Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 95, pp. 1–15.

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

DOI: 10.58997/ejde.2025.95

UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS WITH

CARATHEODORY NONLINEARITIES AT THE INTERIOR

AND ON THE BOUNDARY

EDGAR ANTONIO, MARTÍN P. ÁRCIGA-ALEJANDRE,

ROSA PARDO, JORGE SÁNCHEZ-ORTIZ

Abstract. We establish an explicit uniform a priori estimate for weak solutions to slightly sub-

critical elliptic problems with nonlinearities simultaneously at the interior and on the boundary.
Our explicit L∞(Ω) a priori estimates are in terms of powers of their H1(Ω) norms. To prove

our result, we combine a De Giorgi-Nash-Moser iteration scheme together with elliptic regularity
and the Gagliardo-Nirenberg interpolation inequality.

1. Introduction

Let us consider the nonlinear boundary value problem of semilinear eliptic equations

−∆u+ u = f(x, u), x ∈ Ω,

∂u

∂η
= fB(x, u), x ∈ ∂Ω,

(1.1)

where Ω ⊂ RN , (N > 2), is an open, connected, bounded domain with C2 boundary, ∂/∂η = η ·∇
is the (unit) outer normal derivative, and the functions f : Ω × R → R, and fB : ∂Ω × R → R,
are both slightly subcritical Carathéodory functions. In (H1)–(H4) below, we give the precise
statement of the hypotheses on the nonlinearities at the interior, and on the boundary.

Our goal is to establish explicit L∞(Ω) a priori estimates for weak solutions to (1.1), in terms
of powers of their H1(Ω) norms (see Theorem 2.2). Our estimates are independent of the sign of
the solutions. Consequently, any sequence of solutions to (1.1), uniformly bounded in the H1(Ω)
norm, is also uniformly bounded in the L∞(Ω) norm.

Our techniques are based on an iterative process due to Moser, in the elliptic regularity theory,
and in the Gagliardo-Nirenberg interpolation inequality.

For the homogeneous Dirichlet boundary conditions, by a Moser’s type procedure, it is well
known that weak solutions to a subcritical or even critical elliptic problem are in Lq(Ω) for all
1 < q < ∞ (see [11, Lemma 1], see also [4, Section 2.2], [15, Lemma B.3]. Moreover, by elliptic
regularity, the solutions are in L∞(Ω).

Moser’s results can be extended to the case of nonlinear boundary conditions, and also to a
general quasilinear problem, which includes in particular (1.1), see, for instance, [9, Theorem 3.1].
In [9] the authors state that weak solutions to some quasilinear problem are in L∞(Ω)∩L∞(∂Ω).
By elliptic regularity, weak solutions to (1.1) are in fact more regular, and in particular, they are
uniformly continuous functions. Indeed, the elliptic regularity theory, applied to weak solutions of
a subcritical or even critical problem implies that they are in C(Ω), see estimate (5.2) in Theorem
5.1. So, in that case,

∥u∥L∞(∂Ω) ≤ ∥u∥C(Ω) = ∥u∥L∞(Ω). (1.2)

2020 Mathematics Subject Classification. 35B45, 35J66, 35B33, 35J75, 35J25.
Key words and phrases. A priori estimates; slightly subcritical non-linearities; L∞ a priori bounds;

nonlinear boundary conditions.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 24, 2025. Published October 13, 2025.

1



2 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

The type of L∞(Ω) estimates given by (2.12) are known for slightly subcritical nonlinearities in
the homogeneous Dirichlet problem with the Laplacian operator, see [13, Theorem 1.5], with the
p-Laplacian operator, see [14, Theorem 1.6], and also with a linear problem at the interior joint
with nonlinear boundary conditions on the boundary of power type, see [3].

In this article, we analyze the combined effect of both nonlinearities simultaneously. We estab-
lish the explicit estimates provided by Theorem 2.2, where both nonlinearities in the interior and
on the boundary are slightly subcritical, not necessarily of power type.

This article is organized in the following way. Section 2 contains the statement of our main
result, Theorem 2.2; we also give an application to finite energy solutions. The proof of Theorem
2.2 is achieved in Section 3. By the sake of completeness, we include two appendices. In Appendix
4, we recall the regularity of weak solution to the linear problem with non homogeneous data both
at the interior and on the boundary, see Theorem 4.1. Appendix 5 deals with further regularity
of weak solutions to (1.1), see Theorem 5.1.

2. Main result

For p > 1, we define the trace operator Γ :W 1,p(Ω) → Lp(∂Ω), in the following way

(1) Γu = u|∂Ω if u ∈W 1,p(Ω) ∩ C(Ω),
(2) ∥Γu∥Lp(∂Ω) ≤ C∥u∥W 1,p(Ω),

where C = C(p, |Ω|) is a constant and ∂Ω is C2. From the surjectivity and the continuity of the
trace operator, we obtain

Γ :W 1,p(Ω) →W 1− 1
p ,p(∂Ω) ↪→ Lq(∂Ω), for 1 ≤ q ≤ (N − 1)p

N − p
,

and
∥Γu∥Lq(∂Ω) ≤ C∥u∥W 1,p(Ω), for some C > 0,

this operator is continuous for 1 ≤ q ≤ (N−1)p
N−p , and compact for 1 ≤ q < (N−1)p

N−p (see [6, Theorem

6.4.1] and [2, Lemma 9.9]).
Throughout this article, we use the Sobolev embedding

H1(Ω) ↪→ L2∗(Ω), (2.1)

and the continuity of the trace operator

H1(Ω) ↪→ L2∗(∂Ω),

where

2∗ :=
2N

N − 2
and 2∗ :=

2(N − 1)

N − 2
=

(N − 1)

N
2∗, (2.2)

are the critical Sobolev exponent and the critical exponent in the sense of the trace, respectively.
For 1 < p, pB ≤ ∞, we denote

2∗N/p :=
2∗

p′
= 2∗

(
1− 1

p

)
and 2∗,N/pB

:=
2∗
p′B

= 2∗

(
1− 1

pB

)
, (2.3)

where p′ is the conjugate exponent of p, that is 1
p + 1

p′ = 1.

