Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 34, pp. 1–17.

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

DOI: 10.58997/ejde.2024.34

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR

FRACTIONAL DIFFERENTIAL EQUATIONS WITH

p-LAPLACIAN AT RESONANCE

JOSE VANTERLER DA C. SOUSA, MARIANE PIGOSSI, NEMAT NYAMORADI

Abstract. In this article, we investigate the existence and multiplicity of

solutions for a fractional differential equations with p-Laplacian equation at

resonance in the ψ-fractional space Hα,β;ψp . In addition, we show that the

energy functional satisfies the Palais-Smale condition.

1. Introduction and motivation

Since the first results on fractional calculus were published, there has been a
growing number of researchers who use fractional differential equations to better
describe phenomena in mechanics, chemistry, medicine, etc. [20, 22, 30, 36, 43,
44, 45, 46]. Also, the theory of fractional differential equations gained space and
strength with the consolidation of fractional calculus. Fractional differential equa-
tions have been valuable tools in fields, such as viscoelasticity, engineering, physics
and economics, see [1, 3, 34, 38]. In addition, researchers have studied properties
such as existence, uniqueness, stability, controllability for different types of frac-
tional differential equations; see [13, 30, 52] and the references therein. On the
other hand, we can highlight important works on fractional differential equations
with p-Laplacian at resonance; see [23, 27, 31, 50]. Many of these works are done
through fractional derivatives of Caputo and Riemann-Liouville type.

In 1999 Drabek and Robinson [15] considered the boundary value problem

−∆pu− λ|u|p−2u+ f(x, u) = 0, in Ω

u|∂∆ = 0,

where ∆pu := ∇
(
|∇u|p−2∇u

)
, Ω is a bounded domain in Rn, p > 1, and f :

Ω × R → R is a bounded Caratheodory function. In 2010 Chang and Li [11]
investigated the existence and multiplicity of nontrivial solutions for the semilinear
elliptic Dirichlet boundary value problem

−∆pu = f(x, u), in Ω

u = 0 on ∂∆,

2020 Mathematics Subject Classification. 26A33, 34A08, 35R11, 35J50.
Key words and phrases. Fractional differential equations; p-Laplacian equation; resonance;

existence and multiplicity.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted June 14, 2021. Published April 26, 2024.

1



2 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

where Ω ⊂ Rn (n ≥ 1) is an open bounded domain with smooth boundary ∂Ω
and f ∈ C(Ω× R,R). Other interesting works on the existence and multiplicity of
solutions involving p-Laplacian at resonance, without resonance, can be found in
[8, 9, 14, 16, 17, 33, 47, 51].

Motivated by fractional operators (integrals and derivatives), problems of differ-
ential equations with p-Laplacian at resonance have gained prominence. In princi-
ple, the investigated results generalize the integer cases, and have possible particular
cases other than fractional derivatives. For more readings, we refer the reader to
[2, 5, 24, 25, 26, 28, 37, 49].

Jiang [27] studied the solvability of fractional differential equation with p-Laplacian
at resonance,

Dβ
o+

(
φp(D

α
0+u)

)
(t) + f(t, u(t), Dα−1

0+ u(t), Dα
0+u(t)) = 0

u(0) = Dα
0+(0) = 0,

u(t) =

∫ 1

0

u(t)h(t)dt = 1,

where 0 ≤ β ≤ 1, 1 ≤ α ≤ 2,
∫ 1

0
h(t)tα−1dt = 1, φp(s) = |s|p−2, p > 1, and Dβ

0+(·),
Dα

0+(·) are the Riemann-Liouville fractional derivatives.
In 2017, Hu and Zhang [23] investigated the existence of positive solutions of the

fractional differential equation with periodic boundary value,

Dα
o+u(t) = f(t, u(t)) 0 < t < 1

u(0) = u(1), u′(0) = 1, u′′(0) = u′′(1),

where 2 < α < 3, Dα
o+(·) is the Caputo fractional derivative, and f : [0, 1]×R → R.

p-Laplace equations involving double phase have been studied during the previ-
ous years and their theory is already well developed. They gained prominence from
results involving fractional operators. Since the pioneering work by Landesman and
Lazer [32], several works have been devoted to resonant problems for ordinary and
partial differential equations. However, some challenging and interesting problems
still remain open. Resonance problems for divergence operators have been of inter-
est since the 1970. For the common Laplacian and the p-Laplacian, there are several
classical papers and some recent papers exploring resonant problems in Rn; see [3].
On the other hand, the existence and multiplicity of solutions for boundary value
problems of two non-singular points in resonance have been extensively addressed
in the literature, see [4, 10, 19, 1]. Although there are some works in the literature
involving fractional operators, they are still numerous open questions because of the
difficulty of working with fractional operators. Research on p-Laplacian singulars
at resonance has proceeded very slowly. One of the motivations of this paper is to
provide new results and possible tools for future work.

