Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 50, pp. 1–23.

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

ASYMPTOTIC BEHAVIOR OF COOPERATIVE SYSTEMS

INVOLVING p-LAPLACIAN OPERATORS

PABLO ÁLVAREZ-CAUDEVILLA

Abstract. This work is devoted to the analysis of the asymptotic behavior

of a parameter dependent quasilinear cooperative eigenvalue system when a
parameter in front of some non-negative potentials approaches infinity. In

particular we consider operators of p-Laplacian type. We prove that the eigen-

functions concentrate on the subdomains where those potentials vanish at the
limit, while the eigenvalue approaches an upper bound that will depend on

those subdomains. We also show several properties for the unusual limiting

problems.

1. Introduction

1.1. Models and preliminaries. In this article we study the asymptotic behavior
of the elliptic cooperative eigenvalue problem with potential terms

(−∆p + λa|u|p−2)u− b|u|α−1u|v|βv = τ |u|p−2u,

(−∆q + λd|v|q−2)v − c|u|αu|v|β−1v = τ |v|q−2v,
(1.1)

for (u, v) ∈W 1,p
0 (Ω)×W 1,q

0 (Ω) and when the parameter λ ∈ R approaches infinity.
Moreover, Ω is a smooth bounded domain of RN , N ≥ 1, with smooth boundary
∂Ω, for example of class C2 or Lipschitz. Also, we consider α, β ≥ 0 and p, q > 1
satisfying the relation

α+ 1

p
+
β + 1

q
= 1. (1.2)

Moreover, the operator −∆p stands for the so called p-Laplacian operator so that

−∆pu = −div(|∇u|p−2∇u),

and similarly for −∆q as a q-Laplacian operator. To simplify the notation we might
write the eigenvalue problem (1.1) in matrix form

S(V1, V2)

(
u
v

)
:=

(
(−∆p + V1)u −b|u|α−1u|v|βv
−c|u|αu|v|β−1v (−∆q + V2)v

)
= τ

(
|u|p−2u
|v|q−2v

)
. (1.3)

In particular, for V1 = λa|u|p−2 and V2 = λd|v|q−2 we denote S(V1, V2) as

Sλ := S(λa|u|p−2, λd|v|q−2) and S = S(0, 0). (1.4)

The exponent of the p-Laplacian (respectively the q-Laplacian) and the rest of
the terms in each equation must be consistent so that relation (1.2) is satisfied.

2020 Mathematics Subject Classification. 35J47, 35P20, 49R05.
Key words and phrases. Cooperative systems; p-Laplacian; eigenvalue problems.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted December 22, 2021. Published July 15, 2022.
1



2 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

Otherwise the system does not have the proper structure of an eigenvalue problem
(further details in [17]).

Furthermore, we denote by

τ1[S(V1, V2),Ω] (1.5)

the principal eigenvalue of the operator (1.3) under homogeneous Dirichlet bound-
ary conditions and (ϕ1, ψ1)T the principal eigenfunction associated with the prin-
cipal eigenvalue denoted by (1.5). We will show that such a principal eigenvalue
corresponds to the one with an associated positive eigenfunction (ϕ1, ψ1)T , in the
sense that ϕ1 ≥ 0, ψ1 ≥ 0 and ϕ1 6= 0, ψ1 6= 0. Indeed, the positive eigenfunction
will belong to the positive cone K with positive functions in the interior of the
domain Ω and strictly negative outward normal derivatives on the boundary, i.e.

K :=
{
ϕ ∈W 1,p(Ω) : ϕ(x) > 0 for x ∈ Ω,

∂ϕ(x)

∂n
< 0 for x on ∂Ω

}
,

where n stands for the unitary outward normal vector to the boundary ∂Ω. In
particular, for the eigenvalue problem (1.1) we have that

τ1(λ) = τ1[S(λa|u|p−2, λd|v|q−2),Ω],

as the principal eigenvalue under homogeneous Dirichlet boundary conditions, as-
sociated with the principal eigenfunction (ϕλ, ψλ)T .

Thus, we are interested in understanding the limit limλ→∞ τ1(λ), under certain
assumptions established for the problem (1.1). In particular, we will assume that
a, d are potentials as defined in [1], i.e. as Borel functions

a, d : Ω→ R+ such that a, d ∈ Q(Ω),

with Q(Ω) representing the sets of potentials so that the following two properties
hold:

(1) supx∈Ω a(x) < +∞, supx∈Ω d(x) < +∞;

(2) W 1,p
0 (Ωg0) = {u ∈W 1,p

0 (Ω); u = 0 a.e. on Ωg+}, with g = a, d,

Moreover, those potentials a and d are going to be non-negative so that, a ≥ 0,
d ≥ 0 and we denote the open sets/subdomains of Ω where the potentials a and d
vanish, as

Ωa0 := {x ∈ Ω : a(x) = 0}, Ωd0 := {x ∈ Ω : d(x) = 0}. (1.6)

Also, we denote

Ω0 := Ωa+d
0 = Ωa0 ∩ Ωd0 = {x ∈ Ω : a(x) = d(x) = 0},

so that the following subdomains come into play

Ωa+ := {x ∈ Ω : a(x) > 0} = Ω\Ωa0 , Ωd+ := {x ∈ Ω : d(x) > 0} = Ω\Ωd0,

and, there exists a subdomain of Ω

Ω+ = {x ∈ Ω : a(x) + d(x) > 0},

which is also an open set, with Ω+ ⊂ Ω. Then, we consider a compact set

K0 = (a+ d)−1(0) = Ω \ Ω+.

Note that, the compact set K0 consists of two compact subsets in RN so that

Ωa0 = Int(Ka
0 ) 6= ∅, Ωd0 = Int(Kd

0 ) 6= ∅, (1.7)



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 3

and the subdomains Ωa0 and Ωd0 may not be connected, but with a finite number of
components. However, we suppose that they are Lipschitz domains on the bound-
ary. Maybe these regularity assumptions on the boundary might be relaxed with
extra care on the analysis but for our purposes Lipschitz boundary conditions are
sufficient.

We also suppose that b, c ∈ C(Ω) and (1.1) is strongly cooperative in the sense
that

b(x) > 0 and c(x) > 0, for all x ∈ Ω.

1.2. Motivation. Most of previous results rely on the variational structure of the
problem, for example [16, 20].

Furthermore, focusing on cooperative systems, there are previous works analyz-
ing similar quasilinear systems to (1.1) (see for example [17] and references therein)
where the proofs were based on variational techniques and assuming the partic-
ular case when the cooperative terms are equal, i.e. b = c in (1.1) multiplied by
the proper coefficients depending on the exponents of the coupling terms. Indeed,
assuming that those off-diagonal couple terms are the same having the proper vari-
ational structure, there are several additional works. We would like to stress the
work done by Bozhkov & Mitidieri [7] for a variational system of the form

−∆pu = (α+ 1)c(x)|u|α−1u|v|βv + λa(x)|u|p−2u, in Ω,

−∆qv = (β + 1)c(x)|u|αu|v|β−1v + µb(x)|v|q−2v, in Ω,

(u, v) = (0, 0), on ∂Ω,

(1.8)

where some results of existence and multiplicity of solutions were obtained via the
fibering method. This methodology was introduced by Pohozaev in the 1970s [21,
22], as a convenient generalization of previous versions by Clark [9] and Rabinowitz
[23] of variational approaches, and further developed by Drábek and Pohozaev [14]
and others in the 1980’s to ascertain the existence and multiplicity of solutions
for equations with a variational form (in particular and relevant for our work,
the p-Laplacian) associated with such equation, i.e. potential operator equations,
alternatively with other methods such as bifurcation theory, critical point theory
and so on.