For the nonlinearity f : Ω× R → R, we assume the following hypothesis at the interior:

(H1) f is a Carathéodory function:
(a) f(·, t) is measurable for each t ∈ R;
(b) f(x, ·), is continuous for each x ∈ Ω;

(H2) f is slightly subcritical (at infinity), that is,

|f(x, t)| ≤ |a(x)|f̃
(
|t|
)
, (2.4)

with a(x) ∈ Lr(Ω) for r > N
2 , f̃ : [0,+∞) → [0,+∞) is continuous, non-decreasing,

f̃(t) > 0 for t > 0, and such that

lim
t→+∞

f̃(t)

t
2∗
N/r−1

= 0. (2.5)



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 3

Likewise, for the nonlinearity fB : ∂Ω × R → R, we assume the following hypothesis on the
boundary:

(H3) fB is a Carathéodory function:
(a) fB(·, t) is measurable for each t ∈ R;
(b) fB(x, ·) is continuous for each x ∈ ∂Ω.

(H4) fB is slightly subcritical (at infinity), that is:

|fB(x, t)| ≤ |aB(x)|f̃B(|t|), (2.6)

with aB(x) ∈ LrB (∂Ω) for rB > N − 1, and f̃B : [0,+∞) → [0,+∞) is continuous,

non-decreasing, f̃B(t) > 0 for t > 0, and such that

lim
t→+∞

f̃B(t)

t2∗,N/rB
−1 = 0. (2.7)

We say that u ∈ H1(Ω) is a weak solution to (1.1) if f(·, u) ∈ L(2∗)′(Ω), and fB(·, u) ∈ L(2∗)
′
(∂Ω)

are such that for all ψ ∈ H1(Ω),∫
Ω

∇u∇ψ d+
∫
Ω

uψ d =

∫
Ω

f(x, u)ψ dx+

∫
∂Ω

fB(x, u)ψ dS ,

being (2∗)′ = 2N
N+2 and (2∗)

′ = 2(N−1)
N the conjugate exponents of 2∗ and 2∗, respectively.

Remark 2.1. (i) Let u ∈ H1(Ω). By Sobolev embeddings, for f and fB slightly subcritical, we
have

f̃(|u|) ∈ L
2∗

2∗
N/r

−1
(Ω), where

2∗N/r − 1

2∗
=

1

2
+

1

N
− 1

r
,

f̃B(|u|) ∈ L
2∗

2∗,N/rB
−1 (∂Ω), where

2∗,N/rB − 1

2∗
=

N

2(N − 1)
− 1

rB
.

Hence,

f(·, u) ∈ L(2∗)′(Ω) and fB(·, u) ∈ L(2∗)
′
(∂Ω).

(ii) We can always choose f̃ and f̃B such that f̃(t) > 0 and f̃B(t) > 0 for t > 0. Note that

redefining both functions, f̃(t) and f̃B(t), as max[0,t] f̃ and max[0,t] f̃B , respectively, we can always

choose f̃(t) and f̃B(t) as non-decreasing functions for t > 0.

Now, let us define two new functions,

h(t) :=
t2

∗
N/r−1

f̃(t)
and hB(t) :=

t2∗,N/rB
−1

f̃B(t)
for t > 0. (2.8)

Since the nonlinearities f and fB are both slightly subcritical, it follows that

h(t) → ∞ and hB(t) → ∞ as t→ ∞. (2.9)

Let hm be defined as the minimum of h and a certain power of hB , specifically

hm(t) := min
{
h(t), h

2∗
N/r

−1

2∗,N/rB
−1

B (t)
}
, (2.10)

with h and hB defined in (2.8).
We will denote as aM the maximum of the corresponding norms of a ∈ Lr(Ω) and of aB ∈

LrB (∂Ω), that is

aM := max{∥a∥Lr(Ω), ∥aB∥LrB (∂Ω)}. (2.11)

The next Theorem provies estimates for hm(∥u∥L∞(Ω)) in terms of their H1(Ω) norms.

Theorem 2.2. Assume (H1)–(H4) hold and u is a weak solution to (1.1). Then, for all ε > 0,
there exists Cε > 0 depending of ε, N , |Ω| and |∂Ω|, but independent of u, such that

hm(∥u∥L∞(Ω)) ≤ Cεa
A+ε
M

(
1 + ∥u∥(2

∗
N/r−2)(A+ε)

H1(Ω)

)
, (2.12)



4 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

where hm is defined by (2.10), aM by (2.11), and

A :=


1
2−

N−r
Nr

1
2−

N−1
NrB

if either r ≥ N , or N/2 < r < N and r∗ ≥ NrB
N−1 ,

1 if N/2 < r < N and r∗ ≤ NrB
N−1 .

(2.13)

Remark 2.3. Since (1.2), we have

hm(∥u∥C(Ω)) ≤ Cεa
A+ε
M

(
1 + ∥u∥(2

∗
N/r−2)(A+ε)

H1(Ω)

)
.

Remark 2.4. From the definitions of h and hB given in (2.8), we note that

h(t) =
t2

∗
N/r−1

f̃(t)
and h

2∗
N/r

−1

2∗,N/rB
−1

B (t) =
( t2∗,N/rB

−1

f̃B(t)

) 2∗
N/r

−1

2∗,N/r−1

.

Thus,

hm(t) = min

{
t2

∗
N/r−1

f̃(t)
,

t2
∗
N/r−1

f̃

2∗
N/r

−1

2∗,N/r−1

B (t)

}
.

We apply our result to finite energy solutions of subcritical problems satisfying Ambrosetti-
Rabinowitz condition. A sequence {un} ⊂ H1(Ω) of weak solutions to (1.1) has uniformly bounded
energy if there exists a constant c0 > 0, such that J [un] ≤ c0, where J is the associated energy
functional defined by

J [u] :=
1

2

∫
Ω

(
|∇u|2 + u2

)
−

∫
Ω

F (x, u) dx−
∫
∂Ω

FB(x, u) dσx

with F (x, t) :=
∫ t

0
f(x, s) ds, and FB(x, t) :=

∫ t

0
fB(x, s) ds.

The Ambrosetti–Rabinowitz condition holds if there exist two constants θ > 2, and s0 > 0 such
that

θF (x, s) ≤ sf(x, s), ∀x ∈ Ω, ∀|s| > s0,

θFB(x, s) ≤ sfB(x, s), ∀x ∈ ∂Ω, ∀|s| > s0.
(2.14)

Assuming that (H1)–(H4) and (2.14) hold, a sequence of solutions to (1.1) is uniformly L∞(Ω)
a priori bounded if and only if it has uniformly bounded energy. It can be proved using the same
arguments as in [3, theorem 5.1].

