Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 46, pp. 1–15.

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

DOI: https://doi.org/10.58997/ejde.2023.46

EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO A

FRACTIONAL p-LAPLACIAN ELLIPTIC DIRICHLET PROBLEM

FARIBA GHAREHGAZLOUEI, JOHN R. GRAEF,

SHAPOUR HEIDARKHANI, LINGJU KONG

Abstract. In this article, the authors consider a fractional p-Laplacian elliptic
Dirichlet problem. Using critical point theory and the variational method,

they investigate the existence of at least one, two, and three solutions to the

problem. Examples illustrating the results are interspaced in the paper.

1. Introduction

In this article, we examine the nonlinear elliptic equation involving the fractional
p-Laplacian and depending on a real parameter λ > 0,

(−∆)spu = λf(x, u) + h(u), in Ω,

u = 0, on RN\Ω,
(1.1)

where sp < N , Ω is a bounded open subset of RN with a Lipschitz boundary, the
fractional p-Laplacian operator (−∆)sp is defined by

(−∆)spu(x) = 2 lim
ε↘0

∫
RN\Bε(x)

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
dy, x ∈ RN .

Here, 0 < s < 1 < p < +∞, Bε(x) = {y ∈ RN : |x − y| < ε}, f : Ω × R → R
satisfies a Carathédory condition, and h : R→ R is a Lipschitz continuous function
of order p− 1 with a Lipschitz constant L > 0, i.e.,

|h(ξ1)− h(ξ2)| ≤ L|ξ1 − ξ2|p−1 for all ξ1, ξ2 ∈ R, (1.2)

and such that h(0) = 0.
In recent years, a great deal of attention has been focused on the study of frac-

tional and nonlocal operators of elliptic type, for both pure mathematical research
and concrete real-world applications. Fractional and nonlocal operators appear
in many fields such as optimization, finance, phase transitions, stratified materials,
anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes,
flame propagation, conservation laws, ultra-relativistic limits of quantum mechan-
ics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science,
water waves, and Lévy processes; see, e.g., [2, 8, 14, 17] and the references therein.

2020 Mathematics Subject Classification. 35R11, 35A15, 35J60, 35B38.
Key words and phrases. Fractional p-Laplacian; weak solution; critical points;

variational method.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted April 13, 2023. Published July 3, 2023.

1



2 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

This is one of the reasons why nonlocal fractional problems are widely studied in
the literature in many different contexts.

The application of a mountain pass theorem to Dirichlet problems involving non-
local integro-differential operators of fractional Laplacian type are given in [19, 20].
Wei and Su [21] showed that the fractional Laplacian problem possesses infinitely
many weak solutions. Lehrer et al. [15] investigated the existence of nonnegative
solutions to problem (1.1) in the case h ≡ 0. Their problem is set on a unbounded
domain and compactness issues have to be handled. Iannizzotto et al. [12] studied
existence and multiplicity results for fractional p-Laplacian type problems via Morse
theory. Kim [13] applied abstract critical point results to establish an estimate of
a positive interval for the parameter λ within which the problem (1.1) with h ≡ 0
admits at least one or two nontrivial weak solutions provided the nonlinearity f
satisfies a subcritical growth condition. In addition, under certain conditions, he
established an a priori estimate in L∞(Ω) for any possible weak solution by applying
a bootstrap argument.

In this paper we obtain three different results about the existence of weak solu-
tions to the problem (1.1) by using critical point theorems established in [4, 5, 7].

The first aim of this paper is to provide an estimate of the positive interval
for the parameter λ in which the problem (1.1) possesses at least one nontrivial
weak solution in the case where the nonlinear term f satisfies a subcritical growth
condition. We also wish to consider the existence of two solutions to our problem
by using a result of Bonanno [5, Theorem 3.2]. In a recent paper, Bonanno and
Chinǹı [6] studied the existence of at least two distinct weak solutions to a problem
involving a p(x)-Laplacian by applying critical point theory. Our first main result
will require the (P.S.)[r] condition, while in our second one, we will ask that the
(AR)-condition holds and use it to ensure that the (usual) (PS)-condition is satis-
fied. We refer the reader to the papers [3, 6, 11] where this approach was applied
successfully.

Finally, our third goal is to obtain the existence of three solutions to (1.1); this
problem is less studied by researchers. In this case, we consider problem (1.1) where
the nonlinearity f has subcritical growth, and we apply variational methods and
critical point theory. The main tool used is the critical point theorem of Bonanno
and Marano [7, Theorem 3.6].

