Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 11, pp. 1–13.

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

DOI: 10.58997/ejde.2024.11

DIRICHLET PROBLEMS WITH ANISOTROPIC PRINCIPAL

PART INVOLVING UNBOUNDED COEFFICIENTS

DUMITRU MOTREANU, ELISABETTA TORNATORE

Abstract. This article establishes the existence of solutions in a weak sense

for a quasilinear Dirichlet problem exhibiting anisotropic differential operator
with unbounded coefficients in the principal part and full dependence on the

gradient in the lower order terms. A major part of this work focuses on the

existence of a uniform bound for the solution set in the anisotropic setting.
The unbounded coefficients are handled through an appropriate truncation

and a priori estimates.

1. Introduction

The aim of this article is to study quasilinear elliptic equations driven by an
anisotropic differential operator with unbounded coefficients and that have the re-
action term in the form of convection (i.e., it jointly depends on the solution and
its gradient). Specifically, we state the following Dirichlet problem

−
N∑
i=1

∂i(Gi(u)|∂iu|pi−2∂iu) = F (x, u,∇u) in Ω

u = 0 on ∂Ω

(1.1)

on a bounded domain Ω in RN (N ≥ 2) with a Lipschitz boundary ∂Ω. The notation
∂i stands for the distributional partial derivative with respect to the variable xi, i.e.,
∂i = ∂/∂xi , and ∇u = (∂1, . . . , ∂N ) is the gradient of u. In (1.1) there are given real
numbers pi ∈ (1,+∞) with i = 1, . . . , N , continuous functions Gi : R → [ai,+∞),
with ai > 0 for i = 1, . . . , N , and a Carathéodory function F : Ω × R × RN → R
(i.e., F (·, t, ξ) is measurable on Ω for each (t, ξ) ∈ R×RN and F (x, ·, ·) is continuous
on R × RN for almost all x ∈ Ω). The notation Gi(u) means the composition of
the function Gi : R→ R with the solution u : Ω→ R.

Set −→p := (p1, . . . , pN ) and denote by W 1,−→p
0 (Ω) the completion of the set of

smooth functions with compact support C∞c (Ω) with respect to the norm

‖u‖ :=

N∑
i=1

‖∂iu‖Lpi .

2020 Mathematics Subject Classification. 35J70, 35J92.
Key words and phrases. Anisotropic elliptic equation; anisotropic Sobolev space;

unbounded coefficient; bounded solution; truncation; pseudomonotone operator.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted March 16, 2023. Published January 30, 2024.

1



2 D. MOTREANU, E. TORNATORE EJDE-2024/11

Therefore W 1,−→p
0 (Ω) is a reflexive Banach space. This is the natural underlying

space associated to problem (1.1). We mention that the completion of C∞c (RN )

with respect to the above norm is the important space D1,−→p .
A significant feature of problem (1.1) is that the differential operator driving

the equation is anisotropic, thus admitting to have unequal pi. This causes lack of
homogeneity and lack of radial scaling. In this respect, the simplest case in (1.1) is
when Gi ≡ 1 for all i = 1, . . . , N , which reads

−∆−→p u = F (x, u,∇u) in Ω

u = 0 on ∂Ω.
(1.2)

The operator in the left-hand side of equation (1.2) is the (negative) anisotropic
−→p -Laplacian −∆−→p : W 1,−→p

0 (Ω)→W 1,−→p
0 (Ω)∗ defined by

〈−∆−→p u, v〉 =

N∑
i=1

∫
Ω

|∂iu(x)|pi−2∂iu(x)∂iv(x)dx (1.3)

for all u, v ∈ W 1,−→p
0 (Ω). We emphasize that the operator −∆−→p in (1.3) has prop-

erties essentially different with respect to its isotropic counterpart which is the
(negative) p-Laplacian −∆p : W 1,p

0 (Ω) → W−1,p′(Ω). For example, we point
out the nonexistence of the first eigenvalue in the case of −∆−→p . The known re-
sults on anisotropic elliptic problems concern the case where the driven operator
is −∆−→p and the reaction term does not depend on the gradient of the solution,
i.e., F (x, u,∇u) = f(x, u). Due to these restrictions, a variational approach can
be implemented. For relevant results in this direction we refer to [4, 5, 6, 9, 10].
The recent paper [3] deals with anisotropic elliptic problems with a leading operator
more general than −∆−→p . In addition to the major mathematical interest, there is a
strong physical motivation for such type of problems. We highlight for example the
applications in fluid mechanics involving anisotropic media where the conductivity
depends on the direction (we refer to [1] for a comprehensive description).

