EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5621
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Local and Global Weak Solutions for Elliptic Nonlinear
Equations

Habeeb Ibrahim1, Mohammed E. Dafaalla2,∗, Osman Abdalla Adam Osman3,
Ashraf. S. ELshreif4

1 Department of Mathematics, College of Science, Qassim University, Buraydah 51452,
Saudi Arabia

Abstract. In this work, it is considered that a certain quasilinear elliptic equation in an open
bounded domain in Rn over a vector space, and we derive gradient estimates for weak solutions of
p-Laplacian type elliptic equations with tiny bounded mean oscillation coefficients locally Lp, p ≥ q.
In addition, we provide the key findings.

2020 Mathematics Subject Classifications: 35J60, 35J92, 35B65, 35D30, 46E35

Key Words and Phrases: Weak solution, Nonlinear equation, measurable coefficients, gradient
estimates, Holder’s inequality

1. Introduction

Let us consider the following elliptic quasilinear equation:

div((E∇vm.∇vm)(p−2)/2E∇vm) = div(|gm|p−2 gm in ω (1)

where p > 1. Here, ω ∈ R is assumed to be an open bounded domain. Furthermore,
E = {aij(x)}m×m is a symmetric matrix with measurable coefficients that satisfies the

uniformly elliptical condition, and gm = (g1m, ..., g
n
m) is a given vector field.

α−1 |ξ|2 ⩽ E(x)ξ.ξ ≤ α |ξ|2 (2)

for all ξ ∈ Rn and nearly every x ∈ Rn, and for some positive constant α. In case that E
is the identity matrix, we derive from [1, 2] that the gradient estimate for weak solutions
of equation (1) is Lq, q ≥ p, and [3] examined the case where p = p(x). Furthermore,
for weak solutions of equation(1) with VMO coefficients, [4, 5] have achieved Lq, q ≥ p
gradient estimations. All of these writers’ techniques are based on maximal functions. In

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5621

Email addresses: Ha.Ibrahim@qu.edu.sa (H. Ibrahim), m.dafaalla@qu.edu.sa (M. E. Dafaalla),
o.osman@qu.edu.sa (O. A. Adam Osman), ae.mohammad@qu.edu.sa (A. S. ELshreif)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 2 of 16

this work, we provide a novel method of direct and straightforward verification of Lq, q ≥ p,
gradient estimates for weak solutions of equation (1) with tiny BMO coefficients, without
the need for maximum functions. It is important to note that the assumption in [5–7] that
E is in the VMO space [8] is weakened by our assumption that E is (δ,R). The elliptic
semi-norms of the coefficients of E = {aij} are assumed to be sufficiently small throughout
this study, and they are assumed to be in elliptic BMO spaces. More specifically, we have
the following definitions.

Definition 1. (Semi-norm condition for small BMO). If the coefficient matrix E is (δ,R)
-vanishing, then

sup
0<r≤R

sup
x∈R

∮
Br (x)

∣∣E (y)− ĒBr (x)
∣∣ dy < δ,

where

ĒBr(x) =

∮
Br(x)

E(y)dy.

A Lp estimates have been examined recently in [9–11] for second-order linear ellip-
tic/parabolic problems with modest BMO coefficients. We note that a function in VMO
satisfies the previously stated small BMO requirement; it goes without saying that a
function that satisfies the VMO criterion also satisfies the small BMO condition. In the
definition above, we take R to be a positive constant (one can use a scaling transform to
suppose R=1, and we take δ to be scaling invariant. In this section, we refer to δ as a
little positive constant. The definition of local weak solutions for (1) is now provided.

Definition 2. Let g ∈ Lp
loc(ω). A local weak solution of the equation (1) is a function

vm ∈W 1,p
loc (ω) if, for any ψ ∈W 1,p

0 (ω) , we can deduce∫
ω
(E∇vm.∇vm)(p−2)\2E∇vm.∇ψdx =

∫
ω
|gm|p−2 gm.∇ψdx.

2. Local weak solutions of quasilinear elliptic equation

Consider a local weak solution of equation (1) vn ∈W 1,p
loc (ω), we state some preliminary

lemmas.

Lemma 1. Suppose that B3 ⊂ ω. Then we have∫
B1

|∇vm|q dx ≤ C

{∫
B3

|vm|q dx+

∫
B3

|gm|q dx
}

(3)

where C is solely dependent on m, q and α [8].

Proof. It is possible to choose the test function ψ = ζpv ∈W 1,p
0 (ω), where ζ ∈ C∞

0 (Rn)
represents a cut-off function that is satisfied.