However, for non-variational systems such as (1.1) there are only a few results
involving pq-Laplacian operators. In this case we would like to emphasize the
work done by Clément, Fleckinger, Mitidieri and Thélin [10] where they proved the
existence of radial positive solutions for a quasilinear elliptic system

−∆pu = |u|α|v|β , in Ω,

−∆qv = |u|γ |v|δ, in Ω,

(u, v) = (0, 0), on ∂Ω,

where the exponents α, β, δ, γ are non-negative and imposing the condition βγ > 0,
which maintains the system coupled.

As one of the main motivations to study these kinds of problems, system (1.1) is
an extension of a similar system assuming the Laplacian operators, which represents
the steady states of a parabolic cooperative problem of the form

∂u

∂t
−∆u = λu+ b(x)v − a(x)f(x, u)u, in Ω× (0,∞),



4 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

∂v

∂t
−∆v = c(x)u+ λv − d(x)g(x, v)v, in Ω× (0,∞),

(u, v)T = (0, 0)T on ∂Ω× (0,∞),

(u(·, 0), v(·, 0))T = (u0, v0)T > (0, 0)T in Ω,

where γ ∈ Cµ(Ω), for all γ ∈ {a, b, c, d} and some µ ∈ (0, 1], such that a, d ≥ 0
and b, c > 0 in Ω, λ is a real parameter and the nonlinear functions f, g ∈
Cµ,1+µ(Ω̄× [0,∞)) satisfy that for any h ∈ {f, g}, h(x, 0) = 0 and ∂uh(x, u) > 0 for
all x ∈ Ω and u > 0. This model arises in population dynamics for the analysis of
cooperative species where the environments are heterogeneous (see [4] for further
details). In those parabolic models the principal eigenvalue of the linearised asso-
ciated elliptic problem seems to be crucial in ascertaining the long time behaviour
of the population, i.e. the linear eigenvalue problem

−∆u+ λau− bv = τu,

−∆v + λdv − cu = τv,
(1.9)

for (u, v)T ∈ H1
0 (Ω) × H1

0 (Ω), Indeed, there are several works analyzing that as-
ymptotic behavior, when λ goes to infinity, assuming spatial heterogeneities for the
potentials in front of the parameter λ, for one single equation such as [3], and for
cooperative systems as well such as [2, 12], or for very general spatial heterogeneities
conditions [1].

Thus, the problem under consideration here might represent the steady state
solutions of the parabolic problem

∂u

∂t
+ (−∆p + λa|u|p−2)u− b|u|α−1u|v|βv = τ |u|p−2u, in Ω× (0,∞),

∂v

∂t
+ (−∆q + λd|v|q−2)v − c|u|αu|v|β−1v = τ |v|q−2v, in Ω× (0,∞),

(u, v)T = (0, 0)T on ∂Ω× (0,∞),

(u(·, 0), v(·, 0))T = (u0, v0)T > (0, 0)T in Ω,

(1.10)

where, again, the principal eigenvalue of the elliptic problem (1.1) will play an
important role in the dynamical behavior of the model. In relation to the con-
vergence of similar problems, however assuming fractional operators, in [19] the
authors obtained some multiplicity results and the convergence of those solutions
when a parameter is passed to the limit to a limiting problem with infinitely many
solutions.

1.3. Main results. Here, as mentioned above, we assume a non-variational cooper-
ative system (1.1) assuming that the off-diagonal terms b and c of the pq-Laplacian
system (1.1) are different, so considering a non-self-adjoint operator of the form
(1.3) with

b(x) 6= c(x), for all x ∈ Ω,

and, then, obtaining the convergence of the principal eigenvalue under the very
general heterogenous assumptions (1) and (2) for the non-negative potentials a and
d set up above, and under a less restrictive system as the one assumed in [17]. In
[17] the authors obtained such a convergence for a system with the same coop-
erative coefficients, applying variational arguments and under several constraints.
Consequently, the analysis shown here is completely different from the previous
works.



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 5

Although, our methods will depend on certain convergences, other important
elements are based on operator theory that will provide us with several important
properties of the principal eigenvalue of our problem.

Indeed, these properties will play a crucial role in obtaining the asymptotic
behavior of problem (1.1), as well as having their own interest since they have
never been proved before. Especially due to the non-variational nature of problem
(1.1). Indeed, despite the fact that our problem lacks such a variational structure we
are able to prove the following crucial issue: the principal eigenvalue is the smallest,
simple, positive eigenvalue and, also, isolated. To do so we use different approaches
for linear problems that we adapt for the nonlinear system (1.1) as well as some
arguments shown in [15]. Note that those properties for the eigenvalue problem
(1.1) might be established as well for the unusual limiting problem obtained here;
see details below.

Consequently, under the assumptions established above we state the main result
of this paper.

Theorem 1.1. Let Ω ⊂ RN be an open set and assume that a and d are two
potentials for which conditions (1) and (2) are fulfilled. Then

lim
λ→+∞

τ1(λ) = τ1, (1.11)

where τ1(λ) is the principal eigenvalue of the pq-Laplacian system (1.1), and the
limiting principal eigenvalue τ1 is the one corresponding to the limiting pq-Laplacian
system

−∆pu− b|u|α−1u|Pv|βPv = τ1|u|p−2u,

−∆qv − c|Pu|αPu|v|β−1v = τ1|v|q−2v,
(1.12)

for (u, v)T ∈ W 1,p
0 (Ωa0) ×W 1,q

0 (Ωd0) and where P stands for the projection on the
subdomain where both potentials vanish at the same time, i.e.

Pw = χΩa0∩Ωd0
w, with w = u, v.

In fact, if the intersection of the subdomains where the potentials a and d vanish
were empty

Ωa0 ∩ Ωd0 = ∅,
P is defined to be zero and the limiting principal eigenvalue τ1 will be the infimum
among the principal eigenvalues corresponding to the uncoupled system

−∆pu = τ1|u|p−2u in Ωa0 ,

−∆qv = τ1|v|q−2v in Ωd0,
(1.13)

under homogeneous Dirichlet boundary conditions and such that

τ1 = inf{τ1[−∆p,Ω
a
0 ], τ1[−∆q,Ω

d
0]}.

In addition, any sequence of normalized eigenfunctions {(ϕλ, ψλ)T } associated with

τ1(λ) admits a subsequence that converges strongly in W 1,p
0 (Ω) ×W 1,q

0 (Ω) to the
normalized eigenfunction (ϕ∗, ψ∗)

T in Lp(Ω) × Lq(Ω) associated with τ1, in the
sense that ∫

Ω

|ϕ∗|p +

∫
Ω

|ψ∗|q = 1.



6 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

Remark 1.2. The principal eigenvalue τ1(`) is simple, isolated and positive (see

details below) so the whole sequence {(ϕλ, ψλ)T } converges strongly in W 1,p
0 (Ω)×

W 1,q
0 (Ω) to the corresponding normalized eigenfunction (ϕ∗, ψ∗)

T ∈ W 1,p
0 (Ωa0) ×

W 1,q
0 (Ωd0) (e.g. [2]). We will prove this fact as part of Theorem 1.1.
Moreover, we cannot forget that for this quasilinear system (1.1) we have the

possibility of existence of semi-trivial solutions.

