω)) is

given by Lemma 3.4. Since v is a weak solution of (1.1) and satisfies ‖v+‖Lr′p(Ω) = σ
2

then v ≤ 1 a.e., which gives

u ≤ 2

σ
‖u+‖Lr′p(Ω) a.e.

If u− is not identically zero, we apply the same argument to −u, which is a weak
solution of (1.1), to find that

u ≥ − 2

σ
‖u−‖Lr′p(Ω) a.e.

and estimate (3.5) follows.



8 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

Finally, the continuity of u results from [6, Theorem 3.13], which derives from
[23, Theorem 1.5]. �

Remark 3.5. To our knowledge, it is not known if the solutions are continuous
up to the boundary of Ω or class C0,α in the case sp ≤ N and V ′ unbounded.
Indeed, when V ′ is bounded then f = V ′|u|p−2u ∈ L∞(Ω) and, by the results by
Iannizzotto et al. [20], u ∈ Cα(Ω) for some α > 0.

4. Existence of principal eigenvalues with indefinite weights

Let us assume in this section that V and m satisfy conditions (C1) and (C2) and
consider the eigenvalue problem

(−∆p)
su+ V ′|u|p−2u = µ|u|p−2u in Ω,

u = 0 in RN \ Ω,
(4.1)

with V ′ = (V − λm) and µ an eigenvalue parameter depending on the real λ.
According to [12], problem (4.1) admits a unique principal eigenvalue which we will
denote µ(λ). Moreover, µ(λ) is simple and can be characterized as

µ(λ) = inf
{
K(1− s)[u]p

W s,p(RN )
+

∫
Ω

(V (x)− λm(x))|u|p dx : u ∈ W̃ s,p(Ω),

‖u‖Lp(Ω) = 1
}
.

(4.2)

Note that λ0 is a principal eigenvalue of our problem (1.1) if and only if µ(λ0) = 0.
Our aim here is to give reasonable assumptions on V and m so that the curve of
the function λ 7→ µ(λ) intersects the x-axis.

We introduce the sets

G0 :=
{
u ∈ W̃ s,p(Ω) : ‖u‖

Lp(Ω)
= 1,

∫
Ω

m(x)|u|pdx = 0
}
,

G :=
{
u ∈ W̃ s,p(Ω) : ‖u‖

Lp(Ω)
= 1
}
.

(4.3)

The following proposition gives useful properties on the function λ 7→ µ(λ). We
will denote here

Ω+ := {x ∈ Ω, m(x) > 0}, Ω− := {x ∈ Ω, m(x) < 0}, Ω0 := {x ∈ Ω, m(x) = 0},
and ϕλ the unique positive eigenfunction of Lp(Ω)-norm equal to 1 associated with
µ(λ).

Proposition 4.1. (i) µ : R→ R is concave and differentiable, with

µ′(λ) = −
∫

Ω

m(x)ϕpλdx ∀λ ∈ R.

(ii) (a) limλ→+∞ µ(λ) = −∞.
(b) If |Ω−| > 0 then limλ→−∞ µ(λ) = −∞.

(iii) If |Ω−| = 0 then µ is strictly decreasing on R and, if moreover |Ω−∪Ω0| =
0, then

lim
λ→−∞

µ(λ) = +∞.

(iv) supλ∈R µ(λ) = α(V,m) where

α(V,m) := inf {EV (u), u ∈ G0} . (4.4)

Moreover, α(V,m) is finite if and only if |Ω+| < |Ω|.



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 9

Proof. (i) We prove that µ : R → R is concave. Let λ and β be two distinct real
numbers. Let t ∈ [0, 1] and set θt = tλ+ (1− t)β. Let u ∈ G. Since

−EV−θtm(u) = t(−EV−λm(u)) + (1− t)(−EV−βm(u)),

we have
−µ(θt) 6 t (−µ(λ)) + (1− t) (−µ(β)) ,

which means that the function −µ is convex.
Let λ ∈ R and (λk)k be a sequence converging to λ. Let ϕk and ϕλ be eigen-

functions associated with µ(λk) and µ(λ) respectively with Lp(Ω)-norm equal to
1. By the results of [12, Theorem 2.9], the eigenfunction ϕλ is > 0 a.e. in Ω (see
Remark 4.3). By Lemma 4.4 below we have, for some C1 > 0 and C2 > 0,[

ϕk
]p
Ws,p(RN )

≤ C1EV−λkm(ϕk) + C2

∫
Ω

|ϕk|pdx = C1µ(λk) + C2,

so
lim sup
k→∞

[
ϕk
]p
Ws,p(RN )

≤ C1µ(λ) + C2

and therefore the sequence (ϕk)k is bounded in W̃ s,p(Ω). Hence there exists

ϕ0 ∈ W̃ s,p(Ω) and some subsequence, written again (ϕk)k, such that ϕk ⇀ ϕ0

in W̃ s,p(Ω), ϕk → ϕ0 in Lp(Ω) and in Lr
′p(Ω). In particular ‖ϕ0‖Lp(Ω)

= 1. Since

µ(λ) = limk→+∞ µ(λk), it follows that

µ(λ) > lim
k→+∞

EV−λkm(ϕk) > EV−λm(ϕ0) > µ(λ)

and hence µ(λ) = EV−λm(ϕ0). Using the simplicity of the principal eigenvalue
of problem (4.1) and the fact that ‖ϕ0‖Lp(Ω) = 1 and ϕ0 ≥ 0, we conclude that
ϕ0 = ϕλ. Moreover,

µ(λk) = EV−λkm(ϕk)

= EV−λm(ϕk) + (λ− λk)

∫
Ω

m(x)|ϕk|pdx

> µ(λ) + (λ− λk)

∫
Ω

m(x)|ϕk|pdx

and, by replacing λk by λ and ϕk by ϕλ in the inequality above, we obtain:

µ(λ) > µ(λk) + (λk − λ)

∫
Ω

m(x)|ϕλ|pdx.

Putting together this two inequalities we obtain

(λ− λk)

∫
Ω

m(x)|ϕk|pdx 6 µ(λk)− µ(λ) 6 (λ− λk)

∫
Ω

m(x)|ϕλ|pdx

from which we conclude that

µ′(λ) = −
∫

Ω

m(x)|ϕλ|pdx.

(ii) Since |Ω+| > 0 by (C1), there exists a function ξ ∈ W̃ s,p(Ω) such that∫
Ω
m(x)|ξ|pdx > 0,

∫
Ω
|ξ|pdx = 1 and therefore

µ(λ) 6 K(1− s)
∫
R2N

|ξ(x)− ξ(y)|p

|x− y|N+sp
dx dy +

∫
Ω

V (x)|ξ|pdx− λ
∫

Ω

m(x)|ξ|pdx.

Thus limλ→+∞ µ(λ) = −∞. Similarly, if |Ω−| > 0 then limλ→−∞ µ(λ) = −∞.



10 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

(iii) If |Ω−| = 0, then µ is strictly decreasing on R because−µ′(λ) =
∫

Ω
m(x)|ϕλ|pdx >

0 for all λ ∈ R. If |Ω−∪Ω0| = 0 assume by contradiction that the function λ 7→ µ(λ)
is bounded. Let (λk)k be such that λk → −∞ and write ϕk = ϕλk . Using Lemma
4.4, we have

sup
λ∈R

µ(λ) ≥ µ(λk) = E(V−λkm)(ϕk)

= EV (ϕk)− λk
∫

Ω

m(x)|ϕk|pdx︸ ︷︷ ︸
60

≥ EV (ϕk) >
1

C1

([
ϕk
]p
Ws,p(RN )

− C2

)
so (ϕk)k is a bounded sequence in W̃ s,p(Ω). Thus, there exist ϕ ∈ W̃ s,p(Ω) and

some subsequence (ϕk)k such that ϕk ⇀ ϕ in W̃ s,p(Ω) and ϕk → ϕ in Lpr
′
(Ω)

and in Lp(Ω). As ϕk is Lp-normalized, then ‖ϕ‖
Lp(Ω)

= 1 and
∫

Ω
m(x)|ϕ|p dx =

limk→∞
∫

Ω
m(x)|ϕk|p dx > 0. Then

−∞ = lim
k→∞

λk

∫
Ω

m(x)|ϕk|p dx ≥ EV (ϕ)− sup
λ∈R

µ(λ) > −∞,

a contradiction.
(iv). If |Ω− ∪ Ω0| = 0 then G0 = ∅ and using (ii) and (iii) we obtain

α(V m) = +∞ = lim
λ→−∞

µ(λ) = sup
λ∈R

µ(λ).