Motivated by works and open questions above, we consider the fractional bound-
ary value problem

HDα,β;ψ
T

(
|HDα,β;ψ

0+ ξ(x)|p−2HDα,β;ψ
0+ ξ(x)

)
= f(x, ξ)

I
β(β−1);ψ
0+ ξ(0) = I

β(β−1);ψ
T ξ(T ) = 0,

(1.1)

where Ω = [0, T ] is a bounded domain in R, HDα,β,ψ
0+ (·) and HDα,β,ψ

T (·) are the

ψ-Hilfer fractional derivatives of order α ( 1p < α ≤ 1) and type 0 ≤ β ≤ 1,



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 3

f ∈ C(Ω × R,R), and 1 < p < ∞. Besides that, I
β(β−1)
0+ (·) and I

β(β−1)
T (·) are the

ψ-Riemann-Liouville fractional integrals of order β(β − 1).
Equation (1.1) is called a resonant problem at the first eigenvalue if

lim
|ξ|→∞

f(x, ξ)

|ξ|p−2ξ
= λ1, uniformly for x ∈ Ω. (1.2)

Jiu and Su [29] obtained the existence of multiple solutions of (1.1) with (1.2) and
the non-quadratic condition

lim
|ξ|→∞

(ξf(x, ξ)− pF(x, ξ)) = −∞, uniformly for x ∈ Ω. (1.3)

With other versions of the non-quadratic conditions, several papers have studied
the case

λ1 ≤ a(x) = lim inf
|ξ|→∞

f(x, ξ)

|ξ|p−2ξ
≤ lim sup|ξ|→∞

f(x, ξ)

|ξ|p−2ξ
= b(x) < λ2, (1.4)

uniformly for x ∈ Ω. Here we assume that

max
|s|⩽R

f(x, s) ∈ Lp(Ω), ∀R > 0. (1.5)

We also assume that some uniformity holds in (1.4): for each ε > 0 there exists
η(ε) > 0 such that

λ1 → ε ⩽
f(x, s)

|s|p−2s
, ∀|s| ⩾ η(ε) a.e. in [0, T ], (1.6)

and for each ε > 0 there exists η(ε) > 0 such that

λ1 → f(x, s)

|s|p−2s
⩽ λ2 + ε, ∀|s| ⩾ η(ε) a.e. in [0, T ].

Remark 1.1. Note that (1.5) and (1.6) imply the growth condition

|f(x, s)| ⩽ a|s|p−1 + b(x), ∀s ∈ R, a.e. in [0, T ] (1.7)

where a > 0 and b(·) ∈ Lp
′
.

Remark 1.2. Inequalities (1.5) and (1.6) also imply that for each ε > 0 there

exists bε ∈ Lp
′
such that

|s|p(λ1 − ε)− bε(x) ≤ sf(x, s) ≤ |s|p(λ2 + ε) + bε(x), ∀s ∈ R a.e. in [0, T ]. (1.8)

Let λ1(a) be the first eigenvalues of the equation

HDα,β,ψ
T

(
|HDα,β,ψ

0+ ξ(x)|p−2 HDα,β,ψ
0+ ξ(x)

)
− a(x)|ξ|p−2ξ = λ|ξ|p−2ξ

with Dirichlet boundary condition. It is well know that λ1(a) is simple and isolated.
Then the second eigenvalue is well defined as

λ2(a) = inf
{
λ > λ1(a) : λ is eigenvalue of HDα,β,ψ

T

(
|HDα,β,ψ

0+ ξ(x)|p−2 HDα,β,ψ
0+ ξ(x)

)
− a(x) ∈ Hα,β,ψp

}
.

By the monotonicity of λ1(a) and λ2(b), condition (1.4) implies

λ1(a) ⩽ 0 ⩽ λ2(b) .



4 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

For the first eigenfunctions φ1(a) > 0, if we let V = span{φ1(a)}, then

V ⊥ =
{
ξ ∈ Hα,β,ψp :

∫ T

0

(φ(a))p−1ξdx = 0
}
.

Also we have

Hα,β,ψp = V ⊕ V ⊥. (1.9)

From [7], we know that there exists λ(a) ∈ (λ1(a), λ2(a)] such that∫ T

0

(∣∣∣HDα,β,ψ
0+ ξ(x)

∣∣∣p − a(x)|ξ|p
)
dx ⩾ λ(a)

∫ T

0

|ξ|pdx (1.10)

for any ξ ∈ V ⊥. Similarly, we can define λ1(b), φ1(b) and λ(b).
Before presenting our the main results, we list some assumptions.

(A1) For 1 < p <∞ we assume that

f ∈ C([0, T ]× R,R) (1.11)

and satisfies the growth condition

|f(x, t)| ⩽ c(1 + |t|q−1) for all x ∈ [0, T ], t ∈ R

for some c > 0 and q ∈ [1, p∗), where p∗α = p
1−αp if p < 1, and p∗α = ∞ if

1 ⩽ p.
(A2) There exists a constant M > 0 such that

a(x) ⩽
f(x, ξ)

|ξ|p−2ξ
⩽ b(x), for |ξ| ⩾M, x ∈ Ω,

where a and b are continuous functions.
(A3) lim|ξ|→∞

∫ T
0

(
F(x, ξ)− 1

pb(x)|ξ|
p
)
dx = −∞.

(A4) lim|ξ|→∞
(
ξf(x, ξ)− pF(x, ξ)

)
= −∞, where F(x, ξ) =

∫ ξ
0
f(x, t)dt.

In this article, we investigate the existence and multiplicity of solutions for (1.1)
under the conditions (A1)–(A4) in two steps. In the first step, we obtain the
following result.

Theorem 1.3. Assume that (A1) and (A2) hold. If one of the following condition
is satisfied

(a) λ1(b) > 0,
(b) λ1(b) ⩾ 0 and (A3) holds,

(c) λ1(a) < 0 < λ(b),

(d) λ1(a) ⩽ 0 ⩽ λ(b) and (A4) holds,

then (1.1) has at least one solution.