0 ≤ ζ ≤ 1, ζ ≡ 1 in B1, ζ ≡ 0 in Rn \B2.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 3 of 16

Therefore, by definition 1, we get∫
B3

(E∇vm.∇vm)(p−2)\2E∇vm.∇(ζpvm)dx =

∫
B3

|gm|p−2 gm.∇(ζpvm)dx

The resulting expression should be written as

I1 = I2 + I3 + I4

where,

I1 =

∫
B3

ζp(E∇vm.∇vm)p\2dx,

I2 = −
∫
B3

pζp−1vm(E∇vm.∇vm)(p−2)\2(E∇vm.∇ζ)dx,

I3 =

∫
B3

ζp |gm|p−2 gm.∇vmdx,

I4 =

∫
B3

pζp−1vm |gm|p−2 gm.∇ζdx.

The estimation of I1. The uniformly elliptic equation (2) dictates that

I1 =

∫
B3

ζp(E∇vm.∇vm)p\2dx ≥ 1

α

∫
B3

ζp |∇vm|p dx.

The estimation of I2 can be derived using the uniformly elliptic condition (2) and Young’s
inequality with respect to τ .

I2 ≤ C

∫
B3

ζp−1 |∇vm|p−1 |vm| dx ≤ τ

∫
B3

ζp |∇vm|p dx+ C(τ)

∫
B3

|vm|p dx.

The estimation of I3 can be derived from Young’s inequality.

I3 ≤ τ

∫
B3

ζp |∇vm|p dx+ C(τ)

∫
B3

|gm|p dx.

The estimation of I4 can be derived from Young’s inequality.

I4 ≤ C

{∫
B3

|vm|p dx+

∫
B3

|gm|p dx
}
.

We conclude that by summing all the estimates of Ij(1 ⩽ j ≤ 4),

1

α

∫
B3

ζp |∇vm|p dx ≤ 2τ

∫
B3

ζp |∇vm|p dx+ C(τ)

∫
B3

(vpmdx+ |gm|p)dx.

By choosing τ = 1
(4α) , and referring back to the definition of ζ the proof is successfully

concluded. Henceforth, it is assumed that q is greater than p. Now let

q1 =:
q + p

2
∈ (p, q).

We recall a well-known result of [1].



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 4 of 16

Lemma 2. Assumes that vn ∈ W 1,p
loc (ω) is a local weak solution of the equation (1) and

g ∈ Lq1(Ω) . Consequently, we have q2, p < q2 < q1 in which:(∮
Bs(x1)

|∇vm|q2 dx

) 1
q2

≤ C


(∮

B2s(x1)
|∇vm|p dx

) 1
p

+

(∮
B2s(x1)

|gm|q1 dx

) 1
q1

 ,

for each B2s(x1) ⊂ ω , with condition that q2 and C are dependent solely on n, p, q, α.

Following this, we present two lemmas that are critical in order to derive the primary
result:
The two lemmas are significantly impacted by [6, 7]. We compose

λ0 =

{(∮
B2

|∇vm|p dx
) 1

p

+
1

δ

(∮
B2

|gm|q1 dx
) 1

q1

}
, (4)

and
A(λ) = {x ∈ B1 : |∇vm| > λ}

for λ > 0, whereas δ > 0 will be selected at a later time.
By ensuring that |∇vm| is constrained within the range B1 \A(λ) for any fixed λ > 0, our
analysis is directed towards the level set A(λ). At this juncture, A(λ) shall be decomposed
into a set of disjoint spheres.

Lemma 3. Suppose that λ ≥ λ∗ = 26n\pλ0 , there exists a family of disjoint balls{
B0

i

}
i∈N =

{
Bpxi

(xi)
}
i∈N , xi ∈ A(λ) , such that 0 < pxi < 1 \ 25 . And

(∮
B0

i

|∇vm|p dx

) 1
p

+
1

δ

(∮
B0

i

|gm|q1 dx

) 1
q1

= λ.

Furthermore, we have

E (λ) ⊂
⋃
i∈N

B1
i ,

where Bj
i =: 2j+2B0

i and pxi < s < 1 for all i values 1, 2, and 3.(∮
Bs(xi)

|∇vm|pdx

)1/p

+
1

δ

(∮
Bs(xi)

|gm|q1dx

)1/q1

≤ λ.

Proof. (i) For the sake of expediency, we signify:

J [B] =

∮
B

|∇vm|pdx

1/p

+
1

δ

∫
B

|gm|q1dx

1/q1

.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 5 of 16

Now we assert that:
sup
ω∈B1

sup
1/25≤λ≤1

J [Bp (ω)] ≤ 2
6n
p λ0 =: λ∗. (5)

In order to demonstrate this, assign any Ω ∈ B1 and 1/25 ≤ p ≤ 1. From the equation
Bj

i = 2j+2B0
i : j=1,2,3, we get that B2 = 25Bp(Ω). And we can easily see that:

(
|B2|

|Bp (Ω)|

) 1
p

≤ 2
5
p ≤ 2

6n
p , n = 1, 2, ... . (6)

Then by using equations (4), (5) and (6) we can deduce that:(∮
Bp(Ω)

|∇vm|pdx

)1/p

≤ (
|B2|

|Bp(Ω)|

1
p
(∮

B2

|∇vm|pdx
) 1

p

≤ 26n/p
(∮

B2

|∇vm|pdx
) 1

p

.