3. L∞(Ω) a priori explicit estimates

Our method combines elliptic regularity with the Gagliardo-Nirenberg interpolation inequality.
Let u be an arbitrary solution to (1.1). First, we find estimates of the nonlinearities in terms of
products of the H1(Ω)-norm of u and their L∞(Ω)-norm. With it, using elliptic regularity (see
Theorem (4.1)), we obtain estimates of the W 1,m(Ω)-norm, with m > N , of the solutions to (1.1).
Finally, applying the Gagliardo-Nirenberg interpolation inequality, (see [12]), we obtain an explicit
estimate of the L∞(Ω)-norm of u in terms of the H1(Ω) norm of u.

Proof of Theorem 2.2. Let u ∈ H1(Ω) be a weak solution to (1.1). By Theorem 5.1, u ∈ H1(Ω)∩
L∞(Ω).

Firstly, we will estimate both nonlinearities (the interior and the boundary nonlinearities) in
terms of the H1(Ω)-norm and the L∞(Ω)-norm of u.

Step 1. W 1,m(Ω) estimates for m > N . By hypothesis, f̃ and f̃B are both increasing. By (1.2)
we denote

M := f̃(∥u∥L∞(Ω)) = max
[0,∥u∥L∞(Ω)]

f̃ ,

MB := f̃B(∥u∥L∞(Ω)) = max
[0,∥u∥L∞(Ω)]

f̃B .
(3.1)

Along this proof, we will use the obvious fact that for any γ > 0, there exist two constants C1

and C2, only dependent on γ, such that

C1(1 + xγ) ≤ (1 + x)γ ≤ C2(1 + xγ), for all x ≥ 0. (3.2)



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 5

Throughout this proof, C denotes several constants independent of u.
By the growth condition (2.4) and the definition given in (3.1), we have that∫

Ω

|f(x, u)|qdx ≤
∫
Ω

|a(x)|q f̃(|u|)q−t+t dx ≤ CMq−t

∫
Ω

|a(x)|q f̃(|u|)t dx, (3.3)

for all t < q, and all

q ∈
(N
2
,min{r,N}

)
. (3.4)

Using Hölder’s inequality, for all 1 < s <∞, we can write∫
Ω

|a(x)|q f̃(|u|)t dx ≤
(∫

Ω

|a(x)|qsdx
)1/s(∫

Ω

f̃(|u|)ts
′
dx

)1/s′

, (3.5)

where s′ is such that 1
s + 1

s′ = 1. Choosing s and t < q, so that qs = r and ts′ = 2∗

2∗
N/r

−1 , we have

t :=
2∗

2∗N/r − 1

(
1− q

r

)
< q

⇐⇒ 1

q
− 1

r
<

2∗N/r − 1

2∗
= 1− 1

r
− 1

2
+

1

N

⇐⇒ q >
2N

N + 2
,

(3.6)

since q > N
2 > 2N

N+2 .

On the other hand, by subcriticality, see (2.5), and the Sobolev embeddings, see (2.1), we have∫
Ω

|f̃(|u|)|
2∗

2∗
N/r

−1
dx ≤ C

∫
Ω

(
1 + |u|2

∗
)
dx

≤ C
(
1 + ∥u∥2

∗

L2∗ (Ω)

)
(3.7)

≤ C
(
1 + ∥u∥2

∗

H1(Ω)

)
. (3.8)

Substituting (3.8) in the second factor on the right-hand side of (3.5),∫
Ω

|a(x)|q f̃(|u|)t dx ≤ C
(∫

Ω

|a(x)|qsdx
)1/s(

1 + ∥u∥2
∗

H1(Ω)

)1/s′

. (3.9)

Finally, substituting (3.9) in (3.3) and since 1/(qs′) = 1/q − 1/r, we obtain(∫
Ω

|f(x, u)|qdx
)1/q

≤ CM1− t
q ∥a∥Lr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )

H1(Ω)

)
. (3.10)

Likewise, by the condition (2.6) and the subcriticality (2.7), we obtain∫
∂Ω

|fB(x, u)|qB dS ≤
∫
∂Ω

|aB(x)|qB f̃B(|u|)qB−tB+tB dS

≤ CMqB−tB
B

∫
∂Ω

|aB(x)|qB f̃B(|u|)tB dS,
(3.11)

for all tB < qB , and all

qB ∈ (N − 1, rB). (3.12)

Using Hölder’s inequality, for all 1 < sB <∞, we obtain∫
∂Ω

|aB(x)|qB f̃B(|u|)tB dS ≤
(∫

∂Ω

|aB(x)|qBsBdS
)1/sB(∫

∂Ω

f̃B(|u|)tBs′BdS
)1/s′B

, (3.13)



6 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

where s′B is such that 1
sB

+ 1
s′B

= 1. Choosing, as before, sB , tB < qB , so that qBsB = rB , and

tBs
′
B = 2∗

2∗,N/rB
−1 ; thus,

tB :=
2∗

2∗,N/rB − 1

(
1− qB

rB

)
< qB

⇐⇒ 1

qB
− 1

rB
<

2∗,N/rB − 1

2∗
= 1− 1

rB
− N − 2

2(N − 1)

⇐⇒ 1

qB
<

N

2(N − 1)

⇐⇒ qB >
2(N − 1)

N
,

(3.14)

and the last inequality is satisfied since qB > N − 1 and N > 2.
On the other hand, again by subcriticality, see (2.6) and (2.7), we have∫

∂Ω

|f̃B(|u|)|
2∗

2∗,N/rB
−1 dx ≤ C

∫
∂Ω

(
1 + |u|2∗

)
dS

≤ C
(
1 + ∥u∥2∗L2∗ (∂Ω)

)
(3.15)

≤ C
(
1 + ∥u∥2∗H1(Ω)

)
, (3.16)

Since tBs
′
B = 2∗/(2∗,N/rB − 1), and substituting (3.16) in the second factor on the right-hand

side of (3.13),∫
∂Ω

|aB(x)|qB f̃B(|u|)tB dS ≤ C
(∫

∂Ω

|aB(x)|qBsBdS
)1/sB (

1 + ∥u∥2∗H1(Ω)

)1/s′B
, (3.17)

Finally, substituting (3.17) into (3.11), and since 1/(qBs
′
B) = 1/qB − 1/rB , we obtain(∫

∂Ω

|fB(x, u)|qBdx
)1/qB

≤ CM
1− tB

qB

B ∥aB∥LrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)

H1(Ω)

)
. (3.18)

Now, using elliptic regularity, we estimate the norm ∥u∥W 1,m(Ω) in terms of the corresponding
norms of the nonlinearities, see Theorem 4.1, Equation (4.2). Specifically, using (3.10) and (3.18),
we obtain

∥u∥W 1,m(Ω) ≤ C
[
M1− t

q ∥a∥Lr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )

H1(Ω)

)
+M

1− tB
qB

B ∥aB∥LrB (∂Ω)