On the second step we use the assumption

(A5) f(x, 0) = 0 and there is a function ℓ(x) such that, lim|ξ|→∞
pF(x,ξ)
∥ξ∥p ≤ ℓ(x)

with λ1(ℓ) > 0, x ∈ [0, T ],

and obtain the following result.

Theorem 1.4. Assume that (c) or (d) of Theorem 1.3, and (A5) hold. Then (1.1)
has a nontrivial solution.



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 5

Remark 1.5. Obviously, (A5) is weaker than the condition

lim
|ξ|→0

pF(x, ξ)

∥ξ∥p
= ℓ(x) ⩽ λ1, x ∈ [0, T ]

which implies that 0 is a local minimum of I.

Note that, taking α = 1 and ψ(t) = t in (1.1), we have the integer case problem,

−
(
|ξ′|p−2 ξ′

)′
= f(x, ξ)

ξ(0) = ξ(T ) = 0.

As 0 < α ≤ 1 and with the freedom of chosing ψ(t), we have a wide class of
possible cases for probem (1.1).

The rest of this article is organized as follows: In Section 2, we present some
preliminary results of fractional calculus, i.e., the Riemann-Liouville fractional in-
tegral with respect to another function and the ψ-Hilfer fractional derivative, and
the ψ-fractional space Hα,β,ψp with its respective norm. In fact, we verify that the

space Hα,β,ψp is uniformly convex. In section 3, we prove the main results of this
article, i.e., the existence and multiplicity of solutions for (1.1).

2. Mathematical background and auxiliary results

Let X be a real Banach space and Ψ ∈ C1(X,R) satisfying the Palais-Smale
condition. Let A = {u ∈ X : Ψ′(u) = 0} be the critical set of Ψ. Let u ∈ A be
an isolated critical point with Ψ(u) = b ∈ R, and U be an isolated neighborhood
of u. The group C∗(Ψ, u) = H∗(Ψ

c ∩ U,Ψc ∩ U |u), ∗ = 0, 1, . . . is called the *-th
critical group of Ψ at u, where Ψc = {u ∈ X : Ψ(u) ≤ c}, H∗(·, ·) are the singular
relative homology groups with a coefficients group G. By the excision property
of the homology groups, the critical groups are independent of the choices of U ,
then they are well defined. In particular, if u, v are the critical points of Ψ and
Cq(Ψ, u) ̸= Cq(Ψ, v) for some q then u ̸= v, see [12].

Let (a, b) (−∞ ≤ a < b ≤ ∞) be a finite or infinite interval of the real line R
and α > 0. Also let ψ (·) be an increasing and positive monotone function on [a, b],
having a continuous derivative ψ′(x) ̸= 0 on (a, b). The fractional integrals of a
function f with respect to another function ψ on [a, b] is defined in [43],

Iα;ψa+ ξ(x) =
1

Γ(α)

∫ x

a

ψ′(t)
(
ψ(x)− ψ(t)

)α−1
ξ(t) dt. (2.1)

Analogously, we define Iα;ψb− (·).
Let n − 1 < α < n, with n ∈ N, I = [a, b] be the interval such that −∞ ≤ a <

b ≤ ∞, and let two functions f, ψ ∈ Cn([a, b],R) be such that ψ is increasing and

ψ′(x) ̸= 0, for all x ∈ I. Them the ψ-Hilfer fractional derivatives HDα,β;ψ
a+ (·) of

order α and type 0 ≤ β ≤ 1 are defined by, [43],

HDα,β;ψ
a+ ξ(x) = I

β(n−α);ψ
a+

( 1

ψ′(x)

d

dx

)n
I
(1−β)(n−α);ψ
a+ ξ(x). (2.2)

Analogously, we define HDα,β;ψ
b− (·).

Next, we present the integration by parts of the ψ-Riemann-Liouville fractional
integral and ψ-Hilfer fractional derivative.



6 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

As shown in [40], the equality∫ b

a

(
Iα;ψa+ ξ(t)

)
θ(t) dt =

∫ b

a

ξ(t)ψ′(t)Iα;ψb−
( θ(t)
ψ′(t)

)
dt (2.3)

holds if θ ∈ Lp, ξ ∈ Lq, 1
p + 1

q ≤ 1 + α, p ≥ 1, q ≥ 1, with p ̸= 1, q ̸= 1 in the case
1
p +

1
q = q + α.

Theorem 2.1 ([40]). Let ψ(·) be an increasing and positive monotone function
on [a, b], having a continuous derivative ψ′(·) ̸= 0 on (a, b). If 0 < α ≤ 1 and
0 ≤ β ≤ 1, then∫ b

a

(H

Dα,β;ψ
a+ ξ(t)

)
θ(t) dt =

∫ b

a

ξ(t)ψ′(t)HDα,β;ψ
b−

(
θ(t)

ψ′(t)

)
dt (2.4)

for ξ ∈ AC1 and θ ∈ C1 satisfying the boundary conditions ξ(a) = 0 = ξ(b).