We define the spaces W 1,p
0 (E) for a measurable set E ⊆ RN in terms of the

capacity. In other words,

W 1,p
0 (E) := {u ∈W 1,p(RN ) so that u = 0 q.e. on RN\E}, (1.14)

where q.e. means “quasi everywhere” (with respect to a set of capacity zero), E ⊆
RN is any measurable subset (not necessarily open). Observe, that particularly for

any open set Ω ⊂ RN there is a nice characterization of W 1,p
0 (Ω) using capacity

(see for instance [8, Theorem 4.1.2] or [24]), namely we have

u ∈W 1,p
0 (Ω)⇔ (u ∈W 1,p(RN ) and u = 0 q.e. on RN\Ω).

Note that W 1,p
0 (E) is a closed subspace of W 1,p(RN ) and inherits its structure.

Consequently, the imbedding W 1,p
0 (E) into Lp(E) remains compact. Notice also

that according to our definition W 1,p
0 (E) is never empty, because it always contains

the function identically equal to 0. That could be also deduced from the condition
(1.7). It is also clear from the definition that W 1,p

0 (E) = {0} when the capacity of
E is zero.

However, when the potentials are continuous functions we must recall that the
definition for the space W 1,p

0 (Ω) denoted above by (1.14) is equivalent to

W 1,p
0 (Ω) := {u ∈W 1,p(RN ) so that u = 0 a.e. on RN\Ω},

since the set Ω will be open and the zero set has capacity zero. This implies that
part of the zero set of the potentials a and d with measure zero could be inside Ωa+
and Ωd+ respectively. Thus, under hypothesis (1.6) we know that

W 1,p
0 (Ka) := W 1,p(RN ) ∩ {u = 0 q.e. in RN \Ka}

= W 1,p(RN ) ∩ {u = 0 a.e. in RN \Ka}.

Indeed, the only kind of regularity assumption that is contained in condition (2)

which can be understood as a stability-type property for W 1,p
0 (resp. W 1,q

0 ).

2. p-Laplacian eigenvalue problem

We assume that the Banach space W 1,p
0 (Ω) with 1 < p < ∞, for a bounded

domain Ω in RN , is equipped with the norm

‖u‖W 1,p
0 (Ω) :=

(∫
Ω

|∇u|p
)1/p

.

Thanks to Poincaré’s inequality this norm is equivalent to the standard one for the
Sobolev spacesW 1,p

0 (Ω) in bounded domains. The pair (u, v)T ∈W 1,p
0 (Ω)×W 1,q

0 (Ω)



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 7

is called a weak solution of the problem (1.1) if the following integral identities hold∫
Ω

|∇u|p−2∇u∇ν1 + λ

∫
Ω

a|u|p−2u ν1 −
∫

Ω

b|u|α|v|β+1ν1

= τ(λ)

∫
Ω

|u|p−2u ν1,∫
Ω

|∇v|q−2∇v∇ν2 + λ

∫
Ω

d|v|q−2v ν2 −
∫

Ω

c|u|α+1|v|βν2

= τ(λ)

∫
Ω

|v|q−2v ν2,

(2.1)

for any (ν1, ν2)T ∈ W 1,p
0 (Ω) × W 1,q

0 (Ω). In addition, a real number τ is called
an eigenvalue under homogeneous Dirichlet boundary conditions and (u, v)T is its
associated eigenfunction of the system

−∆pu− b|u|α−1u|v|βv = τ |u|p−2u, in Ω,

−∆qv − c|u|αu|v|β−1v = τ |v|q−2v, in Ω,

(u, v) = (0, 0), on ∂Ω,

(2.2)

if ∫
Ω

|∇u|p−2∇u∇ν1 +

∫
Ω

|∇v|q−2∇v∇ν2

= τ
(∫

Ω

|u|p−2u ν1 +

∫
Ω

|v|q−2v ν2

)
+

∫
Ω

b|u|α|v|β+1ν1 +

∫
Ω

c|u|α+1|v|βν2,

(2.3)

for every (ν1, ν2)T ∈W 1,p
0 (Ω)×W 1,q

0 (Ω).

2.1. Operator properties. We first obtain the semicontinuity and monotonicity
of the operators for the single equation (2.8). Indeed, assuming the problem

(−∆p + λa|u|p−2)u = f, in Ω,

u = 0, on ∂Ω,
(2.4)

for p > 1 and f in the dual space of Y = W 1,p
0 (Ω), denoted by Y ′ = W−1,p′(Ω)

with 1 = 1
p + 1

p′ we obtain the following result.

Lemma 2.1. Let f ∈W−1,p′(Ω) be for problem (2.4). Then, the operator

B(·) = −∆p(·) + λa| · |p−2(·), (2.5)

is continuous and monotone.

Proof. Take a bounded sequence {un} in W 1,p
0 (Ω). Then, because of the compact-

ness we can find a subsequence, again labeled {un}, which converges weakly in

W 1,p
0 (Ω) and strongly in Lp(Ω) to a certain u. Moreover,

‖B(un)− B(u)‖M(Y ) = sup
‖h‖Y ≤1

‖〈B(un)− B(u), h〉‖Y

= sup
‖h‖Y ≤1

∥∥∫
Ω

[|∇un|p−2∇un − |∇u|p−2∇u] · ∇h

+ λ

∫
Ω

a[|un|p−2un − |u|p−2u]h
∥∥
Y
,



8 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

with M(Y ) denoting the space of bounded operators from Y to Y . Applying
Hölder’s inequality yields

‖B(un)− B(u)‖M(Y )

≤
(∫

Ω

∣∣|∇un|p−2∇un − |∇u|p−2∇u
∣∣p′)1/p′(∫

Ω

|∇h|p
)1/p

+ λ
(∫

Ω

∣∣a[|un|p−2un − |u|p−2u]
∣∣p′)1/p′(∫

Ω

|h|p
)1/p

≤ K
[( ∫

Ω

∣∣|∇un|p−2∇un − |∇u|p−2∇u
∣∣p′)1/p′

+ `
(∫

Ω

∣∣a[|un|p−2un − |u|p−2u]
∣∣p′)1/p′]

.

(2.6)

Note that
1

p
+

1

p′
= 1, so that p′ =

p

p− 1
.

Thus, we find that (2.6) certainly converges to zero because of the convergence of
the sequence {un} in Lp(Ω) and, also, by the continuity of the Nemytski operator

M(s) = |s|p−2s, from Lp(Ω) to Lp
′
(Ω).

Now, we see that the operator B is also a monotone operator. Indeed, for u 6= v

〈B(u)− B(v), u− v〉

=

∫
Ω

[|∇u|p−2∇u− |∇v|p−2∇v] · (∇u−∇v)

+ λ

∫
Ω

a[|u|p−2u− |v|p−2v](u− v)

≥
∫

Ω

|∇u|p −
∫

Ω

|∇v|p−2∇v · ∇u+

∫
Ω

|∇v|p −
∫

Ω

|∇u|p−2∇u · ∇v

≥
∫

Ω

|∇u|p −
(∫

Ω

|∇v|p
′
)1/p′(∫

Ω

|∇u|p
)1/p

+

∫
Ω

|∇v|p −
(∫

Ω

|∇u|p
′
)1/p′(∫

Ω

|∇v|p
)1/p

= [‖u‖p−1 − ‖v‖p−1][‖u‖ − ‖v‖] ≥ 0,

since s 7→ |s|p−1 is an increasing function on (0,+∞). Therefore, the operator B is
monotone and continuous. �

Remark 2.2. Consequently, by Lemma 2.1 we have the existence of solutions for
a problem of the form (2.4) from Browder’s Theorem [13, Th. 5.3.22], since the
operator is also semicontinuous, bounded and coercive.

