Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 34, pp. 1–8.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

DIRICHLET (p, q)-EQUATIONS WITH GRADIENT DEPENDENT

AND LOCALLY DEFINED REACTION

ZHENHAI LIU, NIKOLAOS S. PAPAGEORGIOU

Abstract. We consider a Dirichlet (p, q)-equation, with a gradient dependent

reaction which is only locally defined. Using truncations, theory of nonlinear
operators of monotone type, and fixed point theory (the Leray-Schauder Al-

ternative Theorem), we show the existence of a positive smooth solution.

1. Introduction

Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this article we
study the (p, q)-equation with gradient dependence (convection)

−∆pu(z)−∆qu(z) = f(z, u(z), Du(z)) inΩ,

u|∂Ω = 0, u > 0, 1 < q < p.
(1.1)

Given r ∈ (1,+∞) by ∆r we denote the r-Laplace differential operator by

∆ru = div(|Du|r−2Du) for all u ∈W 1,r
0 (Ω).

In problem (1.1) we have the sum of two such operators ((p, q)-equation). So
the differential operator (left hand side) of the problem is not homogeneous. The
reaction term (right hand side) of (1.1), depends also one the gradient of u (con-
vection). This classifies the problem as non-variational and for this reason our
method of proof is topological and uses the fixed point theory (in particular, the
Leray-Schauder Alternative Principle). Our aim is to obtain positive solutions.

Recently such problems were studied by Faraci-Motreanu-Puglisi [5], Gasiński-
Papageorgiou [7], Hu-Papageorgiou [9], Liu-Motreanu-Zeng [12], Papageorgiou-
Vetro-Vetro [17], Papageorgiou-Zhang [18] (problems with Laplacian or p-Laplacian),
Bai [2], Bai-Gasinski-Papageorgiou [4], Gasinski-Winkert [8], Liu-Papageorgiou [13]
(nonlinear nonhomogeneous problems), and Bai-Gasinski-Papageorgiou [3], Papa-
georgiou-Radulescu-Repovs [15], Papageorgiou-Zhang [19], (problems with singular
and convection terms). In all the aforementioned works, it is required that the
reaction is nonnegative and/or it satisfies a restrictive growth condition involving
the principal eigenvalue of the Dirichlet p-Laplacian (see, for example, [5, 7, 8]). In
contrast here the reaction term is sign-changing and exhibits an oscillatory behav-
ior near zero (namely the reaction function starts positive and at a certain point
becomes strictly negative). Moreover, f(z, ·, y) is only locally defined (near zero).

2010 Mathematics Subject Classification. 35J20, 35J60,35J92.
Key words and phrases. (p, q)-differential operator; convection; fixed point; nonlinear;

regularity; positive solution; Leray-Schauder alternative theorem.
c©2021 Texas State University.

Submitted October 17, 2020. Published April 30, 2021.

1



2 Z. H. LIU, N. S. PAPAGEORGIOU EJDE-2021/34

Our approach differs from the above works which employed the so-called “frozen
variable method” (see Liu-Papageorgiou [13]). Here instead, we use the theory of
nonlinear operators of monotone type.

2. Mathematical background-hypotheses

Let X be a Banach space and g : X → X a map. We say that g(·) is compact,
if it is continuous and maps bounded sets to relatively compact sets. We will use
the Leray-Schauder Alternative Principle that asserts the following.

Theorem 2.1. If X is a Banach space, g : X → X is a compact map and D =
{x ∈ X : x = tg(x) for some 0 < t < 1}, then one of the following statements holds

(a) D is unbounded, or
(b) g admits a fixed point.