Definition 2.2 ([40]). Let 0 < α ≤ 1, 0 ≤ β ≤ 1 and 1 < p < ∞. Let ξ be a
weight, [0, T ] ⊂ R be open, with ξ ̸= 0 a.e. in [0, T ]. The ψ-fractional derivative
weight space Hα,β;ψp := Hα,β;ψp ([0, T ],R) is defined as the closure of C∞

0 ([0, T ],R),
and is given by

Hα,β;ψp =
{
ξ ∈ Lp ([0, T ],R) : HDα,β;ψ

0+ ξ ∈ Lp ([0, T ],R) ,

I
β(β−1)
0+ ξ(0) = I

β(β−1)
T ξ(T ) = 0

}
= C∞

0 ([0, T ],R)

(2.5)

with the norm
∥ξ∥Hα,β;ψp

=
(
∥ξ∥pLp + ∥HDα,β;ψ

0+ ξ∥pLp
)1/p

, (2.6)

where HDα,β;ψ
0+ (·) is the ψ-Hilfer fractional derivative with 0 < α ≤ 1 and 0 ≤ β ≤ 1.

Choosing p = 2, in (2.5), we have the ψ-fractional derivative weight space Hα,β;ψp

defined on C∞
0 ([0, T ],R) with respect to the norm

∥ξ∥Hα,β;ψ2
=

(∫ T

0

|ξ(t)|2 dt+
∫ T

0

|HDα,β;ψ
0+ ξ(t)|2 dt

)1/2

.

The space Hα,β;ψ2 is a Hilbert space with the norm

∥ξ∥Hα,β;ψ2
=

(∫ T

0

|HDα,β;ψ
0+ ξ(t)|2 dt

)1/2

with 0 < α ≤ 1 and 0 ≤ β ≤ 1.

Lemma 2.3 ([40]). Let 0 < α ≤ 1, 0 ≤ β ≤ 1, and 1 ≤ p < ∞. For each
ξ ∈ Lp([0, T ],R), we have

∥Iα,ψ0+ ∥Lp[0,T ] ≤
(ψ(T )− ψ(0))α

Γ(α+ 1)
∥ξ∥Lp[0,T ]

for all t ∈ [0, T ].

Proposition 2.4 ([40]). Let 0 < α ≤ 1, 0 ≤ β ≤ 1, and 1 < α < ∞. For all
ξ ∈ Hα,β;ψp , if 1− α ≥ 1

p or α > 1
p , we have

∥ξ∥Lp ≤ (ψ(T )− ψ(0))α

Γ(α+ 1)
∥HDα,β;ψ

0+ ξ∥Lp . (2.7)



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 7

Moreover, if α > 1
p and 1

p +
1
q = 1, then

∥ξ∥∞ ≤ (ψ(T )− ψ(0))α−
1
p

Γ(α)((α− 1)q + 1)1/q
∥HDα,β;ψ

0+ ξ∥Lp (2.8)

where ∥ξ∥∞ = supt∈[0,T ] |ξ(t)|.
From (2.8), we also have, [40],

∥ξ∥∞ ≤ (ψ(T )− ψ(0))α−
1
p

Γ(α)((α− 1)q + 1)1/q
∥HDα,β;ψ

0+ ξ∥Hα,β;ψp
,

that is Hα,β;ψp is continuously injected into C([0, T ]) for α > 1/p.

According to (2.7) we can consider Hα,β;ψp with respect to the equivalent norm

∥ξ∥ = ∥HDα,β;ψ
0+ ξ∥Lp .

Under condition (1.11), it is well know that the weak solutions of (1.1) correspond
to the critical points of the functional I : Hα,β;ψp → R defined by

I(ξ) =
1

p

∫ T

0

∣∣∣HDα,β,ψ
0+ ξ(x)

∣∣∣p dx−
∫ p

0

F(x, ξ)dx, (2.9)

where F(x, ξ) =
∫ ξ
0
f(x, t)dt. The next result on convexity is important for the

main results of this paper.

Theorem 2.5. The space (Hα,β,ψp , ∥ · ∥Hα,β,ψp
) is uniformly convex.

Proof. First, let p ∈ [2,∞). For each z, w ∈ R, it holds

|z + w

2
|p + |z − w

2
|p ⩽ 1

2
(|z|p + |w|p).

Let ξ, ν ∈ Hα,β,ψp satisfy ∥ξ∥Hα,β,ψp
= ∥ν∥Hα,β,ψp

= 1, and ∥ξ−ν∥Hα,β,ψp
⩾ ε, ε ∈ (0, 2].

We have

∥ξ + ν

2
∥p
Hα,β,ψp

+ ∥ξ − ν

2
∥p
Hα,β,ψp

=

∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x) +H Dα,β,ψ

0+ ν(x)p

2

∣∣)dx
+

∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x) +H Dα,β,ψ

0+ ν(x)p

2

∣∣)dx
⩽

∫ T

0

1

2

(
|HDα,β,ψ

0+ ξ(x)|p + |HDα,β,ψ
0+ ν(x)|p

)
dx

=
1

2

(
∥ξ∥Hα,β,ψp

+ ∥ν∥Hα,β,ψp

)
= 1

which yields

∥ξ + ν

2
∥p
Hα,β,ψp

⩽ 1−
(ε
2

)p
. (2.10)

On the other hand, if p ∈ (1, 2) then for each z, w ∈ R it holds

|z + w

2
|p

′
+ |z − w

2
|p

′
⩽

(1
2
(|z|p + |w|p)

) 1
p−1

. (2.11)

Straight forward computations show that if v ∈ Hα,β,ψp , then ∥ |HDα,β,ψ
0+ ν|p∥Hα,β,ψp−1

=

∥ν∥p
′

Hα,β,ψp
.