The degree of difficulty regarding problem (1.1) is even higher due to the fact
that the variable coefficients Gi(u) may be unbounded. Isotropic problems with
unbounded coefficients complying with (1.1) have been considered in [7] and [8].
Given a real number p ∈ (1,+∞), in [7] it is investigated the problem

−div(G(u)|∇u|p−2∇u) = F (x, u,∇u) in Ω

u = 0 on ∂Ω

for a continuous function G : R → [a0,+∞) with a0 > 0, whereas [8] is concerned
with the weighted problem

−div(a(x)g(|u|)|∇u|p−2∇u) = f(x, u,∇u) in Ω,

u = 0 on ∂Ω

for a positive weight a ∈ L1
loc(Ω) and a continuous function g : [0,+∞)→ [a0,+∞)

with a0 > 0.
In this article we focus on equation (1.1) where the anisotropic leading operator

incorporates the unbounded coefficients Gi(u). By a weak solution to problem (1.1)

we mean any element u ∈ W 1,−→p
0 (Ω) such that Gi(u(x))|∂iu(x)|pi−2∂iu(x)∂iv(x),



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 3

with i = 1, . . . , N , and F (x, u(x),∇u(x))v(x) are integrable on Ω, and it holds

N∑
i=1

∫
Ω

Gi(u(x))|∂iu(x)|pi−2∂iu(x)∂iv(x)dx =

∫
Ω

F (x, u(x),∇u(x))v(x)dx (1.4)

for all v ∈W 1,−→p
0 (Ω).

The main contribution of this work is to build a coherent approach allowing
for the first time to study equations that are driven by an anisotropic differential
operator with unbounded coefficients and that exhibit a convection term (meaning
to have full dependence on the solution and its gradient). Our main result is
Theorem 2.3 below that provides under verifiable hypotheses the existence of a
weak solution to problem (1.1) in the sense of (1.4). Furthermore, we prove in
Theorem 3.2 the existence of a uniform bound for the solution set of problem (1.1).
An essential fact is that the uniform bound does not depend on the coefficients Gi

entering (1.1) except on the lower bound ai of Gi for every i = 1, . . . , N . The proof
of the main result relies on a truncation argument dropping the unboundedness of
the coefficients Gi(u) as well as on related a priori estimates. Another important
tool is an auxiliary problem for which the theory of pseudomonotone operators can
be applied.

The rest of this article is organized as follows. Section 2 presents the hypotheses
and the main result. In Section 3 it is carried out the proof that the solution set
to problem (1.1) is uniformly bounded. Section 4 deals with an auxiliary truncated
problem and an associated operator. Section 5 is devoted to the proof of the main
result that provides the existence of solutions to problem (1.1).

2. Statements of hypotheses and main result

For the rest of the paper we assume that

N∑
i=1

1

pi
> 1. (2.1)

Recall the critical exponent

p∗ :=
N∑N

i=1
1
pi
− 1

. (2.2)

If pi = p for all i = 1, . . . , N , then p∗ in (2.2) becomes the ordinary Sobolev
critical exponent when N > p. Under assumption (2.1), there are the continuous
embeddings

W 1,−→p
0 (Ω) ↪→ Lq(Ω) (2.3)

provided 1 ≤ q ≤ p∗, which are compact if 1 ≤ q < p∗ (see [6, Theorem 1]). We set

p := max{p1, . . . , pN} and p := min{p1, . . . , pN}

and further assume that

p < p∗. (2.4)

In view of (2.3), there is a constant θ > 0 such that

‖u‖pLp ≤ θ‖u‖p , ∀u ∈W 1,−→p
0 (Ω). (2.5)



4 D. MOTREANU, E. TORNATORE EJDE-2024/11

To simplify the presentation, for any real number r > 1 we denote r′ := r/(r−1)
(the Hölder conjugate of r). The strong and weak convergence are denoted by →
and ⇀, respectively.

The Carathéodory function F : Ω × R × RN → R describing the reaction term
in (1.1) is subject to the following hypotheses:

(H1) There exist constants c1 ≥ 0, c2 ≥ 0, c3 ≥ 0, and r ∈ (p, p∗) such that

|F (x, t, ξ)| ≤ c1
( N∑

i=1

|ξi|pi

)1/r′

+ c2|t|r−1 + c3

for a.e. x ∈ Ω, all t ∈ R, and ξ = (ξ1, . . . , ξN ) ∈ RN .
(H2) There exist constants d1 ≥ 0 and d2 ≥ 0 with d1 + Np−1d2θ < ai for all

i = 1, . . . , N , and a function σ ∈ L1(Ω) such that

F (x, t, ξ)t ≤ d1

N∑
i=1

|ξi|pi + d2|t|p + σ(x)

for a.e. x ∈ Ω, all t ∈ R, and ξ = (ξ1, . . . , ξN ) ∈ RN .

Next we focus on the Nemytskii operator determined by the function F .

