Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 73, pp. 1–12.

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

DOI: 10.58997/ejde.2024.73

EXISTENCE OF WEAK SOLUTIONS FOR NONLOCAL

DIRICHLET PROBLEMS VIA YOUNG MEASURE THEORY

MOUAD ALLALOU, MOHAMED EL OUAARABI, ABDERRAHMANE RAJI

Abstract. This article investigates the existence of weak solutions for a class
of nonlocal problems with Dirichlet boundary conditions. The proof of the ex-

istence result relies on Galerkin’s approximation and Young’s measure theory.

1. Introduction

Let D be a bounded domain in Rn with smooth boundary ∂D, and 1 < p <∞.
We prove the existence of weak solutions for the nonlocal problem with Dirichlet
boundary conditions,

−g
(∫

D
E(υ) dz

)
div[a(z,∇υ) + |∇υ|p−2∇υ] + |υ|p−2υ = f(z, υ) in D,

υ = 0 on ∂D,
(1.1)

where

E(υ) =
∫
D

(
A(z,∇υ) + 1

p
|∇υ|p

)
dz.

The functions f : D × R → R, g, A : D × Rm → R, and a : D × Rm → Rm are
subject to conditions specified below.

The study of nonlinear boundary value problems has garnered significant atten-
tion over the past few decades, driven by advancements in fields such as elastic
mechanics, electrorheological fluids, and image restoration; see [1, 7, 9, 17, 20, 23].

Transmission problems appear in various applications in physics and biology (see
[4, 8, 16]). Recently, in [15], the authors investigated the existence of ground-state
solutions for a class of Kirchhoff-type transmission problems.

This work aims to explore the existence of weak solutions to problem (1.1) by
employing the principles of Young measures theory. Notably, our problem cannot
be addressed with a variational framework because of the specific functions g and
f . These functions introduce significant technical challenges, necessitating the use
of alternative tools, such as Young measures, which facilitate the identification
of weak limits. To the best of our knowledge, this is the first study to approach
problem (1.1) using this theoretical framework. For an exploration of closely related
topics, readers are encouraged to consult references [10, 12] and additional sources

2020 Mathematics Subject Classification. 35J60, 35J25, 35D30.
Key words and phrases. Weak solution; Dirichlet boundary conditions; existence;

Young measures; Galerkin’s approximation; Sobolev space.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted October 5, 2024. Published November 17, 2024.

1



2 M. ALLALOU, M. EL OUAARABI, A. RAJI EJDE-2024/73

cited therein. For a thorough discussion on the steady-state case employing Young
measures theory, we refer to [2, 3, 13, 22].

A weak solution for (1.1) is defined as a function υ ∈ W 1,p
0 (D) satisfying the

following equation for all Φ ∈W 1,p
0 (D),

g
(∫

D
F(υ) dz

)∫
D

(
a(z,∇υ) + |∇υ|p−2∇υ

)
· ∇Φdz +

∫
D
|υ|p−2υ.Φdz

=

∫
D
f(z, υ)Φ dz.

In this article we use the following assumptions:

(A1) a : D × Rm → Rm and f : D × R → R are Carathéodory functions which
implies their measurability with respect to z ∈ D and continuity with re-
spect to the other variables. Additionally, the mapping ξ 7→ a(z, ξ) is both
a C1 function and monotonic, i.e.,

(a(z, ξ)− a(z, ξ′)) · (ξ − ξ′) ≥ 0 ∀ξ, ξ′ ∈ Rm.

(A2) We can find elements α1, α2 ∈ Lp′
(D), along with α3 ∈ L1(D) and positive

constants c0, c1 > 0, such that

|a(z, ξ)| ≤ c0(α1(z) + |ξ|p−1),

|f(z, w)| ≤ α2(z) + |w|q,
pA(z, ξ) ≥ a(z, ξ) · ξ ≥ c1|ξ|p − α3(z)

for all w ∈ R and 0 ≤ q < p− 1.
(A3) A : D × Rm → R is a Carathéodory function within the context of (A1).