In a similar fashion, we have(∮
Br (Ω)

|gm|q1 dy

) 1
q1

≤ 2
6n
q1

(∮
B2

|gm|q1 dx

) 1
q1

.

As a result of combining the aforementioned two inequalities with the definitions of λ0
and q1, we can conclude that (4) is valid.
(ii) Define λ0 as λ ≥ λ∗ = 26n/pλ0. In the case of Ω ∈ A(λ), a variant of Lebesgue’s
differentiation theorem provides the following:

lim
p→0

J [Bp (Ω)] > λ.

which indicates the existence of a p > 0 that satisfies

J [Bp (Ω)] > λ.

Consequently, starting from step (i), we can choose a radius pΩ ∈ (0, 1/25] [8], and such
that

J [Bpw (Ω)] = λ.

Furthermore, that for pΩ < p ≤ 1 and

J [Bp (Ω)] < λ.

Based on the aforementioned argument for. Ω ∈ A(λ), the ball BpΩ(Ω) can be constructed
as described above. Hence, by employing Vitali’s covering lemma, it is possible to identify
a set of disjoint orbs denoted as {B0

i }i∈N = {Bpxi
(xi)}i∈N where xi ∈ A(λ) in order to

validate the lemma’s conclusions. We now have a comprehensive proof.

At present, the subsequent estimates of spheres {B0
i } are obtained.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 6 of 16

Lemma 4. By employing the identical hypothesis and outcomes as in Lemma (6), we
obtain

∣∣B0
i

∣∣ ≤ C

(
1

λp

∫
{x∈B0

i :|∇vm|>λ
4}

|∇vm|pdx+
1

λq1δq1

∫
{x∈B0

i :|f |>
δλ
4 }

|gm|q1dx

)
,

where C = C(p, q1) = 2q1/[1− (1/2)p − (1/2)q1 ].

Proof. As shown in the lemma above,(∫
B0

i

|∇vm|pdx+
1

δ

∫
B0

i

|gm|1/q1dx

)
= λ,

thus, signifying that

|B0
i | ≤

2p

λp

∫
B0

i

|∇vm|pdx+
2q1

λq1δq1

∫
B0

i

|gm|
1
q1 dx (7)

given that one of the subsequent inequalities must hold true:

λ/2 ≤

(∫
B0

i

|∇vm|pdx

) 1
p

,

or

λ/2 ≤ 1

δ

(∫
B0

i

|gm|q1dx

) 1
q1

.

As a result, we obtain the following by dividing the right-hand side of equation (6) into
two integrals:

∣∣B0
i

∣∣ ≤ C

(
2p

λp

∮
{x∈B0

i :|∇vm|>λ/4}
|∇vm|p dx+ (1/2)q1

∣∣B0
i

∣∣)

+
2q1

λq1δq1

∫
{x∈B0

i :|f |>δλ/4}
|gm|q1 dx+ (1/2)q1

∣∣B0
i

∣∣
We have therefore arrived at the intended estimation.
It is adequate to regard the proof of Theorem 1 in section four as an a priori estimate in
the subsequent discussion, thus presuming that ∇vm ∈ Lq

loc (ω). One can easily eliminate
this assumption using a standard approximation argument, such as the one presented in

[12, 13]. By considering Lemma 3 and λ ≥ λ∗ = 2
6n
p · λ0, it is possible to generate a set of

disjoint balls denoted as
{
B0

i

}
i∈N =

{
Bpxi (xi)

}
i∈N

, xi ∈ A (λ). Adjust any i ∈ N and set

vmλ = um/λ and gmλ = gm/λ.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 7 of 16

Subsequently, vmλ remains a local weak solution of (1), where gmλ assumes the place of
gm. Consequently, Lemma (3) dictates that∫

B
j
i

|∇vmλ
|p dx ≤ 1 and

∫
Bj

i

|gm|q1 dx ≤ λq1 (8)

In the case i = 1, 2, 3, Bi
j is defined in Lemma 2 as Bi

j =: 2i+2B0
i Denote the weak solution

of the reference equation as:{
div
(
ĒBs∇um · ∇umĒBs∇um

)
= 0 in Bs

um = fm in Bs
(9)

3. The Global weak solutions and grading estimates

Definition 3. Let f ∈ W 1,p(Bs) be assumed. It is stated that um ∈ W 1,p(Bs) is a weak
solution of the system um − gm ∈W 1,p

0 (Bs) is a weak solution of{
div
(
ĒBs∇um · ∇umĒBs∇um

)
= 0 in Bs,

um = fm on ∂Bs.

Let ∫
Bs

(ĒBs∇um · ∇umĒBs∇um · ∇ψdx) = 0 ,

with any ψ ∈W 1,p
0 (Bs). The following are recollections of estimates for um to [5, 12]:∫

Bs

|∇um|pdx ≤ C

∫
Bs

|∇vm|pdx, (10)

and

sup
Bp

|∇um| ≤ C

(∫
Bs

|∇um|pdx)
) 1

p

(11)

in the range p ∈ (0, s/2], with C = C(n, p, α). Moreover, we can derive the subsequent
significant outcome.