Proposition 2.1. Assuming (2.1), (2.4) and (H1), the map N : W 1,−→p
0 (Ω) →

Lr′(Ω) given by

Nu = F (x, u,∇u), ∀u ∈W 1,−→p
0 (Ω),

is well defined, bounded (in the sense it maps bounded sets into bounded sets) and
continuous.

Proof. By (H1) and the convexity of the function t 7→ tr
′

for t > 0, we obtain the
estimate∫

Ω

|F (x, u,∇u)|r
′
dx ≤ C

( N∑
i=1

∫
Ω

|∂iu|pidx+

∫
Ω

|u|rdx+ 1
)
, ∀u ∈W 1,−→p

0 (Ω),

with a constant C > 0. Since ∂iu ∈ Lpi(Ω) for all i = 1, . . . , N and u ∈ Lr(Ω)

(note r < p∗) whenever u ∈ W 1,−→p
0 (Ω), we obtain that Nu ∈ Lr′(Ω). The obtained

estimate shows that the mapping N : W 1,−→p
0 (Ω) → Lr′(Ω) is well defined and

bounded.
To show the continuity of the mappingN , let un → u inW 1,−→p

0 (Ω). The definition

of the space W 1,−→p
0 (Ω) and the continuous embedding W 1,−→p

0 (Ω) ⊂ Lr(Ω) imply that
∂i(un) → ∂iu in Lpi(Ω) for i = 1, . . . , N , and un → u in Lr(Ω). Since r′ > 1, the
growth condition in assumption (H1) yields

|F (x, t, ξ)| ≤ c1
N∑
i=1

|ξi|
pi
r′ + c2|t|r−1 + c3

for a.e. x ∈ Ω, all t ∈ R, and ξ = (ξ1, . . . , ξN ) ∈ RN . Taking into account
that un → u in Lr(Ω) and ∂i(un)→ ∂iu in Lpi(Ω) for i = 1, . . . , N , Krasnoselkii’s
classical theorem concerning the continuity of a Nemytskii operator guarantees that
F (x, un,∇un)→ F (x, u,∇u) in Lr′(Ω). The stated conclusion follows. �



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 5

Corollary 2.2. Assume that conditions (2.1), (2.4) and (H1) are fulfilled. If un →
u in W 1,−→p

0 (Ω), then

lim
n→∞

〈N (un), un − u〉 = 0.

Proof. We note that by Hölder’s inequality,

|〈N (un), un − u〉| =
∣∣ ∫

Ω

F (x, un,∇un)(un − u)dx
∣∣

≤ ‖F (x, un,∇un)‖Lr′‖un − u‖Lr .

By the continuous embedding (2.3) we know that un → u in Lr(Ω) while Propo-

sition 2.1 entails that the sequence F (x, un,∇un) is bounded in Lr′(Ω). Conse-
quently, the thesis is valid. �

Now we state our main result providing the existence of a bounded weak solution
to problem (1.1).

Theorem 2.3. Assume that conditions (2.1) and (2.4) hold, Gi : R → [ai,+∞),
with ai > 0 for i = 1, . . . , N , are continuous functions, and F : Ω×R×RN → R is
a Carathéodory function satisfying hypotheses (H1) and (H2). Then problem (1.1)

has at least a weak solution u ∈ W 1,−→p
0 (Ω) in the sense of (1.4). Moreover, the

solution u is bounded.

The proof of Theorem 2.3 is given in Section 5.

3. Uniformly bounded solution set

Our first objective is to estimate the solutions in W 1,−→p
0 (Ω).

Lemma 3.1. Assume that conditions (2.1), (2.4), (H1) and (H2) hold. Then the

set of solutions to problem (1.1) is bounded in W 1,−→p
0 (Ω) with a bound that depends

on the function Gi only through the lower bound ai of Gi for i = 1, . . . , N .

Proof. Let u ∈ W 1,−→p
0 (Ω) be a weak solution of (1.1). Equality (1.4) with v = u

gives
N∑
i=1

∫
Ω

Gi(u(x))|∂iu(x)|pidx =

∫
Ω

F (x, u,∇u)u dx.

Then hypothesis (H2) yields

N∑
i=1

ai‖∂iu‖pi

Lpi ≤ d1

N∑
i=1

‖∂iu‖pi

Lpi + d2‖u‖
p

Lp + ‖σ‖L1 .

Using (2.5), we obtain

N∑
i=1

(ai − d1)‖∂iu‖pi

Lpi ≤ d2θ(

N∑
i=1

‖∂iu‖Lpi )p + ‖σ‖L1

≤ Np−1d2θ

N∑
i=1

‖∂iu‖
p

Lpi + ‖σ‖L1

≤ Np−1d2θ(N +

N∑
i=1

‖∂iu‖pi

Lpi ) + ‖σ‖L1 .