The remainder of this paper is organized as follows. First, in Section 2, we recall
briefly some basic results for fractional Sobolev spaces. In Section 3, we obtain the
existence of at least one, two, or three nontrivial weak solutions to the problem
(1.1) provided the parameter λ belongs to a positive interval to be determined.

2. Preliminaries

This section is devoted to the definition of the fractional Sobolev spaces and
related properties that will be used in the next section.

For s ∈ (0, 1) and p ∈ (1,+∞), the fractional Sobolev space W s,p(RN ) is defined
as

W s,p(RN ) :=
{
u ∈ Lp(RN ) :

∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy < +∞

}
,

which is an interpolation Banach space between Lp(RN ) and W 1,p(RN ). The norm
for this space is

‖u‖W s,p(RN ) := (‖u‖p
Lp(RN )

+ |u|p
W s,p(RN )

)1/p,



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 3

where

‖u‖p
Lp(RN )

:=

∫
RN
|u|p dx and |u|p

W s,p(RN )
:=

∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy.

It is known (see [1]) that W s,p(RN ) is a separable and reflexive Banach space and
that C∞0 (RN ) is dense in W s,p(RN ), i.e., W s,p

0 (RN ) = W s,p(RN ).
For our problem we consider the subspace of W s,p(RN ) given by

Xp
s (Ω) = {u ∈W s,p(RN ) : u(x) = 0 a.e. in RN\Ω}

with the norm

‖u‖Xps (Ω) := (‖u‖pLp(Ω) + |u|p
W s,p(RN )

)1/p, (2.1)

which is known to be a uniformly convex Banach space (see [22, Lemma 2.4]). We
will need the following lemmas to prove our main theorems.

Lemma 2.1 ([13, Lemma 2.1]). Let Ω be a bounded open set in RN , s ∈ (0, 1),
and p ∈ [1,+∞). Then

‖u‖pLp(Ω) ≤
sp|Ω|sp/N

2ω
sp
N +1

N

|u|p
W s,p(RN )

for any u ∈ W̃ s,p(RN ). Here, |Ω| is the Lebesgue measure of Ω, ωN denotes the

volume of the N -dimensional unit ball, and W̃ s,p(RN ) is the space of all u ∈ Xp
s (Ω)

such that ũ ∈W s,p(RN ), where ũ is the extension by zero of u.

Remark 2.2. In view of Lemma 2.1, it is clear from (2.1) that there is an equiva-
lence between the norms in W s,p(RN ) and Xp

s (Ω).

Lemma 2.3 ([16]). Let s ∈ (0, 1) and p ∈ [1,+∞) be such that sp < N . Then, for
any u ∈W s,p(RN ),

‖u‖p
Lp
∗
s (Ω)
≤ Cp∗s |u|

p
W s,p(RN )

,

where

Cp∗s =
(N + 2p)3ppp+22(N+1)(N+2)s(1− s)

N
p
p∗s |SN−1| spN +1(N − sp)p−1

.

Here, |SN−1| denotes the surface area of the (N − 1)-dimensional unit sphere and

p∗s is the fractional critical Sobolev exponent, that is, p∗s = pN
N−sp .

Remark 2.4. Recall that for each s ∈ (0, 1) and p ∈ [1,+∞) such that sp < N ,
from [9, Theorem 4.54], we have the continuous embedding

Xp
s (Ω) ↪→ Lq(Ω) for all q ∈ [1, p∗s].

In particular, the space Xp
s (Ω) is compactly embedded in Lq(Ω) for any q ∈ [1, p∗s).

In fact, according to Lemma 2.3, for each u ∈ Xp
s (Ω), there exists Cq > 0 such that

‖u‖Lq(Ω) ≤ C1/p
q |u|W s,p(RN ).

The constant Cq is important in obtaining an interval on λ in which (1.1) has one
or more nontrivial weak solutions.

Definition 2.5 ( [4, p. 2993], [5, p. 210]). Let Φ and Ψ be two continuously
Gâteaux differentiable functionals defined on a real Banach space X and fix r ∈ R.
The functional I = Φ−Ψ is said to satisfy the Palais-Smale condition cut off upper
at r, denoted by (P.S.)[r] if any sequence {un}n∈N in X such that



4 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

(1) {I(un)} is bounded,
(2) limn→∞ ‖I ′(un)‖X∗ = 0, and
(3) Φ(un) < r for each n ∈ N,

has a convergent subsequence.
If only conditions (1) and (2) hold, then I = Φ−Ψ is said to satisfy the (usual)

Palais-Smale (P.S.) condition.

We next wish to define what is meant by a weak solution of our problem.