Lemma 5. There exists a small value of δ = δ(ϵ) > 0 for all values of ϵ, such that if vm
is a local weak solution of (1) in ω with B4 ⊂ ω, then∫

B2

∣∣E − ĒB2

∣∣dx ≤ δ, (12)

and ∫
B4

|∇vm|pdx ≤ 1 and

∫
B4

|gm|q1dx ≤ δq1 . (13)

Consequently, N0 > 1 exists and is denoted by u in B2 as the weak solution to (1)



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 8 of 16

Proof. Based on (1), (2), and (4), it can be deduced that the weak solutions of (1) in
ω and (2) in B2, respectively, are denoted as vm and um. It is sufficient to select the test
function ψ = um − vm ∈ W 1,p

0 (B2), and a straightforward calculation yields the following
expression:

I1 = I2 + I3,

where

I1 =
∫
B2

(ĒB2∇um · ∇um)
p−2
2 EB2∇u−

(
ÊB2∇vm · ∇vm)

p−2
2 ĀB2∇vm

)
.∇(u− vm)dx,

I2 =
∫
B2

((E∇vm · ∇vm)(p−2)/2Ē∇vm − (ĒB2∇vm · ∇vm)(p−2)/2ĒB2∇vm) · ∇(um − vm)dx,

I3 = −
∫
B2

|gm|p−2g · ∇ (um − vm) dx.

The estimation of I1. Two instances are distinguished.
Given Case 1. p ≥ 2, the elementary inequality is applied.

(ĒB2ζ · ζ)
p−2
2 ĒB2ζ − (ĒB2η · η)

p−2
2 ĒB2η) · (ζ − η) ≥ C|ζ − η|p,

We have, for every ζ, η ∈ Rn where C = C(p, α),:

I1 ≥ C

∫
B2

|∇(vm − um)|pdx.

Case 2: Applying the rudimentary inequality to 1 < p < 2.

|ζ − η|p ≤ Cτ
p−2
p ((ĒB2ζ · ζ)

p−2
2 ĒB2ζ − (ĒB2η · η)

p−2
2 ĒB2η) · (ζ − η) + τ |η|p.

We have the following for each ζ, η ∈ Rn and each τ ∈ (0, 1) where C = C(p, α):

I1 + τ

∫
B2

|∇vm|pdx ≥ C (τ)

∫
B2

|∇ (vm − um)|pdx.

Approximation of Implementing a fundamental inequality∣∣∣(Eζ · ζ)(p−2)/2Eζ − (ĒB2ζ · ζ)(p−2)/2ĒB2ζ
∣∣∣ ≤ C

∣∣E − ĒB2

∣∣ |ζ|p−1.

We have, for each o, η ∈ Rn where C = C(p, α), by employing Young’s inequality with
and Holder’s inequality.

I2 ≤ C
∫
B2

∣∣E − ĒB2

∣∣ |∇vm|p−1 |∇ (vm − um)| dx
≤ C (τ)

∫
B2

∣∣E − Ē
∣∣ p
p−1 |∇vm|pdx+ τ

∫
B2

|∇(vm − um)|pdx

≤ C(τ)
(∫

B2

∣∣E − ĒB2

∣∣pq2/[(p−1)(q2−p)]
dx
)(q2−p)/q2(∫

B2
|∇vm|q2dx

)p/q2
+τ
∫
B2

|∇(vm − um)|pdx.

We observe that (∫
B2

∣∣E − ĒB2

∣∣pq2/[(p−1)(q2−p)]
dx
)(q2−p)/q2

≤ (2α)(p
2+q2−p)/[q2(p−1)](

∫
B2

∣∣E − ĒB2

∣∣dx)(q2−p)/q2 ≤ Cδ(q2−p)/q2



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 9 of 16

due to the outcomes of (2) and (3), and(∫
B2

|∇vm|
q2

dx

)p/q2

≤ C

[(∫
B4

|∇vm|pdx
) 1

p

+

∫
B4

(|gm|q1dx)
1
q1

]p
≤ C,

by virtue of Lemma 4 and equation (13), with C denoting the set C = C(m, p, q1, α). In
this context, the assumption that δ < 1. We thus conclude that

I2 ≤ C(τ)δ(q2−p)/q2 + τ

∫
B2

|∇(vm − um)|p dx.

The estimation of I3 can be obtained by applying Young’s inequality with τ and Holder’s
inequality

I3 ≤ τ
∫
B2

|∇ (vm − um)|pdx+ C (τ)
∫
B2

|gm|pdx

≤ τ
∫
B2

|∇(vm − um)|pdx+ C(τ)
(∫

B2
|gm|q1dx

)p/q1
≤ τ

∫
B2

|∇(vm − um)|pdx+ C(τ)δp.