Furthermore, the mapping ξ 7→ A(z, ξ) is both convex and C1-function and
it fulfills the relation a(z, ξ) = ∇ξA(z, ξ) = (∂A/∂ξ)(z, ξ).

(A4) g :W 1,p(D) → (0,+∞) are continuous and bounded on any bounded subset
of W 1,p(D) such that there are constants g0, g1 > 0 satisfying

g0 ≤ g(s) ≤ g1.

Our main result in this article reads as follows.

Theorem 1.1. Assume that (A1)–(A4) hold. Then, problem (1.1) has a weak

solution in W 1,p
0 (D).

The structure of this paper is as follows: Section 2 offers a concise overview
of essential aspects of Young measures. In Section 3, we focus on developing the
approximate solutions and establishing preliminary estimates. The final section
addresses various convergence outcomes and outlines the demonstration of the pri-
mary theorem.

2. Preliminaries

2.1. Fundamentals of Young measures. In this section, we provide a succinct
summary of the fundamental concepts behind generalized Young measures and
revisit relevant findings that will be employed in subsequent discussions. Our ap-
proach is influenced by the works of [18, 14], with additional insights available in
[19] for a more thorough introduction.

We denote by C0(Rm) the closure of the space comprising continuous functions
on Rm with compact support concerning the | · |∞-norm. Its dual space can be



EJDE-2024/73 WEAK SOLUTIONS FOR NONLOCAL DIRICHLET PROBLEMS 3

identified as M(Rm) the space encompassing signed Radon measures with finite
mass. The duality pairing for σ : D → M(Rm) is defined as

⟨σ, ψ⟩ =
∫
Rm

ψ(η) dσ(η).

Lemma 2.1 ([19]). Assume that the sequence {yµ}µ≥1 is bounded in L∞(D;Rm).
Then there exist a subsequence still denoted {yµ}µ and a Borel probability measure
σz on Rm for a.e. z ∈ D, such that for almost each h ∈ C(Rm) we have

ψ(yµ) →∗ ψ̄ weakly in L∞(D),

where

ψ̄(z) =

∫
Rm

ψ(η)dσz(η).

Definition 2.2. We call σ = {σz}z∈D the family of Young measures associated
with the subsequence {yµ}µ. It is shown in [6], that if for all R > 0

lim sup
µ→∞

|{z ∈ D ∩BR(0) : |yµ(z)| ≥ L}| = 0,

then for any measurable D′ ⊂ D,

ψ(z, yµ) → ⟨σz, ψ(z, .)⟩ =
∫
Rm

ψ(z, η)dσz(η) weakly in L1(D)

for any Carathéodory function ψ : D′ × Rm → R such that ψ(z, yµ) is equi-
integrable.

If we consider yµ = ∇wµ, where wµ : D → R, the above properties remain true,
and the following lemma can be proved in a similar way as in [5, Lemma 4.1].

Lemma 2.3. Let (∇wµ) be a bounded sequence in Lp(D;Rm). Then the Young
measure σz generated by ∇wµ in Lp(D;Rm) satisfies:

(1) |σz|M(Rm) = 1 for a.e. z ∈ D, i.e., σz is a probability measure.

(2) The weak L1-limit of ∇wµ is given by ⟨σz, id⟩ =
∫
Rm η · dσz(η).

(3) σz satisfies ⟨σz, id⟩ = ∇υ(z) for a.e. z ∈ D.

We will need the following Fatou-type inequality.

Lemma 2.4. Let ψ : D × Rm → R be a Carathéodory function and wµ : D → R a
sequence of measurable functions such that ∇wµ generates the Young measure σz
with |σz|g(Rm) = 1 for almost every z ∈ D. Then

lim inf
µ→∞

∫
D
ψ(z,∇wµ) dz ≥

∫
D

∫
Rm

ψ(z, η) dσz(η) dz

provided that the negative part of ψ(z,∇wµ) is equi-integrable.