Definition 2.6. Let 0 < s < 1 < p < +∞. We say that u ∈ Xp
s (Ω) is a weak

solution of problem (1.1) if∫
RN

∫
RN

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

|x− y|N+sp
dx dy

= λ

∫
Ω

f(x, u)v dx+

∫
Ω

h(u)v dx

for all v ∈ Xp
s (Ω).

We define Φ : Xp
s (Ω)→ R by

Φ(u) :=
1

p

∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy −

∫
Ω

H(u) dx for all u ∈ Xp
s (Ω), (2.2)

where H(t) =
∫ t

0
h(ξ)dξ for t ∈ R. The functional Φ is Fréchet differentiable and

its Fréchet derivative is given by

〈Φ′(u), v〉 =

∫
RN

∫
RN

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

|x− y|N+sp
dx dy

−
∫

Ω

h(u)v dx

for any v ∈ Xp
s (Ω).

We will need the condition

(H1) there exist nonnegative functions α, β ∈ L∞(Ω) such that

|f(x, t)| ≤ α(x) + β(x)|t|q−1 for all (x, t) ∈ Ω× R,

where 1 < q < p∗s.

Define the function F : Ω× R→ Ω× R by

F (x, t) =

∫ t

0

f(x, ξ)dξ for (x, ξ) ∈ Ω× R

and the functionals Ψ, Iλ : Xp
s (Ω)→ R by

Ψ(u) :=

∫
Ω

F (x, u) dx,

Iλ(u) = Φ(u)− λΨ(u)

(2.3)

for all u ∈ Xp
s (Ω). In what follows, we will assume that the Lipschitz constant

L > 0 belonging to the function h in (1.2) satisfies

L <
21−pω

sp
N +1

N

ps|Ω|ps/N
, (2.4)



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 5

from which we see that

0 < L <
21−pω

sp
N +1

N

ps|Ω|ps/N
<

2ω
sp
N +1

N

ps|Ω|ps/N
. (2.5)

3. Main results

We begin by presenting a result that guarantees the existence of at least one
solution to problem (1.1). We will need the constant

µ :=
[ 22p+N−sp

(p− sp)(N + p− sp)
+

21+sp

sp(p− sp+ 1)
+

1

sp(N − sp)

]
ω2
NN

2.

Theorem 3.1. Let p ≥ 2, f : Ω × R → R be a Carathédory function satisfying
(H1), and assume that there exist three real positive constants τ , ρ, and δ such that:

(H2)

2ω2
Nρ

N−spδpµ
2ω

sp
N +1

N + Lsp|Ω|sp/N

2ω
sp
N +1

N − Lsp|Ω|sp/N
< τp;

(H3)

ωN

(
‖α‖∞|Ω|1/p

′
(
ps|Ω|sp/N

2ω
sp
N

+1

N

)1/p

τ + q−1C
q/p
q ‖β‖∞τ q

)
(

2ω
sp
N +1

N − Lsp|Ω|sp/N
)
τp

<
ρsp ess infx∈Ω F (x, δ)

2N+1δpµ
(

2ω
sp
N +1

N + Lsp|Ω|sp/N
) ,

where 1/p+ 1/p′ = 1;
(H4) F (x, t) ≥ 0 for each (x, t) ∈ Ω× R+.

Then, for each

λ ∈ Λw :=
(2N (2ω

sp
N +1

N + Lsp|Ω|sp/N )δpµ

pρspω
sp/N
N ess infx∈Ω F (x, δ)

,

(2ω
sp
N +1

N − Lsp|Ω|sp/N )τp

2pω
sp
N +1

N (‖α‖∞|Ω|1/p′(ps|Ω|
sp/N

2ω
sp
N

+1

N

)1/pτ + q−1C
q/p
q ‖β‖∞τ q)

)
,

(3.1)

problem (1.1) admits at least one nontrivial solution uλ ∈ Xp
s (Ω).

Proof. Our goal is to apply [5, Theorem 2.3] to problem (1.1). To this end, we take
the real Banach space Xp

s (Ω) with the norm as defined in Section 2, and Φ and Ψ
to be the functionals defined in (2.2) and (2.3). Taking into account that h is a
Lipschitz continuous function of order p− 1 with Lipschitz constant (see (2.4))

0 < L <
21−pω

sp
N +1

N

ps|Ω|ps/N

and h(0) = 0, we have

1

p
|u|p

W s,p(RN )
− L

p
‖u‖pLp(Ω) ≤ Φ(u) ≤ 1

p
|u|p

W s,p(RN )
+
L

p
‖u‖pLp(Ω),



6 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

namely,

2ω
sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

|u|p
W s,p(RN )

≤ Φ(u)

≤
2ω

sp
N +1

N + Lps|Ω|sp/N

2pω
sp
N +1

N

|u|p
W s,p(RN )

.