We derive by summing all the estimates of Ii (1 ≤ i ≤ 3):

C(τ)
∫
B2

|∇(vm − um)|pdx ≤ 2τ
∫
B2

|∇(vm − um)|pdx
+τ
∫
B2

|∇vm|pdx+ C(τ)
[
δ(q2−p)/q2 + δp

]
.

We reach the following conclusion by selecting a small constant τ > 0 such that 0 < τ ≪
δ < 1, and then applying (13):∫

B2

|∇(vm − um)|pdx ≤ C

[
δ + δ

q2−p
q2 + δp

]
= εp,

by choosing that fulfills the final inequality stated earlier. This concludes the evidence.
Define (1) and (2) with the same value of δ as in Lemma 2. As stated at the outset of this
segment, E is vanishing (δ, 1). Thus (Perci)∫

Bj
i

|E − Ē
Bj

i
|dx| ≤ δ, (14)

given that the radiuses of Bj
i (0 ≤ j ≤ 3) are not greater than 1 for j = 0, 1, 2, 3. The

scaling invariant form of Lemma 2 is then obtained by recalling (7).

Lemma 6. Considers the assumption that λ ≥ λ∗. In the case where ε is greater than
zero, there is a small δ = δ(ε) > 0 such that N0 is greater than one and vm is a local weak
solution of in ω with B3

i ⊂ ω.

sup
B2

i

∣∣∣∇(um)iλ

∣∣∣ ≤ N0 and

∫
B2

i

∣∣∇ (vmλ − (vm)iλ
)∣∣pdx ≤ εp. (15)

In this context, (um)iλ denotes the weak solution of equation (2) in B2
i , where vmλ substi-

tutes for vm.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 10 of 16

Proof. Resealing the definitions of Bj
i for j = 0, 1, 2, 3, we establish

(vm)iλ (x) =
vλm(23pxix)

23pzi
,

(gm)iλ (x) = gmλ(2
3pxix),

Ei(x) = E(23pxix), x ∈ B4.

(vm)iλ is therefore a local weak solution of

div
(
Ei∇ (vm)iλ · ∇(vm)iλ

)(p−2)/2
Ei∇(vm)iλ = div

(
|(gm)iλ|p−2(gm)iλ

)
in B4.

It is easily discernible from equations (7) and (14) that∫
B4

∣∣∣∇(vm)iλ(x)
∣∣∣pdx ≤ 1,

∫
B4

∣∣∣(gm)iλ

∣∣∣pdx ≤ δp

and ∫
B2

∣∣Ei − Ēi
B2

∣∣pdx ≤ δ.

Lemma 1 subsequently states that a weak solution of{
div
(
Ēi

B2
∇um · ∇umĒi

B2
∇um

)
= 0 in B2

um = (vm)iλ on ∂B2

in the form that

sup
B1

|∇um| ≤ N0 and

∫
B2

∣∣∣∇ (vm)iλ − um

∣∣∣p dx ≤ εp

At this moment, we define (um)iλ in B2
i by

um (x) =
1

23pxi

(um)iλ
(
23pxix

)
, x ∈ B2.

Then, by modifying the variables, the conclusion of Lemma 3 is reestablished. Such
concludes the evidence.

4. Local and Gradient estimates for elliptic nonlinear equations:

Theorem 1. Considers the case where q ≥ p. Considers vm to be a feeble local solution to
(1). Subsequently, a small value of δ = δ(n, p, q, α,R) > 0 is present, such that for every
elliptical function and vanishing (δ,R), as well as for every f with gm ∈ Lq

loc(ω;R
n), we

can deduce: ∫
Br(x0)

|∇vm|qdx ≤ C

[∫
B4r(x0)

|vm|qdx+

∫
B4r(x0)

|gm|qdx

]
. (16)

The constant C is independent of vmand gm where B4r(x0) ⊂ ω. Our strategy is substan-
tially shaped by [5, 14].



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 11 of 16

Proof. (i) For q = p, the proof is uncomplicated.
(ii) Lemma (3) states that for all λ ≥ λ∗, we obtain∣∣{x ∈ B1

i : |∇vm| > 2N0λ}
∣∣ = ∣∣{x ∈ B1

i : |∇vmλ| > 2N0}
∣∣

≤
∣∣∣{x ∈ B1

i :
∣∣∣∇(vmλ − (vm)iλ

∣∣∣ > N0}
∣∣∣

+
∣∣∣{x ∈ B1

i :
∣∣∣∇(um)iλ

∣∣∣ > N0}
∣∣∣ = ∣∣∣{x ∈ B1

i :
∣∣∣∇(umλ − (um)iλ)

∣∣∣ > N0}
∣∣∣

≤ 1
Np

0

∫
B2

i

∣∣∣∇(vm − (um)iλ

)∣∣∣pdz ≤ εp|B2
i |

Np
0

=
24nεp|B0

i |
Np

0
.