8 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

Let ν1, ν2 ∈ Hα,β,ψp then |HDα,β,ψ
0+ ν1|p

′
, |HDα,β,ψ

0+ ν2|p
′ ∈ Lp−1([0, T ]) with 0 <

p− 1 < 1 and from

∥ |HDα,β,ψ
0+ ν1|p

′
+ |HDα,β,ψ

0+ ν2|p
′
∥Hα,β,ψp−1

⩾ ∥ |HDα,β,ψ
0+ ν1|p

′
∥Hα,β,ψp−1

+ ∥ |HDα,β,ψ
0+ ν2|p

′
∥Hα,β,ψp−1

,
(2.12)

we have

∥ν1 + ν2
2

∥p
Hα,β,ψp

+ ∥ν1 − ν2
2

∥p
Hα,β,ψp

= ∥
∣∣HDα,β,ψ

0+

(ν1 + ν2
2

)∣∣p′∥Hα,β,ψp−1
+ ∥ big|HDα,β,ψ

0+

(ν1 − ν2
2

)∣∣p′∥Hα,β,ψp−1

⩽ ∥
∣∣HDα,β,ψ

0+

(ν1 + ν2
2

)∣∣p′ + ∣∣HDα,β,ψ
0+

(ν1 − ν2
2

)∣∣p′∥Hα,β,ψp−1

=
[ ∫ T

0

(∣∣HDα,β,ψ
0+ ν1 +

H Dα,β,ψ
0+ ν2

2

∣∣p′ + ∣∣HDα,β,ψ
0+ ν1 −H Dα,β,ψ

0+ ν2

2

∣∣p′)p−1
dx

] 1
p−1

⩽
[1
2

∫ T

0

(∣∣HDα,β,ψ
0+ ν1

∣∣p + ∣∣HDα,β,ψ
0+ ν2

∣∣p)dx] 1
p−1

=
(1
2
∥ν1∥pHα,β,ψp

+
1

2
∥ν2∥pHα,β,ψp

) 1
p−1 .

For ξ, ν ∈ Hα,β,ψp with ∥ξ∥Hα,β,ψp
= ∥ν∥Hα,β,ψp

= 1 and ∥ξ − ν∥Hα,β,ψp
⩾ ε ∈ (0, 2], we

obtain

∥ξ + ν

2
∥p

′
⩽ 1−

(ε
2

)p′
. (2.13)

From (2.10) and (2.13) in either case there exists δ(ε) > 0 such that ∥ξ+ν∥Hα,β,ψp
⩽

2(1− δ(ε)). □

Definition 2.6. A functional I is said to satisfy the Palais-Smale condition at the
level c ∈ R ((PS)c for short) if every sequence {ξn} ⊂ Hα,β,ψp with

I(ξn) → c, (∥ξn∥+ 1)I(ξn) → 0, as n→ ∞ (2.14)

possesses a convergent subsequence. Furthermore, I satisfies the (PS) if I satisfies
(PS)c at each c ∈ R.

Lemma 2.7. The sequence

gn =
f(x, nwn)

np−1

is bounded in Lp
′
([0, T ]) and consequently a subsequence gn converges weakly to a

function g in Lp
′
([0, T ]).

The proof of the above lemma is an immediate consequence of (1.7).

Lemma 2.8 ([6]). The g obtained in Lemma 2.7 satisfies g = 0 almost everywhere
in [0, T ]\A, where A = {x ∈ [0, T ] : z(x) ̸= 0}.

Lemma 2.9 ([6]). Set

m(x) =

{
g(x)

|z(x)|p−2z(x) , on A,

β, on [0, T ]\A,
where β is a fixed number with λ1 < β < λ2. Then

λ1 ⩽ m(x) < λ2 a.e. in [0, T ]. (2.15)



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 9

Lemma 2.10 ([47]). Let E be a vector space such that for subspace X and Y ,
E = X ⊕ Y . If Y is a finite dimensional and Z is a subspace of E such that
X ∩ Z = {0} and dimZ = dimY , then E = X ⊕ Z.

3. Palais-Smale condition

Lemma 3.1. Under the assumptions of Theorem 1.3, the functional I satisfies the
(PS) condition.

Proof. We consider the following 4 cases.

Case 1: Assume (A1), (A2) and (a). We show that I is coercive on Hα,β,ψp . Since

λ1(b) > 0 and b ∈ C([0, T ],R) (see inequality (1.10)), we have∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx
=

∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ|p
)
dx+

∫ T

0

b(x)|ξ(x)|p dx

⩽
∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ|p
)
dx+ c

∫ T

0

|ξ(x)|pdx

⩽
∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ(x)|p
)
dx

+
c

λ1(b)

∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ(x)|p
)
dx

=
(
1 +

c

λ1(b)

) ∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx− 2 +

∫ T

0

b(x)|ξ(x)|pdx

⩽ c̃

∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ|p
)
dx.

Then there exists a constant δ > 0 such that∫ T

0

(∣∣HDα,β,ψ
0+ ξ(x)

∣∣p − b(x)|ξ|p
)
dx ⩾ δ

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx (3.1)

with ξ ∈ Hα,β,ψp . Using conditions (A1) and (A2), it follows that

F(x, ξ) ⩽
1

p
b(x)|ξ(x)|p + c. (3.2)

Then using the inequalities (3.1) and (3.2) yields

I(ξ) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx−
∫ T

0

F(x, ξ)dx ⩾
δ

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx− c.

Then, we have I(ξ) → ∞ as ∥ξ∥ → ∞.