(3.2)

From the first inequality in (3.2), it follows that Φ is coercive. It is also clear
that Φ ∈ C1(Xp

s (Ω),R). To show that Φ′ admits a continuous inverse, in view of
[23, Theorem 26.A(d)], it suffices to show that Φ′ is coercive, hemicontinuous, and
uniformly monotone.

By Lemma 2.1, it is clear that for any u ∈ Xp
s (Ω), we have

〈Φ′(u), u〉
‖u‖Xps (Ω)

≥ 1

‖u‖Xps (Ω)

(∫
RN

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))2

|x− y|N+sp
dx dy −

∫
Ω

h(u)u dx
)

≥
2ω

sp
N +1

N

(2ω
sp
N +1

N + sp|Ω|sp/N )|u|W s,p(RN )

(
|u|p

W s,p(RN )
− L‖u‖pLp(Ω)

)
≥

2ω
sp
N +1

N − Lsp|Ω|sp/N

2ω
sp
N +1

N + sp|Ω|sp/N
|u|p−1

W s,p(RN )
.

Since L <
2ω

sp
N

+1

N

ps|Ω|ps/N , this implies

lim
‖u‖Xps (Ω)→∞

〈Φ′(u), u〉
‖u‖Xps (Ω)

=∞,

i.e., Φ′ is coercive. The fact that Φ′ is hemicontinuous can be shown using standard
arguments (see, for example, [18]).

Finally, we show that Φ′ is uniformly monotone. First recall the inequality that
for any ξ, ψ ∈ R,

(|ξ|r−2ξ − |ψ|r−2ψ)(ξ − ψ) ≥ 2−r|ξ − ψ|r, if r ≥ 2. (3.3)

In view of (2.4) and Lemma 2.1, for every u, v ∈ Xp
s (Ω), there exists a positive

constant k1 such that

〈Φ′(u)− Φ′(v), u− v〉

=

∫
RN

∫
RN

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

)
|x− y|N+sp

× ((u− v)(x)− (u− v)(y)) dx dy −
∫

Ω

(h(u)− h(v))(u− v) dx

≥ 2−p|u− v|p
W s,p(RN )

− L‖u− v‖pLp(Ω)

≥
(

2−p − Lps|Ω|ps/N

2ω
sp
N +1

N

)
|u− v|p

W s,p(RN )

≥ k1‖u− v‖pXps (Ω)
.



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 7

From condition (H1) and Remark 2.4, the functional Ψ belongs to C1(Xp
s (Ω),R)

and has a compact derivative. This ensures that the functional Iλ = Φ−λΨ satisfies
(P.S.)[r] for each r > 0 (see [4, Proposition 2.1]).

To apply [5, Theorem 2.] to the functional Iλ, first note that infXps (Ω) Φ =
Φ(0) = Ψ(0) = 0. We need to show that there is an r > 0 and v̄ ∈ Xp

s (Ω) with

0 < Φ(v̄) < r such that
supΦ(u)≤r Ψ(u)

r < Ψ(v)
Φ(v) . To this end, set

r :=
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

τp

with L satisfying (2.5), and define w by

w(x) =


0, if x ∈ RN \BN (0, ρ),

δ, if x ∈ BN (0, ρ2 ),
2δ
ρ (ρ− |x|), if x ∈ BN (0, ρ) \BN (0, ρ2 ).

(3.4)

We take Bρ = BN (0, ρ); then

Φ(w) ≤
2ω

sp
N +1

N + Lps|Ω|sp/N

2pω
sp
N +1

N

∫
RN

∫
RN

|w(x)− w(y)|p

|x− y|N+sp
dx dy

=
2ω

sp
N +1

N + Lps|Ω|sp/N

2pω
sp
N +1

N

(∫
Bρ\Bρ/2

∫
Bρ\Bρ/2

|w(x)− w(y)|p

|x− y|N+sp
dx dy

+ 2

∫
Bρ\Bρ/2

∫
RN\Bρ

|w(x)− w(y)|p

|x− y|N+sp
dx dy

+ 2

∫
Bρ/2

∫
Bρ\Bρ/2

|w(x)− w(y)|p

|x− y|N+sp
dx dy

+ 2

∫
RN\Bρ

∫
Bρ/2

|w(x)− w(y)|p

|x− y|N+sp
dx dy

)
=

2ω
sp
N +1

N + Lps|Ω|sp/N

2pω
sp
N +1

N

(R1 + 2R2 + 2R3 + 2R4).