If |Ω− ∪ Ω0| > 0, as G0 ⊂ G and

µ(λ) 6 EV−λm(u) = EV (u)− λ
∫

Ω

m(x)|u|pdx = EV (u) ∀u ∈ G0.

then supλ∈R µ(λ) 6 α(V,m). To obtain the reverse inequality observe that, by (i)
and (ii), the function µ possesses a global maximum, that is, supλ∈R µ(λ) is reached
at some λ0 ∈ R, which in particular implies that

0 = µ′(λ0) =

∫
Ω

m(x)|ϕλ0
|pdx.

Consequently, ϕλ0
∈ G0 and then α(V,m) 6 EV (ϕλ0

). But

µ(λ0) = EV−λ0m(ϕλ0
) = EV (ϕλ0

) and µ(λ0) = sup
λ∈R

µ(λ),

then α(V,m) 6 supλ∈R µ(λ). Thus we obtain

α(V,m) = sup
λ∈R

µ(λ).

The proof that α(V,m) is achieved whenever it is finite, is standard and we omit
it. �

As a consequence of this proposition we have the following result.

Theorem 4.2. Assume that V and m satisfy the hypotheses (C1) and (C2).

(i) If |Ω−| = 0, then (1.1) admits a principal eigenvalue if and only if α(V,m) >
0. In this case the principal eigenvalue is unique and it is character-

ized by the equation λ1(V,m) = minMEV , where M :=
{
u ∈ W̃ s,p(Ω) :∫

Ω
m(x)|u|pdx = 1

}
.



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 11

(ii) If |Ω−| > 0, then (1.1) admits a principal eigenvalue if and only if α(V,m) >
0. More precisely,
(a) if α(V,m) > 0, then (1.1) admits exactly two principal eigenvalues

λ−1(V,m) = −min
M−

EV < λ1(V,m) = min
M

EV ,

where

M− :=
{
u ∈ W̃ s,p(Ω) :

∫
Ω

m|u|pdx = −1
}

;

(b) if α(V,m) = 0, then (1.1) admits a unique principal eigenvalue λ1(V,m)
given by

λ1(V,m) = inf
M
EV = − inf

M−
EV .

These infima are not achieved. In addition, any function u ∈ W̃ s,p(Ω)\
{0} such that EV (u) =

∫
Ω
m|u|pdx = 0 is an eigenfunction associated

with λ1(V,m).
(iii) In the case α(V,m) > 0 any function u ∈ M satisfying EV (u) = λ1(V,m)

is an eigenfunction associated with λ1(V,m) and it is sign definite. Same
result for u ∈M− satisfying EV (u) = λ−1(V,m).

Proof. The proof given in [8] can be easily adapted here as a corollary of Proposition
4.1. We only give the proof (b) of ii. to show how to use Picone’s inequality stated
in Lemma 7.1. If α(V,m) = 0, then there exists a real λ0 such that µ(λ0) = 0 so
λ0 is a principal eigenvalue of (1.1). Let us show that

λ0 = inf
M
EV = − inf

M−
EV .

We only give the proof of the first identity, the proof of the second one is similar.
As α(V,m) = supλ∈R µ(λ) then µ′(λ0) = 0 = −

∫
Ω
m(x)|ϕλ0

|pdx.
Let u ∈M be such that u > 0. For any T > 0 define uT := min{u, T} and take

ϕλ0
+ ε with ε > 0 Let us prove that z :=

upT(
ϕλ0

+ε
)p−1 ∈ W̃ s,p(Ω). Indeed, for any

(x, y) ∈ RN × RN we have

|z(x)− z(y)| ≤
∣∣∣uT (x)p − uT (y)p

(ϕλ0
(x) + ε)p−1

∣∣∣+ |uT (y)|p
∣∣∣ (ϕλ0

(y) + ε)p−1 − (ϕλ0
(x) + ε)p−1

(ϕλ0
(y) + ε)p−1(ϕλ0

(x) + ε)p−1

∣∣∣
(4.5)

and using for all (a, b) ∈ R+ × R+ and q > 0 the trivial inequality

|aq − bq| ≤ q
(
|a|q−1 + |b|q−1

)
|a− b|

with q = p or q = p− 1 we have

|z(x)− z(y)| ≤ 2pT p−1

εp−1
|u(x)− u(y)|+ 2(p− 1)T p

εp
|ϕλ0

(x)− ϕλ0
(y)| (4.6)

and therefore

[z]p
W s,p(RN )

≤ C
(

[u]p
W s,p(RN )

+ [ϕλ0
]p
W s,p(RN )

)
<∞.

An application of Picone’s inequality to functions uT and ϕλ0 +ε, and the fact that
upT(

ϕλ0
+ε
)p−1 ∈ W̃ s,p(Ω), imply that

0 6
∫
R2N

L(uT , ϕλ0 + ε)(x, y)

|y − x|N+sp
dx dy



12 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

=
[
uT
]p
Ws,p(RN )

−
∫
R2N

∣∣ϕλ0
(y)− ϕλ0

(x)
∣∣p−2(

ϕλ0
(y)− ϕλ0

(x)
)

|x− y|N+sp

×
( upT (y)(
ϕλ0

+ ε
)p−1

(y)
−

upT (x)(
ϕλ0

+ ε
)p−1

(x)

)
dx dy

=
[
uT
]p
Ws,p(RN )

+
1

K(1− s)

(
− λ0

∫
Ω

m(x)
∣∣ϕλ0

(x)
∣∣p−1 upT(

ϕλ0
+ ε
)p−1 dx

+

∫
Ω

V (x)
∣∣ϕλ0

∣∣p−1 upT(
ϕλ0 + ε

)p−1 dx
)
.

So when ε→ 0, by the Lebesgue convergence theorem

0 ≤ K(1− s)[uT ]p
W s,p(RN )

− λ0

∫
Ω

m(x)|uT |pdx+

∫
Ω

V (x)upT dx

for all T > 0. Moreover, since as T → +∞ we obtain uT = u. Then by Fatou’s
lemma,

0 6 K(1− s)[u]p
W s,p(RN )

− λ0 +

∫
Ω

V (x)|u|pdx. (4.7)

So λ0 6 infMEV . To prove the reverse inequality let us show that there exists a
sequence of functions ofM whose energy EV converges to λ0. Let ψ ∈ C∞(Ω) such

that ψ > 0,
∫

Ω
m(x)ψpdx > 0 and

∫
Ω
m(x)ϕp−1

λ0
ψdx > 0. Let the sequence (uk)k

be of the form

uk =
ϕλ0

+ ψ
k( ∫

Ω
m(x)

∣∣ϕλ0 + ψ
k

∣∣pdx)1/p
.

It is straightforward that all elements of this sequence are in manifoldM, and when
k is big enough uk > 0. Furthermore, because the functions t 7→ EV

(
ϕλ0

+ tψ
)

and s 7→
∣∣ϕλ0

+ sψ
∣∣p are continuous and at least once differentiable on

[
0, 1

k

]
, then

there exist 0 < tk, sk < 1/k such that

EV
(
ϕλ0

+
ψ

k

)
=

1

k
〈E′V

(
ϕλ0

+ tkψ
)
, ψ〉,∫

Ω

m(x)
∣∣ϕλ0 +

1

k
ψ
∣∣pdx =

p

k

∫
Ω

m(x)
∣∣ϕλ0 + skψ

∣∣p−1
ψdx.

As a result,

EV (uk) =
1∫

Ω
m(x)

∣∣ϕλ0 + 1
kψ
∣∣pdxEV (ϕλ0 +

ψ

k

)
=

k

p
∫

Ω
m(x)

∣∣ϕλ0
+ skψ

∣∣p−1
ψdx

×
〈E′V

(
ϕλ0

+ tkψ
)
, ψ〉

k
.

So when k tends to infinity we find that EV (uk) −→ λ0. Thus we can conclude that

λ0 = inf
M
EV . (4.8)

But ϕλ0 is an eigenfunction associated with µ(λ0) = 0, that means that EV (ϕ0) =

λ0

∫
Ω
m(x)

∣∣ϕλ0

∣∣pdx = 0 which implies λ0 is not achieved. Finally, if u ∈ W̃ s,p(Ω) \
{0} satisfies EV (u) =

∫
Ω
m(x)|u|pdx = 0 we have

sup
λ∈R

µ(λ) = 0 = EV (u) = EV−λ0m(u) > µ(λ0)

∫
Ω

|u|pdx = 0, (4.9)



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 13

and therefore u is a function where the infimum µ(λ0) of equation (4.2) is achieved.
Then, as the eigenvalue µ(λ0) is simple, there exists c > 0 such that u = cφλ0 so
then λ0 = λ1(V,m). �

Remark 4.3. One can prove, as at the beginning of the previous proof, that if

0 ≤ u ∈ W̃ s,p(Ω) ∩ L∞(Ω) and v ∈ W̃ s,p(Ω) satisfies v ≥ c > 0 a.e. for some c > 0

then up

vp−1 ∈ W̃ s,p(Ω) ∩ L∞(Ω).