6 D. MOTREANU, E. TORNATORE EJDE-2024/11

From the condition d1 +Np−1d2θ < ai for all i = 1, . . . , N , imposed in assumption
(H2), the conclusion is achieved. �

Now we show that the solution set of problem (1.1) is uniformly bounded.

Theorem 3.2. If conditions (2.1), (2.4), (H1) and (H2) are satisfied, then the
solution set of problem (1.1) is uniformly bounded, which means that there exists

a constant C0 > 0 such that ‖u‖L∞ ≤ C0 for all weak solutions u ∈ W 1,−→p
0 (Ω) to

problem (1.1). The uniform bound C0 depends on the function Gi only through its
lower bound ai for i = 1, . . . , N .

Proof. Let u ∈W 1,−→p
0 (Ω) be a weak solution to problem (1.1). Writing u = u+−u−

with u+ = max{u, 0} (the positive part of u) and u− = max{−u, 0} (the negative
part of u), we are going to prove the uniform boundedness for u+ and u−. Since
the arguments are similar, we only give the proof for u+.

Given an arbitrary number h > 0 we pose uh := min{u+, h}. Corresponding to

any number k > 0 and any integer 1 ≤ j ≤ N , we note that u+(uh)kpj ∈W 1,−→p
0 (Ω).

This follows from

|∂i(u+(uh)k)| = |(uh)k∂i(u
+) + k(uh)k−1u+∂i(uh)|

≤ (k + 1)(uh)k|∂i(u+)|, for i = 1, . . . , N.
(3.1)

Using u+(uh)kpj as test function in (1.4) implies

N∑
i=1

∫
Ω

Gi(u)|∂iu|pi−2∂iu∂i(u
+(uh)kpj )dx =

∫
Ω

F (x, u,∇u)u+(uh)kpjdx. (3.2)

The following estimate of the left-hand side of (3.2) holds

N∑
i=1

∫
Ω

Gi(u)|∂iu|pi−2∂iu∂i(u
+(uh)kpj )dx

=

N∑
i=1

∫
Ω

Gi(u)|∂iu|pi−2∂iu(∂i(u
+)(uh)kpj + kpj(uh)kpj−1u+∂i(uh))dx

≥
N∑
i=1

ai

∫
Ω

(uh)kpj |∂i(u+)|pidx, for j = 1, . . . , N.

(3.3)

Using (H1) we estimate the right-hand side of (3.2) as follows∫
Ω

F (x, u,∇u)u+(uh)kpjdx

≤ c1
∫

Ω

(( N∑
i=1

|∂iu|pi

)1/r′

(uh)
kpj
r′
)(

(uh)
kpj
r u+

)
dx

+ c2

∫
Ω

|u|r−1(uh)kpju+dx+ c3

∫
Ω

(uh)kpju+dx.

Then Young’s inequality under the first integral with any ε > 0 provides a constant
c(ε) > 0 such that∫

Ω

F (x, u,∇u)u+(uh)kpjdx



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 7

≤ ε
N∑
i=1

∫
Ω

|∂i(u+)|pi(uh)kpjdx+ c(ε)

∫
Ω

(uh)kpj (u+)rdx

+ c2

∫
Ω

(uh)kpj (u+)rdx+ c3

(∫
Ω

(uh)kpj (u+)rdx+ |Ω|
)
,

where |Ω| denotes the Lebesgue measure of Ω. Consequently, we obtain the estimate∫
Ω

F (x, u,∇u)u+(uh)kpjdx

≤ ε
N∑
i=1

∫
Ω

|∂i(u+)|pi(uh)kpjdx+ b
(∫

Ω

(uh)kpj (u+)rdx+ 1
)
,

(3.4)

with a constant b > 0.
Combining (3.2), (3.3) and (3.4) yields

N∑
i=1

(ai − ε)
∫

Ω

(uh)kpj |∂i(u+)|pidx ≤ b
(∫

Ω

(uh)kpj (u+)rdx+ 1
)
.

If we choose ε > 0 small enough, the preceding inequality reads

N∑
i=1

∫
Ω

(uh)kpj |∂i(u+)|pidx ≤ b0
(∫

Ω

(uh)kpj (u+)rdx+ 1
)
, (3.5)

with a constant b0 > 0.
By (3.1) and (3.5), for each j = 1, . . . , N we infer that

‖∂j(u+(uh)k)‖Lpj ≤ b1/pj

0 (k + 1)
(∫

Ω

(uh)kpj (u+)rdx+ 1
)1/pj

. (3.6)

Since r ∈ (p, p∗) (see hypothesis (H1)), we are able to choose q ∈ (pj , r) with

(r − pj)q
q − pj

< p∗, for j = 1, . . . , N. (3.7)

By means of Hölder’s inequality, (2.3), (3.7) and Lemma 3.1 we then derive the
existence of a constant K > 0 such that∫

Ω

(uh)kpj (u+)rdx =

∫
Ω

(u+)r−pj )((uh)ku+)pjdx

≤
(∫

Ω

(u+)
(r−pj)q

q−pj dx
) q−pj

q
(∫

Ω

(
u+(uh)k

)q
dx
)pj/q

≤ K‖u+(uh)k‖pj

Lq .