The conclusions established above for the operators B (2.5) corresponding to the
single equation (2.4) can be extended to a pq-Laplacian cooperative system

(−∆p + λa|u|p−2)u− b|u|α−1u|v|βv = f, in Ω,

(−∆q + λd|v|q−2)v − c|u|αu|v|β−1v = g, in Ω,

u = v = 0, on ∂Ω,

(2.7)



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 9

for p > 1, q > 1. Thus, we prove, following similar arguments and under certain
extra assumptions, that the operator corresponding to the cooperative system is
continuous. However, in general, we must stress that for large cooperative terms b
and c the monotonicity is not true. Hence, only the continuity can be obtained.

Lemma 2.3. Assume a cooperative system of the form (2.7) such that f belongs

to the dual space of Y1 = W 1,p
0 (Ω) denoted by

Y ′1 = W−1,p′(Ω) with 1 =
1

p
+

1

p′
,

g belongs to the dual space of Y2 = W 1,q
0 (Ω) denoted by

Y ′2 = W−1,q′(Ω) with 1 =
1

q
+

1

q′
,

and condition (1.2) is also satisfied. Then, the operator Sλ acting in (1.3) is con-
tinuous.

Proof. Taking a bounded sequence

{(un, vn)T } ∈W 1,p
0 (Ω)×W 1,q

0 (Ω),

by compactness we can find a subsequence, again labeled {(un, vn)T }, which con-

verges weakly in W 1,p
0 (Ω) ×W 1,q

0 (Ω) and strongly in Lp(Ω) × Lq(Ω) to a certain
(u, v)T . Moreover,∥∥∥Sλ(unvn

)
− Sλ

(
u
v

)∥∥∥
M(W )

= sup
‖hi‖Yi≤1
i=1,2

∥∥∥〈Sλ(unvn
)
− Sλ

(
u
v

)
,

(
h1

H2

)〉∥∥∥
W

= sup
‖hi‖Yi≤1
i=1,2

∥∥∥∫
Ω

[|∇un|p−2∇un − |∇u|p−2∇u] · ∇h1

+

∫
Ω

[|∇vn|q−2∇vn − |∇v|q−2∇v] · ∇h2

+ λ
(∫

Ω

a[|un|p−2un − |u|p−2u]h1 +

∫
Ω

d[|vn|q−2vn − |v|q−2v]h2

)
−
∫

Ω

b|u|α−1u|v|βv h1 −
∫

Ω

c|u|αu|v|β−1v h2

∥∥∥
W
,

where W := Y1×Y2 = W 1,p
0 (Ω)×W 1,q

0 (Ω). It is now clear that it converges to zero,
once we apply the Hölder’s inequality and the cooperative character of the system.
Indeed,∥∥∥Sλ(unvn

)
− Sλ

(
u
v

)∥∥∥
M(W )

≤
(∫

Ω

∣∣|∇un|p−2∇un − |∇u|p−2∇u
∣∣p′)1/p′(∫

Ω

∣∣∇h1

∣∣p)1/p

+
(∫

Ω

∣∣|∇vn|q−2∇vn − |∇v|q−2∇v
∣∣q′)1/q′(∫

Ω

|∇h2|q
)1/q



10 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

+ λ
(∫

Ω

∣∣a[|un|p−2un − |u|p−2u]
∣∣p′)1/p′(∫

Ω

|h1|p
)1/p

+ λ
(∫

Ω

∣∣d[|vn|q−2vn − |v|q−2v]
∣∣q′)1/q′(∫

Ω

|h2|q
)1/q

≤ K
((∫

Ω

∣∣|∇un|p−2∇un − |∇u|p−2∇u
∣∣p′)1/p′

+ λ
(∫

Ω

∣∣a[|un|p−2un − |u|p−2u]
∣∣p′)1/p′(∫

Ω

∣∣|∇vn|q−2∇vn − |∇v|q−2∇v
∣∣q′)1/q′

+ `
(∫

Ω

∣∣d[|vn|q−2vn − |v|q−2v]
∣∣q′)1/q′)

.

Therefore, the operator Sλ is continuous. �

2.2. Principal eigenvalue properties. In this section we establish a result which
is a counterpart of the already existent one corresponding to the single equation
(see e.g. [5, 6, 20]) of the form

−∆pu = τ |u|p−2u, (2.8)

under homogeneous Dirichlet boundary conditions. For which it is known that
it admits, a unique positive first eigenvalue τ1 with a non-negative eigenfunction.
Moreover, the principal eigenvalue of problem (2.8) is also isolated and simple as a
consequence of its variational characterization.

For cooperative systems we can consider the variational cooperative eigenvalue
problem

−∆pu− (α+ 1)B|u|α−1u|v|βv = τ0|u|p−2u, in Ω,

−∆qv − (β + 1)B|u|αu|v|β−1v = τ0|v|q−2v, in Ω,

(u, v) = (0, 0) on ∂Ω,

(2.9)

where τ0 = τ0[S0,Ω] stands for the principal eigenvalue for the symmetric operator

S0

(
u
v

)
:=

(
−∆pu −(α+ 1)B|u|α−1u|v|βv

−(β + 1)B|u|αu|v|β−1v −∆qv

)
.

Furthermore, since problem (2.9) has a variational structure, in this case we have
an expression for the first eigenvalue based on the Rayleigh quotient, i.e.

τ0 = inf
u∈W 1,p

0 (Ω),

v∈W 1,q
0 (Ω)

∫
Ω
|∇u|p +

∫
Ω
|∇v|q − (α+ β + 2)

∫
Ω
B|u|α+1|v|β+1∫

Ω
up +

∫
Ω
vq

. (2.10)

Hence, applying the results obtained by Kawohl & Lindqvist [16] and Lindqvist [20]
we can find that the first eigenvalue of the problem (2.9) is unique, positive and
isolated for any arbitrary domain Ω in RN . Moreover, its associated eigenfunction is
unique and positive in Ω and with maximal regularity C1,α(Ω), for some 0 < α < 1
by elliptic regularity. Those facts can be proved following the arguments nicely
shown in [16] for problem (2.9); see also [17, Lemma 2.3].

Thus, we state a similar result for the cooperative system of the form (2.2) prov-
ing that there is no positive eigenvalue below τ1 (denoting the smallest eigenvalue)
as well as to being an isolated eigenvalue from above, associated with a positive
eigenfunction (ϕ1, ψ1)T ∈ W 1,p

0 (Ω) ×W 1,q
0 (Ω), so that ϕ1 > 0 and ψ1 > 0, with

negative outward normal derivatives on the boundary. However, we must note that



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 11

our problem is not variational so a different approach must be followed. The results
actually rely on the strong maximum principle for p-Laplacian problems.