Lemma 4.4. Let ω be a function satisfying (C1) and let Z be a bounded subset of
Lr(Ω). If ω > 0 a.e. is a function on Lr(Ω) for some 1 ≤ r < p∗s, then there are
two strictly positive constants C1 and C2 such that[

u
]p
Ws,p(RN )

6 C1EV (u) + C2

∫
Ω

ω(x)|u|pdx (4.10)

for all functions V ∈ Z and for all u ∈ W̃ s,p(Ω).

Proof. This proof is a partial adaptation of Lemma 2 of [8]. Let T be a positive

real such that ‖V ‖Lr(Ω) 6 T for all V ∈ Z. Let ε > 0 fixed such that ε < K(1−s)
T .

According to Hölder inequality and the hypothesis (C1), we can write∣∣∣ ∫
Ω

V (x)|u|pdx
∣∣∣ 6 ‖V ‖Lr(Ω)‖u‖pLpr′ (Ω)

.

Claim. For all ε > 0, there exists Mε > 0 such that

‖u‖p
Lpr
′
(Ω)
6 ε
[
u
]p
Ws,p(RN )

+Mε

∫
Ω

ω(x)|u|pdx (4.11)

for all u ∈ W̃ s,p(Ω).
Indeed, suppose by contradiction that there exists ε0 > 0, and sequence (uk)k of

W̃ s,p(Ω) such that

‖uk‖
Lpr
′
(Ω)

= 1 and ε0

[
u
]p
Ws,p(RN )

+ k

∫
Ω

ω(x)|uk|pdx < 1.

Then (uk)k is bounded W̃ s,p(Ω), so there exists u0 ∈ W̃ s,p(Ω) and sub-sequence

also denoted by (uk)k of W̃ s,p(Ω) such that uk ⇀ u0 in W̃ s,p(Ω) and uk → u0 in

L
pr′

(Ω) (see [10, Theorem 2.16]). So, we have on one hand

lim
k→+∞

‖uk‖
Lpr
′
(Ω)

= ‖u0‖
Lpr
′
(Ω)

= 1,

and therefore u0 6≡ 0 in Ω. Moreover, using once again the inequality of the
hypothesis we have ∫

Ω

ω(x)|uk|pdx <
1

k
.

Then passing to the limit we find by Fatou’s lemma that∫
Ω

ω(x)|u0|pdx 6 0,

which is a contradiction since ω > 0 in Ω and u0 6≡ 0 in Ω. We have proved the
claim.

By applying the inequality (4.11) for 0 < ε < K(1−s)
T there is a positive real Mε

such that∣∣ ∫
Ω

V (x)|u|pdx
∣∣ 6 ε‖V ‖Lr(Ω)

[
u
]p
Ws,p(RN )

+ ‖V ‖Lr(Ω)Mε

∫
Ω

ω(x)|u|pdx



14 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

6 εT
[
u
]p
Ws,p(RN )

+ TMε

∫
Ω

ω(x)|u|pdx.

So we obtain[
u
]p
Ws,p(RN )

6
1

K(1− s)− εT
EV (u) +

TMε

K(1− s)− εT

∫
Ω

ω(x)|u|pdx.

The lemma follows by setting

C1 =
1

K(1− s)− εT
and C2 =

TMε

K(1− s)− εT
.

�

As an application of Picone’s inequality of Lemma 7.1 we can prove the simplicity
and the uniqueness of the principal eigenvalues λ±1(V,m).

Proposition 4.5. Assume that α(V,m) ≥ 0. Let u > 0 a.e. be an eigenfunction
of problem (1.1) associated with λ1(V,m) and let v ≥ 0 a.e. be an eigenfunction
associated with an eigenvalue λ ≥ λ1(V,m). Then there exists c ∈ R such that
u = cv a.e. and λ = λ1(V,m).

Similarly, if u is an eigenfunction of problem (1.1) associated with λ−1(V,m)
with u > 0 a.e. and v is eigenfunction associated with an eigenvalue λ ≤ λ1(V,m)
with v > 0 a.e. then there exists c ∈ R such that u = cv a.e. and λ = λ−1(V,m).

Proof. Let us apply Picone’s inequality given in Lemma 7.1 to the functions u and
v + ε with ε > 0. By Remark 4.3,

0 6
∫
RN

∫
RN

L(u, v + ε)(x, y)

|x− y|N+sp
dx dy

= [u]p
Ws,p(RN )

−
∫
RN

∫
RN

× |v(y)− v(x)|p−2 (v(y)− v(x))

|x− y|N+sp

( up(y)

(v(y) + ε)p−1
− up(x)

(v(x) + ε)p−1

)
dx dy

=
1

K(1− s)

(
λ1(V,m)

∫
Ω

m(x)|u|pdx−
∫

Ω

V (x)|u|pdx
)

− 1

K(1− s)

(
λ

∫
Ω

m(x)|v|p−1 up

(v + ε)p−1
dx−

∫
Ω

V (x)|v|p−1 up

(v + ε)p−1
dx
)
.

By using the Lebesgue dominated convergence theorem and passing to the limit we
have

0 6
∫
RN

∫
RN

L(u, v)(x, y)

|x− y|N+sp
dx dy 6

λ1(V,m)− λ
K(1− s)

∫
Ω

m(x)|u|pdx.

Therefore if α(V,m) > 0 and λ > λ1(V,m), as we have
∫

Ω
m(x)|u|pdx > 0, we

conclude from the previous inequality that λ1(V,m) = λ and L(u, v) = 0. Hence, by
Picone’s inequality there is a constant c > 0 such that u = cv. In the case α(V,m) =
0 we have

∫
Ω
m(x)|u|pdx = 0 and we can conclude from the previous calculation that

L(u, v) = 0. Hence, by the conclusions of Picone’s inequality, there is a constant
c > 0 such that u = cv from which we deduce that EV (v) =

∫
Ω
m(x)|v|pdx = 0.

Thus, according to the result (ii)(b) of Theorem 4.2, v is an eigenfunction associated
with λ1(V,m), and therefore one must have λ = λ1(V,m). �



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 15

5. Nodal domains and isolation of the principal eigenvalues

5.1. Measure of the nodal domains of non principal eigenvalues. By a nodal

domain of a function v ∈ W̃ s,p(Ω) ∩ C(Ω), v 6≡ 0, we mean a maximal connected
open subset of either {x ∈ Ω : v(x) > 0} or {x ∈ Ω : v(x) < 0}.

Theorem 5.1. Let v be an eigenfunction of (1.1) associated with an eigenvalue λ
different from λ1(V,m) and λ−1(V,m). Then there exists constant C = C(s, p,N,Ω) >
0 such that, if N is a nodal domain of v, then

|N | ≥
(
C‖V − λm‖Lr(Ω)

)−γ
> 0, (5.1)

for

γ =
r′q

q − r′p
with


q =∞ if N < sp

q ≥ p if N = sp

q = p∗s if N > sp.

Proof. Let N be a nodal domain, and assume for instance that v < 0 on N . Let

us take ϕ = v−.χN as test function in (1.1). Notice that trivially ϕ ∈ W̃ (s,p)(Ω).
Thus

K(1− s)
∫
RN

∫
RN

|v(y)− v(x)|p−2 (v(y)− v(x)) (ϕ(y)− ϕ(x))

|x− y|N+sp
dx dy

=

∫
N

(λm− V )|v−|pdx

so

K(1− s)
[
ϕ
]p
Ws,p(RN )

=

∫
N

(V − λm)|v−|pdx ≤ ‖V − λm‖Lr(Ω)

(∫
N
|v−|pr

′
dx
)1/r′

.

Let us start with the case N > ps. By the previous Sobolev embedding theorem,
for some constant c > 0, we have

c‖ϕ‖p
Lp
∗
s (Ω)
≤
[
ϕ
]p
Ws,p(RN )

.

Hence

cK(1− s)‖ϕ‖p
Lp
∗
s (Ω)
≤ K(1− s)

[
ϕ
]p
Ws,p(RN )

≤ ‖V − λm‖Lr(Ω)

(∫
N
|v−|pr

′
dx
)1/r′

≤ ‖V − λm‖Lr(Ω)‖v−‖pLp∗s (Ω)
|N |

1
r′−

p
p∗s ,

and the estimate (5.1) follows.
If N = sp there exists some c > 0 such that for all q ≥ p,

cK(1− s)‖ϕ‖pLq(Ω) ≤ K(1− s)
[
ϕ
]p
Ws,p(RN )

≤ ‖V − λm‖Lr(Ω)

(∫
N
|v−|pr

′
dx
)1/r′

≤ ‖V − λm‖Lr(Ω)‖v−‖pLq(Ω)|N |
1
r′−

p
q ,

and the estimate (5.1) follows.
In the case N < sp, there exists some c > 0 such that

cK(1− s)‖ϕ‖pL∞(Ω) ≤ K(1− s)
[
ϕ
]p
Ws,p(RN )



16 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

≤ ‖V − λm‖Lr(Ω)

(∫
N
|v−|pr

′
dx
)1/r′

≤ ‖V − λm‖Lr(Ω)‖v−‖pL∞(Ω)|N |
1/r′ ,

and the estimate (5.1) follows. �

The following statement is a straightforward consequence of the above theorem.