In view of (3.6) we find a constant b1 > 0 for which

‖∂j(u+(uh)k)‖Lpj ≤ b1(k + 1)(‖u+(uh)k‖Lq + 1),

thus

‖u+(uh)k‖ ≤ b1N(k + 1)(‖u+(uh)k‖Lq + 1).

From the continuous embedding (2.3) we obtain

‖u+(uh)k‖Lp∗ ≤ b2(k + 1)(‖u+‖k+1
Lq(k+1) + 1),

with a constant b2 > 0. Through Fatou’s lemma, letting h→ 0 results in

‖u+‖k+1
Lp∗(k+1) = ‖(u+)k+1‖Lp∗ ≤ b2(k + 1)(‖u+‖k+1

Lq(k+1) + 1). (3.8)



8 D. MOTREANU, E. TORNATORE EJDE-2024/11

We note that if there is a sequence kn → +∞ with ‖u+‖Lp∗(kn+1) ≤ 1 for
all n, then it holds ‖u+‖L∞ ≤ 1 and we are done. It remains to examine two
situations: (a) we have ‖u+‖Lp∗(k+1) > 1 for all k > 0; (b) there is k0 > 0 such that
‖u+‖Lp∗(k0+1) ≤ 1 and ‖u+‖Lp∗(k+1) > 1 for all k > k0.

If case (a) occurs, (3.8) reduces to

‖u+‖Lp∗(k+1) ≤ (2b2)
1

k+1 (k + 1)
1

k+1 ‖u+‖Lq(k+1) . ∀k > 0.

Taking into account that the function k 7→ (k + 1)1/
√
k+1 is bounded on (0,+∞),

the preceding inequality entails

‖u+‖Lp∗(k+1) ≤ C1/
√
k+1‖u+‖Lq(k+1) , ∀k > 0, (3.9)

with a constant C > 0. We are in a position to implement the following Moser
iteration:

(kn + 1)q = (kn−1 + 1)p∗, ∀n ≥ 2, (3.10)

starting with (k1 + 1)q = p∗. The successive application of (3.9) produces

‖u+‖Lp∗(kn+1) ≤ C
∑n

i=1
1√

ki+1 ‖u+‖Lp∗ .

The definition of (kn) ensures that kn → +∞ and the series
∑∞

n=1 1/
√
kn + 1

converges. The application of Lemma 3.1 and letting n → ∞ show that ‖u‖L∞ ≤
C0, with a constant C0 > 0 independent on the solution u, thus reaching the desired
conclusion.

If case (b) holds, (3.8) results in

‖u+‖Lp∗(k+1) ≤ (2b2)
1

k+1 (k + 1)
1

k+1 ‖u+‖Lq(k+1) , ∀k > k0,

‖u+‖Lp∗(k0+1) ≤ (2b2)
1

k0+1 (k0 + 1)
1

k0+1 .

Arguing as above, we obtain

‖u+‖Lp∗(k+1) ≤ C1/
√
k+1‖u+‖Lq(k+1) , ∀k > k0, (3.11)

with a constant C > 0. Now we carry out the Moser iteration (3.10) starting with
(k1 + 1)q = p∗(k0 + 1). Then the repeated application of (3.11) leads to

‖u+‖Lp∗(kn+1) ≤ C
∑n

i=1
1√

ki+1 ‖u+‖Lp∗(k0+1) .

The same reasoning as in case (a) enables us to conclude that ‖u‖L∞(Ω) ≤ C0 with
a constant C0 > 0 independent of the solution u.

Summarizing, we have shown that one can find a constant C0 > 0 as stated in
the theorem. A careful reading of the preceding proof reveals that the constant
C0 does not depend on Gi except on its lower bound ai for i = 1, . . . , N , which
completes the proof. �

4. Truncated problem and associated operator

A major difficulty in handling problem (1.1) consists in the fact that the co-
efficients Gi are unbounded. This issue is resolved by truncation. In the case of
isotropic problems (possibly with weights) the idea appears in [7] and [8].



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 9

Fix a real number R > 0. For each i = 1, . . . , N , we truncate the function Gi

entering problem (1.1) as follows

GiR(t) =


Gi(t) if |t| ≤ R
Gi(R) if t > R

Gi(−R) if t < −R.
(4.1)

Notice that GiR : R→ R is a continuous and bounded function and has the range
in [ai,+∞) as the function Gi does.