To do so we introduce the following notation

−∆pu− b|u|α−1u|v|βv = A1w, −∆qv − c|u|αu|v|β−1v = A2w,

where w = (u, v) and Ai, for i = 1, 2 the operator involved in each equation so that
(2.2) becomes

A1w = τ(Dw)1,

A2w = τ(Dw)2.
(2.11)

with

τ |u|p−2u = τ(Dw)1, τ |v|q−2v = τ(Dw)2.

According to Lemma 2.3 the operator S is continuous in the positive cone K cor-
responding to the Banach space W 1,p

0 (Ω) ×W 1,q
0 (Ω). From condition (1.2) on the

exponents we find that

either p− 1 < α+ β + 1 < q − 1,

or q − 1 < α+ β + 1 < p− 1,
(2.12)

depending on the assumption of p ≤ q or q ≤ p. Then assuming (2.12) we introduce
a “pseudo-homogenous” condition for the operator S saying that

S(tw) ≥ tr−1S(w), (2.13)

where w = (u, v)T , with r = p or r = q, depending on the different possible
situations, and if t ≥ 0. In particular, it follows that

A1(tw) ≥ tr−1A1w, A2(tw) ≥ tr−1A2w.

The last property is similar to the equivalent one for linear operators

L(µw) = µL(w),

which means that the operator is homogeneous of degree one or simply homoge-
neous.

Furthermore, thanks to [15, Lemma 5.5] and (2.12) we actually have that weak
solutions are uniformly bounded in L∞(Ω) × L∞(Ω). Thus, due to [18, Theorem
1, page 1203] the solutions of system (2.2) belong to C1,α(Ω) × C1,α(Ω), for some
0 < α < 1.

It is important to point out that the next result is also valid for the limiting
problem (1.12) under the heterogenous assumptions established at the beginning of
this work.

Lemma 2.4. Let τ1 be the smallest eigenvalue of the problem (2.2), under homo-
geneous Dirichlet boundary conditions and denoted by

τ1 := τ1[S,Ω] .

Moreover, τ1 is algebraically simple, isolated and it possesses a unique positive
eigenfunction (up to a multiplicative constant), denoted by (ϕ1, ψ1)T . Further-
more, (ϕ1, ψ1)T is strictly positive and there is not any other eigenvalue τ of (2.2)
satisfying τ < τ1.



12 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

Proof. Positivity of the first eigenvalue. To prove this we claim that if it exists
(u, v)T ∈ K \ {0} so that

S
(
u
v

)
≥ µ

(
|u|p−2u
|v|q−2v

)
,

with µ > 0, then, there is τ1 ≥ µ and (ϕ1, ψ1)T ∈ K \ {0} such that

S
(
ϕ1

ψ1

)
= τ1

(
|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
.

In other words, the topological degree on the positive cone K for the operator(
diag(|u|p−2u, |v|q−2v)− 1

µ
S
(
u
v

))
,

in the unit ball changes. Hence, the problem(
|u|p−2u
|v|q−2v

)
=

1

µ
S
(
u
v

)
+ t

(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
, (2.14)

has no solution in K if t > 0. To prove this, assume that if (u, v)T is a solution of
(2.14) in K \ {0} we have that(

|u|p−2u
|v|q−2v

)
≥ t
(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
and

(
|u|p−2u
|v|q−2v

)
≥ t̃
(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
,

where t̃ = s̃r−1 is the maximum among all the t’s. Then(
|u|p−2u
|v|q−2v

)
≥ 1

µ
S
(
t̃ϕ1

t̃ψ1

)
+ t

(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
,

and, hence, since by (2.13) and assuming (2.12),

S
(
t̃ϕ1

t̃ψ1

)
≥ s̃r−1S

(
ϕ1

ψ1

)
,

and, by definition of (ϕ1, ψ1)T , we find that(
|u|p−2u
|v|q−2v

)
≥ s̃r−1

µ
µ

(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
+ t

(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
= (t̃+ t)

(
|ϕ1|p−2ϕ1

|ψ|q−2ψ1

)
,

which is a contradiction with the maximality of t̃. Therefore, there is no solution
to equation (2.14) if t > 0 or the topological degree on the positive cone K of the
operator (

diag(|u|p−2u, |v|q−2v)− 1

µ
S
(
u
v

))
,

in the unit ball is zero if t > 0. Consequently, t = 0 and, hence,

either, there exists a positive solution, or the topological degree on
the positive cone K changes.

In the first situation, we have that the eigenvalue is positive. On the other hand, for
the second situation we have that the degree is 1 and we find a positive eigenvalue.
Also, the degree is 1 if τ1 = 0. However, if that is the case we will find that if there
exists λ ∈ [0, 1

µ ] such that (
|u|p−2u
|v|q−2v

)
= λS(w),



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 13

where w = (u, v)T , with normalized w ∈ K, w = (u, v)T , then we actually have
that λ = 1

µ . To prove it, let us take a sequence λn → 1
µ , as n→∞ and wn positive

solutions so that

wn = λnS(wn), with wn → w0, as n→∞,

where w0 is a positive solution. Thus,

wn = λnS(wn) =
1

µ
S(wn) +

(
λn −

1

µ

)
S(wn).

Therefore, passing to the limit we obtain only positive solutions if λ = 1/µ, proving
that the first eigenvalue is strictly positive.

Non-existence of positive eigenvalue smaller (or bigger) than τ1. Next
we prove that there is no other positive eigenvalue smaller than τ1, assuming that
for the first eigenvalue its associated eigenfunction (ϕ1, ψ1)T has both components
non-negative, i.e.(

|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
=

1

τ1
S
(
ϕ1

ψ1

)
, where (ϕ1, ψ1)T ∈ K.

Subsequently, to prove that τ1 is the smallest positive eigenvalue we argue by con-
tradiction. Thus, let us assume that there exists a positive eigenfunction (u, v)T so
that u > 0 and v > 0 and satisfying the eigenvalue problem (2.2), i.e.(

|u|p−2u
|v|q−2v

)
=

1

τ
S
(
u
v

)
, where (u, v)T ∈ K, (2.15)

and with 0 < τ < τ1. In other words, there is another positive eigenvalue smaller
than τ1 with an associated positive eigenfunction. Then, by definition we have that(

|u|p−2u
|v|q−2v

)
=

1

τ
S
(
u
v

)
=

1

τ1
S
(
u
v

)
+
(1

τ
− 1

τ1

)
S
(
u
v

)
=

1

τ1
S
(
u
v

)
+
(1

τ
− 1

τ1

)
τ

(
|u|p−2u
|v|q−2v

)
≥ 1

τ1
S
(
u
v

)
+
(

1− τ

τ1

)r−1
(
|u|p−2u
|v|q−2v

)
.

Note that

0 < 1− τ

τ1
< 1, since τ < τ1.