(
1 + ∥u∥

2∗
(

1
qB

− 1
rB

)
H1(Ω)

)]
,

(3.19)

where m = min{q∗, NqB
N−1} (q∗ := Nq

N−q ), whenever 1 ≤ q < N , see Theorem 4.1. Fixing

qB :=
(N − 1)q∗

N
=⇒ m = q∗ =

NqB
N − 1

> N, (3.20)

(in the forthcoming Remark 3.1, we explain the necessity of the election for qB), moreover, we
have the following equivalences

qB :=
(N − 1)q∗

N
⇐⇒ 2∗

qB
=

2∗

q∗
⇐⇒ 2∗,N/qB = 2∗N/q. (3.21)

Indeed, we only have to notice that, using the definitions (2.2), (2.3) and (3.20), we can conclude
that

2∗,N/qB = 2∗ −
2∗
qB

= 2∗ +
2∗

N
− 2∗

q
= 2∗ − 2∗

q
= 2∗N/q.

With that election of qB , we also need to restrict q in order to satisfy (3.12). Specifically

q ∈
(N
2
,min

{
r,

NrB
N − 1 + rB

})
. (3.22)



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 7

Note that, because of the definition of qB , see (3.20), and their restriction, (3.12), the following
inequality has to be satisfied

N − 1 <
(N − 1)q∗

N
= qB < rB .

By (3.4), we obtain that q∗ > N so (N−1)q∗

N > N − 1. Thus, we only need to check that

q∗ <
NrB
N − 1

⇐⇒ 1

q
− 1

N
>
N − 1

NrB
⇐⇒ 1

q
>
N − 1

NrB
+

1

N
⇐⇒ q <

NrB
N − 1 + rB

,

from which, using (3.4), and that
NrB

N − 1 + rB
< N, (3.23)

we conclude (3.22).

Step 2. Gagliardo-Nirenberg interpolation inequality. The Gagliardo-Nirenberg interpolation
inequality (see [12]), implies that there exists a constant C = C(N, q, |Ω|), such that

∥u∥L∞(Ω) ≤ C∥u∥σW 1,q∗ (Ω)∥u∥
1−σ
L2∗ (Ω)

, (3.24)

where
1

σ
= 1 + 2∗

( 2

N
− 1

q

)
. (3.25)

From (3.21), by the definition of 2∗N/q, see (2.3), it is easy to check that

1

σ
= 1 + 2∗

[(2−N

N

)
+

(
1− 1

q

)]
= 2∗N/q − 1. (3.26)

Substituting the estimate of ∥u∥W 1,m(Ω), see 3.19, and using (3.2) in the inequality (3.24), we
obtain

∥u∥L∞(Ω) ≤ C
[
M1− t

q ∥a∥Lr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )

H1(Ω)

)
+M

1− tB
qB

B ∥aB∥LrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)

H1(Ω)

)]σ
∥u∥(1−σ)

L2∗ (Ω)

≤ C
[
M (1− t

q )σ∥a∥σLr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )σ

H1(Ω)

)
+M

(1− tB
qB

)σ

B ∥aB∥σLrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)σ

H1(Ω)

)]
∥u∥(1−σ)

L2∗ (Ω)
,

(3.27)

We now look closely at the exponents of ∥u∥L∞(Ω) in the right-hand side, in order to achieve
our estimates. Taking into account the definitions of M and MB , see (3.1), that f and fB are
non-decreasing, and the definitions of the functions h and hB , see (2.8), we can write the following
relation between them,

M =
∥u∥2

∗
N/r−1

L∞(Ω)

h(∥u∥L∞(Ω))
and MB =

∥u∥2∗,N/rB
−1

L∞(Ω)

hB(∥u∥L∞(Ω))
. (3.28)

Moreover, using the definitions of t, see (3.6), and of 2∗N/p, see (2.3), we obtain

1− t

q
= 1− 2∗

2∗N/r − 1

(1
q
− 1

r

)
=

2∗N/q − 1

2∗N/r − 1
. (3.29)

Thus, because of the expression (3.29), we deduce(
2∗N/r − 1

)(
1− t

q

)
= (2∗,N/q − 1),

and because of the definition of σ, see (3.26),(
2∗N/r − 1

)(
1− t

q

)
σ = 1. (3.30)



8 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

Similarly, from the definitions of tB , see (3.14), and of 2∗,N/qB , see (2.3), we obtain

1− tB
qB

= 1− 2∗
2∗,N/rB − 1

( 1

qB
− 1

rB

)
=

2∗,N/qB − 1

2∗,N/rB − 1
. (3.31)

Likewise, since (3.31), the definition of σ, see (3.26), and the equivalences (3.21), we obtain(
2∗,N/rB − 1

) (
1− tB

qB

)
σ =

2∗,N/qB − 1

2∗,N/q − 1
= 1. (3.32)

Now, we divide both sides of the inequality (3.27) by ∥u∥L∞(Ω). Using the definitions of M and
MB , also the two expressions concerning σ; (3.30), (3.32), and the definition of aM , see (2.11), we
obtain

1 ≤ CaσM

( (
1 + ∥u∥2

∗( 1
q−

1
r )σ

H1(Ω)

)
h

1
2∗
N/r

−1
(∥u∥L∞(Ω))

+

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)σ

H1(Ω)

)
h

1
2∗,N/rB

−1

B (∥u∥L∞(Ω))

)
∥u∥(1−σ)

L2∗ (Ω)
. (3.33)

The definition of hm (see (2.10)), implies that

1

h

1
2∗
N/r

−1

m (∥u∥L∞(Ω))

= max

{
1

h
1

2∗
N/r

−1
(∥u∥L∞(Ω))

,
1

h
1

2∗,N/rB
−1

B (∥u∥L∞(Ω))

}
.