Next, we estimate R1, R2, R3, and R4 by direct calculations. Recall that if g
is a continuous radial function (i.e., g(x) = g̃(|x|)) on a closed ball Bγ of radius γ,
then ∫

Bγ

g(x) dx = NωN

∫ γ

0

g̃(r)rN−1dr.

We then have

R1 =

∫
Bρ\Bρ/2

∫
Bρ\Bρ/2

|w(x)− w(y)|p

|x− y|N+sp
dx dy

≤ 2pδp

ρp

∫
Bρ\Bρ/2

∫
Bρ\Bρ/2

|x− y|p

|x− y|N+sp
dx dy

≤ 2pδpωNN

ρp

∫
Bρ\Bρ/2

∫ ρ+|y|

0

rp−sp−1drdy

≤ 2pδpωNN

ρp

∫
Bρ\Bρ/2

(ρ+ |y|)p−sp

p− sp
dy



8 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

=
2pδpω2

NN
2

(p− sp)ρp

∫ 2ρ

3
2ρ

rp+N−sp−1dr

=
2pδpω2

Nρ
N−spN2

(p− sp)(p+N − sp)

(
2p+N−sp −

(3

2

)p+N−sp)
.

R2 =

∫
Bρ\Bρ/2

∫
RN\Bρ

|w(x)− w(y)|p

|x− y|N+sp
dx dy

=
2pδp

ρp

∫
Bρ\Bρ/2

∫
RN\Bρ

|ρ− |y||p

|x− y|N+sp
dx dy

=
2pδpωNN

ρp

∫
Bρ\Bρ/2

∫ +∞

ρ−|y|

|ρ− |y||p

rsp+1
drdy

=
2pδpωNN

ρpsp

∫
Bρ\Bρ/2

|ρ− |y||p−spdy

=
2pδpω2

NN
2

ρpsp

∫ ρ
2

0

rp−sp(ρ− r)N−1dr

≤ δpρN−spω2
NN

2

21−spsp(p− sp+ 1)
.

R3 =

∫
Bρ/2

∫
Bρ\Bρ/2

|w(x)− w(y)|p

|x− y|N+sp
dx dy

=
2pδp

ρ2

∫
Bρ/2

∫
Bρ\Bρ/2

| − ρ
2 + |x||p

|x− y|N+sp
dx dy

=
2pδp

ρp

∫
Bρ\Bρ/2

∫
Bρ/2

| − ρ
2 + |x||p

|x− y|N+sp
dy dx

=
2pδpωNN

ρp

∫
Bρ\Bρ/2

∣∣∣−ρ
2

+ |x|
∣∣∣p ∫ |x|+ ρ

2

|x|− ρ2

1

rsp+1
dr dx

≤ 2pδpωNN

ρpsp

∫
Bρ\Bρ/2

∣∣∣−ρ
2

+ |x|
∣∣∣p−sp dx

=
2pδpω2

NN
2

ρpsp

∫ ρ
2

0

rp−sp
(
r +

ρ

2

)N−1

dr

≤ ρN−spδpω2
NN

2

21−spsp(p− sp+ 1)
.

R4 =

∫
Bρ/2

∫
RN\Bρ

|w(x)− w(y)|p

|x− y|N+sp
dx dy

= δp
∫
Bρ/2

∫
RN\Bρ

1

|x− y|N+sp
dx dy

= δpωNN

∫
Bρ/2

∫ ∞
ρ−|y|

r−sp−1drdy

= δpωNN

∫
Bρ/2

1

sp(ρ− |y|)sp
dy



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 9

=
δpω2

NN
2

sp

∫ ρ

ρ/2

rN−sp−1dr

=
δpω2

NN
2ρN−sp

sp(N − sp)

(
1− 1

2N−sp

)
≤ δpω2

NN
2ρN−sp

sp(N − sp)
.

Then, we have w ∈ Xp
s (Ω) and

Φ(w) ≤
2ω

sp
N +1

N + Lsp|Ω|sp/N

pω
sp
N −1

N

δpρN−spµ. (3.5)

Hence, it follows from (H2) that 0 < Φ(w) < r. From (H4), we have

Ψ(w) ≥
∫
Bρ/2

F (x,w) dx ≥ ess infx∈Ω F (x, δ)
ωNρ

N

2N
. (3.6)

By (3.2), the estimate Φ(u) ≤ r implies that |u|p
W s,p(RN )

≤ τp. From Lemma 2.1,

for every u ∈ Φ−1(−∞, r] we have

‖u‖Lp(Ω) ≤
(ps|Ω|sp/N

2ω
sp
N +1

N

)1/p

τ.