Next, we show that for a certain k (to be determined below) it follows that

S
(
u
v

)
≥ S

(
kϕ1

kψ1

)
. (2.16)

To prove so we use the weak formulation of the eigenvalue problem (2.3) i.e.∫
Ω

|∇u|p−2∇u · ∇ν1 − kp−1

∫
Ω

|∇ϕ1|p−2∇ϕ1 · ∇ν1 −
∫

Ω

b|u|α|v|β+1ν1

+ kα+β+1

∫
Ω

b|ϕ1|α|ψ1|β+1ν1

= τ

∫
Ω

|u|p−2u ν1 − kp−1

∫
Ω

|ϕ1|p−2ϕ1 ν1,



14 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50∫
Ω

|∇v|q−2∇v · ∇ν2 − kq−1

∫
Ω

|∇ψ1|q−2∇ψ1 · ∇ν2 −
∫

Ω

c|u|α+1|v|βν2

+ kα+β+1

∫
Ω

c|ϕ1|α+1|ψ1|βν2

= τ

∫
Ω

|v|q−2v ν2 − kq−1

∫
Ω

|ψ1|q−2ψ1 ν2,

for a test function (ν1, ν2)T . Since the eigenfunctions are positive and the system
is cooperative (b(x) > 0 and c(x) > 0) taking

B = max
x∈Ω
{b(x), c(x)},

we have that (we also add α + 1 in the first equation and b + 1 in the second, for
the coupling terms)∫

Ω

|∇u|p−2∇u · ∇ν1 − kp−1

∫
Ω

|∇ϕ1|p−2∇ϕ1 · ∇ν1 −
∫

Ω

b|u|α|v|β+1ν1

+ kα+β+1

∫
Ω

b|ϕ1|α|ψ1|β+1ν1

≥
∫

Ω

|∇u|p−2∇u · ∇ν1

− (α+ 1)

∫
Ω

B|u|α|v|β+1ν1 − kp−1

∫
Ω

|∇ϕ1|p−2∇ϕ1 · ∇ν1,∫
Ω

|∇v|q−2∇v · ∇ν2 − kq−1

∫
Ω

|∇ψ1|q−2∇ψ1 · ∇ν2 −
∫

Ω

c|u|α+1|v|βν2

+ kα+β+1

∫
Ω

c|ϕ1|α+1|ψ1|βν2

≥
∫

Ω

|∇v|q−2∇v · ∇ν2

− (β + 1)

∫
Ω

B|u|α+1|v|βν2 − kq−1

∫
Ω

|∇ψ1|q−2∇ψ1 · ∇ν2.

Thanks to the variational cooperative eigenvalue problem (2.9) and the Rayleigh
quotient of the first eigenvalue τ0 (2.10) it follows that∫

Ω

|∇u|p−2∇u · ∇ν1 − kp−1

∫
Ω

|∇ϕ1|p−2∇ϕ1 · ∇ν1 −
∫

Ω

b|u|α|v|β+1ν1

+ kα+β+1

∫
Ω

b|ϕ1|α|ψ1|β+1ν1

≥ τ0
∫

Ω

|u|p−2u ν1 − kp−1

∫
Ω

|∇ϕ1|p−2∇ϕ1 · ∇ν1,∫
Ω

|∇v|q−2∇v · ∇ν2 − kq−1

∫
Ω

|∇ψ1|q−2∇ψ1 · ∇ν2 −
∫

Ω

c|u|α+1|v|βν2

+ kα+β+1

∫
Ω

c|ϕ1|α+1|ψ1|βν2

≥ τ0
∫

Ω

|v|q−2v ν2 − kq−1

∫
Ω

|∇ψ1|q−2∇ψ1 · ∇ν2.



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 15

Hence, to arrive at the inequality (2.16) we need that

k ≤ min
{( τ0

∫
Ω
|u|p−2u ν1∫

Ω
|∇ϕ1|p−2∇ϕ1 · ∇ν1

) 1
p−1

,
( τ0

∫
Ω
|v|q−2v ν2∫

Ω
|∇ψ1|q−2∇ψ1 · ∇ν2

) 1
q−1
}
, (2.17)

with u, ϕ1, ν1 ∈ W 1,p
0 (Ω) and v, ψ1, ν2 ∈ W 1,q

0 (Ω). The maximal value for k that
satisfies (2.17) is denoted by

κ = kmax.

In particular, expression (2.17) provides us with the condition

(|u|p−2u, |v|q−2v)T ≥ κr−1(|ϕ1|p−2ϕ1, |ψ1|q−2ψ1)T , (2.18)

with the appropriate r = p or r = q depending on (2.12). Consequently, thanks to
(2.15) and (2.18) we find that(

|u|p−2u
|v|q−2v

)
≥ 1

τ1
S
(
u
v

)
+
(

1− τ

τ1

)r−1
(
|u|p−2u
|v|q−2v

)
≥ 1

τ1
S
(
κϕ1

κψ1

)
+
(

1− τ

τ1

)r−1

κr−1

(
|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
≥ κr−1

(
|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
+
(

1− τ

τ1

)r−1

κr−1

(
|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
=
(
κr−1 +

(
1− τ

τ1

)r−1

κr−1
)(|ϕ1|p−2ϕ1

|ψ1|q−2ψ1

)
,

which contradicts the maximality of κr−1 and proving that there is no positive
eigenvalue below τ1.

Similar arguments show that there is no eigenvalue bigger that τ1 which has a
positive associated eigenfunction.

Simplicity. To show that the principal eigenvalue τ1 is a simple eigenvalue we
assume that, apart from the eigenfunction (ϕ1, ψ1)T ∈ K, there exists another
eigenfunction (u, v)T ∈ K. Arguing as above we arrive at (u, v)T ≥ K1(ϕ1, ψ1)T

and, also, the opposite inequality (u, v)T ≤ K2(ϕ1, ψ1)T , so that K1K1 = 1, show-
ing that both eigenfunctions are proportional. Therefore, the principal eigenvalue
is simple.

Isolated principal eigenvalue τ1. To prove it, we assume a sequence of eigen-
values {τn} of problem (2.2) such that τn → τ1, as n → ∞. Moreover, the se-
quence of eigenfunctions {(un, vn)T } belongs to the positive cone K, normalized
‖(un, vn)T ‖L∞(Ω×L∞(Ω) = 1. Thanks to regularity theory and maximal principles

we actually have that the sequence is bounded in C1,α, with α ∈ (0, 1). Hence, we
have the convergence of such a sequence in C1,α, up to a subsequence, i.e.

(un, vn)T → (ϕ1, ψ1)T , as n→∞,
where (ϕ1, ψ1)T is the associated eigenfunction with the principal eigenvalue τ1.
Therefore, since the elements of the sequence will belong to the positive cone, and
τ1 is the only eigenvalue with a positive eigenfunction we arrive at a contradiction,
proving that such an eigenvalue is isolated. �

Remark 2.5. Lemma 2.4 is also true for the system with

(−∆p + λa|u|p−2)u− b|u|α−1u|v|βv = A1(λ)w,

(−∆q + λd|v|q−2)v − c|u|αu|v|β−1v = A2(λ)w.



16 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

Also, thanks to the monotonicity of the principal eigenvalue with respect to the
potential and with respect to the domain we know that the function τ1(λ) is con-
tinuous and increasing with respect to the operator S(V1, V2) denoted by (1.3) (cf.
[17]) so that the limit limλ→∞ τ1(λ) exists. Indeed, under the spatial heterogenous
conditions for the potentials a and d this limit is bounded above. Otherwise the
limit could be possibly ±∞.

Furthermore, for the limiting system we consider the problem

−∆pu− b|u|α−1u|v|βv = τ1|u|p−2u,

−∆qv − c|u|αu|v|β−1v = τ1|v|q−2v,

for (u, v)T ∈W 1,p
0 (Ωa0)×W 1,q

0 (Ωd0). We say that (u, v)T is a solution of this system
when each equation is satisfied in the sense/framework

(−∆pu+ λa|u|p−2)u = f, u ∈W 1,p
0 (A),

so that A satisfies the spatial heterogeneous conditions under consideration in this
paper and f in the dual space W−1,p′(A)

3. Proof of the main results

To prove Theorem 1.1, we do the follow in 3 steps. First we prove the convergence
of the eigenfunctions in W 1,p

0 (Ω)×W 1,q
0 (Ω).