So, substituting this maximum in the inequality (3.33), we obtain

h

1
2∗
N/r

−1

m (∥u∥L∞(Ω)) ≤ CaσM

(
1 + ∥u∥2

∗( 1
q−

1
r )σ

H1(Ω) + ∥u∥
2∗(

1
qB

− 1
rB

)σ

H1(Ω)

)
∥u∥(1−σ)

L2∗ (Ω)
. (3.34)

The right-hand side in the above inequality is bounded above by a term with the largest exponent
of both addends. Let us denote this maximum by

EM := max
{
2∗
(1
q
− 1

r

)
, 2∗

( 1

qB
− 1

rB

)}
. (3.35)

From inequality (3.34), definition (3.35) and Sobolev’s embedding, we obtain

hm(∥u∥L∞(Ω)) ≤ CaθM

(
1 + ∥u∥βH1(Ω)

)
, (3.36)

where

θ :=
(
2∗N/r − 1

)
σ =

2∗N/r − 1

2∗N/q − 1
, (3.37)

β :=
(
EM +

1− σ

σ

)
θ. (3.38)

Now, we look closely at the definition of EM . Firstly by definitions of 2∗ and of 2∗, see (2.3),
secondly by election of qB , see (3.20), and finally rearranging terms, we observe that

2∗
(1
q
− 1

r

)
≥ 2∗

( 1

qB
− 1

rB

)
⇐⇒ 1

q
∓ 1

N
− 1

r
≥ N − 1

N

( 1

qB
− 1

rB

)
⇐⇒ 1

r
− 1

N
≤ N − 1

NrB
.

(3.39)

If r ≥ N , then 1/r − 1/N ≤ 0, and the last inequality holds. Moreover, if N/2 < r < N , then the
last inequality holds if and only if

(1/r − 1/N)
−1

=: r∗ ≥ NrB/(N − 1).

Observe that

r∗ ≥ NrB/(N − 1) ⇐⇒ 2∗N/r ≥ 2∗,N/rB .



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 9

On the contrary, the reverse inequality to (3.39), will be satisfied whenever N/2 < r < N , and
r∗ ≤ NrB/(N − 1). Hence

EM =

{
2∗
(
1
q − 1

r

)
if r ≥ N or N/2 < r < N and r∗ ≥ NrB

N−1 ,

2∗
(

1
qB

− 1
rB

)
if N/2 < r < N and r∗ ≤ NrB

N−1 .
(3.40)

Consequently, we have two cases in the search for the optimum exponents θ and β varying q, see
(3.22),

Case (I): Either r ≥ N , or N/2 < r < N and r∗ ≥ NrB
N−1 . Using the definition of β, (3.38), the

first equality for EM in (3.40), and the expression for σ, see (3.25), and for 2∗N/r, see (2.3), we

have

β =
[
2∗
(1
q
− 1

r

)
+

1− σ

σ

]
θ (3.41)

=
[
2∗
(1
q
− 1

r

)
+ 2∗

( 2

N
− 1

q

)]
θ = (2∗N/r − 2)θ. (3.42)

The function θ : q 7→ θ(q), defined by (3.37) is decreasing. We look for the infimum for q in the
interval (3.4).

Assume r ≥ N . Since (3.22)–(3.23), we deduce that q ∈
(
N
2 ,

NrB
N−1+rB

)
.

Assume N/2 < r < N and r∗ ≥ NrB
N−1 . Note that

r∗ ≥ NrB
N − 1

⇐⇒ 1

r
− 1

N
≤ N − 1

NrB
⇐⇒ r ≥ NrB

N − 1 + rB
, (3.43)

and q ∈
(
N
2 ,

NrB
N−1+rB

)
.

Hence, in case I,

inf
q∈(N

2 ,
NrB

N−1+rB
)

θ(q) = θ
( NrB
N − 1 + rB

)
=

1
2 − N−r

Nr
1
2 − N−1

NrB

. (3.44)

Case (II): N/2 < r < N and r∗ ≤ NrB
N−1 . Likewise, using the definition of β, (3.38), the second

equality for EM in (3.40), and the expressions for σ (3.25), for 2∗N/p, 2∗,N/pB
(2.3), and the

equivalence (3.21), we have

β =
[
2∗

( 1

qB
− 1

rB

)
+

1− σ

σ

]
θ

=
[
2∗

( 1

qB
∓ 1− 1

rB

)
+ 2∗

( 2

N
∓ 1− 1

q

)]
θ

=
[
2∗,N/rB − 2∗,N/qB − 2 + 2∗N/q

]
θ

= [2∗,N/rB − 2]θ.

(3.45)

For q satisfying (3.22), thanks to (3.43), we deduce that q ∈ (N2 , r), hence

inf
q∈(N

2 ,r)
θ(q) = θ(r) = 1. (3.46)

Finally, we introduce into the inequality (3.36), the infima of θ and β given by (3.44) and (3.42)
respectively in case I, and by (3.46) and (3.45), in case II. Since these infima are not attained in
the set where q belongs, for any ε > 0, there exists a constant Cε > 0 such that

hm(∥u∥L∞(Ω)) ≤ Cεa
A+ε
M

(
1 + ∥u∥(2

∗
N/r−2)(A+ε)

H1(Ω)

)
,

where A is defined in (2.13), and Cε = Cε(ε,N, |Ω|, |∂Ω|) and it is independent of u. □

In the following Remark, we state the necessity of the election for qB , see (3.20).



10 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

Remark 3.1. Assume that (3.20) does not hold and, to fix ideas, that

qB <
(N − 1)q∗

N
=⇒ m =

NqB
N − 1

> N.

We also have the equivalence

qB <
(N − 1)q∗

N
⇐⇒ 2∗

q∗
<

2∗
qB

⇐⇒ 2∗,N/qB < 2∗N/q.

Indeed, the first equivalence is obvious. With respect to the second one, notice that, due to the
definitions of 2∗N/q and of 2∗,N/qB , see (2.3), we can conclude that

2∗,N/qB = 2∗ −
2∗
qB

< 2∗ −
2∗

q∗
= 2∗ − 2∗

q
= 2∗N/q.

Now, in the Gagliardo-Nirenberg interpolation inequality, see (3.24), the parameter σ is given by

1

σ
= 1 + 2∗

( 1

N
− 1

m

)
= 1 + 2∗

( 1

N
∓ 1− N − 1

NqB

)
= 2∗,N/qB − 1

< 2∗N/q − 1.

And the expression (3.30) becomes(
2∗N/r − 1

)(
1− t

q

)
σ =

2∗,N/q − 1

2∗,N/qB − 1
> 1.

The above inequality implies that the exponent of ∥u∥L∞(Ω) in the right-hand side will dominate
1, which is the exponent of ∥u∥L∞(Ω) in the LHS, and the bounds can not be reached.

Likewise, if qB > (N−1)q∗

N , then m = q∗ and it can be proved that

1

σ
= 1 + 2∗

( 2

N
∓ 1− 1

q

)
= 2∗N/q − 1 < 2∗,N/qB − 1,

and so (
2∗,N/rB − 1

) (
1− tB

qB

)
σ = (2∗,N/qB − 1)σ =

2∗,N/qB − 1

2∗N/q − 1
> 1,

concluding that necessarily, qB has to be chosen as in (3.20).