We consider the nonlinear eigenvalue problem

−∆qu(z) = λ̂|u(z)|q−2u(z) in Ω, u|∂Ω = 0. (2.1)

An “eigenvalue” of (2.1), is a number λ̂ ∈ R such that problem (2.1) admits

a nontrivial solution û ∈ W 1,q
0 (Ω), called an “eigenfunction” corresponding to the

eigenvalue λ̂. Nonlinear regularity theory (see, for example, Gasinski-Papageorgiou
[6, Section 6.2]) implies that û ∈ C1(Ω̄). We know that problem (2.1) admits a

smallest eigenvalue λ̂1(q) > 0 such that

• λ̂1(q) is isolated (that is, we can find ε > 0 such that (λ̂1(q), λ̂1(q) + ε)
contains no eigenvalue);

• λ̂1(q) is simple (that is, if û, v̂ ∈ C1
0 (Ω̄) are eigenfunctions corresponding to

λ̂1(q), then û = θv̂ for some θ ∈ R \ {0});
•

λ̂1(q) = inf[
‖Du‖qq
‖u‖qq

: u ∈W 1,q
0 (Ω), u 6= 0]. (2.2)

In (2.2) the infimum is realized on the corresponding one dimensional eigenspace.
From the above properties it follows that the elements of this eigenspace do not

change sign. For every other eigenvalue λ̂ 6= λ̂1(q) the corresponding eigenfunctions
are nodal (sign-changing).

We will also need the following weighted version of the eigenvalue problem (2.1)

−∆qu(z) = λ̃m(z)|u(z)|q−2u(z) in Ω, u|∂Ω = 0. (2.3)

Here m ∈ L∞(Ω),m(z) ≥ 0 for a.a. z ∈ Ω,m 6≡ 0. Then (2.3) has a small-

est eigenvalue λ̃1(m, q) > 0, which is isolated, simple and admits the variational
characterization

λ̃1(m, q) = inf
[ ‖Du‖qq∫

Ω
m(z)|u|qdz

: u ∈W 1,q
0 (Ω), u 6= 0

]
. (2.4)

Again the infimum in (2.4) is realized on the corresponding one dimensional
eigenspace, the elements of which have fixed sign and belong in C1

0 (Ω̄). Let C+ =
{u ∈ C1

0 (Ω̄) : u(z) ≥ 0 for all z ∈ Ω̄} (the positive (order) cone of C1
0 (Ω̄)). This

cone has a nonempty interior given by

inf C+ =
{
u ∈ C+ : u(z) > 0 for all z ∈ Ω,

∂u

∂n

∣∣
∂Ω

< 0
}



EJDE-2021/34 DIRICHLET (p, q)-EQUATIONS WITH GRADIENT DEPENDENT 3

with n(·) being the outward unit normal on ∂Ω. The nonlinear maximum principle
(see Gasiński-Papageorgiou [6],p.738), implies that the eigenfunctions correspond-

ing to λ̃1(m, q) > 0 are in intC+ or in − intC+.
Using all these properties, we infer the following strict monotonicity property for

the map m→ λ̃1(m, q).

Proposition 2.2. If m,m′ ∈ L∞(Ω), 0 ≤ m(z) ≤ m′(z) for a.a. z ∈ Ω, m 6= 0,

and m 6= m′, then λ̃1(m′, q) < λ̃1(m, q).

Our conditions on the reaction term f(z, x, y) are the following:

(H1) f : Ω×R× RN → R is a Carathéodory function such that f(z, 0, 0) = 0 for
a.a. z ∈ Ω and
(i) if |f(z, x, y)| ≤ a(z)[1 + xp−1 + |y|p−1] for a.a. z ∈ Ω, all x ≥ 0, all

y ∈ RN with a ∈ L∞(Ω);
(ii) there exist M > 0 and δ > 0 such that

f(z,M, y) < 0 for a.a. z ∈ Ω, all |y| ≤ δ;

(iii) there exist δ0 > 0 and η ∈ L∞(Ω) such that

λ̂1(q) ≤ η(z) for a.a. z ∈ Ω, η 6= λ̂1(q),

η(z)xq−1 ≤ f(z, x, y) for a.a. z ∈ Ω, all 0 ≤ x ≤ δ0, all y ∈ RN ,

lim sup
x→0+

f(z, x, y)

xq−1
≤ ĉθ uniformly for a.a. z ∈ Ω, all |y| ≤ θ.