Corollary 5.2. Any weak solution of (1.1) has a finite number of nodal domains.

Proof. Let Nj be a nodal domain of a certain eigenfunction associated with an
eigenvalue. Let us assume by contradiction that there exists an infinity of nodal
domains (Nj)j≥1 of this eigenfunction. We know that according to (5.1) there exists
a positive constant c > 0 such that we have

|Nj | > c ∀j .

Thus

|Ω| >
∑
j

|Nj | > c
∑
j

1 ,

which is a contradiction. �

5.2. Isolation of λ1(V,m) and λ−1(V,m). The following theorem states that the
eigenvalues λ±1(V,m) are isolated provided α(V,m) ≥ 0. Notice that if α(V,m) >
0, there are no eigenvalues in the interval (λ−1(V,m), λ1(V,m)).

Theorem 5.3. Let α(V,m) ≥ 0. The eigenvalues λ±1(V,m) are isolated in the
spectrum of (1.1), that is to say that there exists δ± > 0 such that there are no eigen-
values in the intervals (λ1(V,m), λ1(V,m) + δ+) and (λ−1(V,m)− δ−, λ−1(V,m)).

Proof. We only prove the result for λ1(V,m) by arguing by contradiction. Let us
assume that there exists a sequence (λk)k of eigenvalues such that

λk > λ1(V,m) and lim
k→∞

λk = λ1(V,m).

Denote by uk a positive eigenfunction associated with λk. Replacing uk by uk/[uk]
W̃ s,p(Ω)

if necessary, we can assume that the sequence (uk)k is bounded. By the results on
compact embeddings, there exists a subsequence (still denoted (uk)k) converging

to some u ∈ W̃ s,p(Ω) weakly in W̃ s,p(Ω), strongly in Lr
′p(Ω), a.e. and in measure

in Ω such that

lim
k→∞

∫
Ω

V (x)|uk|p dx =

∫
Ω

V (x)|u|p dx, lim
k→∞

∫
Ω

m(x)|uk|p dx =

∫
Ω

m(x)|u|p dx.

Since uk is an eigenfunction associated with λk we have

EV (uk) = K(1− s)
[
uk
]p
W s,p(RN )

+

∫
Ω

V (x)|uk|p dx

= K(1− s) +

∫
Ω

V (x)|uk|p dx = λk

∫
Ω

m(x)|uk|p dx.

Thus passing to the limit and using that EV is weakly lower semi-continuous we
obtain

EV (u) = K(1− s)[u]p
W s,p(RN )

+

∫
Ω

V (x)|u|p dx



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 17

≤ K(1− s) +

∫
Ω

V (x)|u|p dx = λ1(V,m)

∫
Ω

m(x)|u|p dx.

In particular u 6≡ 0 and

EV (u) ≤ λ1(V,m)

∫
Ω

m(x)|u|p dx. (5.2)

Assume first that α(V,m) > 0. Then (5.2) implies that
∫

Ω
m(x)|u|p dx 6= 0. In fact

we have
∫

Ω
m(x)|u|p dx > 0 otherwise, by taking v = u/

(
−
∫

Ω
m(x)|u|p dx

)1/p ∈
M− we will have from the definition of λ−1(V,m) that

−λ−1(V,m) ≤ EV (v) =
EV (u)

−
∫

Ω
m(x)|u|p dx

=⇒ λ−1(V,m)

∫
Ω

m(x)|u|p dx ≤ EV (u)

which, jointly with the inequality (5.2) will give λ−1(V,m) ≥ λ1(v,M), a contra-
diction. Since we have proved that

∫
Ω
m(x) |u|p dx > 0 we then have, by definition

of λ1(V,m), that

λ1(V,m)

∫
Ω

m(x) |u|p dx ≤ EV (u)

and therefore λ1(V,m)
∫

Ω
m(x) |u|p dx = EV (u). Thus, u is an eigenfunction as-

sociated with the principal eigenvalue λ1(V,m) and it must be either positive a.e.
or negative a.e. in Ω. On the other hand, if for each k we denote N+

k := {x ∈
Ω : uk(x) > 0} and N−k := {x ∈ Ω : uk(x) < 0}, by Theorem 5.1, we obtain the

existence of a constant c > 0 such that |N+
k | > c and |N−k | > c. However, if we

assume that u > 0 (the case u < 0 is analogous) it follows from the convergence in
measure that |N−k | → 0, which is a contradiction.

Assume now that α(V,m) = 0. We claim that
∫

Ω
m(x)|u|p dx = 0. Indeed, if for

instance
∫

Ω
m(x)|u|p dx > 0 then we will have, by definition of λ1(V,m), that

λ1(V,m)

∫
Ω

m(x)|u|p dx ≤ EV (u)

that, jointly with equation (5.2) will give that the infimum λ1(V,m) is achieved, a
contradiction. If

∫
Ω
m(x)|u|p dx < 0 then we will have instead

λ−1(V,m)

∫
Ω

m(x) |u|p dx ≤ EV (u)

and, since λ1(V,m) = λ−1(V,m), we again get a contradiction. We have just
proved that

∫
Ω
m(x)|u|p dx = 0. Hence, by equation (5.2) EV (u) ≤ 0, it must be

EV (u) = 0 by the definition of α(V,m) = 0. Thus u is an eigenfunction associated
with λ1(V,m) so u must be either > 0 a.e. or < 0 a.e. in Ω and we obtain a
contradiction as in the previous case. �

6. Regularity of the principal eigenvalues with respect to s

Now we study the behaviour of the first eigenvalues λ±1(V,m) with respect to s.
As we want to vary s, then to simplify the study, we now impose conditions on V and
m which are independent of s. So we assume V,m in Lr(Ω) with r > max{1, Np }.

We start by proving a lemma in the behaviour of sequences (1 − s)[us]pW s,p(RN )

as s varies.



18 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

Lemma 6.1. Let s0 ∈ (0, 1] and (sn)n be a sequence in (0, 1) converging to s0. Let

(un) be a sequence of functions such that, for all n ∈ N, un ∈ W̃ sn,p(Ω) and

(1− sn)[un]p
W sn,p(RN )

≤ L

for some L ≥ 0. Let q ∈ [1, p∗). Then there exists a function u ∈ W̃ s0,p(Ω) such
that, up to a subsequence,

(1) [u]p
W s0,p(RN )

≤ lim infn→∞[un]p
W sn,p(RN )

if s0 < 1 and∫
Ω
|∇u|p dx ≤ lim infn→∞(1− sn)K[un]p

W sn,p(RN )
if s0 = 1.

(2) un → u in Lq(Ω) and un → u in Lp(Ω).

Proof. First of all, by Poincaré’s inequality,

‖un‖pp ≤ C(N, p)(diam(Ω))snp(1− sn)[un]p
W sn,p(RN )

≤ C(N, p)(diam(Ω)snpL ≤ C

for some constant depending only N, p, diam(Ω), s0 and L. Assume first that s0 < 1
and let ε > 0 be small enough. Observe that since q < p∗ it follows that q < p∗s0−ε
if ε is small enough. Hence, if s0− ε < sn, using property 2 of Proposition 2.1, and
the previous estimate we have

[un]p
W s0−ε,p(RN )

≤ [un]p
W sn,p(RN )

+ C(N, p)
( 1

(s0 − ε)p
− 1

snp

)
≤ C (6.1)

for some C independent of n. Then there exists u ∈ W̃ s0−ε,p(Ω) and a subsequence,
still denoted by (un)n, such that

un ⇀ u in W s0−ε,p(RN ),

un → u in Lq(Ω),

un → u in Lp(Ω),

where we have used the compact imbedding of W 1−ε,p(Ω) into Lq(Ω) and into
Lp(Ω). Hence for all ε > 0, using (6.1),

[u]p
W s0−ε,p(RN )

≤ lim inf
n→∞

[un]p
W s0−ε,p(RN )

≤ lim inf
n→∞

[un]p
W sn,p(RN )

+ C(N, p)
( 1

(s0 − ε)p
− 1

s0p

)
.

Letting ε→ 0 and using Fatou’s lemma the conclusion 1 is reached.
If s0 = 1, by Lemma 3.10 of [5] we infer the existence of u ∈W 1,p

0 (Ω) such that,
up to a subsequence, un → u in Lp(Ω). Moreover, using property 3 of Proposition
2.2 and the hypothesis we obtain that, since 1− ε < sn if n is large enough,

ε[un]pW 1−ε,p(Ω) ≤ (1− sn)2(1−sn+ε)p diam(Ω)(sn−1+ε)p[un]pW sn,p(Ω) ≤ C (6.2)

for some C independent of n. Thus, the sequence (un) is bounded in W 1−ε,p(Ω) .