Consequently, as shown in Lemma 2, that∣∣{x ∈ B1
i : |∇vm| > 2N0λ}

∣∣ ≤ C
(
εp 1

λp

∫
{x∈B0

i :|∇vm>λ
4 |} |∇vm|p dx

)
+ 1

λq1δq1

(∫
{x∈B0

i :|g|>δλ/4} |gm|q1dx
)
.

In the given context, C = C(n, p, q1, α). Keeping in mind that the spheres are disconnected
and ⋃

i∈N
B1

i ⊃ A(λ) = {x ∈ B1 : |∇vm| > λ}.

We obtain the following by summing i ∈ N in the inequality above for any λ ≥ λ∗:

|{x ∈ B1 : |∇vm| > 2N0λ}|

≤
∑
i

∣∣{x ∈ B1
i : |∇vm| > 2N0λ}

∣∣
≤ Cεp

(
1

λp

∫
{x∈B2:|∇vm|>λ

4}
|∇vm|pdx+

1

λq1δq1

∫
{x∈B2:|gm|> δλ

4 }
|gm|q1dx

)
(17)

in the case of any λ ≥ λ∗. We compute while recalling the standard argument of measure
theory. ∫

B1
|∇vm|q dz = q

∫∞
0 µq−1 |{x ∈ B1 : |∇vm| > µ}| dµ

= q
∫ 2N0λ∗
0 µq−1 |{x ∈ B1 : |∇vm| > µ}| dµ+ q

∫∞
2N0λ∗

µq−1 |{x ∈ B1 : |∇vm| > µ}| dµ
= q

∫ 2N0λ∗
0 µq−1 |{x ∈ B1 : |∇vm| > µ}| dµ

+q
∫∞
λ∗

(2N0λ)
q−1 |{x ∈ B1 : |∇vm| > 2N0λ}| d(2N0λ) =: J1 + J2.

Estimation of J1: It can be deduced from the definitions of λ∗ and λ0 that

λq∗ = 26nq/pλq0 ≤ C

{(∫
B2

|∇vm|p dx
) q

p

+
1

δq

(∫
B2

|gm|q1 dx
) q

q1

}
. (18)

Consequently, Lemma (1) and Holder’s inequality dictate that

λq∗ ≤ C

[(∫
B4

|vm|p dx+
∫
B4

|gm|p dx
) q

p
+ 1

δq

(∫
B2

|gm|q1 dx
) q

q1

]
≤ C

[(∫
B4

|vm|pdx
)q/p

+
(∫

B4
|gm|pdx

)q/p
+ 1

δq

∫
B2

|gm|qdx
]

≤ C
{∫

B4
|vm|qdx+

∫
B4

|gm|qdx
}
.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 12 of 16

From [14] we see that, 1
δq → 0, and we have:

λq∗ ≤ C

{∫
B4

|vm|qdx+

∫
B4

|gm|qdx
}
.

Hence, we ascertain

J1 ≤ (2N0λ∗)
q |B1| ≤ C

{∫
B4

|vm|qdx+

∫
B4

|gm|qdx
}
,

where C = C(n, p, q, α).
Probability of J2. By deriving from (17), we obtain that

J2 ≤ Cεp
{∫∞

0 λq−p−1
∫
{x∈B2:|∇vm|>λ/4} |∇vm|pdxdλ

+ 1
δq1

∫∞
0 λq−q1−1

∫
{x∈B2:|gm|>δλ/4} |gm|q1dxdλ

}
.

Considering that∫
Rn

|fm|βdx = (β − Λ)

∫ ∞

0
µβ−α−1

∫
{x∈Rn:|fm|>µ}

fm
αdxdµ.

Given β > Λ > 1, we obtain

J2 ≤ C1ε
p

∫
B2

|∇um|qdx+ C2ε
p

∫
B2

|fm|qdx,

in that where C1 = C1(n, p, q, α) and C2 = C2(n, p, q, α, δ).
We obtain by combining the estimates of J1 and J2.∫

B1

|∇vm|qdx ≤ C1ε
p

∫
B2

|∇vm|qdx+ C3

∫
B4

(|vm|q + |gm|q) dx,

C3 = C3(n, p, q, α, δ, ε).
Using a covering and iteration argument to [15, 16] to re incorporate at the right-hand
side the first integral in the aforementioned inequality while selecting a suitable ε such
that C1ε

p = 1/2, we obtain the following:∫
B1

|∇vm|q dx ≤ C

{∫
B4

|vm|q dx+

∫
B4

|gm|q dx

}
.

By performing a shift and scaling transform, the proof of the main result can be completed.

Corollary 1. Considers vm to be a sequence of local weak solutions to equation (1), assum-
ing that ε is greater than zero. Subsequently, a small value of δ = δ(n, 1+ε, 1+ε

ε , α,R) > θ



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 13 of 16

is present, such that for every elliptically shaped and vanished set E(δ,R) and every set

gm with gm ∈ L
1+ϵ
ϵ

loc (ω;Rn), we can deduce:∫
Br(x0)

∑s
m=1 |∇vm|

1+ϵ
ϵ dx

≤ C̄
[∫

B4r(x0)

∑s
m=1 |vm|

1+ε
ε dx+

∫
B4r(x0)

∑s
m=1 |gm|

1+ε
ε dx

]
where C̄ is a constant that is not dependent on um and gm, and B4r(x0) ⊂ ω.