3. Proof of Theorem 1.1

Let us consider the functional L(υ) :W 1,p
0 (D) → R given by

Φ 7→ g
(∫

D

(
A(z,∇υ) + 1

p
|∇υ|p dz

))[ ∫
D
a(z,∇υ) · ∇Φdz

+

∫
D
|∇υ|p−2∇υ · ∇Φdz

]
+

∫
D
|υ|p−2υ.Φdz −

∫
D
f(z, υ)Φ dz,

for arbitrary υ ∈W 1,p
0 (D) and Φ ∈W 1,p

0 (D).



4 M. ALLALOU, M. EL OUAARABI, A. RAJI EJDE-2024/73

Lemma 3.1. The functional L(υ) is well defined, linear and bounded.

Proof. Firstly, utilizing Hölder inequality and conditions (A2)–(A4), we establish

|Λ1| :=
∣∣∣g(∫

D
(A(z,∇υ) + 1

p
|∇υ|p) dz

)
×
[ ∫

D
a(z,∇υ)∇Φdz +

∫
D
|∇υ|p−2∇υ∇Φdz

]∣∣∣
≤ g1

(∫
D
|a(z,∇υ)| · |∇Φ|dz +

∫
D
|∇u|p−1 · |∇Φ|dz

)
≤ g1

(∫
D
c0(α1(z) + |∇υ|p−1)|∇Φ|dz

)
+ ∥∇υ∥p−1

p ∥∇Φ∥p

≤ C(∥α1∥p′ + |∇υ∥p−1
p )∥∇Φ∥p + ∥∇υ∥p−1

p ∥∇Φ∥p
≤ C∥∇Φ∥p.

Conversely, we can also infer, based on the growth condition of f in (A2) and the
Hölder inequality, that

|Λ2| :=
∣∣ ∫

D
f(z, υ)Φ dz

∣∣
≤

∫
D
|f(z, υ)Φ|dz

≤ (∥α2∥p′ + ∥υ∥p−1
p )∥Φ∥p

≤ λ(∥α2∥p′ + λp−1∥∇υ∥p−1
p )∥∇Φ∥p,

with λ denoting the constant in Poincare’s inequality, there exists a positive con-
stant λ such that

∥Φ∥p ≤ λ∥∇Φ∥p ∀Φ ∈W 1,p
0 (D). (3.1)

On the other hand,

|Λ3| := |
∫
D
|υ|p−2υΦdz|

≤
∫
D
|υ|p−1|Φ|dz

≤
( 1
p′

+
1

p

)
∥υ∥p−1

p ∥∇Φ∥p.

As the estimates of Λi for i = 1, 2, 3 are finite, L(υ) is well defined. Moreover, L(υ)
is linear and for all Φ ∈W 1,p

0 (D), the inequality

|⟨L(υ),Φ⟩| ≤ |Λ1|+ |Λ2|+ |Λ3| ≤ C∥∇Φ∥p
holds, indicating that L(υ) is bounded. □

By Lemma 3.1, we can define the operator L : W 1,p
0 (D) → W−1,p′

(D), that
satisfies the following result.

Proposition 3.2. The restriction of L to a finite dimensional linear subspace O
of W 1,p

0 (D) is continuous.

Proof. Let O be a finite linear subspace of W 1,p
0 (D). Suppose (υµ) is a sequence in

O that converges to υ in O.
Firstly, υµ → υ and ∇υµ → ∇υ almost everywhere.