Hence there exists u ∈ W̃ 1−ε,p(Ω) and a subsequence, still denoted by (un)n, such
that

un ⇀ u in W 1−ε,p(Ω),

un → u in Lq(Ω),

un → u in Lp(Ω).

Thus, letting n→∞ in (6.2) and using that un ⇀ u in W 1−ε,p(RN ) we obtain

ε[u]pW 1−ε,p(Ω) ≤ lim inf
n→∞

ε[un]pW 1−ε,p(Ω) ≤ 2εp diam(Ω)εp lim inf
n→∞

(1− sn)[un]p
W sn,p(RN )

.



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 19

Finally, letting ε→ 0 and using Corollary 2 of [7] we obtain the result of 1. �

Our next result concerns the hypothesis on α(V,m) that allow us to have prin-
cipal eigenvalues. As we want to study the sign of α(V,m) as s varies, for s ∈ (0, 1]
let us write

α(s) = inf
{

(1− s)K[u]p
W s,p(RN )

+

∫
Ω

V (x)|u|p dx : u ∈ W̃ s,p(Ω), ‖u‖p = 1, and∫
Ω

m(x)|u|p dx = 0
}

if s 6= 1, and

α(s) = inf
{∫

Ω

|∇u|p dx+

∫
Ω

V (x)|u|p dx : u ∈W 1,p
0 (Ω), ‖u‖p = 1 and∫

Ω

m(x)|u|p dx = 0
}

if s = 1.

Proposition 6.2. Let s0 ∈ (0, 1] and assume that α(s0) > 0. Then there exists
ε > 0 such that α(s) > 0 for all s ∈ (s0 − ε, s0 + ε) ∩ (0, 1].

Proof. Assume by contradiction that there exists a sequence sn → s0 and a function

un ∈ W̃ sn,p(Ω) such that

(1− sn)K[un]p
W sn,p(RN )

+

∫
Ω

V (x)|un|pdx ≤ 0, ‖un‖p = 1,

∫
Ω

m(x)|un|p dx = 0.

Let tn = ‖un‖r′p and distinguish two cases.
Case (a): the sequence (tn)n is bounded. Then the sequence (1−sn)K[un]p

W sn,p(RN )

is bounded.
Case (b): the sequence (tn)n tends to +∞. Then taking vn = un/tn we have

(1− sn)K[vn]p
W sn,p(RN )

+

∫
Ω

V (x)|vn|p dx ≤ 0

and the sequence (1−sn)K[vn]p
W sn,p(RN )

is bounded. Let us write zn = un if case (a)

occurs and zn = vn if case (b) occurs. Let us now distinguish the cases 0 < s0 < 1
and the case s0 = 1.
1. Case 0 < s0 < 1. It follows from Lemma 6.1 with q = r′p that there exists

z ∈ W̃ s0,p(Ω) such that, in case (a),

(1− s0)K[z]p
W s0,p(RN )

+

∫
Ω

V (x)|z|pdx

≤ lim inf
n→∞

(
(1− sn)K[zn]p

W sn,p(RN )
+

∫
Ω

V (x)|zn|pdx dx
)
≤ 0, ‖z‖p = 1,

and the same inequality holds in case (b) with ‖z‖r′p = 1. Since
∫

Ω
m(x)|z|p = 0

we have a contradiction with α(s0) > 0.
2. Case s0 = 1. We obtain similarly that the sequence (1 − sn)K[zn]p

W sn,p(RN )

is bounded, with either ‖zn‖p = 1 or ‖zn‖r′p = 1. By Lemma 6.1 there exists

z ∈W 1,p
0 (Ω) such that∫

Ω

|∇u|p dx+

∫
Ω

V (x)|z|p dx ≤ lim inf
n→∞

(1−sn)K[zn]p
W sn,p(RN )

+

∫
Ω

V (x)|zn|p dx ≤ 0.



20 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

Notice that again ‖z‖q = 1, with either q = p or q = r′p, and
∫

Ω
m|(x)z|p = 0.

Thus α(1) ≤ 0, a contradiction. �

Let us now write

λ±1(s) := λ±1(V,m) = ± inf
{
K(1− s)

∫
R2N

|u(x)− u(y)|p dx dy
|x− y|N+sp

+

∫
Ω

V |u|pdx :

u ∈ W̃ s,p(Ω),

∫
Ω

m|u|pdx = ±1
}

and

λ±1 := ± inf
{∫

Ω

(
|∇u|p + V (x)|u|p

)
dx : u ∈W 1,p

0 (Ω),

∫
Ω

m(x)|u|pdx = ±1
}
.

We have the following result that generalizes, for indefinite weights, [10, Lemma
4.12].

Proposition 6.3. Assume that for some s0 ∈ (0, 1], α(s0) > 0. Then

lim
s→s0

λ±1(s) = λ±1(s0).

Proof. We only give the proof for λ1(s). By Proposition 6.2 α(s) > 0 for s close to
s0, so λ1(s) is a principal eigenvalue associated with problem (1.1). Let (sn)n be a
sequence in (0, 1] converging to s0 ∈ (0, 1]. Let us show that

lim
n→+∞

λ1(sn) = λ1(s0). (6.3)

1. Case s0 ∈ (0, 1). By definition of the first eigenvalue, we know that if ϕ ∈ C∞0 (Ω)
and

∫
Ω
m(x)|ϕ|p = 1, then

λ1(sn) ≤ K(1− sn)

∫
R2N

|ϕ(x)− ϕ(y)|p

|x− y|N+snp
dx dy +

∫
Ω

V (x)|ϕ|p dx

for all n ∈ N. Therefore, by dominated convergence theorem, we obtain lim supn→+∞ λ1(sn) ≤
λ1(s0). To prove the reverse inequality let (snk)k be a subsequence of (sn)n such
that

lim
k→+∞

λ1(snk) = lim inf
n→+∞

λ1(sn).

Let 0 ≤ unk ∈ W̃ sk,p(Ω) be an eigenfunction associated with λ1(snk) such that∫
Ω
m|unk |p dx > 0 and

[unk ]p
W
snk

,p
(RN )

= 1,

then in particular, using unk as test function in equation (1.1) for λ = λ(snk), we
have

λ1(snk)

∫
Ω

m(x)|unk |pdx = K(1− snk) +

∫
Ω

V (x)|unk |pdx. (6.4)

By Lemma 6.1 there exists u ∈ W̃ s0,p(Ω) such that, up to a subsequence,

[u]p
W s0,p(RN )

≤ lim inf
k→∞

[unk ]p
W
snk

,p
(RN )

= 1, unk → u in Lr
′p(Ω) and unk → u in Lp(Ω).

(6.5)
Hence, using (6.4) we find on the one hand that

lim inf
k→∞

λ1(snk)

∫
Ω

m(x)|u|pdx = K(1− s0) +

∫
Ω

V (x)|u|p dx (6.6)



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 21

and, on the other hand using (6.4) and (6.6),

K(1− s0)[u]p
W s0,p(RN )

+

∫
Ω

V (x)|u|p dx

≤ K(1− s0) + lim inf
k→∞

∫
Ω

V (x)|unk |p dx

= lim inf
k→∞

λ1(snk)

∫
Ω

m(x)u|p dx.

(6.7)

It remains to prove that
∫

Ω
m(x)|u|p dx > 0 to conclude from the previous inequality

that

λ1(s0) ≤ lim inf
k→∞

λ(snk)

(notice that the function v = u/
( ∫

Ω
m(x)|u|p dx

)1/p
will be then admissible in the

definition of λ1(s0)) and the proof of the proposition is completed.
To prove that

∫
Ω
m(x)|u|p dx > 0, remember first that

∫
Ω
m(x)|unk |p dx > 0 for

all k ∈ N and assume by contradiction that
∫

Ω
m(x)|u|p dx = 0. Using (6.6) we

infer that u 6≡ 0 and, using (6.7) we obtain

K(1− s0)[u]p
Ws0,p(RN )

+

∫
Ω

V (x)|u|p dx ≤ 0,

a contradiction with the hypothesis α(s0) > 0.
2. Case s0 = 1. Let ϕ ∈ C∞0 (Ω) such that

∫
Ω
m(x)|ϕ|p dx = 1. Then for any n ∈ N,

λ1(sn) ≤ K(1− sn)

∫
R2N

|ϕ(x)− ϕ(y)|p

|x− y|N+snp
dx dy +

∫
Ω

V (x)|ϕ|pdx.