Throughout that proof, we have explicit estimates of hm(∥u∥L∞(Ω)) expressed in their L2∗(Ω)

norm and L2∗(∂Ω) norm (see (3.7) and (3.15)). Previously, we unify those estimates in their
H1(Ω) norm to simplify the expression. In the next Corollary, we split those estimates in terms
of the L2∗(Ω) norm and the L2∗(∂Ω) norm.

Corollary 3.2. Assume that the hypotheses of Theorem 2.2 hold. Then, for all ε > 0, there exists
Cε > 0 depending of ε, N , |Ω| and |∂Ω|, but independent of u, such that

hm(∥u∥L∞(Ω)) ≤ CaA+ε
M

(
1 + ∥u∥A1+ε

L2∗ (Ω)
+ ∥u∥A2+ε

L2∗ (∂Ω)∥u∥
A3+ε
L2∗ (Ω)

)
,

where A is defined in (2.13), with

A1 := (2∗N/r − 2)A, A2 := 0, A3 := (2∗,N/rB − 2)A,

if either r ≥ N or N/2 < r < N and r∗ ≥ NrB
N−1 ; and

A1 := 2∗N/r − 2, A2 := 2∗,N/rB − 2∗N/r, A3 := 2∗N/r − 2,

if N/2 < r < N and r∗ ≤ NrB
N−1 .



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 11

Proof. The proof is similar to the proof of the Theorem (2.2).

Step 1. W 1,m(Ω) estimates for m > N . Substituting (3.7) in the second factor on the right-hand
side of (3.5), and this is (3.3), we obtain(∫

Ω

|f(x, u)|qdx
)1/q

≤ CM1− t
q ∥a∥Lr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )

L2∗ (Ω)

)
, (3.47)

with M defined in (3.1), t in (3.6) and q in (3.4), respectively. See the analogy with (3.10).
On the other hand, replacing (3.15) in the second factor on the right-hand side of (3.13), and

this in (3.11), we obtain(∫
∂Ω

|fB(x, u)|qBdx
)1/qB

≤ CM
1− tB

qB

B ∥aB∥LrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)

L2∗ (∂Ω)

)
, (3.48)

with MB defined in (3.1), tB in (3.14), and qB in (3.12) respectively. See the analogy with (3.18).
By elliptic regularity, we estimate the norm ∥u∥W 1,m(Ω) in terms of (3.47) and (3.48), see

Theorem 4.1, obtaining

∥u∥W 1,m(Ω)

≤ C
[
M1− t

q ∥a∥Lr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )

L2∗ (Ω)

)
+M

1− tB
qB

B ∥aB∥LrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)

L2∗ (∂Ω)

)]
,

(3.49)

with m > N . See also the analogy with (3.19).

Step 2. Gagliardo-Nirenberg interpolation inequality. Substituting (3.49) in the Gagliardo-
Nirenberg inequality (3.24) and using the inequality (3.2), we obtain

∥u∥L∞(Ω) ≤ C
[
M (1− t

q )σ∥a∥σLr(Ω)

(
1 + ∥u∥2

∗( 1
q−

1
r )σ

L2∗ (Ω)

)
+M

(1− tB
qB

)σ

B ∥aB∥σLrB (∂Ω)

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)σ

L2∗ (∂Ω)

)]
∥u∥(1−σ)

L2∗ (Ω)
,

(3.50)

Using the definitions of M and MB , see (3.28), using also (3.30), (3.32), the definition of aM (see
(2.11)), and dividing both sides of the inequality (3.50) by ∥u∥L∞(Ω), we obtain

1 ≤ CaσM

( (
1 + ∥u∥2

∗( 1
q−

1
r )σ

L2∗ (Ω)

)
h

1
2∗
N/r

−1
(∥u∥L∞(Ω))

+

(
1 + ∥u∥

2∗(
1

qB
− 1

rB
)σ

L2∗ (∂Ω)

)
h

1
2∗,N/rB

−1

B (∥u∥L∞(Ω))

)
∥u∥(1−σ)

L2∗(Ω) .

Then

h

1
2∗
N/r

−1

m (∥u∥L∞(Ω)) ≤ CaσM

(
2 + ∥u∥2

∗( 1
q−

1
r )σ

L2∗ (Ω)
+ ∥u∥

2∗(
1

qB
− 1

rB
)σ

L2∗ (∂Ω)

)
∥u∥(1−σ)

L2∗ (Ω)
.

where hm is defined in (2.10). See the analogy with (3.34). Clearing, we obtain

hm(∥u∥L∞(Ω)) ≤ Ca
σ(2∗N/r−1)

M

(
1+∥u∥2

∗( 1
q−

1
r )(2

∗
N/r−1)σ

L2∗ (Ω)
+∥u∥

2∗(
1

qB
− 1

rB
)(2∗N/r−1)σ

L2∗ (∂Ω)

)
∥u∥(1−σ)(2∗N/r−1)

L2∗ (Ω)
.

Substituting in the exponents the parameter θ (see (3.37)), we obtain

hm(∥u∥L∞(Ω)) ≤ CaθM

(
1 + ∥u∥[2

∗( 1
q−

1
r )+

1−σ
σ ]θ

L2∗ (Ω)
+ ∥u∥

2∗(
1

qB
− 1

rB
)θ

L2∗ (∂Ω) ∥u∥
1−σ
σ θ

L2∗ (Ω)

)
. (3.51)

Let us define the function θ1 = θ1(q) as the first exponent inside the brackets. Using the
definitions of 2∗N/q, see (2.3), and of σ, see (3.26), we obtain

θ1(q) :=
[
2∗
(1
q
− 1

r

)
+

1− σ

σ

]
θ(q) = (2∗N/r − 2)θ(q),

note that this value is equal to β in case I of Theorem (2.2), see (3.41)–(3.42).
We define the function θ2 = θ2(q) as the second exponent. By the definition of 2∗,N/qB , see

(2.3), and the equivalence (3.21),

θ2(q) := 2∗

( 1

qB
∓ 1− 1

rB

)
θ(q) = (2∗,N/rB − 2∗N/q)θ(q).



12 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

We define the function θ3 = θ3(q) as the third exponent. Using the expression (3.26) for σ,

θ3(q) :=
(1− σ

σ

)
θ(q) = (2∗N/q − 2)θ(q).

As before, let qB , θ be defined by (3.20), and (3.37) respectively. The function

(θ1 + θ2 + θ3)(q) = (2∗N/r + 2∗,N/rB − 4)θ(q),

is decreasing, and we look for their infimum for q in the interval (3.22). Thus, as before, we
consider the previous two cases.