EJDE-2024/73 WEAK SOLUTIONS FOR NONLOCAL DIRICHLET PROBLEMS 5

Secondly, ∫
D
|υµ − υ|p dz → 0 and

∫
D
|∇υµ −∇υ|p dz → 0,

since υµ → υ strongly inO. Hence, there exist Q1, Q2 ∈ L1(D) such that |υµ−υ|p ≤
Q1 and |∇υµ −∇υ|p ≤ Q2. We know that for γ > 1

|t1 + t2|γ ≤ 2γ−1(|t1|γ + |t2|γ).
Then

|υµ|p = |υµ − υ + υ|p ≤ 2p−1(|υµ − υ|p + ∥υ∥p) ≤ 2p−1(Q1 + ∥υ∥p).
Like in the demonstration of |υµ|p, it follows that |υµ|p and |∇υµ|p are bounded by
a constant C. Thus, the continuity condition in (A1), (A3) and (A4) permits to
deduce that

g
(∫

Ω

(A(z,∇υk) +
1

p
|∇υk|p)dz

)(∫
Ω

a(x,∇υk)∇Φ(z) dz

−
∫
Ω

|∇υk|p−2∇υk∇Φ(z) dz
)
+

∫
Ω

|υk|p−2υk.Φdz

converges to

g
(∫

Ω

(A(z,∇υ) + 1

p
|∇υ|p)dz

)(∫
Ω

a(x,∇υ)∇Φ(z) dz

−
∫
Ω

|∇υ|p−2∇υ∇Φ(z) dz
)
+

∫
Ω

|υ|p−2υ.Φdz

and

f(z, υµ)Φ(z) → f(z, υ)Φ(z)

almost everywhere as k → ∞. Indeed, if D′ is a measurable subset of D, and
Φ ∈W 1,p

0 (D), then∫
Ω′

|a(z,∇υk) · ∇Φ− |∇υk|p−2∇υk · ∇Φ|dz

≤
∫
Ω′
c0(α1(z) + |∇υk|p−1)|∇Φ|dz +

∫
Ω′

|∇υk|p−1 · |∇Φ|dz

≤
(
c0|α1|p′ + (c0 + 1) ∥∇υk∥p−1

p︸ ︷︷ ︸
≤C

)(∫
Ω′

|∇Φ|p dz
)1/p

and (without loss of generality, we can assume q = p− 1)∫
D′

|f(z, υµ)Φ(z)|dz ≤
∫
D′
(α2(z) + |υµ|p−1)|Φ|dz

≤ λ(|α2|p′ + ∥υµ∥p−1
p︸ ︷︷ ︸

≤C

)
(∫

D′
|∇Φ|p dz

)1/p

,

Using Hölder’s and Poincaré inequalities, along with (3.1). Moreover, we have

g(

∫
D′

(
A(z,∇υµ) +

1

p
|∇υµ|p

)
dz) ≤ g1 <∞,

by (A4) and the boundedness of |∇υµ|p. By utilizing the Vitali Theorem, we can
establish the continuity of L. □



6 M. ALLALOU, M. EL OUAARABI, A. RAJI EJDE-2024/73

Remark 3.3. In this section, we have used only the condition q ≤ p − 1. Thus
Lemma 3.1 and Proposition 3.2 are still valid as q = p− 1.

Now, the problem (1.1) is equivalent to find a solution υ ∈W 1,p
0 (D) such that

⟨L(υ),Φ⟩ = 0 for all Φ ∈W 1,p
0 (D).

To find such a solution we apply a Galerkin scheme. Since W 1,p
0 (D) is separable

there exists a sequence (Oµ) of finite dimensional subspaces such that ∪µ≥1Oµ is

dense in W 1,p
0 (D). Let {x1, . . . , xr} be a basis of Oµ where dimOµ = r. Next, Let

us define

G : Rr → Rr,

(di)i=1,...,r → (⟨L(dixi), xj⟩)j=1,...,r. .

Proposition 3.4. L is continuous and G(d) · d→ ∞ as |d|Rr → ∞.

Proof. G is trivially continuous, by the continuity of L restricted to Oµ (see Propo-
sition 3.2 if necessary). Consider d ∈ Rr and υ = dixi ∈ Oµ (with conventional
summation). The condition |d|Rr → ∞ is equivalent to ∥υ∥1,p → ∞, and we have

G(d) · d = ⟨L(υ), υ⟩.