Thus, by Proposition 2.2,

lim sup
n→+∞

λ1(sn) ≤
∫

Ω

|∇ϕ|pdx+

∫
Ω

V (x)|ϕ|pdx.

As ϕ is arbitrary, we have

lim sup
n→+∞

λ1(sn) ≤ λ1(1).

As in the previous case, let us prove that

lim inf
n→+∞

λ1(sn) ≥ λ1(1).

Let (snk)k be a subsequence of (sn)n such that

lim
k→+∞

λ1(snk) = lim inf
n→+∞

λ1(sn). (6.8)

Let uk be an eigenfunction associated with λ1(snk) such that

K(1− sk)[uk]p
Wsk,p(RN )

= 1. (6.9)

Then, as unk is an eigenfunction we have

λ1(snk)

∫
Ω

m(x)|unk |pdx = K(1− sk)[unk ]p
Wsk,p(RN )

+

∫
Ω

V (x)|unk |pdx

= 1 +

∫
Ω

V (x)|unk |pdx.
(6.10)



22 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

By Lemma 6.1 there exists u ∈W 1,p
0 (Ω) such that∫

Ω

|∇u|p dx+

∫
Ω

V (x)|u|p dx

≤ lim inf
k→∞

(
(1− snk)K[unk ]p

W sk,p(RN )
+

∫
Ω

V (x)|unk |p dx
)

= lim inf
k→∞

λ1(snk)

∫
Ω

m(x)|u|pdx.

(6.11)

Thus, if
∫

Ω
m(x)|u|p dx > 0, we can conclude, rescaling the previous inequality, that

λ1 ≤ lim inf
k→+∞

λ1(snk)

and the proof of the proposition is complete. To prove that
∫

Ω
m(x)|u|p dx >

0 we argue as before using now equations (6.10), (6.11) and that α(1) > 0 by
hypothesis. �

7. Appendix A

The following two results are, essentially, consequence of the convexity of the
function t 7→ |t|p−2t.

Lemma 7.1. A discrete version of Picone’s inequality [1] Let p ∈ (1,+∞). For
all functions ξ and φ defined on RN such that ξ > 0, and φ > 0, we have

L(ξ, φ) > 0 on RN × RN

with

L(ξ, φ)(x, y) := |ξ(y)−ξ(x)|p−|φ(y)−φ(x)|p−2 (φ(y)− φ(x))
( ξp(y)

φ(y)p−1
− ξp(x)

φ(x)p−1

)
,

for all (x, y) ∈ RN × RN . Moreover, we have

L(ξ, φ) = 0 ⇐⇒ ∃k ∈ R s.t. φ = kξ.

Proof. For sake of completeness we give the proof of this inequality. It uses the
following convexity inequality due to [1]. Fix x, y in RN and put a = ξ(y), b = ξ(x),

t = φ(x)
φ(y) and assume that 0 < b < a. It suffices to prove that for any p > 1 and

0 < t < 1, one has

|a− b|p ≥ ap(1− t)p−1 − bp
(1

t
− 1
)p−1

which is equivalent to say that

(1− t)
( |a− b|p

(1− t)p
)

+ t
bp

tp
> ap

which follows from the convexity of the function f(x) = |x|p. Notice that the
equality on this inequalities arrives if and only if t = b/a, that is, L(ξ, φ)(x, y) = 0
for all x, y ∈ RN if and only if ξ/φ = cte. �

Let us quote without proof the following second estimate.

Lemma 7.2 ([6, Lemma A1]). Let 1 < p <∞ and g : R→ R be a convex function,
then

|a− b|p−2(a− b)
[
A|g′(a)|p−2g′(a)−B|g′(b)|p−2g′(b)

]
≥ |g(a)− g(b)|p−2

(
g(a)− g(b)

)(
A−B

)
,

(7.1)



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 23

for every a, b ∈ R, and every A,B ≥ 0.

8. Appendix B

For completeness we give the following regularity result for the nonlocal non-
homogeneous problem

(−∆p)
su+ V (x)|u|p−2u = f(x) in Ω, u = 0 in RN \ Ω. (8.1)

Proposition 8.1. Assume sp < N , f ∈ Lq(Ω), V ∈ Lq(Ω) for some q ≥ N/sp,

and u ∈ W̃ s,p(Ω) is a solution of (8.1). Then for any t ∈ [1,+∞), u ∈ Lt(Ω) and
there exists a constant Ct depending on t and on Ω, ‖V ‖N/sp, ‖f‖N/sp, N , s, p
such that

‖u‖Lt(Ω) ≤ Ct. (8.2)

Proof. We borrow some ideas from [6, 18]. For every 0 < ε � 1 and any positive
function ϕ ∈ C∞0 (Ω) we define the smooth convex Lipschitz function

gε(t) = (ε2 + t2)1/2,

and choose the test function ψ = ϕ|g′ε(u)|p−2g′ε(u) in the variational formulation of
(8.1). Then we obtain

K(1− s)
∫
R2N

∣∣u(x)− u(y)
∣∣p−2(

u(x)− u(y)
)

|x− y|N+sp

(
ϕ(x)|g′ε

(
u(x)

)
|p−2g′ε

(
u(x)

)
− ϕ(y)|g′ε

(
u(y)

)
|p−2g′ε

(
u(y)

))
dx dy

≤
∫

Ω

|f(x)ϕ(x)|g′ε
(
u(x)

)
|p−2g′ε

(
u(x)

)
|dx

+

∫
Ω

|V (x)|u(x)|p−1ϕ(x)|g′ε
(
u(x)

)
|p−2g′ε

(
u(x)

)
|dx.

By using (7.1) with a = u(x), b = u(y), A = ϕ(x) and B = ϕ(y) we have

K(1− s)
∫
R2N

∣∣gε(u(x)
)
− gε

(
u(y)

)∣∣p−2(
gε
(
u(x)

)
− gε

(
u(y)

))
|x− y|N+sp

(
ϕ(x)− ϕ(y)

)
dx dy

≤
∫

Ω

|f(x)|ϕ(x)|g′ε
(
u(x)

)
|p−1dx+

∫
Ω

|V (x)‖u(x)|p−1ϕ(x)|g′ε
(
u(x)

)
|p−1dx.

By observing that gε converges to g(t) := |t| as ε→ 0, |g′ε(t)| ≤ 1 and using Fatou’s
Lemma, we obtain

K(1− s)
∫
R2N

∣∣∣∣u(x)
∣∣− ∣∣u(y)

∣∣∣∣p−2(∣∣u(x)
∣∣− ∣∣u(y)

∣∣)
|x− y|N+sp

(
ϕ(x)− ϕ(y)

)
dx dy

≤
∫

Ω

|f(x)|ϕ(x)dx+

∫
Ω

|V (x)‖u(x)|p−1ϕ(x)dx.

(8.3)

By the density of C∞0 (Ω) in W̃ s,p(Ω) (see Proposition 2.2), the same inequality

remains true for any positive ϕ ∈ W̃ s,p(Ω).
For k > 0 and t ≥ p we define uk and ϕk as follows:

uk := min{|u|, k} and ϕk(u) :=
tp

pp(t− p+ 1)
ut−p+1
k .



24 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

By definition, ϕk(u) ∈ W̃ s,p(Ω), and then by relation (8.3), we can write

K(1− s)
∫
R2N

∣∣∣∣u(x)
∣∣− ∣∣u(y)

∣∣∣∣p−2(∣∣u(x)
∣∣− ∣∣u(y)

∣∣)
|x− y|N+sp

(ϕk(u)(x)− ϕk(u)(y)) dx dy

≤
∫

Ω

|f(x)|ϕk(u(x))dx+

∫
Ω

|V (x)‖u(x)|p−1ϕk(u(x))dx.

(8.4)
For M > 0, set ΩM := {x ∈ Ω : |V (x)| > M}. We have∫

Ω

|V (x)‖u|p−1ϕk(u)dx ≤M
∫

Ω\ΩM
|u(x)|p−1|ϕk(u)|dx

+ ‖V ‖LN/sp(ΩM )

(∫
Ω

‖u(x)|p−1ϕk(u)|N/(N−sp)dx
)N−sp/N

.