Evidently hypotheses (H1)(ii) is satisfied if f(z,M, 0) ≤ −ĉ < 0 for a.a. z ∈ Ω.
Hypotheses (H1)(ii) and (H1)(iii) imply the oscillatory behavior of f(z, ·, y) near
zero, mentioned in the Introduction.

As examples of functions satisfy (H1) we have following, (For the sake of sim-
plicity we drop the z-dependence).

f(x, y) = c0[xq−1 − xp−1] + c1|y|p−1

for all x ≥ 0, all y ∈ RN, with c0 > λ̂1(q), c1 > 0; and

f(x, y) = c2x
q−1[1− xτ−q lnx] + x|y|p−1

for all x ≥ 0, all y ∈ RN, with c2 > λ̂1(q), τ ≥ q.
In what follows, pM : R → R denotes the truncation function at level M , that

is,

pM (x) =

{
x if x ≤M,

M if M < x.

Evidently pM (·) is Lipschitz continuous.

Also for x ∈ R, we denote x± = max{±x, 0}. For u ∈ W 1,p
0 (Ω) we define

u±(z) = u(z)± for all z ∈ Ω. Then u± ∈ W 1,p
0 (Ω), u = u+ − u−, |u| = u+ + u−.

Finally for r ∈ (1,∞), by Ar : W 1,r
0 (Ω)→W−1,r′(Ω) = W 1,r

0 (Ω)∗ (with 1
r+ 1

r′ = 1),
we denote the nonlinear map

〈Ar(u), h〉 =

∫
Ω

|Du|r−2(Du,Dh)RNdz for all u, h ∈W 1,p
0 (Ω).

This map is bounded (maps bounded sets to bounded ones), continuous, strictly
monotone (hence maximal monotone) and of type (S)+ (see [16, p. 157]).



4 Z. H. LIU, N. S. PAPAGEORGIOU EJDE-2021/34

3. Positive solutions

In this section using the theory of nonlinear operators of monotone type and
fixed point arguments based on Theorem 2.1, we show the existence of a positive
smooth solution for problem (1.1).

Let V : W 1,p
0 (Ω)→W−1,p′(Ω) (with 1

p + 1
p′ = 1) be defined by

V (u) = Ap(u) +Aq(u) for all u ∈W 1,p
0 (Ω).

Proposition 3.1. V −1 : W−1,p′(Ω)→ W 1,p
0 (Ω) exists and is bounded and contin-

uous.

Proof. The map V (·) is continuous, strictly monotone (hence maximal monotone
too) and coercive (since 〈V (u), u〉 = ‖Du‖pp+‖Du‖qq). If follows that V (·) is surjec-
tive (see Papageorgiou-Rădulescu-Repovš [16, Corollary 2.8.7, p. 135]. Therefore

V −1 : W−1,p′(Ω) → W 1,p
0 (Ω) is well-defined and on account of the coercivity of

V (·), V −1(·) is bounded (maps bounded sets to bounded ones). We examine the

continuity of V (·). So, let u∗n → u∗ in W−1,p′(Ω) and set un = V −1(u∗n) ∈W 1,p
0 (Ω)

for all n ∈ N. Then u∗n = V (un) for all n ∈ N which implies {un}n∈N ⊆W 1,p
0 (Ω) is

bounded, using the coercivity of V (·)).
So, we may assume that

un
w−→ u in W 1,p

0 (Ω) as n→∞.
We have that 〈V (un), un − u〉 = 〈u∗n, un − u〉 → 0, which implies ‖Dun‖p →
‖Du‖p, which in turn implies un → u in W 1,p

0 (Ω), by the Kadec-Klee property of

W 1,p
0 (Ω)); this implies V (u) = u∗ which in turn implies u = V −1(u∗) and so V −1(·)

is continuous. �

For ε > 0, let f̂εM : Ω× R× RN → R be the Carathéodory function defined by

f̂εM (z, x, y) = f(z, pM (x) + ε, y).