With the truncated coefficients GiR in (4.1) we state the auxiliary problem

−
N∑
i=1

∂i(GiR(u)|∂iu|pi−2∂iu) = F (x, u,∇u) in Ω

u = 0 on ∂Ω.

(4.2)

We associate to problem (4.2) an operator AR : W 1,−→p
0 (Ω)→W 1,−→p

0 (Ω)∗ defined by

〈AR(u), v〉 =

N∑
i=1

∫
Ω

GiR(u)|∂iu|pi−2∂iu∂ivdx, ∀u, v ∈W 1,−→p
0 (Ω). (4.3)

The next proposition lists the main properties of the operator AR in (4.3).

Proposition 4.1. Assume conditions (2.1) and (2.4). For each R > 0, the map-

ping AR : W 1,−→p
0 (Ω) → W 1,−→p

0 (Ω)∗ is well defined, bounded, continuous, and fulfills

the S+-property, meaning that if un ⇀ u in W 1,−→p
0 (Ω) and

lim sup
n→∞

〈AR(un), un − u〉 ≤ 0, (4.4)

then un → u in W 1,−→p
0 (Ω).

Proof. By Hölder’s inequality, for all u, v ∈ W 1,−→p
0 (Ω) and i = 1, . . . , N , we note

that ∣∣ ∫
Ω

GiR(u)|∂iu|pi−2∂iu∂ivdx
∣∣ ≤ max

|t|≤R
Gi(t)‖∂iu‖pi−1

Lpi ‖∂iv‖Lpi . (4.5)

We infer for all u ∈ W 1,−→p
0 (Ω) that AR(u) ∈ W 1,−→p

0 (Ω)∗, so AR is well defined.
Moreover, (4.5) shows that the mapping AR is bounded.

Now we verify that AR is continuous. To this end, let un → u in W 1,−→p
0 (Ω). We

have
‖AR(un)−AR(u)‖

W 1,−→p
0 (Ω)∗

≤
N∑
i=1

‖(GiR(un)−GiR(u))|∂i(un)|pi−2∂i(un)‖
L

pi
pi−1

+

N∑
i=1

‖GiR(u)(|∂i(un)|pi−2∂i(un)− |∂i(u)|pi−2∂i(u))‖
L

pi
pi−1

.

(4.6)

Let us notice that

‖(GiR(un)−GiR(u))|∂i(un)|pi−2∂i(un)‖
pi

pi−1

L
pi

pi−1

≤
∫

Ω

|GiR(un)−GiR(u)|
pi

pi−1 |∂i(un)|pidx.



10 D. MOTREANU, E. TORNATORE EJDE-2024/11

By the continuity and boundedness of the function GiR, in conjunction with un →
u in W 1,−→p

0 (Ω), we are able to apply Lebesgue’s dominated convergence theorem
obtaining

lim
n→∞

‖(GiR(un)−GiR(u))|∂i(un)|pi−2∂i(un)‖
L

pi
pi−1

= 0, (4.7)

lim
n→∞

‖GiR(u)(|∂i(un)|pi−2∂i(un)− |∂i(u)|pi−2∂i(u))‖
L

pi
pi−1

= 0. (4.8)

Combining (4.6), (4.7) and (4.8) shows that AR(un)→ AR(u) in W 1,−→p
0 (Ω)∗, which

establishes the continuity of AR.

It remains to prove the S+-property. Let un ⇀ u in W 1,−→p
0 (Ω) such that (4.4) is

satisfied. Hence it is assured the validity for

lim sup
n→∞

〈AR(un)−AR(u), un − u〉 ≤ 0. (4.9)

We observe that

〈AR(un)−AR(u), un − u〉

≥
N∑
i=1

ai

∫
Ω

(|∂i(un)|pi−2∂i(un)− |∂i(u)|pi−2∂i(u))∂i(un − u)dx

+

N∑
i=1

∫
Ω

(GiR(un)−GiR(u))|∂i(u)|pi−2∂i(u)∂i(un − u)dx.

(4.10)

As for (4.7) we can prove that

lim
n→∞

∫
Ω

(GiR(un)−GiR(u))|∂i(u)|pi−2∂i(u)∂i(un − u)dx = 0. (4.11)

Then (4.9), (4.10), (4.11) and Hölder’s inequality imply

lim
n→∞

(‖∂i(un)‖Lpi − ‖∂i(u)‖Lpi )
(
‖∂i(un)‖pi−1

Lpi − ‖∂i(u)‖pi−1
Lpi

)
= 0

for all i = 1, . . . , N , from which it follows

lim
n→∞

‖∂i(un)‖Lpi = ‖∂iu‖Lpi , ∀i = 1, . . . , N.