Note that

Λ4 := g
(∫

D
(A(z,∇υ) + 1

p
|∇υ|p) dz

)[ ∫
D
a(x,∇υ) · ∇υ dz +

∫
D
|∇υ|p dz

]
≥ g0

[ ∫
D
a(x,∇υ) · ∇υ dz +

∫
D
|∇υ|p dz

]
(by (A4))

≥ g0

(∫
D
c1|∇υ|p dz −

∫
D
α3(z) dz

)
+ g0

∫
D
|∇υ|p dz

≥ Cmin

∫
D
|∇υ|p dz − C ′

∫
D
α3(z) dz,

since β ≥ 1. Finally, from the growth condition (A2) and (3.1) we have

|Λ5| := |
∫
D
f(z, υ)υ dz|

≤
∫
D
|f(z, υ)υ|dz

≤
∫
D
(α2(z) + ∥υ∥p−1)∥υ∥ dz

≤ λ∥α2∥p′∥∇υ∥p + λp+1∥∇υ∥p+1
p .

Hence

⟨L(υ), υ⟩ ≥ Λ4 − Λ5

≥ Cmin∥∇υ∥pp − C ′∥α3∥p′ − λ∥α2|∥p′∥∇u∥p − λp+1∥∇υ∥p+1
p → ∞

as ∥υ∥1,p → ∞, since Cmin, C
′ > 0 and (p > max(1, q + 1)). □

Proposition 3.5. For all k ∈ N there exists υµ ∈ Oµ such that

⟨L(υµ),Φ⟩ = 0 for all Φ ∈ Oµ.



EJDE-2024/73 WEAK SOLUTIONS FOR NONLOCAL DIRICHLET PROBLEMS 7

Proof. By Proposition 3.4, there exists R > 0 such that for all d ∈ ∂BR(0) ⊂ Rr we
have G(d) · d > 0, and the usual topological argument [24, Proposition 2.8], there
exists z ∈ BR(0) such that G(z) = 0. Hence, for all k ∈ N there exists υµ ∈ Oµ

such that ⟨L(υµ), φ⟩ = 0 for all φ ∈ Oµ. □

Proposition 3.6. The constructed sequence (υµ) in Proposition 3.5 is uniformly
bounded, i.e., there is a constant R > 0 such that ∥υµ∥1,p ≤ R for all k ∈ N.

Proof. By Proposition 3.4 there exists R > 0 with the property that ⟨L(υ), υ⟩ > 1
whenever ∥υ∥1,p > R. Hence, for the sequence of Galerkin approximations υµ ∈ Oµ

which satisfy ⟨L(υµ), υµ⟩ = 0 by the Proposition 3.5, we get the uniform bounded-

ness of (υµ) in W
1,p
0 (D). □

4. Proofs and properties for the convergence

In this section, we present general convergence findings pertaining to the func-
tions denoted as a(·), A(·), and f(·). Given that the sequence (υµ) remains within

bounded limits in the space W 1,p
0 (D), as established in Propositions 3.4, 3.5, and

3.6, we can infer, based on the assertions of Lemma 2.4, the existence of a Young
measure denoted as σx. This measure is generated by the gradients of υµ within
the space Lp(D;Rm).

We define

ã(z,∇υ) = a(z,∇υ) + |∇υ|p−2∇υ,

where ã adheres to conditions (A1)-(A3) with both coercivity and growth rate set
to p, specifically,

ã(z, ξ).ξ ≥ |ξ|p,
|ã(z, ξ)| ≤ |ξ|p−1 + S(c3, p, q).

(4.1)

Lemma 4.1. The Young measure σz generated by ∇υµ satisfies

(ã(z, η)− ã(z,∇υ)) · (η −∇υ) = 0 on suppσz,

where suppσz is the support of σz for a.e. z ∈ D.

Proof. We consider the sequence

eµ := (ã(z,∇υµ)− ã(z,∇υ)) · (∇υµ −∇υ)
= ã(z,∇υµ) · (∇υµ −∇υ)− ã(z,∇υ) · (∇υµ −∇υ)
= eµ,1 + eµ,2.