Step 1. Convergence of the eigenfunctions {(ϕλ, ψλ)T } inX := W 1,p
0 (Ω)×W 1,q

0 (Ω).
Let {`n}n≥1 be any increasing unbounded sequence, i.e. 0 < λn < λm if n < m,
and limn→∞ `n =∞. Then, for every n ≥ 1 we consider a sequence {(ϕλn , ψλn)T }
of normalized solutions in Y := Lp(Ω)× Lq(Ω) for system (1.1) in the sense that∫

Ω

|ϕλn |p +

∫
Ω

|ψλn |q = 1,

associated with τ1(λn) for problem (1.1). Then, multiplying (1.1) by (ϕλn , ψλn)T

and integrating by parts yields∫
Ω

|∇ϕλn |p +

∫
Ω

|∇ψλn |q + λn

∫
Ω

(a|ϕλn |p + d|ψλn |q)

= τ1(λn) +

∫
Ω

(b+ c)|ϕλn |α+1|ψλn |β+1.

It is now clear that by Hölder’s inequality, condition (1.2), and the cooperative
assumptions on the coefficients b and c, we find that∫

Ω

(b+ c)|ϕλn |α+1|ψλn |β+1 ≤ C
(∫

Ω

|ϕλn |p
)α+1

p
(∫

Ω

|ψλn |q
) β+1

q

,

for a positive constant C > 0, so that by construction we have∫
Ω

|∇ϕλn |p ≤ K,
∫

Ω

|∇ψλn |q ≤ K, λn

∫
Ω

(aϕpλn + dψqλn) ≤ K, (3.1)

for a positive constant K. We point out that the principal eigenvalue τ1(λ) is
bounded above, thanks to the monotonicity of the principal eigenvalue with re-
spect to the domain, by the principal eigenvalue for the operator S under Dirichlet
homogeneous boundary conditions in the subdomain Ω0 and denoted by (1.4), i.e.

τ1(λ) ≤ τ1[S; Ω0].



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 17

Hence, the sequence {(ϕλn , ψλn)T } is bounded in X := W 1,p
0 (Ω) ×W 1,q

0 (Ω). As

the imbedding W 1,r
0 (Ω) ↪→ Lr(Ω) with r = p, r = q, is compact, we can extract a

subsequence, again labeled {(ϕλn , ψλn)T }, weakly convergent in X and strongly in
Y to some function (ϕ∗, ψ∗) ∈ Y , i.e.

lim
λ→∞

‖(ϕλn , ψλn)T − (ϕ∗, ψ∗)
T ‖Y = 0, Y = Lp(Ω)× Lq(Ω).

In fact, we will prove in the sequel that the sequence {(ϕλn , ψλn)T } is actually a
Cauchy sequence in X. In other words, we have the strong convergence of that
subsequence in U , this implies that

lim
λ→∞

‖(ϕλn , ψλn)T − (ϕ∗, ψ∗)
T ‖X = 0, X = W 1,p

0 (Ω)×W 1,q
0 (Ω).

with (ϕ∗, ψ∗)
T ∈ X and such that∫

Ω

|ϕ∗|p +

∫
Ω

|ψ∗|q = 1.

Subsequently, fix n < m so that 0 < λn < λm and set

Dn,m :=

∫
Ω

∣∣∇(ϕλn − ϕλm)
∣∣p +

∫
Ω

∣∣∇(ψλn − ψλm)
∣∣q. (3.2)

To obtain the convergence we consider the so-called Clarkson’s inequality i.e.

|u1 − u2|p

2p−1
+
|u1 + u2|p

2p−1
≤ |u1|p + |u2|p. (3.3)

Also, thanks to the strict convexity of the mapping u 7→ |u|p it follows that

|u2|p > |u1|p + p|u1|p−2u1(u2 − u1), for points in RN and u1 6= u2, p > 1. (3.4)

Hence, according to (3.4) for u1 + u2,

|u1 + u2|p

2p
≥ |u1|p +

1

2
p|u1|p−2u1(u2 − u1). (3.5)

Consequently, combining both inequalities (3.3), (3.5) we find that

|u1 − u2|p

2p−1
≤ |u2|p − |u1|p − p|u1|p−2u1(u2 − u1),

and, hence, if ∇(ϕλn − ϕλm) = u1 − u2 it yields

|∇ϕλn −∇ϕλm |p

2p−1
≤ |∇ϕλm |p − |∇ϕλn |p − p|∇ϕλn |p−2∇ϕλn · (∇ϕλm −∇ϕλn).

Indeed, integrating ver Ω,

1

2p−1

∫
Ω

|∇ϕλn −∇ϕλm |p

≤
∫

Ω

|∇ϕλm |p −
∫

Ω

|∇ϕλn |p −
∫

Ω

p|∇ϕλn |p−2∇ϕλn · (∇ϕλm −∇ϕλn).

Then

Dn,m ≤
∫

Ω

|∇ϕλm |p −
∫

Ω

|∇ϕλn |p −
∫

Ω

p|∇ϕλn |p−2∇ϕλn · (∇ϕλm −∇ϕλn)

+

∫
Ω

|∇ψλm |q −
∫

Ω

|∇ψλn |q −
∫

Ω

q|∇ψλn |q−2∇ψλn · (∇ψλm −∇ψλn).



18 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

Now, applying the weak formulation (2.1) for the eigenvalue problem (1.1) with the
principal eigenfunctions {(ϕλn , ψλn)T } (respectively {(ϕλm , ψλm)T }) and rearrang-
ing terms we find that

Dn,m

≤ τ1(λm)

∫
Ω

|ϕλm |p − λm
∫

Ω

a|ϕλm |p +

∫
Ω

b|ϕλm |α+1|ψλm |β+1

+ τ1(λm)

∫
Ω

|ψλm |q − λm
∫

Ω

d|ψλm |q +

∫
Ω

c|ϕλm |α+1|ψλm |β+1

+ (p− 1)
[
τ1(λn)

∫
Ω

|ϕλn |p − λn
∫

Ω

a|ϕλn |p +

∫
Ω

b|ϕλn |α+1|ψλn |β+1
]

+ (q − 1)
[
τ1(λn)

∫
Ω

|ψλn |q − λn
∫

Ω

d|ψλn |q +

∫
Ω

c|ϕλn |α+1|ψλn |β+1
]

− p
[
τ1(λn)

∫
Ω

|ϕλn |p−1ϕλm − λn
∫

Ω

a|ϕλn |p−1ϕλm +

∫
Ω

b|ϕλn |α|ψλn |β+1ϕλm

]
− q
[
τ1(λn)

∫
Ω

|ψλn |q−1ψλm − λn
∫

Ω

a|ψλn |q−1ψλm +

∫
Ω

c|ϕλn |α+1|ψλn |βψλm
]
.