Let Nf̂εM
: W 1,p

0 (Ω) → Lp
′
(Ω) be the corresponding Nemytskii (superposition)

operator, defined by

Nf̂εM
(u)(·) = f(·, pM (u(·)) + ε,Du(·)) for all u ∈W 1,p

0 (Ω).

On account of hypothesis (H1)(i) and using Krasnoselskii’s theorem (see, for exam-
ple, Gasiński-Papageorgiou [6, Theorem 3.4.4, p. 407]), we have that

Nf̂εM
: W 1,p

0 (Ω)→ Lp
′
(Ω) is continuous . (3.1)

Also, let i+ : W 1,p
0 (Ω)→W 1,p

0 (Ω) be defined by

i+(u) = u+ for all u ∈W 1,p
0 (Ω). (3.2)

We introduce the map N̂ε : W 1,p
0 (Ω)→ Lp

′
(Ω) defined by

N̂ε(u) = (Nf̂εM
◦ i+)(u) for all u ∈W 1,p

0 (Ω).

From (3.1) and (3.2) we see that

N̂ε(·) is bounded and continuous. (3.3)

We set Kε = V −1 ◦ N̂ε.

Proposition 3.2. If (H1) holds, then Kε : W 1,p
0 (Ω)→W 1,p

0 (Ω) is compact.



EJDE-2021/34 DIRICHLET (p, q)-EQUATIONS WITH GRADIENT DEPENDENT 5

Proof. From Proposition 3.1 and (3.3), we infer that Kε(·) is continuous. Let

B ⊆W 1,p
0 (Ω) be bounded. From (3.3) we have that

N̂ε(B) ⊆ Lp
′
(Ω) is bounded . (3.4)

From the Sobolev embedding theorem, we know that W 1,p
0 (Ω) ↪→ Lp(Ω) compactly

and densely.
Invoking [6, Lemma 2.2.27, p. 141 ] and Schauder’s Theorem (see Gasinski-

Papageorgiou [6, Theorem 3.1.22, p. 275]) we have that

Lp
′
(Ω) = Lp(Ω)∗ ↪→W−1,p′(Ω) = W 1,p

0 (Ω)∗ compactly and densely.

Then from (3.4) it follows that N̂ε(B) ⊆W−1,p′(Ω) is relatively compact, Therefore,

N̂ε(·) is a compact map. �

Let 0 < ε ≤ δ0 and define

Dε = {u ∈W 1,p
0 (Ω) : u = tKε(u) for some 0 < t < 1}.

Proposition 3.3. If (H1) holds, and 0 < ε ≤ δ0, then Dε ⊆W 1,p
0 (Ω) is bounded.

Proof. Let u ∈ Dε. We have

1

t
u = Kε(u) = (V −1 ◦ N̂ε)(u),

which implies V ( 1
tu) = N̂ε(u); therefore,

− 1

tp−1
∆p(u)− 1

tq−1
∆q(u) = f(z, pM (u+) + ε,Du+) in Ω. (3.5)

On account of hypothesis (H1)(iii) and since 0 < ε ≤ δ0, from (3.5) we see that

u 6= 0. On (3.5) we act with −u− ∈W 1,p
0 (Ω) and obtain

1

tp−1
‖Du−‖pp +

1

tq−1
‖Du−‖qq =

∫
Ω

f(z, ε, 0)(−u−)dz ≤ 0

(see hypothesis (H1)(iii)). This implies u ≥ 0, u 6= 0.
From (3.5) and Ladyzhenskaya-Uraltseva [10, Theorem 7.1, p. 286], we have that

u ∈ L∞(Ω). Then the regularity theory of Lieberman [11] implies that u ∈ C+\{0}.
In fact on account of hypotheses (H1)(i) and (H1)(iii), given r ∈ (p, p∗), we can
find c3 = c3(r) > 0 such that

f(z, x, y) ≥ η(z)xq−1 − c3xr−1 for a.a. z ∈ Ω, all x ≥ 0, all y ∈ RN .