Hence, condition (H1), Hölder’s inequality, and the content of Remark 2.4 imply
that, for each u ∈ Φ−1(−∞, r],

Ψ(u) =

∫
Ω

F (x, u) ≤
∫

Ω

|α(x)||u(x)| dx+ q−1

∫
Ω

|β(x)||u(x)|q dx (3.7)

≤ ‖α‖∞|Ω|1/p
′
‖u‖Lp(Ω) + q−1‖β‖∞‖u‖qLq(Ω) (3.8)

≤ ‖α‖∞|Ω|1/p
′
(ps|Ω|sp/N

2ω
sp
N +1

N

)1/p

τ + q−1Cq/pq ‖β‖∞τ q.

In view of (3.5), (3.6), the above inequality, and (H3), we obtain

supu∈Φ−1(−∞,r] Ψ(u)

r
≤
‖α‖∞|Ω|1/p

′
(
ps|Ω|sp/N

2ω
sp
N

+1

N

)1/p

τ + q−1C
q/p
q ‖β‖∞τ q

r

<
ess infx∈Ω F (x, δ)ωNρ

N

2N

r
≤ Ψ(w)

Φ(w)
,

(3.9)

which means that
supΦ(u)≤r Ψ(u)

r < Ψ(v)
Φ(v) holds for some v̄ ∈ Xp

s (Ω). Hence, for each

λ ∈
(

Φ(w)
Ψ(w) ,

r
supΦ(u)≤r Ψ(u)

)
the functional Iλ admits at least one critical point uλ

with
0 < Φ(uλ) < r,

which in turn is a nontrivial solution of problem (1.1). �

Remark 3.2. Condition (H3) in Theorem 3.1 can be replaced by the less general
but more easily verifiable condition

‖α‖∞|Ω|1/p
′
(ps|Ω|sp/N

2ω
sp
N +1

N

)1/p

τ + q−1Cq/pq ‖β‖∞τ q <
ωNρ

N

2N
ρsp ess infx∈Ω F (x, δ).



10 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

As an illustration of Theorem 3.1, we have the following example.

Example 3.3. On the domain Ω = {(x1, x2) : x2
1 + x2

2 < 1} ⊂ R2, consider the
problem

(−∆)
1/4
2 u = λf(x, u) + sin(u), in Ω,

u = 0, on RN\Ω.
Here we have N = 2, p = 2, p′ = 2, and s = 1/4. For t ∈ R, let

f(t) =

{
et, t ≤ 1,

e, t > 1.

From the definition of f , we have

F (t) =

{
et − 1, t ≤ 1,

et− 1, t > 1.

By choosing α(x) = e, β(x) = 10−13, and q = 2, we see that the function f satisfies
condition (H1). Choosing δ = 1, ρ = 4, and τ = 280, simple calculations show that
the remaining conditions in Theorem 3.1 also hold. Hence, for every

λ ∈
(12(4π + 1)

e− 1
,

70(4π − 1)

πe+ 0.169× 10−2

)
,

the above problem admits at least one nontrivial weak solution.

Our second aim in this paper is to obtain a result on the existence of two distinct
solutions to problem (1.1). The following theorem is obtained by applying [5,
Theorem 3.2].

Theorem 3.4. Let f : Ω × R → R be a Carathéodory function satisfying (H1).
Moreover, assume that

(H5) (Ambrosetti-Rabinowitz Condition) there exist ν > 2p+1
2p−1p and R > 0 such

that

0 < νF (x, t) < tf(x, t) for all x ∈ Ω and |t| ≥ R.
Then, for each

λ ∈ Λr :=
(

0,
(2ω

sp
N +1

N − Lsp|Ω|sp/N )τp

2pω
sp
N +1

N (α‖∞|Ω|1/p′(ps|Ω|
sp/N

2ω
sp
N

+1

N

)1/pτ + q−1Cqq‖β‖∞τ q)

)
,

problem (1.1) admits at least two nontrivial solutions.

Proof. Let Φ and Ψ be the functionals defined in (2.2) and (2.3). Notice that they
satisfy all regularity assumptions required in [5, Theorem 3.2]). Arguing as in the
proof of Theorem 3.1, choosing

r :=
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

τp

with L as in (2.5), for each λ ∈ Λr we obtain

supu∈Φ−1(−∞,r] Ψ(u)

r
≤
‖α‖∞|Ω|1/p

′
(ps|Ω|

sp/N

2ω
sp
N

+1

N

)1/pτ + q−1Cqq‖β‖∞τ q

r
<

1

λ



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 11

(see (3.9)). Now, from condition (H5), a straight forward calculation shows that
there are positive constants m and C such that

F (x, t) ≥ m|t|ν − C for all x ∈ Ω and t ∈ R. (3.10)

Hence, for every λ ∈ Λr, u ∈ Xp
s (Ω) \ {0} and t > 1, we obtain

Iλ(tu(x)) = Φ(tu(x))− λ
∫

Ω

F (x, tu) dx

≤
2ω

sp
N +1

N + Lps|Ω|sp/N

2pω
sp
N +1

N

tp‖u‖p
Xps (Ω)

−mλtν
∫

Ω

|u|ν dx+ λC|Ω|.