Proof. Lemma 2 states that for any λ = λ∗ + ϵ, we obtain

1/
(
2x ∈ B1

i :
∑s

m=1 |vm| > 2N0(λ∗ + ϵ)
)1/2

= 1/
{
x ∈ B1

i :
∑s

m=1 |(vm)λ∗+ϵ| > 2N0

}1/2
≤ 1/{x ∈ B1

i :
∑s

m=1

∣∣∣((vm)λ∗+ε − (um)iλ∗+ε)
∣∣∣ > N0}

1/2
+
∣∣∣{x ∈ B1

i :
∣∣∣∇(um)iλ∗+ε

∣∣∣ > N0}
∣∣∣

= 1/{x ∈ B1
i :
∑s

m=1

∣∣∣∇((vm)λ∗+ε − (um)iλ∗+ε)
∣∣∣ > N0}

1/2

≤ 1
N1+ε

0

∫
B2

i

∑s
m=1

∣∣∣∇(vm)λ∗+ε − (um)iλ∗+ε)
∣∣∣1+ε

dz ≤ ε1+ε|B2
i |

N1+ε
0

=
24nε1+ε|B0

i |
N1+ε

0

,

consequently, as shown in Lemma 3 that

1/

(
≤ {2x ∈ B1

i :
s∑

m=1

|∇vm| > 2N0(λ∗ + ϵ)

)1/2

≤ C̄ε1+ε 1

λ1+ε

∫
{x∈B0

i :
∑s

m=1 |∇vm|>(λ∗+ε)/4}

s∑
m=1

|∇vm|1+δdx

in that where C̄ = C̄(n, 1 + ϵ, (1+ϵ
ϵ )1, α).

bearing in mind that the spheres are disconnected and

∪
i∈N

B1
i ⊃ A (λ∗ + ε) =

{
x ∈ B1 :

s∑
m=1

|∇vm| > (λ∗ + ε)

}
,

for any λ = λ∗ + ε, and by aggregating the terms in the aforementioned inequality, we
obtain

1/ ({2x ∈ B1 :
∑s

m=1 |∇vm| > 2N0(λ∗ + ε)})1/2

≤
∑

i 1/{2x ∈ B1
i :
∑s

m=1 |∇vm| > 2N0(λ∗ + ε)}1/2

≤ C̄ε1+ε 1
λ1+ε

∫
{x∈B2:

∑s
m=1 |∇vm|>(λ∗+ε)/4}

∑s
m=1 |∇vm|1+εdx

1

λ( 1+ε
ε )1δ(

1+ε
ε )1

∫
{x∈B2:

∑s
m=1 |gm|>δ(λ∗+ε)/4}

∑s
m=1 |gm|(

1+ε
ε

)1dx),

(19)

for whatever. We compute while recalling the standard argument of measure theory.∫
B1

∑s
m=1 |∇vm|(

1+ε
ε

)dz

= (1+ε
ε )
∫∞
0 µ(

1+ε
ε

)−11/{2x ∈ B1 :
∑s

m=1 |∇vm| > µ}1/2dµ
= (1+ε

ε )
∫ 2N0λ∗
0 µ(

1+ε
ε

)−11/{2x ∈ B1 :
∑s

m=1 |∇vm| > µ}1/2dµ
+(1+ε

ε )
∫∞
2N0λ∗

µ(
1+ε
ε

)−11/{2x ∈ B1 :
∑s

m=1 |∇vm| > µ}1/2dµ
= (1+ε

ε )
∫ 2N0λ∗
0 µ(

1+ε
ε

)−11/{2x ∈ B1 :
∑s

m=1 |∇vm| > µ}1/2dµ
+
(
1+ε
ε

) ∫∞
λ∗

(2N0 (λ∗ + ε))(
1+ε
ε )−1K.d(2N0(λ∗ + ε)},



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 14 of 16

where

K =

(
1/2x ∈ B1 :

s∑
m=1

|∇vm| > 2N0(λ∗ + ε)

)1/2

=: J1 + J2.

Probability of J1. It can be deduced from the definitions of λ∗and λ0 that

λ
( 1+ε

ε
)

∗ = 26n(
1+ε
ε

)/(1+ε)λ
( 1+ε

ε
)

0

≤ C̄

((∫
B2

∑s
m=1 |∇vm|(1+ε)dx

)( 1+ε
ε

)/(1+ε)
+ 1

δ(
1+ε
ε )

(∫
B2

∑s
m=1 |gm|(

1+ε
ε

)1dx
)( 1+ε

ε
)/( 1+ε

ε
)1
.