Then from (3.5) we have

∆pu+ tp−q∆qu ≤ c3(M + δ0)r−pur−p in Ω;

therefore, u ∈ intC+, see Pucci-Serrin [20, pp. 111,120].

Claim: 0 ≤ u(z) ≤ M for all z ∈ Ω̄. Arguing by contradiction, suppose that the
assertion of the Claim is not true. Then we can find z0 ∈ Ω such that u(z0) =
max

Ω̄
u > M . Then we can find an open neighborhood Ω0 of z0, with Lipschitz

boundary and Ω̄0 ⊆ Ω such that

Du(z0) = 0,
∂u

∂n

∣∣
∂Ω0

< 0, f(z,M + ε,Du(z)) ≤ 0, for a.a. z ∈ Ω0, (3.6)

see hypothesis (H1)(iii).



6 Z. H. LIU, N. S. PAPAGEORGIOU EJDE-2021/34

Recall that by (3.5),

−∆pu(z)− tp−q∆qu(z) = tp−1f(z,M + ε,Du(z)) for a.a. z ∈ Ω0

Acting with u and using the nonlinear Green’s identity (see Papageorgiou-Rădulescu-
Repovš [16, p.35]), by (3.6) we have

0 ≤
∫

Ω0

|Du|pdz +

∫
Ω0

|Du|qdz

= tp−1

∫
Ω0

f(z,M + ε,Du)dz + tp−1

∫
∂Ω0

∂u

∂n
[|Du|p−2 + |Du|q−2]udσ < 0 .

This contradiction contradiction proves the Claim.
From (3.5), the Claim, and 0 < t < 1, we have

‖Du‖pp ≤M
∫

Ω

|f(z, u+ ε,Du)|dz,

which implies

‖Du‖pp ≤ c4[1 + ‖Du‖p−1
p ]

for some c4 > 0, see hypothesis (H1)(i)). Therefore, Dε ⊆W 1,p
0 (Ω) is bounded. �

Propositions 3.2 and 3.3 permit the use of Theorem 2.1 (the Leray-Schauder

Alternative Principle). So, for 0 < ε ≤ δ0, we can find uε ∈ W 1,p
0 (Ω) such that

uε = Kε(uε). Therefore,

−∆puε −∆quε = f(z, pM (u+
ε ) + ε,Du+

ε ) in Ω, uε|∂Ω = 0.

From the proof of Proposition 3.3, we have

uε ∈ intC+ and 0 ≤ uε(z) ≤M for all z ∈ Ω̄.

Then it follows that

−∆puε(z)−∆quε(z) = f(z, uε(z) + ε,Duε(z)) in Ω, (3.7)

which implies that

{uε}0<ε≤δ0 ⊆W
1,p
0 (Ω) is bounded. (3.8)

We let ε→ 0+ to obtain a positive solution for problem (1.1).

Theorem 3.4. If (H1) holds, then (1.1) admits a positive solution ū ∈ intC+.

Proof. Let εn = 1/n for n ∈ N and let un = uεn ∈ intC+ from (3.7). From (3.8)
and the nonlinear regularity theory of Lieberman [11], we know that there exists

α ∈ (0, 1) such that {un}n∈N ⊆ C1,α
0 (Ω̄) is bounded.

Since C1,α
0 (Ω̄) ↪→ C1

0 (Ω̄) compactly, we may assume that

un → ū in C1
0 (Ω̄). (3.9)

From (3.7) and (3.9), if follows that

−∆pū(z)−∆qū(z) = f(z, ū(z), Dū(z)) in Ω, ū|∂Ω = 0.