Since ν > p, this condition guarantees that Iλ is unbounded from below. We recall
that Iλ is a Gâteaux differentiable functional whose Gâteaux derivative at the point
u ∈ Xp

s (Ω) is the functional I ′λ(u) ∈ (Xp
s (Ω))∗ given by

〈I ′λ(u), v〉 =

∫
RN

∫
RN

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

|x− y|N+sp
dx dy

− λ
∫

Ω

f(x, u)v dx−
∫

Ω

h(u)v dx,

for every v ∈ Xp
s (Ω).

To show that Iλ satisfies the (PS)-condition, let {un}n∈N ⊂ Xp
s (Ω) be a sequence

such that {Iλ(un)}n∈N is bounded and I ′λ(un)→ 0 in (Xp
s (Ω))∗ as n→ +∞. Then,

there exists a positive constant s0 such that

|Iλ(un)| ≤ s0 and ‖I ′λ(un)‖ ≤ s0quadfor all n ∈ N.

Using condition (H5), Lemma 2.1, (2.5), and the definition of I ′λ, we see that for
all n ∈ N, there exists D > 0 such that

νs0 + s0‖un‖Xps (Ω) ≥ νIλ(un)− 〈I ′λ(un), un〉

≥ ν

p
|un|pW s,p(RN )

− νL

p
‖un‖pLp(Ω) − |un|

p
W s,p(RN )

− L‖un‖pLp(Ω)

+ λ

∫
Ω

(f(x, un)un − νF (x, un)) dx

≥
(ν
p
− 1
)
|un|pW s,p(RN )

− L
(ν
p

+ 1
)
‖un‖pLp(Ω) −D

≥
(ν
p
− 1
)
|un|pW s,p(RN )

− L
(ν
p

+ 1
)ps|Ω|sp/N

2ω
sp
N +1

N

|un|pW s,p(RN )
−D

≥
((ν

p
− 1
)
− 2−p

(ν
p

+ 1
))
|un|pW s,p(RN )

−D.

Since ν > 2p+1
2p−1p, the equivalence in Remark 2.2 shows that the sequence {un}n∈N

is bounded.
Since Xp

s (Ω) is a reflexive Banach space, we have, up to taking a subsequence if
necessary,

un ⇀ u in Xp
s (Ω).

By the fact that I ′λ(un)→ 0 and un ⇀ u in Xp
s (Ω), we obtain

(I ′λ(un)− I ′λ(u))(un − u)→ 0.



12 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

Furthermore, ∫
Ω

(f(x, un)− f(x, u))(un − u) dx→ 0 as n→ +∞,∫
Ω

(h(un)− h(u))(un − u) dx→ 0 as n→ +∞.

An easy computation shows that

〈I ′λ(un)− I ′λ(u), un − u〉

=

∫
RN

∫
RN

|(un − u)(x)− (un − u)(y)|p−2((un − u)(x)− (un − u)(y))2

|x− y|N−sp
dx dy

−
∫

Ω

(h(un)− h(u))(un − u) dx− λ
∫

Ω

(f(x, un)− f(x, u))(un − u) dx

≥ k3‖un − u‖pXps (Ω)
−
∫

Ω

(h(un)− h(u))(un − u) dx

− λ
∫

Ω

(f(x, un)− f(x, u))(un − u) dx,

where k3 is a positive constant. This implies that the sequence {un}n∈N converges
strongly to u in Xp

s (Ω). Therefore, Iλ satisfies the (PS)-condition and so all con-
ditions of [5, Theorem 3.2]) are satisfied. Hence, for each λ ∈ Λr, the function
Iλ admits at least two distinct critical points that are solutions of the problem
(1.1). �

In our final result, we discuss the existence of at least three solutions to problem
(1.1).

Theorem 3.5. Let p > q and f : Ω×R→ R be a Carathéodory function satisfying
(H1) and let (H3), (H4) hold. In addition, assume that there exist three positive
constants τ , ρ, and δ, such that

(H6)
ω2
Nδ

pρN−spN2

2N−sp−1sp(N+p−sp) > τp.

Then, if (3.1) holds, problem (1.1) admits at least three distinct weak solutions.