Case (I) Either r ≥ N , or N/2 < r < N and r∗ ≥ NrB
N−1 . In this case, q ∈ (N2 ,

NrB
N−1+rB

). For A

defined in (2.13), the exponents are

A′
1 := θ1

( NrB
N − 1 + rB

)
= (2∗N/r − 2)A,

A′
2 := θ2

( NrB
N − 1 + rB

)
=

2

N − 2

(N − 1 + rB
rB

− 1− N − 1

rB

)
A = 0,

and

A′
3 := θ3

( NrB
N − 1 + rB

)
=

(
2∗
(
1− N − 1 + rB

NrB

)
− 2

)
A

=
(2(N − 2)

N − 2

(rB − 1

rB

)
− 2

)
A

= (2∗,N/rB − 2)A.

Hence, inequality (3.51) can be rewritten as

hm(∥u∥L∞(Ω)) ≤ CaA+ε
M

(
1 + ∥u∥(2

∗
N/r−2)A+ε

L2∗ (Ω)
+ ∥u∥εL2∗ (∂Ω)∥u∥

(2∗,N/rB
−2)A+ε

L2∗ (Ω)

)
,

where A is defined in (2.13).

Case (II) N/2 < r < N and r∗ ≤ NrB
N−1 . In that case, q ∈ (N2 , r) (see (3.22)). The exponents are

A′′
1 := θ1 (r) = 2∗N/r − 2,

A′′
2 := θ2(r) =

2

N − 2

(N
r

− 1∓N − N − 1

rB

)
= 2∗,N/rB − 2∗N/r,

A′′
3 := θ3(r) = 2∗N/r − 2.

Therefore, inequality (3.51) is rewritten as

hm(∥u∥L∞(Ω)) ≤ Cεa
1+ε
M

(
1 + ∥u∥2

∗
N/r−2+ε

L2∗ (Ω)
+ ∥u∥2∗,N/rB

−2∗N/r+ε

L2∗ (∂Ω) ∥u∥2
∗
N/r−2+ε

L2∗ (Ω)

)
. □

The next corollary proves that any sequence {uk} ⊂ H1(Ω) of weak solution to (1.1), uniformly
bounded in the L2∗(Ω) norm and in the L2∗(∂Ω) norm, is also uniformly bounded in the C(Ω)-
norm.

Corollary 3.3. Assume (H1)–(H4) hold and let {uk} ⊂ H 1(Ω) be a sequence of weak solutions
to (1.1) satisfying that, there exists C0 > 0, such that

∥uk∥L2∗ (Ω) ≤ C0 and ∥uk∥L2∗ (∂Ω) ≤ C0.

Then, there exists C > 0 such that,
∥uk∥C(Ω) ≤ C.

Proof. We proceed by contradiction, assuming that ∥uk∥L∞(Ω) → ∞. By the Theorem (2.2) and
the remark 3.1, we obtain

hm(∥uk∥C(Ω)) ≤ C, for C > 0, (3.52)

where, hm is defined in (2.10). Using (2.9), we deduce that hm(∥uk∥C(Ω)) → ∞ as k → ∞, which

contradicts (3.52). □

Corollary 3.4. Assume (H1)–(H4) hold and let {uk} ⊂ H 1(Ω) be a sequence of weak solutions
to (1.1). Then the following statements are equivalent:



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 13

(i) ∥uk∥L2∗ (Ω) ≤ C1 and ∥uk∥L2∗ (∂Ω) ≤ C1,

(ii) ∥uk∥C(Ω) ≤ C3,

(iii) ∥uk∥H1(Ω) ≤ C2.

for some constants Ci independent of k, i = 1, 2, 3.

Proof. We prove that (i) ⇒ (ii) ⇒ (iii) ⇒ (i).
The proof of (i) ⇒ (ii) follows directly from the Corollary 3.3. Now, using the elliptic regularity

result, see the estimate (5.1) in the Theorem 5.1, and the Gagliardo-Nirenberg interpolation, the
proof of (ii) ⇒ (iii) is done. Finally, Sobolev’s embedding and the continuity of the trace operator,
proves that (iii) ⇒ (i). □

4. Appendix: Regularity for the Neumann non homogeneous linear problem

In this appendix, we recall the regularity of weak solution to the linear problem with non homo-
geneous data both at the interior and on the boundary. Let us consider the linear nonhomogeneous
Neumann problem

−∆u+ u = g(x), x ∈ Ω,

∂u

∂η
= gB(x), x ∈ ∂Ω,

(4.1)

where Ω ⊂ RN , (N > 2), is an open, connected and bounded domain with C2 boundary.

Theorem 4.1. Let us consider the problem (4.1), there exists a positive constant C > 0 indepen-
dent of u, h and gB such that the following holds:

(i) If ∂Ω ∈ C0,1, g ∈ Lq(Ω) and gB ∈ LqB (∂Ω) with q ≥ 1 and qB ≥ 1, then there exists a
unique u ∈W 1,m(Ω) and

∥u∥W 1,m(Ω) ≤ C
(
∥g∥Lq(Ω) + ∥gB∥LqB (∂Ω)

)
, (4.2)

where m = min{ Nq
N−q ,

NqB
N−1} whenever 1 ≤ q < N , or m = min{q, NqB

N−1} whenever q ≥ N .

Furthermore, if q > N
2 and qB > N − 1, then

∥u∥Cν(Ω) ≤ C
(
∥g∥Lq(Ω) + ∥gB∥LqB (∂Ω)

)
,

where ν = 1− N
m , (m > N).

(ii) If ∂Ω ∈ C1,1, g ∈ Cν(Ω) ∩ Lq(Ω) and gB ∈ LqB (∂Ω) with q > N
2 and qB > N − 1, then

there exists a unique u ∈ Cν(Ω) ∩ C2,ν(Ω).
(iii) If ∂Ω ∈ C2,ν , g ∈ Cν(Ω) and gB ∈ C1,ν(∂Ω) with ν ∈ (0, 1), then there exists a unique

u ∈ C2,ν(Ω) and

∥u∥C2,ν(Ω) ≤ C
(
∥g∥Cν(Ω) + ∥gB∥C1,ν(∂Ω)

)
,

where C is a positive constant independent of u, g and gB.