Given the growth condition of ã in (4.1) and the weak convergence described in
Lemma 2.3, it follows that

lim inf
µ→∞

∫
D
eµ,2 dz =

∫
D
ã(z,∇υ) ·

(∫
Rm

ηdσz(η)︸ ︷︷ ︸
=:∇υ(z)

−∇υ
)
dz = 0.

Thus

e := lim inf
µ→∞

∫
D
eµdz = lim inf

µ→∞

∫
D
eµ,1dz.



8 M. ALLALOU, M. EL OUAARABI, A. RAJI EJDE-2024/73

By the growth condition on ã in (4.1), (ã(z,∇υµ) · ∇υ) is equi-integrable. Let us
fix an arbitrary measurable subset D′ ⊂ D. Then, the coercivity condition in (4.1)
implies ∫

D′
|min(ã(z,∇υµ) · ∇υµ, 0)|dz ≤

∫
D′

|∇υµ|p dz <∞, (4.2)

This shows the equi-integrability of (ã(z,∇υµ)·∇υµ). Therefore (ã(z,∇υµ)·(∇υµ−
∇υ)) is also equi-integrable, and by Lemma 2.4, this yields∫

D

∫
Rm

ã(z, η) · (η −∇υ)dσz(η) dz ≤ lim inf
µ→∞

∫
D
ã(z,∇υµ) · (∇υµ −∇υ) dz = e.

Now, let us show that e ≤ 0. By Propositions 3.5, we can write

g
(∫

Ω

(A(z,∇υk) +
1

p
|∇υk|p)dz

)
×
(∫

Ω

a(x,∇υk) · (∇υk −∇υk) dz −
∫
Ω

|∇υk|p−2∇υk · (∇υk −∇υ)dz
)

=

∫
Ω

f(z, υk)(υk − υ) dz −
∫
Ω

|υk|p−2υk.(υk − υ) dz.

By (A4) we have

g0

∫
D
a(z,∇υµ) · (∇υµ −∇υ) dz

≤
∫
D
f(z, υµ)(υµ − υ) dz −

∫
D
|υµ|p−2υµ(υµ − υ) dz

+ g0

∫
D
|∇υµ|p−2∇υµ · (∇υµ −∇υ) dz.

Then ∫
D
a(z,∇υµ) · (∇υµ −∇υ) dz

≤ 1

g0

(∫
D
f(z, υµ)(υµ − υ) dz −

∫
D
|υµ|p−2υµ.(υµ − υ) dz

)
−
∫
D
|∇υµ|p−2∇υµ · (∇υµ −∇υ)dz

≤ Cm

∫
D
f(z, υµ)(υµ − υ) dz

≤ Cm(∥d2∥p′ + ∥υµ∥p−1
p︸ ︷︷ ︸

≤C

)∥υµ − υ∥p → 0 as k → ∞.

This is achieved by Hölder’s inequality and the fact that υµ → υ in W 1,p
0 (D).

Hence, we have ∫
D

∫
Rm

ã(z, η) · (η −∇υ)dσz(η)dz ≤ 0.

In conclusion, we can infer from this and Equation (4.2) that∫
D

∫
Rm

(
ã(z, η)− ã(z,∇υ)

)
· (η −∇υ) dσz(η) dz ≤ 0.



EJDE-2024/73 WEAK SOLUTIONS FOR NONLOCAL DIRICHLET PROBLEMS 9

The function ã being monotonic, the integral above evaluates to zero with respect
to the product measure dσz(η)⊗ dz, meaning that∫

D

∫
Rm

(
ã(z, η)− ã(z,∇υ)

)
· (η −∇υ) dσz(η)⊗ dz = 0.

Consequently, we obtain(
ã(z, η)− ã(z,∇υ)

)
· (η −∇υ) = 0 on supp σz. □

Proposition 4.2. For almost every z ∈ D, the support of σz is contained within

the set where à coincides with the supporting hyper-plane L defined as

L := {(η, Ã(z,∇υ) + ã(z,∇υ) · (η −∇υ))},

that is

suppσz ⊂ Kz = {η ∈ Rm : Ã(z, η) = Ã(z,∇υ) + ã(z,∇υ) · (η −∇υ)}.