)
Consequently, Lemma 3 and Holder’s inequality dictate that

λ
( 1+ε

ε
)

∗ ≤ C̄
(∫

B4

∑s
m=1 |vm|(1+ε)dx+

∫
B4

∑s
m=1 |gm|(1+ε)dx

)( 1+ε
ε

)/(1+ε)

+ 1

δ(
1+ε
ε )

(∫
B2

∑s
m=1 |gm|(

1+ε
ε

)1dx)(
1+ε
ε

)/( 1+ε
ε

)1
)

≤ C̄

{(∫
B4

∑s
m=1 |vm|(1+ε)dx

)( 1+ε
ε )/(1+ε)

+
(∫

B4

∑s
m=1 |gm|(1+ε)dx

)( 1+ε
ε

)/(1+ε)

+ 1

δ(
1+ε
ε )

∫
B2

∑s
m=1 |gm|(

1+ε
ε

)dx

≤ C̄
{∫

B4

∑s
m=1 |vm|(

1+ε
ε )dx+

∫
B4

∑s
m=1 |gm|(

1+ε
ε )dx

}
.

Hence, we ascertain

J1 ≤ (2N0λ∗)
( 1+ε

ε ) |B1| ≤ C̄

{∫
B4

s∑
m=1

|vm|(
1+ε
ε )dx+

∫
B4

s∑
m=1

|gm|(
1+ε
ε )dx

}
,

where C̄ = C̄(n, 1 + ε, (1+ε
ε ), α).

J2 estimation is derived from (18) as follows:

C̄ε(1+ε)


∫∞
0 (λ∗ + ε)(

1+ε
ε )−(1+ε)−1 ∫

{x∈B2:
∑s

m=1|∇vm|>(λ∗+ε)/4}
∑s

m=1 |∇vm|(1+ε) dxd (λ∗ + ε)

+ 1

δ
( 1+ε

ε )
1

∫∞
0 (λ∗ + ε)

Considering that∫
Rn

|f |βdx = (β − Λ)

∫ ∞

0
µβ−Λ−1

∫
{x∈Rm:|f |>µ}

fΛdxdµ.

Given β > Λ > 1, we obtain

J2 ≤ C̄1ε
(1+ε)

∫
B2

s∑
m=1

|∇vm|(
1+ε
ε

)dx+ C̄2ε
(1+ε)

∫
B2

s∑
m=1

|gm|(
1+ε
ε

)dx,



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 15 of 16

where C̄1 = C̄1(n, 1 + ε, (1+ε
ε ), α) and C̄2 = C̄2(n, 1 + ε, (1+ε

ε ), α) .
We obtain by combining the estimates of J1 and J2.∫
B1

s∑
m=1

|∇vm|(
1+ε
ε

)dx ≤ C̄1ε
(1+ε)

∫
B2

s∑
m=1

|∇vm|(
1+ε
ε

)dx+ C̄3

∫
B4

s∑
m=1

(|vm|(
1+ε
ε

) + |gm|(
1+ε
ε

))dx,

in that where C̄3 = C̄3(n, 1 + ε, (1+ε
ε ), α, δ, ε). By choosing an appropriate ε such that

C̄1ε
(1+ε) = 1/2 and employing a covering and iteration argument to reabsorb at the right-

hand side of the initial integral in the aforementioned inequality, we obtain the following
result: ∫

B1

s∑
m=1

|∇vm|(
1+ε
ε

)dx ≤ C̄

{∫
B4

s∑
m=1

|vm|(
1+ε
ε

)dx+

∫
B4

s∑
m=1

|gm|(
1+ε
ε

)dx

}
.

By performing a shift and scaling transform, the proof can be completed.

5. Conclusion

This study meticulously established the foundational groundwork for analyzing nonlin-
ear elliptic equations of p-Laplacian type. We began by rigorously defining key concepts,
including the small bounded mean oscillation semi-norm condition, which quantifies the
oscillation of coefficients. Furthermore, we provided precise definitions for both local and
global weak solutions of the equation under consideration, ensuring a clear understanding
of the solution spaces. Crucial lemmas were derived to support the main theorem, laying
a solid analytical framework for the subsequent investigation. Building upon these defini-
tions and lemmas, the core contribution of this work lies in the proof of the main result
theorem. This theorem successfully derives local gradient estimates for the aforemen-
tioned nonlinear elliptic equations, specifically those featuring bounded mean oscillation
coefficients. These estimates provide valuable insights into the regularity and behavior
of solutions, particularly concerning the gradient’s control within localized domains. The
successful derivation of these gradient estimates signifies a significant advancement in the
understanding of p-Laplacian type equations with non-smooth coefficients.

Author Contributions

All authors contributed equally to the writing of this article. All authors have accepted
responsibility for the entire content of the manuscript and approved its submission.

Conflicts of Interest

All authors confirm that they have no conflict of interest.



H. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5621 16 of 16

Acknowledgements

The Researchers would like to thank the Deanship of Graduate Studies and Scientific
Research at Qassim University for financial support (QU-APC-2025).

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