Moreover, thanks to [4, Lemma C.2] we have∣∣∣∣u(x)
∣∣− ∣∣u(y)

∣∣∣∣p−2(∣∣u(x)
∣∣− ∣∣u(y)

∣∣)(ut−p+1
k (x)− ut−p+1

k (y))

≥ (t− p+ 1)pp

tp
∣∣uk(x)t/p − uk(y)t/p

∣∣p. (8.5)

Thus, by (8.4) and (8.5), the relation

(1− s)K
∫
R2N

∣∣uk(x)t/p − uk(y)t/p
∣∣p

|x− y|N+sp
dx dy

≤ Mtp

(t− p+ 1)pp

∫
Ω

|u(x)|p−1|ϕk(u(x))|dx

+
tp‖V ‖LN/sp(ΩM )

(t− p+ 1)pp

(∫
Ω

‖u(x)|p−1ϕk(u(x))|p
∗
s/pdx

)p/p∗s
+

tp

(t− p+ 1)pp

∫
Ω

|f(x)|ϕk(u(x))dx

holds and by the Sobolev’s embedding of W̃ s,p(Ω) into Lp
∗
s (Ω), there exists a con-

stant SN,s,p such that

SN,s,p

(∫
Ω

uk(x)
tp∗s
p

)p/p∗s
≤ (1− s)K

∫
R2N

∣∣uk(x)t/p − uk(y)t/p
∣∣p

|x− y|N+sp
dx dy

≤ Mtp

(t− p+ 1)pp

∫
Ω

|u|p−1|ϕk(u)|dx+
tp‖V ‖LN/sp(ΩM )

(t− p+ 1)pp

(∫
Ω

‖u|p−1ϕk(u)|p
∗
s/pdx

)p/p∗s
+

tp

(t− p+ 1)pp

∫
Ω

|f(x)|ϕk(u)dx.

If we choose M such that ‖V ‖LN/sp(ΩM ) ≤
(t−p+1)ppSN,s,p

2tp and use the definition
of ϕk and Hölder’s inequality for the last term of right-hand side of the previous
inequality, we obtain

SN,s,p
( ∫

Ω

|uk(x)|
tp∗s
p
)p/p∗s ≤ Mtp

(t− p+ 1)pp

∫
Ω

|u(x)|tdx+
SN,s,p

2

(∫
Ω

|uk(x)|
tp∗s
p

)p/p∗s
+

tp

(t− p+ 1)pp
‖f‖LN/sp(Ω)

(∫
Ω

|u(x)|p
∗
s(t+1−p)/pdx

)p/p∗s



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 25

and (∫
Ω

|u(x)|p
∗
s(t+1−p)/pdx

)p/p∗s
≤ |Ω|p(p−1)/p∗st

(∫
Ω

|u(x)|tp
∗
s/p
)p(t+1−p)/tp∗s

.

Then for all t = p we obtain

(t− p+ 1)ppSN,s,p
2tp

(∫
Ω

|uk(x)|tN/(N−sp)
)(N−sp)/N

≤M
∫

Ω

|u(x)|tdx

+ ‖f‖LN/sp(Ω)|Ω|p(p−1)/p∗st
(∫

Ω

|u(x)|tN/(N−sp)
) (t+1−p)(N−sp)

tN

.

(8.6)

Let us set t0 = p. Since by definition, u ∈ Lt0(Ω), it follows that u ∈ Lt0N/(N−sp)(Ω),
and thus, thanks to Fatou’s Lemma we have

(t0 − p+ 1)ppSN,s,p
2tp0

‖u‖t0
Lt0N/(N−sp)

≤M‖u‖t0Lt0 (Ω) + |Ω|p(p−1)/p∗st0‖f‖LN/sp‖u‖
t0+1−p
Lt0N/N−sp

.

Therefore using Young’s inequality we obtain

‖u‖Lt0 ≤ C1,t0‖u‖Lt0 (Ω) + C2,t0‖f‖
1/(p−1)

LN/sp
,

where C1,t0 and C2,t0 depend on M , N , s, p, t0, and |Ω|.
Now if we take t1 = t0N

N−sp ≥ p and since u ∈ Lt1(Ω), it follows that u ∈
Lt1N/(N−sp)(Ω) and we let k → +∞, by Fatou’s Lemma, and using Young’s in-
equality we obtain

‖u‖Lt1N/(N−sp) ≤ C1,t1‖u‖Lt(Ω) + C2,t1‖f‖
1/(p−1)

LN/sp
,

where C1,t1 and C2,t1 depend on M , N , s, p, t1, and |Ω|.
Thus as a consequence, if we define the sequence (tl)l∈N by

t0 = p, tl =
( N

N − sp

)l
p, l ∈ N∗

we find that u ∈ Ltl(Ω) for any l ∈ N∗. Since 1 < p ≤ tl for all l ∈ N and
tl −→
l→+∞

+∞, we conclude that u ∈ Lt(Ω) for any t > 1, and (8.2) follows. �

When V ∈ Lr(Ω) and f ∈ Lr(Ω) with r > N
sp , a better estimate holds.

Proposition 8.2. Assume that V ∈ Lr(Ω) and f ∈ Lr(Ω) with r > N
sp . Let

u ∈ W̃ p,s(Ω) be a solution of (8.1). Then u ∈ L∞(Ω) and there exists C =

C
(
s, p,N,Ω, ‖f‖Lr(Ω), ‖V ‖Lr(Ω), ‖u‖Lp∗s (Ω)

)
such that

‖u‖L∞(Ω) ≤ C. (8.7)

Proof. By Proposition 8.1, u ∈ Lt(Ω) for any t > 1 and therefore V |u|p−2u ∈ Lt(Ω)
for any t ∈

(
N
sp , r

)
∩
(

1
p−1+ 1

r

,+∞
)
. Moreover, by Holder’s inequality,

‖V |u|p−2u‖Lt(Ω) ≤ ‖V ‖Lr(Ω)C (p−1)tr
r−t

.



26 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

Thus, by replacing f by f +V |u|p−2u we can assume that V ≡ 0 in equation (8.1).
Let us assume first that u ≥ 0. For any k > 0 take uk defined as above and define
now for any α > 0,

φα,k :=
(
uk
)αp+1 ∈ W̃ s,p(Ω) ∩ L∞(Ω).

Using φα,k as test function, one obtains

K(1− s)
∫
R2N

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp

(
upα+1
k (x)− upα+1

k (y)
)
dx dy

=

∫
Ω

f(x)upα+1
k dx.

Thanks to Lemma 7.2 one has

K(1− s)(αp+ 1)

(α+ 1)p

∫
R2N

∣∣uα+1
k (x)− uα+1

k (y)
∣∣p

|x− y|N+sp
dx dy ≤

∫
Ω

|f(x)|upα+1
k dx

≤
∫

Ω

|f(x)|upα+1dx.

(8.8)

By Holder’s inequality we have[
uα+1
k

]
W s,p(RN )

≤
( (α+ 1)p

K(1− s)(αp+ 1)

)1/p(
‖f‖Lr(Ω)

(∫
Ω

|u|r
′(αp+1)

)1/r′)1/p

.

(8.9)

Since uα+1
k ∈ W̃ s,p(Ω), by the Sobolev’s embedding theorem there exists C1 > 0

such that

‖uk‖Lp∗s (α+1)(Ω) = ‖uα+1
k ‖

1
α+1

Lp
∗
s (Ω)
≤ C

1
α+1

1

[
uα+1
k

] 1
α+1

W s,p(RN )
.

Then, by (8.9), we obtain

‖uk‖Lp∗s (α+1)(Ω)

≤ C
1

α+1

1

( (α+ 1)p

K(1− s)(αp+ 1)

) 1
p(α+1)

(
‖f‖Lr(Ω)

(∫
Ω

ur
′(αp+1)

)1/r′) 1
p(α+1)

.

So, denoting C2 = ‖f‖Lr(Ω), we have

‖uk‖Lp∗s (α+1)(Ω) ≤ C
1

α+1

1

( C2(α+ 1)p

K(1− s)(αp+ 1)

) 1
p(α+1)

(∫
Ω

ur
′(αp+1)

) 1
pr′(α+1)

.

On the other hand there exists C3 > 0 such that( C2(α+ 1)p

K(1− s)(αp+ 1)

) 1
p
√
α+1 ≤ C3 for all α > 0.

Consequently, we obtain that

‖uk‖Lp∗s (α+1)(Ω) ≤ C
1

α+1

1 C
1√
α+1

3 ‖u‖
αp+1

(α+1)p

L(pα+1)r′ (Ω)

≤ C
1

α+1

1 C
1√
α+1

3 |Ω|
p−1

p2(1+α)2r′ ‖u‖
αp+1

(α+1)p

L(α+1)pr′ (Ω)
.

(8.10)

Choosing α = α1 in (8.10) such that (α1 + 1)pr′ = p∗s we obtain

‖uk‖Lp∗s (α1+1)(Ω) ≤ C
1

α1+1

1 C
1√
α1+1

3 |Ω|
p−1

p2(1+α1)2r′ ‖u‖
α1p+1

(α1+1)p

L(α1+1)pr′ (Ω)
.



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 27

Next we choose α = α2 in (8.10) such that (1 + α2)pr′ = (1 + α1)p∗s and obtain

‖uk‖Lp∗s (1+α2)(Ω) ≤ C
1

1+α2
1 C

1√
1+α2

3 |Ω|
p−1

p2(1+α2)2r′ ‖u‖
α2p+1

(α2+1)p

L(α1+1)p∗s (Ω)
.