Proof. Let η ∈ suppσz. By Lemma 4.1 implies for all t ∈ [0, 1], we have

(1− t)
(
ã(z, η)− ã(z,∇υ)

)
· (η −∇υ) = 0. (4.3)

Therefore, by the monotonicity condition and (4.3), we obtain

0 ≤ (1− t)
(
ã(z, η)− ã

(
z,∇υ + t(η −∇υ)

))
· (η −∇υ)

= (1− t)
(
ã(z,∇υ)− ã

(
z,∇υ + t(η −∇υ)

))
· (η −∇υ).

(4.4)

Using the monotonicity condition, we have(
ã(z,∇υ)− ã

(
z,∇υ + t(η −∇υ)

))
· t(∇υ − η) ≥ 0,

and since t ∈ [0, 1], we deduce that(
ã(z,∇υ)− ã

(
z,∇υ + t(η −∇υ)

))
· (1− t)(∇υ − η) ≥ 0. (4.5)

Combining (4.4) and (4.5) we find that(
ã(z,∇υ)− ã

(
z,∇υ + t(η −∇υ)

))
· (η −∇υ) = 0.

It follows from (A3) that

Ã(z, η) = Ã(z,∇υ) +
∫ 1

0

ã
(
z,∇υ + t(η −∇υ)

)
· (η −∇υ) dz

= Ã(z,∇υ) + ã(z,∇υ) · (η −∇υ).

Hence η ∈ Kz, i.e., supp σz ⊂ Kz for almost every z ∈ D. □

Now, we establish the proof of our main result.

Proof of Theorem 1.1. Since ξ 7→ Ã(z, ξ) is convex, we can represent it as

Ã(z, η) =: H(η) ≥ Ã(z,∇υ) + ã(z,∇υ) · (η −∇υ) =: R(η)

for all η ∈ Rm. Assuming that η 7→ H(η) is a C1-function, as specified in the
hypothesis, we obtain the following relationships for any ξ ∈ Rm, t ∈ R

H(η + tξ)−H(η)

t
≥ R(η + tξ)−R(η)

t
for t > 0,



10 M. ALLALOU, M. EL OUAARABI, A. RAJI EJDE-2024/73

H(η + tξ)−H(η)

t
≤ R(η + τξ)−R(η)

t
for t < 0.

Consequently, we can deduce that ∇ηH = ∇ηR, which implies

ã(z, η) = ã(z,∇υ) for all η ∈ Kz ⊃ suppσz. (4.6)

Since ã(z,∇υµ) is equi-integrable, by (4.6) and Lemma 2.3, its weak L1-limit sat-
isfies

ā(z) =

∫
Rm

ã(z, η) dσz(η)

=

∫
suppσz

ã(z, η) dσz(η)

=

∫
suppσz

ã(z,∇υ) dσz(η)

= ã(z,∇υ).

(4.7)

Next, if we consider the following Carathéodory function

B(z, η) = |ã(z, η)− ā(z)|, η ∈ Rm,

then, since a(z,∇υµ) is equi-integrable, we conclude that Bµ(z) := B(z,∇υµ) is
also equi-integrable, and its weak L1-limit is

Bµ → B̄ in L1(D),

where

B̄(z) =
∫
Rm

|ã(z, η)− ā(z)|dσz(η) =
∫
suppσz

|ã(z, η)− ā(z)|dσz(η) = 0,

by (4.5) and (4.6). Notably, the convergence of Bµ is strong since Bµ ≥ 0.
Applying (4.1), we derive theinequalities

Ã(z,∇υµ) ≥
1

p
ã(z,∇υµ) · ∇υµ ≥ 1

p
|∇υµ|p;

thus ∫
D′

|min(Ã(z,∇υµ), 0)|dz <∞.