(iv) If ∂Ω ∈ C2, g ∈ Lp(Ω) and gB ∈W 1− 1
p ,p(∂Ω), then u ∈W 2,p(Ω) and

∥u∥W 2,p(Ω) ≤ C
(
∥g∥Lp(Ω) + ∥gB∥

W
1− 1

p
,p
(∂Ω)

)
,

where C is a positive constant independent of u, g and gB.
(iv) If ∂Ω ∈ C1,ν with ν ∈ (0, 1], g ∈ Cν(Ω) and gB ∈ Cν(∂Ω)∩L∞(∂Ω) then if u is a bounded

weak solution to (4.1), then u ∈ C1,β(Ω) ∩ C2,β(Ω), where β depends on ν and N .

Proof. (i) It follows from [7, Ch.3 Sec. 6] or [10, Lem. 2.2] that there exists a unique u ∈W 1,p(Ω)
solving (4.1). Now if p > N , using the Sobolev embedding theorem, one has u ∈ Cα(Ω). Then by
applying [5, Thm. 6.13] for the corresponding nonhomogeneous Dirichlet problem, we have that
u ∈ C1,α(Ω), see also [10].

(ii) From part (i) we have that u ∈ Cα(Ω). Since ∂Ω ∈ C1,1, Ω satisfies the exterior sphere
condition at every point on the boundary and using the fact that g ∈ Cα(Ω), reasoning as above
it follows from [5, Thm. 6.13] that u ∈ Cα(Ω) ∩ C2,α(Ω).

(iii) See [1, Page 55] or [7, Chap.3 Sec. 3].
(iv) See [1, Page 55] or [7, Chap.3 Sec. 9].



14 E. ANTONIO, M. P. ÁRCIGA-ALEJANDRE, R. PARDO, J. SÁNCHEZ-ORTIZ EJDE-2025/95

(v) By [8, Thm. 2], one has u ∈ C1,β(Ω). Then using the bootstrap for the differential equation
in Ω, we obtain the desired regularity in Ω. □

5. Appendix: Regularity of weak solutions

In this section, we establish auxiliary results on further regularity of weak solutions to (1.1), by
assuming that conditions on the growth of the nonlinearities are subcritical or even critical. Using
a Moser type procedure, it is known that u ∈ Lq(Ω) ∩ Lq(∂Ω) for all q < ∞ (see [9, Theorem
3.1]). Moreover, using elliptic regularity theory, we state the following result that guarantees, in
particular, Hölder regularity of any weak solution to (1.1).

Theorem 5.1. Let Ω ⊂ RN , f : Ω × R → R and fB : ∂Ω × R → R be Carathéodory functions,
such that

|f(x, s)| ≤ |a(x)|
(
1 + |s|2

∗
N/r−1

)
,

|fB(x, s)| ≤ |aB(x)|
(
1 + |s|2∗,N/rB

−1
)
,

where

a(x) ∈ Lr(Ω), with
N

2
< r ≤ ∞,

aB(x) ∈ LrB (∂Ω), with N − 1 < rB ≤ ∞.

Let u ∈ H1(Ω) be a weak solution to (1.1), then u ∈ Lq(Ω)∩Lq(∂Ω) for all 1 ≤ q <∞. Moreover,
u ∈W 1,m(Ω) ∩ Cν(Ω), and the following estimates hold

∥u∥W 1,m(Ω) ≤ C
(
∥f(·, u)∥Lr(Ω) + ∥fB(·, u)∥LrB (∂Ω)

)
, (5.1)

∥u∥Cν(Ω) ≤ C
(
∥f(·, u)∥Lr(Ω) + ∥fB(·, u)∥LrB (∂Ω)

)
, (5.2)

where m = min
{
r∗, NrB

N−1

}
, if 1 ≤ r < N , or m = min

{
r, NrB

N−1

}
, if r ≥ N and ν = 1− N

m . Also

∥u∥L∞(∂Ω) ≤ ∥u∥C(Ω) = ∥u∥L∞(Ω).

Proof. Let u ∈ H1(Ω) be a weak solution to (1.1). Then u ∈ Lq(Ω) ∩ Lq(∂Ω) for all q < ∞ (see
[9, Theorem 3.1]).

Next, we use elliptic regularity theory. By Hölder’s inequality, we have

f(·, u) ∈ Lq(Ω), for every 1 < q < r,

fB(x, u) ∈ LqB (∂Ω), for every 1 < qB < rB ,

By elliptic regularity (see Theorem (4.1)), u ∈W 1,m(Ω) for m = min{q∗, NqB
N−1} whenever 1 ≤ q <

N , or m = min{q, NqB
N−1}, whenever q ≥ N.

Thanks to r > N/2 and rB > N − 1, we can always choose

q ∈ (N/2, r), qB ∈ (N − 1, rB).

Then m > N , so u ∈ Cν(Ω) for ν = 1− N
m .

Moreover, since u ∈ Cν(Ω), a ∈ Lr(Ω) and f̃ ∈ C(Ω), using the Hölder inequality, then, the

product |a(·)|f̃(|u(·)|) ∈ Lr(Ω). Hence, f(·, u(·)) ∈ Lr(Ω).

Similarly, if u ∈ Cν(Ω), aB ∈ LrB (∂Ω) and f̃B ∈ C(∂Ω), by Hölder inequality, then, the product

|aB(·)|f̃B(|u(·)|) ∈ LrB (∂Ω). Hence, we can conclude that fB(·, u(·)) ∈ LrB (∂Ω). Then (5.1) and
(5.2) hold, completing the proof. □

Acknowledgements. Rosa Pardo was supported by the MICINN, Spain, grant PID2022-137074NB-
I00, and by the UCM, Spain, Grupo 920894. Edgar Antonio received support from a CONAHCYT
scholarship, and from the Consortium of Mexican Universities-Ibero-American University Post-
graduate Association (CUMex-AUIP ) for the flight ticket and living expenses at the Universidad
Complutense de Madrid from January 7 to August 2, 2024.



EJDE-2025/95 UNIFORM ESTIMATES FOR ELLIPTIC EQUATIONS 15

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Edgar Antonio
Universidad Complutense de Madrid, Madrid, España.

Universidad Autónoma de Guerrero, Guerrero, México
Email address: eaam020713@gmail.com, eantonio@ucm.es

Mart́ın P. Árciga-Alejandre

Universidad Autónoma de Guerrero, Guerrero, México
Email address: mparciga@uagro.mx

Rosa Pardo

Universidad Complutense de Madrid, Madrid, Spain
Email address: rpardo@ucm.es

Jorge Sánchez-Ortiz
Universidad Autónoma de Guerrero, Guerrero, México

Email address: jsanchez@uagro.mx


	1. Introduction
	2. Main result
	3. L() a priori explicit estimates
	4. Appendix: Regularity for the Neumann non homogeneous linear problem
	5. Appendix: Regularity of weak solutions
	Acknowledgements

	References