Case 2: Assume (A1), (A2), and (b). By contradiction we show that I is coercive
on Hα,β,ψp . Consider the sequence {ξn} ⊂ Hα,β,ψp (without loss of generality assume
it is the whole sequence) and a constant c0 such that

I(ξn) ⩽ c0, as ∥ξn∥ → ∞. (3.3)



10 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

Also, let νn = ξn/∥ξn∥. Then there exists a subsequence {νn} and v ∈ Hα,β,ψp

(without loss of generality assume it is the whole sequence), such that

νn ⇀ v weakly in Hα,β,ψp

νn → v strongly in Lp

νn → v for a.e. x ∈ [0, T ].

(3.4)

Using inequalities (3.2) and (3.3), and dividing by ∥ξn∥p, we obtain

I(ξn) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξn(x)

∣∣pdx−
∫ T

0

F(x, ξn)dx

⩽
1

p

∫ T

0

(∣∣HDα,β,ψ
0+ ξn(x)

∣∣p − b(x)|ξn(x)|p
)
dx+ c

which implies that

c0
∥ξn∥p

⩾
1

p

∫ T

0

(∣∣∣HDα,β,ψ
0+ νn(x)

∣∣∣p − b(x)|νn(x)|p
)
dx+

c

∥ξn∥p
. (3.5)

From this inequality, it follows that

lim sup
n→∞

∫ T

0

∣∣HDα,β,ψ
0+ νn(x)

∣∣pdx ⩽
∫ T

0

b(x)|ν(x)|pdx as n→ ∞. (3.6)

Moreover, since λ1(b) ⩾ 0, from the lower semi-continuity of the norm, we obtain∫ T

0

b(x)|ν(x)|pdx ⩽
∫ T

0

∣∣∣HDα,β,ψ
0+ ν(x)

∣∣∣p dx ⩽ lim inf
n→∞

∫ T

0

∣∣HDα,β,ψ
0+ νn(x)

∣∣pdx,
this together with (3.6) gives ∥νn∥ → ∥ν∥, as n → ∞. Since Hα,β,ψp is uniformly

convex (see Theorem 2.5), we have νn → v in Hα,β,ψp , as n→ ∞ with ∥ν∥ = 1 and∫ T

0

b(x)|ν(x)|pdx =

∫ T

0

∣∣HDα,β,ψ
0+ ν(x)

∣∣pdx.
Without loss of generally, we assume that λ1(b) = 0. Tthen we can take

v = ±φ1(b) which implies that |ξn(x)| → ∞, almost everywhere in [0, T ]. Us-
ing condition (A3) it follows that

lim
n→∞

∫ T

0

(
F(x, ξn)−

1

p
b(x)∥ξn∥p

)
dx = −∞.

Then

I(ξn) =
1

p

∫ T

0

∣∣∣HDα,β,ψ
0+ ξn(x)

∣∣∣p dx−
∫ T

0

F (x, ξn)dx

⩾
∫ T

0

(
F(x, ξn)−

1

p
b(x)|ξn(x)|p

)
dx→ ∞

as n→ ∞. This contradicts (3.3).

Case 3: Assume (A1), (A2), and (c). We show that I is coercive onHα,β,ψp . Assume

that {ξn} ⊂ Hα,β,ψp (without loss of generality assume it is the whole sequence) and
satisfies (2.14), by (A1) it suffices to show that {ξn} is bounded. We prove this by
contradiction. Assume that ∥ξn∥ → ∞ as n → ∞. Let zn = ξn/∥ξn∥. Then there



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 11

exists a subsequence {zn} and z ∈ Hα,β,ψp (without loss of generality assume it is
the whole sequence) such that

zn ⇀ z weakly in Hα,β,ψ0+

zn → z strongly in Lp

zn → z for a.e. x ∈ [0, T ].

Let gn(x) = f(x, ξn)/∥ξn∥p−1, then gn is bounded in Lp with 1
p + 1

p′ = 1. Fur-

thermore, for a subsequence of {gn} without loss of generality assume it is whole
sequence such that

gn ⇀ g weakly in Lp
′
. (3.7)

Next, we discuss 4 claims.

Claim 1: (See Lemma 2.8) g = 0 almost everywhere in [0, T ]\A, where A = {x ∈
[0, T ] : z(x) ̸= 0}.
Claim 2: (See Lemma 2.9) Set

m(x) =

{
g(x)

|z(x)|p−2z(x) , on A

β, on [0, T ]\A

where β is a fixed number with λ1 < β < λ2. Then

λ1 ⩽ m(x) < λ2 a.e. in [0, T ]. (3.8)

Claim 3: zn → z in Hα,β,ψp and z is a nontrivial solution of

HDα,β,ψ
T (|HDα,β,ψ

0+ ξ(x)|p−2 HDα,β,ψ
0+ ξ(x)) = m(x)|ξ|p−2u, in Ω = [0, T ]

I
β(β−1),ψ
0+ u(0) = I

β(β−1),ψ
T u(T ) = 0 on ∂Ω.

(3.9)

From (2.14) for each ϕ ∈ Hα,β,ψp it follows that∫ T

0

|HDα,β,ψ
0+ zn(x)|p−2 HDα,β,ψ

0+ zn(x)
HDα,β,ψ

0+ ϕ(x)dx

−
∫ T

0

f(x, ξn)

∥ξn∥p−1
ϕdx = o(1)∥ϕ∥.