Since the space Lpi(Ω) is uniformly convex, we infer the strong convergence un → u

in W 1,−→p
0 (Ω). The S+-property of the operator AR ensues, which completes the

proof. �

The next result points out the properties of the operator AR −N , with AR and
N introduced in (4.3) and Proposition 2.1, respectively.

Proposition 4.2. Assume (2.1), (2.4), (H1) and (H2). Then, for each real number

R > 0, the mapping AR −N : W 1,−→p
0 (Ω)→W 1,−→p

0 (Ω)∗ has the properties:

(i) AR −N is bounded (i.e., it maps bounded sets into bounded sets).

(ii) AR −N is pseudomonotone, that is, if un ⇀ u in W 1,−→p
0 (Ω) and

lim sup
n→∞

〈(AR −N )(un), un − u〉 ≤ 0, (4.12)

then

lim inf
n→∞

〈(AR −N )(un), un − u〉 ≥ lim inf
n→∞

〈(AR −N )(u), u− v〉 (4.13)

for all v ∈W 1,−→p
0 (Ω).



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 11

(iii) AR −N is coercive, that is,

lim
‖u‖→∞

〈(AR −N )(u), u〉
‖u‖

= +∞. (4.14)

Proof. (i) This is a direct consequence of Propositions 2.1 and 4.1.

(ii) Suppose that un ⇀ u in W 1,−→p
0 (Ω) and (4.12) is satisfied. Corollary 2.2

and (4.12) ensure that (4.4) holds true. We are allowed to apply the S+-property

in Proposition 4.1, which provides the strong convergence un → u in W 1,−→p
0 (Ω).

By Proposition 2.1 and the continuous embedding (2.3) with q = r we know that

N (un) → N (u) in W 1,−→p
0 (Ω)∗. In addition, Proposition 4.1 guarantees AR(un) →

AR(u) in W 1,−→p
0 (Ω)∗. This enables us to conclude that (4.13) holds true.

(iii) According to hypothesis (H2) and (2.5) we have

〈(AR −N )(u), u〉

≥
N∑
i=1

∫
Ω

GiR(u)|∂i(u)|pidx− d1

N∑
i=1

‖∂iu‖pi

Lpi − d2‖u‖
p

Lp − ‖σ‖L1

≥
N∑
i=1

(ai − d1 −Np−1d2θ)‖∂iu‖pi

Lpi −Npd2θ − ‖σ‖L1 .

Since ai − d1 −Np−1d2θ > 0 and pi > 1 for i = 1, . . . , N , the inequality

〈(AR −N )(u), u〉
‖u‖

≥
N∑
i=1

(ai − d1 −Np−1d2θ)‖∂iu‖pi−1
Lpi −

Npd2θ + ‖σ‖L1

‖u‖

allows us to establish (4.14). �

5. Existence of solutions

First we deal with the solvability of auxiliary problem (4.2).

Theorem 5.1. Assume that conditions (2.1) and (2.4) hold, Gi : R → [ai,+∞),
with ai > 0 for i = 1, . . . , N , are continuous functions, and F : Ω × R × RN → R
is a Carathéodory function satisfying hypotheses (H1) and (H2). Then, for every

R > 0, problem (4.2) has at least a weak solution uR ∈W 1,−→p
0 (Ω) which means

N∑
i=1

∫
Ω

GiR(uR(x))|∂iu(x)|pi−2∂iu(x)∂iv(x)dx =

∫
Ω

F (x, u,∇u)vdx (5.1)

for all v ∈ W 1,−→p
0 (Ω). Moreover, the solution set of problem (4.2) is uniformly

bounded with the bound C0 > 0 in Theorem 3.2. In particular, one has ‖uR‖L∞ ≤
C0 for every R > 0, with C0 > 0 in Theorem 3.2.

Proof. Fix R > 0. We note that auxiliary problem (4.2) in W 1,−→p
0 (Ω) is equivalent

to the operator equation
(AR −N )(u) = 0. (5.2)

Proposition 4.2 entails that the operator AR − N : W 1,−→p
0 (Ω) → W 1,−→p

0 (Ω)∗ is
pseudomonotone, bounded and coercive. Hence we are entitled to apply the main
theorem for pseudomonotone operators (see, e.g., [2, Theorem 2.99]) ensuring that

equation (5.2) admits at least a weak solution uR ∈W 1,−→p
0 (Ω). Consequently, uR is

a weak solution of auxiliary problem (4.2).