Consequently, Ã(z,∇υµ) is both bounded and equi-integrable. As a result, its weak

L1-limit is
∫
Rm Ã(z, η) dσz(η). Utilizing Lemma 4.1, we obtain∫

Rm

Ã(z, η) dσz(η) =

∫
suppσz

Ã(z, η) dσz(η) = Ã(z,∇υ),

by Equation (4.2). The continuity of the function g in (A4) and Bµ → 0 in L1(D),
imply that

g
(∫

Ω

(A(z,∇υk) +
1

p
|∇υk|p) dz

)
×
(∫

Ω

a(z,∇υk)∇Φ(z) dz −
∫
Ω

|∇υk|p−2∇υk∇Φ(z) dz
)
+

∫
Ω

|υk|p−2υk dz

converges to

g
(∫

Ω

(A(z,∇υ) + 1

p
|∇υ|p) dz

)



EJDE-2024/73 WEAK SOLUTIONS FOR NONLOCAL DIRICHLET PROBLEMS 11

×
(∫

Ω

a(z,∇υ)∇Φ(z) dz −
∫
Ω

|∇υ|p−2∇υ∇Φ(z)dz
)
+

∫
Ω

∥υ∥p−2υ dz.

To complete the proof, we need to consider the term
∫
D f(z, υµ)Φ(z) dz. We know

that (υµ) is bounded in W 1,p
0 (D) according to Propositions 3.4, 3.5, and 3.6, up to

a subsequence, υµ → υ in Lp(D). For some ϵ positive, we have∫
D
|υµ − υ|p dz ≥

∫
{z∈D:|υµ−υ|≥ϵ}

|υµ − υ|p dz ≥ ϵp|{z ∈ D : |υµ − υ| ≥ ϵ}|,

which implies

|{z ∈ D : |υµ − υ| ≥ ϵ}| ≤ 1

ϵp

∫
D
|υµ − υ|p dz → 0 as k → ∞,

thus υµ → υ in measure and almost everywhere.
The continuity of the function f in (A1) implies

f(z, υµ)Φ(z) → f(z, υ)Φ(z)

almost everywhere. From the growth condition in F and the uniform bound in
Propositions 3.4, 3.5, and 3.6, it follows that f(z, υµ)Φ(z) is equi-integrable. Con-
sequently,

f(z, υµ)Φ(z) → f(z, υ)Φ(z) in L1(D),

and then, by the Vitali convergence theorem, we establish that

lim
µ→∞

∫
D
f(z, υµ)Φ(z) dz =

∫
D
f(z, υ)Φ(z) dz ∀Φ ∈ ∪µ≥1Oµ.

Since ∪µ≥1Oµ is dense in W 1,p
0 (D), it follows that υ is a weak solution of (1.1) as

desired. □

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Mouad Allalou

Applied Mathematics and Scientific Computing Laboratory, Faculty of Science and

Technics, Sultan Moulay Slimane University, Beni Mellal, BP 523, 23000, Morocco
Email address: mouadallalou@gmail.com

Mohamed El Ouaarabi
Fundamental and Applied Mathematics Laboratory, Faculty of Sciences Äın Chock,

Hassan II University, Casablanca, BP 5366, 20100, Morocco.

Applied Mathematics and Scientific Computing Laboratory, Faculty of Science and
Technics, Sultan Moulay Slimane University, Beni Mellal, BP 523, 23000, Morocco

Email address: mohamed.elouaarabi@etu.univh2c.ma, mohamedelouaarabi93@gmail.com

Abderrahmane Raji

Applied Mathematics and Scientific Computing Laboratory, Faculty of Science and

Technics, Sultan Moulay Slimane University, Beni Mellal, BP 523, 23000, Morocco
Email address: rajiabd2@gmail.com


	1. Introduction
	2. Preliminaries 
	2.1. Fundamentals of Young measures

	3. Proof of Theorem ??
	4. Proofs and properties for the convergence
	References