By induction, for all m ∈ N∗ we can show that

‖uk‖Lp∗s (1+αm)(Ω) ≤ C
1

1+αm
1 C

1√
1+αm

3 |Ω|
p−1

p2(1+αm)2r′ ‖u‖
1+αmp
p(1+αm)

L(1+αm−1)p∗s (Ω)
, (8.11)

where (αm)m∈N is a sequence of positive numbers defined by

α0 = 0 and (1 + αm)pr′ = (1 + αm−1)p∗s ∀m ≥ 1.

One easily see that for all m ∈ N, 1 + αm =
(
p∗s
pr′

)m
, and then, by hypothesis,

αm → +∞ as m→ +∞ since r > N
ps . Moreover we have

‖uk‖Lσm (Ω) ≤ C
β2
m

1 Cβm3 |Ω|
β4
m(p−1)

p2r′ ‖u‖δm
Lσm−1 (Ω)

,

with σm = p∗s(αm + 1), βm = 1√
αm+1

, and δm = pαm+1
(αm+1)p . Notice that σm → +∞,

βm → 0 and δm ↑ 1 as m → +∞. Letting k → +∞ and using Fatou’s lemma we
obtain

‖u‖Lσm (Ω) ≤ Cβm4 ‖u‖
δm
Lσm−1 (Ω)

, (8.12)

for some constant C4 > 0. A simple computation gives

‖u‖Lσm (Ω) ≤ C

(
βm+

∑m−1
i=1 βm−i

∏i−1
k=0 δm−k

)
4 ‖u‖

∏m
i=1 δi

Lσ0 (Ω) .

Using that δm ↑ 1 and that βm = ( r
′p
p∗s

)m/2 one can find that

m−1∑
i=1

(
βm−i

i−1∏
k=0

δm−k

)
≤

m∑
i=0

βm−i ≤
1

1− ( r
′p
p∗s

)1/2
<∞

so
‖u‖Lσm (Ω) ≤ C max

{
1, ‖u‖Lp∗s (Ω)

}
and the conclusion follows. If u changes sign one can use instead, as in Proposition
8.1, uk = max{|u|, k}. �

Acknowledgements. O. Asso was financially supporting by the Deutscher Akademis-
cher Austauschdienst (DAAD) through IMSP. This work was partially carried out
while L. Leadi was visiting the LMPA of the Université du Littoral Côte d’Opale
(ULCO). We would like to express our gratitude to these institutions.

References

[1] S. Amgibech; On the discrete version of Picone’s identity, Discrete and applied Mathematics,
156 (2008), no. 1, 1-10.

[2] P. A. Binding, Y. X. Huang; The principal eigencurve for the p-Laplacian, Differential and

Integral equations, 8 (1995), no. 2, 405-414.

[3] P. A. Binding, Y. X. Huang; Existence and nonexistence of positive eigenfunctions for the
p-Laplacian, Procc. American Math. Society, 123 (1995), no. 6, 1833-1838.

[4] L. Brasco, E. Lindgren, E. Parini; The fractional cheeger problem, Interfaces and Free Bound-
aries 16 (2014), no. 3, 419-458.

[5] L. Brasco, E. Parini, M. Squassina; Stability of variational eigenvalues for the fractional p-

Laplacian, Discrete and Contin. Dyn. Sys., 36 (2016), no. 4, 1813-1845.
[6] L. Brasco, E. Parini; The second eigenvalue of the fractional p-Laplacian, Advances in Calculus

of Variations, 9 (2016), no. 4, 323-355.



28 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38

[7] J. Bourgain, H. Brézis, P. Minorescu; Another look at Sobolev spaces. Optimal Control and

PDE (Conference paris 2000 in honour of Prof. Alain Bensoussans’s 60th birthday. Eds. J. L

Menaldi and others. Amsterdam IOS Press 2001, 439-455.
[8] M. Cuesta, R. Q. Humberto; A weighted eigenvalue problem for the p-Laplacian plus a poten-

tial, NoDEA-Nonlinear differential equations and applications, 16 (2009), no. 4, 469-491.

[9] M. Cuesta, L. Leadi; Weighted eigenvalue problems for quasilinear elliptic operators with
mixed Robin-Dirichlet boundary conditions, J. Math. Anal. Appl., 422 (2015), no. 1, 1-26.

[10] L. M. Del Pezzo, A. Quaas; Global bifurcation for fractional p-Laplacian and application, Z.

Anal. Anwend., 35 (2016), no. 4, 411-447.
[11] L. M. Del Pezzo, A. Quaas; Non-resonant Fredholm alternative and anti-maximum principle

for the fractional p-Laplacian, J. Fixed Point Theory Appl., 19 (2017), no. 1, 939-958.

[12] L. M. Del Pezzo, J. Fernández Bonder, L. Lopez Rios; An optimization problem for the
first eigenvalue of the p-fractional Laplacian, Mathematische Nachrichten, 291 (2018), no. 4,

632-651.
[13] L. M. Del Pezzo, J. D. Rossi; Eigenvalues for a nonlocal pseudo p-Laplacian, Discrete and

continuous Dynamical Systems, 36 (2016), no. 12, 6737-6765.

[14] F. Demengel, G. Demengel; Functional spaces for the theory of elliptic partial differential
equations, Translated from the 2007 French original by Reinie Erné, Universitext, Springer,

London, (2012), EDP Sciences, Les Ulis, 2012, xviii + 465 pp, ISBN: 978-1-4471-2806-9, 978-

2-7598-0698-0.
[15] J. Fleckinger, J. Hernández, F. de Thélin; Existence of Multiple Principal Eigenvalues for

some Indefinite Linear Eigenvalue problems, Bollettino della Unione Matematica Italiana Sez.

B Artic. Ric. Mat., (8), 7 (2004), no. 1, 159-188.
[16] G. Franzina, G. Palatucci; Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S), 5 (2014),

no. 2, 373-386.

[17] P. Grisvard; Elliptic problems in nonsmooth domains, Monographs and Studies in Mathe-
matics, 24, Pitman (Advanced Publishing Program), Boston, MA, (1985), xiv+410 p, ISBN:

0-273-08647-2.
[18] M. Guedda, L. Veron; Quasilinear elliptic equations involving critical Sobolev exponents,

Nonlinear Analysis, Theory, Methods & Applications, 13 (1989), no. 8, 879-902.

[19] P. Hess, T. Kato; On some linear and nonlinear eigenvalue problems with an indefinite weight
function. Commun. Partial Diff. Equ., 5 (1980), no. 10, 999-1030.

[20] A. Iannizzotto, S. Mosconi, M. Squassina; Global Hölder regularity for the fractional

p-Laplacian, Revista Matemátic Iberoamericana, 32 (2016), no. 4, 1353-1392. DOI:
10.4171/rmi/921.

[21] L. Leadi, A. Marcos; A weighted eigencurve for Steklov problems with a potential, NoDEA

Nonlinear Differential Equations Appl., 20 (2013), no. 3, 687-713.
[22] R. Servadei, E. Valdinoci; Weak and Viscosity Solutions of the Fractional Laplace Equation,

Publ. Mat., 58 (2014), no. 1, 133-154.

[23] T. Kuusi, G. Mingione, Y. Sire; Nonlocal equations with measure data, Comm. Math. Phys.,
337 (2015), no. 3, 1317-1368.

Oumarou Asso
Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi, 613

Porto-Novo, Bénin
Email address: oumarou.asso@imsp-uac.org

Mabel Cuesta
Université du Littoral Côte d’Opale (ULCO), LMPA, 50 rue F. Buisson 62220 Calais,

France
Email address: mabel.cuesta@univ-littoral.fr

Jonas Têlé Doumatè

Département de Mathématiques, Faculté des Sciences et Techniques, Institut de Mathématiques
et de Sciences Physiques, Université d’Abomey-Calavi, Benin

Email address: jonas.doumate@fast.uac.bj



EJDE-2023/38 EIGENVALUES FOR FRACTIONAL p-LAPLACIAN 29

Liamidi Leadi

Département de Mathématiques, Faculté des Sciences et Techniques, Institut de Mathématiques

et de Sciences Physiques, Université d’Abomey-Calavi, Benin
Email address: leadiare@imsp-uac.org


	1. Introduction
	2. Preliminaries
	2.1. Basic results about fractional Sobolev spaces
	2.2. Embeddings

	3. Weak solutions of the eigenvalue problem and regularity results
	4. Existence of principal eigenvalues with indefinite weights
	5. Nodal domains and isolation of the principal eigenvalues
	5.1. Measure of the nodal domains of non principal eigenvalues
	5.2. Isolation of 1 (V,m) and -1(V,m)

	6. Regularity of the principal eigenvalues with respect to s 
	7. Appendix A
	8. Appendix B
	Acknowledgements

	References