Proof. Here we will apply [7, Theorem 3.6]. We consider the functionals Φ and
Ψ defined in (2.2) and (2.3). Once again, they satisfy the regularity assumptions
needed in [7, Theorem 3.6]. Now, we argue as in the proof of Theorem 3.1 with
w(k) defined in (3.4),

r :=
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

τp,

and 0 < L <
21−pω

sp
N

+1

N

ps|Ω|ps/N . Given that lower bounds for R1, R3, and R4 are greater

than zero, we have

Φ(w) ≥
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

(0 + 2R2 + 2× 0 + 2× 0).

In view of (H6), we have Φ(w) > r > 0. Therefore, from (H3), inequality (3.9)
holds, and so

supΦ(u)≤r Ψ(u)

r
<

Ψ(v)

Φ(v)



EJDE-2023/46 FRACTIONAL ELLIPTIC PROBLEMS 13

holds for some v̄ ∈ Xp
s (Ω).

Now, we prove that for each λ ∈ Λw, the functional Iλ is coercive. Using
condition (H1), Hölder’s inequality, and Remark 2.4, we easily obtain that for all
u ∈ Xp

s (Ω),

Iλ(u) ≥
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

|u|p
W s,p(RN )

− λ
∫

Ω

F (x, u) dx

≥
2ω

sp
N +1

N − Lps|Ω|sp/N

2pω
sp
N +1

N

|u|p
W s,p(RN )

− ‖α‖∞|Ω|1/p
′
‖u‖Lp(Ω) − q−1‖β‖∞‖u‖qLq(Ω)

by (3.7).
Since

L <
2ω

sp
N +1

N

ps|Ω|ps/N

and p > q, we see that Iλ → +∞ as ‖u‖ → +∞, so the functional Iλ is coercive.
Thus, for each λ ∈ Λw, [7, Theorem 3.6] implies that the functional Iλ admits at
least three critical points in Xp

s (Ω) that are solutions of the problem (1.1). �

We conclude this article with an example of Theorem 3.5.

Example 3.6. Let Ω = {(x1, x2) : x2
1 + x2

2 < 1} ⊂ R2, and consider the problem

(−∆)
1/4
2 u = λf(x, u) + tan(u), in Ω,

u = 0, on RN\Ω.

We have N = 2, p = 2, p′ = 2, and s = 1/4. For t ∈ R, let

f(t) =

{
t/2, t ≤ 1,

1/2, t > 1.

From f , we have

F (t) =

{
t2/4, t ≤ 1,
t
2 −

1
4 , t > 1.

By choosing α(x) = 1/2, β(x) = 10−10, and q = 3/2, we see that condition (H1)
holds. If we take δ = 1, ρ = 90, and τ = 64, simple calculations show that all the
conditions in Theorem 3.5 are satisfied. Hence, for every

λ ∈
(

12(4π + 1),
45(4π − 1)

π + .19× 10−2

)
,

the above problem admits at least three nontrivial weak solutions.

Remark 3.7. It would be possible to replace the requirement that α ∈ L∞(Ω)
in condition (H1) by the less restrictive condition that this function belong to the

space L
p
p−1 (Ω) and modifying our calculations. The conclusions we have obtained

would remain true.



14 F. GHAREHGAZLOUEI, J. R. GRAEF, S. HEIDARKHANI, L. KONG EJDE-2023/46

Conclusions

We considered a nonlinear elliptic fractional Dirichlet boundary value problem
involving a p-Laplacian and containing a positive parameter. Our interest was in
obtaining the existence of at least one, two, and three solutions to the problem.
In doing this we estimated an interval for the parameter λ in which problem (1.1)
possesses at least one nontrivial weak solution provided the nonlinear term satisfied
a subcritical growth condition.

To obtain the existence of two solutions, we used a result of Bonanno [5] and
required that the (P.S.)[r] condition or the (AR)-condition holds. In order to obtain
the existence of three solutions, we asked that the nonlinear term has subcritical
growth and used variational methods and a critical point theorem of Bonanno and
Marano [7].

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[19] R. Servadei; Infinitely many solutions for fractional Laplace equations with subcritical non-

linearity, In: Recent Trends in Nonlinear Partial Differential Equations II. Stationary Prob-

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York, 1985.

Fariba Gharehgazlouei
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,

Iran

Email address: f.gharehgazloo@yahoo.com

John R. Graef

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga,
TN 37403, USA

Email address: John-Graef@utc.edu

Shapour Heidarkhani
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149,

Iran
Email address: s.heidarkhani@razi.ac.ir

Lingju Kong

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga,
TN 37403, USA

Email address: Lingju-Kong@utc.edu


	1. Introduction
	2. Preliminaries
	3. Main results
	Conclusions
	References