12 D. MOTREANU, E. TORNATORE EJDE-2024/11

Theorem 3.2 can be applied with GiR in place of Gi for each i = 1, . . . , N because
the same set of assumptions is required to be verified. Since the range of GiR is
contained in [ai,+∞) as it is the case of Gi, we can infer that the solution uR of
(4.2) fulfills the a priori estimate ‖uR‖L∞ ≤ C0, where C0 > 0 is the uniform bound
given in Theorem 3.2. This completes the proof. �

Finally, we will prove that uR ∈W 1,−→p
0 (Ω) found in Theorem 5.1 is a weak solu-

tion of the original problem (1.1) provided R > 0 is sufficiently large. Consequently,
this will establish Theorem 2.3.

Proof of Theorem 2.3. Theorem 3.2 ensures that the solution set of problem (1.1)
is uniformly bounded, thus there is a constant C0 > 0 such that ‖u‖L∞ ≤ C0 for

all weak solutions u ∈W 1,−→p
0 (Ω) to (1.1). As explicitly mentioned in the statement

of Theorem 3.2, the constant C0 does not depend on the function Gi entering
problem (1.1) except on the lower bound ai for i = 1, . . . , N . As seen from (4.1),
ai is a lower bound for each truncation GiR, so the solution set of each auxiliary
problem (4.2) is uniformly bound by the constant C0, which is independent of R.

In particular, the solution uR ∈ W 1,−→p
0 (Ω) to problem (4.2) given by Theorem 5.1

satisfies uR ∈ L∞(Ω) with ‖uR‖L∞ ≤ C0, where C0 > 0 is the uniform bound in
Theorem 3.2.

The preceding reasoning shows that it is allowed to choose R ≥ C0 because C0

is independent of R > 0. With such a choice, there holds |uR(x)| ≤ R almost
everywhere on Ω. In view of (4.1) we obtain

GiR(uR(x)) = Gi(uR(x)) for a.e. x ∈ Ω, and i = 1, . . . , N.

A simple comparison regarding the statements of problems (1.1) and (4.2) confirms

that uR ∈W 1,−→p
0 (Ω) is a bounded weak solution for the original problem (1.1). The

proof is thus complete. �

Acknowledgments. E. Tornatore is member of GNAMPA. This research is par-
tially supported by FFR2023-Tornatore and PRIN 2024 – Progetti di Ricerca di
rilevante Interesse Nazionale – Nonlinear differential problems with applications to
real phenomena (Project N. 2022ZXZTN2)

References

[1] S. N. Antontsev, J. I. Dı́az, S. D. Shmarev; Energy methods for free boundary problems. Appli-

cations to nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations
and their Applications, 48. Birkhäuser Boston, Inc., Boston, MA, 2002.

[2] S. Carl, V. K. Le, D. Motreanu; Nonsmooth variational problems and their inequalities.

Comparison principles and applications, Springer, New York, 2007.
[3] B. Brandolini, F. C. Cirstea; Anisotropic elliptic equations with gradient-dependent lower

order terms and L1 data, Math. Eng., 5 (2023), 1-33.

[4] F. Demengel, T. Dumas; Extremal functions for an embedding from some anisotropic space,
involving the ’one Laplacian’, Discrete Contin. Dyn. Syst., 39 (2019), 1135–1155.

[5] A. El Hamidi, J. M. Rakotoson; Extremal functions for the anisotropic Sobolev inequalities,
Ann. Inst. H. Poincaré C Anal. Non Linéaire, 24 (2007), 741–756.

[6] I. Fragala, F. Gazzola, B. Kawohl; Existence and nonexistence results for anisotropic quasi-

linear elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 21 (2007), 715–734.
[7] D. Motreanu; Nonhomogeneous Dirichlet problems with unbounded coeffiient in the principal

part, Axioms, 11 (2022), Paper No. 739.

[8] D. Motreanu, E. Tornatore; Nonhomogeneous degenerate quasilinear problems with convec-
tion, Nonlinear Anal. Real World Appl., 71 (2023), Paper No. 103800.



EJDE-2024/11 DIRICHLET PROBLEM WITH ANISOTROPIC PRINCIPAL PART 13

[9] J. Vétois; Strong maximum principles for anisotropic elliptic and parabolic equations, Strong

maximum principles for anisotropic elliptic and parabolic equationsAdv. Nonlinear Stud., 12

(2012), 101–114.
[10] J. Vétois; Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic

equations, Adv. Math., 284 (2015), 122–158.

Dumitru Motreanu

Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
Email address: motreanu@univ-perp.fr

Elisabetta Tornatore
Department of Mathematics and Computer Science, University of Palermo, 90123 Palermo,

Italy

Email address: elisa.tornatore@unipa.it


	1. Introduction
	2. Statements of hypotheses and main result
	3. Uniformly bounded solution set
	4. Truncated problem and associated operator
	5. Existence of solutions
	Acknowledgments

	References