(3.10)

Let ϕ = zn − z and note that

lim
n→∞

∫ T

0

f(x, ξn)

∥ξn∥p−1
(zn − z)dx = 0. (3.11)

Using (3.10) and (3.11), it follows that

lim
n→∞

∫ T

0

∣∣HDα,β,ψ
0+ zn(x)

∣∣p−2 HDα,β,ψ
0+ zn(x)

HDα,β,ψ
0+ (zn − z)dx = 0.

From the fact that HDα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
is of type S+, we conclude

that zn → z in Hα,β,ψp with ∥z∥ = 1.
Using (3.7), we have

lim
n→∞

∫ T

0

f(x, ξn)

∥ξn∥p−1
ϕdx =

∫ T

0

gϕ dx.



12 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

Then, from (3.10) and the 3 claims, we obtain∫ T

0

∣∣HDα,β,ψ
0+ zn(x)

∣∣p−2 HDα,β,ψ
0+ zn(x)

HDα,β,ψ
0+ ϕ(x)dx =

∫ T

0

m(x)|z(x)|p−2z(x)dx,

which implies (3.9). On the other hand, using inequality (2.15), the monotonicity
of λ1(a) and λ2(b), it follows that

λ1(m) ⩽ λ1(a) < 0, λ2(m) ⩾ λ2(b) ⩾ λ(b) > 0.

Then 0 is not an eigenvalue of HDα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
−m(x), which

the contradicts (3.9).

Case 4: Assume (A1), (A2), and (d). By contradiction we show that I is coercive
on Hα,β,ψp . We assume that {ξn} ⊂ Hα,β,ψp (without loss of generality assume it is

the whole sequence) and satisfies (2.14), but ∥ξn∥ → ∞ as n→ ∞. Let zn = ξn
∥ξn∥ .

The there exists a subsequence {zn} and z ∈ Hα,β,ψp (without loss of generality
assume it is the whole sequence), such that

zn ⇀ z weakly in Hα,β,ψ0+

zn → z strongly in Lp

zn → z for a.e. x ∈ [0, T ].

Now, using the conditions (A1) and (A4), it follows that

F(x, ξ) ⩽ C|ξ|p + C, (3.12)

with C > 0. Combining (2.14) and (3.12), we obtain

1

p
∥ξn∥p − C∥ξn∥p − C ⩽ C (3.13)

which implies that

1

p
− ∥z∥pp ⩽ 0, so z ̸= 0. (3.14)

If we define Ω′ = {x ∈ [0, T ]|z(x) ̸= 0}, then

meas(Ω′) > 0, |ξn(x)| → ∞, as n→ ∞

with x ∈ Ω′, which implies that

lim
n→∞

(pF(x, ξn)− ξnF(x, ξn)) = ∞, x ∈ Ω′.

By Fatou’s lemma,

lim
n→∞

∫ T

0

(pF(x, ξn)− ξnF(x, ξn)) = ∞.

However, using (2.14), it follows that

lim
n→∞

∫ T

0

(pF(x, ξn)− ξnF(x, ξn)) = −pC.

This contradiction completes the proof. □



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 13

4. Existence and multiplicity

Let φ1(a) and φ1(b) be the eigenfunctions corresponding to λ1(a) and λ1(b),
respectively. If we set E1 = span{φ1(a)} and E2 = span{φ1(b)} then as for (1.9)
we have

Hα,β,ψp = E1 ⊕ E⊥
1 ,

Hα,β,ψp = E2 ⊕ E⊥
2 .

Lemma 4.1. If the continuous functions a and b satisfy a(x) ⩽ b(x) for x ∈ [0, T ],
and

λ1(a) ⩽ 0 ⩽ λ(b),

then Hα,β,ψp = E1 ⊕ E⊥
2 .

Proof. Using Lemma 2.10 we only need to prove that E1∩E⊥
2 = {0}. Without loss

generally, we assume that {x ∈ [0, T ] : a(x) ̸= b(x)} is not empty so it is easy to
see that if

ξ ∈ ker
{H

Dα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
− a(·)

}
∩ ker

{H
Dα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
− b(·)

}
,

then we obtain ξ = 0. For each ξ0 ∈ E1 ∩E⊥
2 , it follows that (see inequality (1.10))

0 ⩾ λ1(a)

∫ T

0

|ξ0|pdx =

∫ T

0

(∣∣HDα,β,ψ
0+ ξ0

∣∣p − a(x)|ξ0|p
)
dx

⩾
∫ T

0

(∣∣HDα,β,ψ
0+ ξ0

∣∣p − b(x)|ξ0|p
)
dx

⩾ λ(b)

∫ T

0

|ξ0|pdx ⩾ 0,

which implies that

ξ0 ∈ ker
{H

Dα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
− a(·)

}
∩ ker

{H
Dα,β,ψ
T

(∣∣HDα,β,ψ
0+ (·)

∣∣p−2 HDα,β,ψ
0+ (·)

)
− b(·)

}
,

thefore ξ0 = 0. □

Proof of Theorem 1.3. For items (a) and (b), since in each case the functional I is
coercive on Hα,β,ψp , the existence of a solution is trivial.

For item (c), we consider the next three sub-items:

(1) I(ξ) → −∞ as ∥ξ∥ → ∞, ξ ∈ E1. Using conditions (A1) and (A2), if we set
G(x, ξ) = F(x, ξ)− 1

pa(x)|ξ|
p, then

G(x, ξ) ⩾ −C. (4.1)

Since λ1(a) < 0 and dim(E1) <∞, (4.1), assuming that (see inequality (1.10))

I(ξ) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx−
∫ T

0

F(x, ξ)dx

=
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx− 1

p

∫ T

0

a(x)|ξ(x)|pdx−
∫ T

0

G(x, ξ)dx



14 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34

≤ λ1(a)

p

∫ T

0

|ξ(x)|pdx+ C

⩽ −C∥ξ∥p + C,

we have that I(ξ) → −∞ as ∥ξ∥ → ∞ and ξ ∈ E1.