So, if we can show that ū 6= 0, then this will be the desired positive solution of
(1.1). We argue indirectly. So, suppose that ū = 0. We set vn = un/‖un‖, n ∈ N.
Then we have that ‖vn‖ = 1, vn ≥ 0 for all n ∈ N and so we may assume that

vn
w−→ v in W 1,p

0 (Ω). (3.10)



EJDE-2021/34 DIRICHLET (p, q)-EQUATIONS WITH GRADIENT DEPENDENT 7

From (3.7) we have

‖un‖p−q〈Ap(vn), h〉+ 〈Aq(vn), h〉 =

∫
Ω

f(z, un + 1
n , Dvn)

‖un‖p−1
hdz (3.11)

for all h ∈W 1,p
0 (Ω).

Let θ = sup
n∈N
‖un‖C1

0 (Ω̄) < ∞ (see (3.9)). On account of hypotheses (H1)(i) and

(H1)(iii), we have
|f(z, x, y)| ≤ c5[xq−1 + xp−1]

for a.a. z ∈ Ω, all x ≥ 0, all y| ≤ θ and some c5 > 0. This implies{f(·, un(·) + 1
n , Dun(·))

‖un‖p−1

}
n∈N ⊆ L

p′(Ω) is bounded. (3.12)

From (3.11) and of Papageorgiou-Rădulescu [14, Proposition 2.10], we can find
c6 > 0 such that ‖vn‖∞ ≤ c6 for all n ∈ N. Then the nonlinear regularity theory
of Lieberman [11, p. 320] implies the existence of α ∈ (0, 1) and c7 > 0 such that

vn ∈ C1,α
0 (Ω̄) = C1,α(Ω̄)∩C1

0 (Ω̄) and ‖vn‖C1,α
0 (Ω̄) ≤ c7 for all n ∈ N. The compact

embedding of C1,α
0 (Ω̄) into C1

0 (Ω̄), implies that we may assume that vn → v in
C1

0 (Ω̄), hence ‖v‖ = 1, v ≥ 0. Also from (3.12) and hypothesis (H1)(iii), we have

f(·, un(·) + 1
n , Dun(·))

‖un‖p−1

w−→ η0(·)v(·)q−1 in Lp
′
(Ω). (3.13)

With η0 ∈ L∞(Ω), η(z) ≤ η0(z) for a.a. z ∈ Ω (see Aizicovici-Papageorgiou-
Staicu [1, Proposition 16]. If in (3.11) we pass to the limit as n → ∞ and use
(3.13), we obtain

〈Aq(v), h〉 =

∫
Ω

η0(z)vq−1hdz for all h ∈W 1,p
0 (Ω),

which implies

−∆qv(z) = η0(z)v(z)q−1 in Ω, v|∂Ω = 0, v ≥ 0. (3.14)

Using Proposition 2.2, we have

λ̃1(η0, q) < λ̃1(λ̂1(q), q) = 1

So, from (3.14) if follows that v = 0 or v is nodal, both cases leading to a contra-
diction. Therefore ū 6= 0 and as before ū ∈ intC+. This is the smooth positive
solution of (1.1). �

Acknowledgments. This research was partially supported by the NNSF of China
Grant No. 12071413, and by the NSF of Guangxi Grant No. 2018GXNSFDA138002.

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Zhenhai Liu (corresponding author)

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization
and Big Data Processing, Yulin Normal University, Yulin 537000, China.

Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Uni-

versity for Nationalities, Nanning, Guangxi, 530006, China
Email address: zhhliu@hotmail.com

Nikolaos S. Papageorgiou
Department of Mathematics, National Technical University, Zografou Campus, 15780

Athens, Greece

Email address: npapg@math.ntua.gr


	1. Introduction
	2. Mathematical background-hypotheses
	3. Positive solutions
	Acknowledgments

	References