(2) I(ξ) is bounded from below on E⊥
2 . Similarly, if we set G1(x, ξ) = F(x, ξ) −

1
pb(x)|ξ|

p , then G(x, ξ) ⩽ C, which implies that, for any ξ ∈ E⊥
2 , we have (see

inequality (1.10))

I(ξ) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx−
∫ T

0

F(x, ξ)dx

=
1

p

∫ T

0

∣∣∣HDα,β,ψ
0+ ξ(x)

∣∣∣p dx− 1

p

∫ T

0

b(x)|ξ(x)|pdx−
∫ T

0

G(x, ξ)dx

⩾
λ(b)

p

∫ T

0

|ξ(x)|pdx− C

⩾ −C,

so I(ξ) is bounded from below on E⊥
2 .

(3) Now, we fix an l such that supξ∈∂C(l)∩E1
I(ξ) ⩽ β − 1 where β = infξ∈E⊥

2
I(ξ),

and C(l) = {ξ ∈ Hα,β,ψp : ∥ξ∥ ⩽ l}. Set

Γ = {γ : C(l) ∩ E1 → Hα,β,ψp : γ(ξ) = i if ξ ∈ E1, ∥ξ∥ = l},
c = inf

γ∈Γ
max
ξ∈C(l)

I(ξ).

Since ∂C(l) ∩ E and E⊥
2 are linking and the (PS) condition holds for I, c ⩾ β is a

critical value of I, so there is a critical point ξ0 ∈ Hα,β,ψp , such that I(ξ0) = c.
For item (d), as for item (c) we only need to prove that I(ξ) → −∞ as ∥ξ∥ →

∞, ξ ∈ E1. Indeed, from G(x, ξ) = F(x, ξ) − 1
pa(x)|ξ|

p−2ξ, g(x, ξ) = f(x, ξ) −
a(x)|ξ|p−2ξ, and (A4), it follows that

lim
|ξ|→∞

G(x, ξ) = ∞, for x ∈ [0, T ]. (4.2)

Then for each ξ ∈ E1, from (4.2) and the fact that dim(E1) <∞, we have

I(ξ) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx−
∫ T

0

F(x, ξ)dx

=
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx− 1

p

∫ T

0

a(x)|ξ(x)|pdx−
∫ T

0

G(x, ξ)dx

=
λ1(a)

p

∫ T

0

|ξ(x)|pdx−
∫ T

0

G(x, ξ)dx→ −∞, as ∥ξ∥ → ∞. □

We are now interested in obtaining multiple nontrivial solutions of (1.1). For
that, we need some results of Morse theory.

Lemma 4.2. Under conditions (A1)–(A5), 0 is a local minimum of the functional
I.



EJDE-2024/334HFIL FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN 15

Proof. Since λ1(ℓ) > 0 there exists a constant ε > 0 such that λ1(ℓ+ ε) > 0. Using
the condition (A5) there exists δ = δ(ε) such that

F(x, t) ⩽
1

p
(ℓ(x) + ε)|t|p, for |t| ⩽ δ, x ∈ [0, T ].

Moreover, for p < s ⩽ p∗ we can find a C > 0 such that

F(x, t) ⩽ C|t|s, for |t| > δ, x ∈ [0, T ].

Then, we obtain

F(x, t) ⩽
1

p
(ℓ(x) + ε)|t|p + C|t|s (4.3)

for t ∈ R, x ∈ [0, T ].
Finally, using (3.1) combining with inequality (4.3) and the embedding theorem,

we have

I(ξ) =
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx−
∫ T

0

F(x, ξ)dx

⩾
1

p

∫ T

0

∣∣HDα,β,ψ
0+ ξ(x)

∣∣pdx− 1

p

∫ T

0

(ℓ(x) + ε)|ξ(x)|p − C

∫ T

0

|ξ(x)|sdx

⩾ C∥ξ∥p − C∥ξ∥s > 0,

as 0 < ∥ξ∥ < 1, which implies that is 0 is a local minimum of I. □

To conclude this article, we prove the second main result.

Proof of Theorem 1.4. From Remark 1.5, we have

C∗(I, 0) = δ0,∗G.

Using a result from [8], the solution ξ0 obtained in Theorem 1.3 satisfies

C1(I, ξ0) ̸= 0.

Hence ξ0 is the nontrivial critical point of I. □

Acknowledgments. The authors are very grateful to the anonymous reviewers
for their useful comments that led to improvements of the original manuscript.

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Jose Vanterler da C. Sousa
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC-UFABC,
Santo André, SP, Brazil

Email address: jose.vanterler@ufabc.edu.br

Mariane Pigossi

Departamento de Matemática, Universidade Federal do Esṕırito Santo, Av. Fernando
Ferrari, 514, UFES, 29075-910, Vitória, ES, Brazil

Email address: mariane.pigossi@ufes.br

Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences Razi University, Kermanshah 67149,
Iran

Email address: neamat80@yahoo.com


	1. Introduction and motivation
	2. Mathematical background and auxiliary results
	3. Palais-Smale condition
	4. Existence and multiplicity
	Acknowledgments

	References