Therefore,

Dn,m ≤ τ1(λm)
(∫

Ω

|ϕλm |p +

∫
Ω

|ψλm |q
)
− τ1(λn)

(∫
Ω

|ϕλn |p +

∫
Ω

|ψλn |q
)

+ pτ1(λn)

∫
Ω

|ϕλn |p−1(ϕλn − ϕλm) + qτ1(λn)

∫
Ω

|ψλn |q−1(ψλn − ψλm)

+ pλn

∫
Ω

a|ϕλn |p−1(ϕλm − ϕλn) + qλn

∫
Ω

d|ψλn |q−1(ψλm − ψλn)

+ λn

∫
Ω

a|ϕλn |p − λm
∫

Ω

a|ϕλm |p + λn

∫
Ω

d|ψλn |q − λm
∫

Ω

d|ψλm |q

+ p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm) + q

∫
Ω

c|ϕλn |α+1|ψλn |β(ψλn − ψλm).

For the terms involving the non-negative potentials a and d and thanks to the
convexity property (3.4), it follows that

λn

∫
Ω

a|ϕλn |p + pλn

∫
Ω

a|ϕλn |p−1(ϕλm − ϕλn)− λm
∫

Ω

a|ϕλm |p

< (λn − λm)

∫
Ω

a|ϕλm |p ≤ 0,

and

λn

∫
Ω

d|ψλn |q + qλn

∫
Ω

d|ψλn |q−1(ψλm − ψλn)− λm
∫

Ω

d|ψλm |q

< (λn − λm)

∫
Ω

d|ψλm |q ≤ 0,

since, by construction n < m and, then λn < λm. Hence, after adding and sub-
tracting some appropriate terms it yields

Dn,m ≤ (τ1(λm)− τ1(λn))
(∫

Ω

|ϕλm |p +

∫
Ω

|ψλm |q
)



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 19

+ τ1(λn)
(∫

Ω

(|ϕλm |p − |ϕλn |p +

∫
Ω

(|ψλm |q − |ψλn |q)
)

+ pτ1(λn)

∫
Ω

|ϕλn |p−1(ϕλn − ϕλm) + qτ1(λn)

∫
Ω

|ψλn |q−1(ψλn − ψλm)

+ p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm) + q

∫
Ω

c|ϕλn |α+1|ψλn |β(ψλn − ψλm).

For the second term we use again the convexity property (3.4), thus

τ1(λn)
(∫

Ω

(|ϕλm |p − |ϕλn |p)
)
≤ τ1(λn)p

∫
Ω

|ϕλm |p−2ϕλm(ϕλm − ϕλn),

τ1(λn)
(∫

Ω

(|ψλm |q − |ψλn |q)
)
≤ τ1(λn)q

∫
Ω

|ψλm |q−2ψλm(ψλm − ψλn).

Consequently, applying Hölder’s inequality and the fact that the eigenfunctions are
bounded in Y = Lp(Ω) × Lq(Ω) to the different terms we have that for a positive
constant C,

(τ1(λm)− τ1(λn))
(∫

Ω

|ϕλm |p +

∫
Ω

|ψλm |q
)
≤ C(τ1(λm)− τ1(λn)),

τ1(λn)

∫
Ω

(|ϕλm |p − |ϕλn |p) ≤ τ1(λn)p

∫
Ω

|ϕλm |p−2ϕλm(ϕλm − ϕλn)

≤ C‖ϕλm − ϕλn‖Lp(Ω),

τ1(λn)

∫
Ω

(|ψλm |q − |ψλn |q) ≤ τ1(λn)q

∫
Ω

|ψλm |q−2ψλm(ψλm − ψλn)

≤ C‖ψλm − ψλn‖Lq(Ω),

pτ1(λn)

∫
Ω

|ϕλn |p−1(ϕλn − ϕλm) ≤ C
(∫

Ω

|ϕλn |p
′
)1/p′(∫

Ω

|ϕλn − ϕλm |p
)1/p

≤ C‖ϕλn − ϕλm‖Lp(Ω),

qτ1(λn)

∫
Ω

|ψλn |q−1(ψλn − ψλm) ≤ C
(∫

Ω

|ψλn |q
′
)1/q′(∫

Ω

|ψλn − ψλm |q
)1/q

≤ C‖ψλn − ψλm‖Lq(Ω).

Also, for the terms with the cooperative coefficients b and c, we have

p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm) and q

∫
Ω

c|ϕλn |α+1|ψλn |β(ψλn − ψλm).

we will apply Young’s and Hölder’s inequalities. Indeed, by Young’s inequality and
assuming the cooperative term b(x) is bounded in Ω, we find that

p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm)

≤ C
∫

Ω

(
α+ 1

p
|ϕλn |

αp
α+1 +

β + 1

q
|ψλn |q

)
(ϕλn − ϕλm).

Moreover, thanks to Hölder’s inequality,∫
Ω

(α+ 1

p
|ϕλn |

αp
α+1 +

β + 1

q
|ψλn |q

)
(ϕλn − ϕλm)

≤ C
(∫

Ω

|ϕλn |
α(p−1)
α+1

)1/p′(∫
Ω

|ϕλn − ϕλm |p
)1/p



20 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

+ C
(∫

Ω

|ψλn |qp
′
)1/p′(∫

Ω

|ϕλn − ϕλm |p
)1/p

.

Thus, by [11, Lemma 2] it follows that

‖ψλn‖
q
qp′ ≤ ε

∫
Ω

|∇ψλn |q +Mε

∫
Ω

ω|ψλn |q,

for ε > 0, a positive constant Mε depending on ε and a bounded positive weight ω.
Then, we finally have that

p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm)

≤ C
(∫

Ω

|ϕλn |
α(p−1)
α+1

)1/p′

‖ϕλn − ϕλm‖Lp(Ω)

+ C
(
ε

∫
Ω

|∇ψλn |q +Mε

∫
Ω

ω|ψλn |q
)
‖ϕλn − ϕλm‖Lp(Ω).

Similarly, for the term with the cooperative coefficient c we find that

q

∫
Ω

c|ϕλn |α+1|ψλn |β(ψλn − ψλm)

≤ C
(∫

Ω

|ψλn |
β(q−1)
β+1

)1/q′(∫
Ω

|ψλn − ψλm |q
)1/q

+ C
(
ε

∫
Ω

|∇ϕλn |p +Mε

∫
Ω

ω|ϕλn |p
)(∫

Ω

|ψλn − ψλm |q
)1/q

,

supposing that

‖ϕλn‖
p
pq′ ≤ ε

∫
Ω

|∇ϕλn |p +Mε

∫
Ω

ω|ϕλn |p,

Hence, since the eigenfunctions are bounded in Y = Lp(Ω)× Lq(Ω),

α(p− 1)

α+ 1
< p and

β(q − 1)

β + 1
< q,

and thanks to (3.1) we finally obtain that

p

∫
Ω

b|ϕλn |α|ψλn |β+1(ϕλn − ϕλm) ≤ C‖ϕλn − ϕλm‖Lp(Ω),

q

∫
Ω

c|ϕλn |α+1|ψλn |β(ψλn − ψλm) ≤ C‖ψλn − ψλm‖Lq(Ω).

Therefore, by the previous inequalities we find that there exists a positive constant
C such that∫

Ω

∣∣∇(ϕλn − ϕλm)
∣∣p +

∫
Ω

∣∣∇(ψλn − ψλm)
∣∣q

≤ C(τ1(λm)− τ1(λn)) + C‖ϕλn − ϕλm‖Lp(Ω) + C‖ψλn − ψλm‖Lq(Ω).

Thus, thanks to the convergence of the sequence {ϕλn} in Lp, the sequence {ψλn}
in Lq and the fact that the function τ1(λ) is convergent, since it is an increasing
function in λ and bounded above, by the monotonicity of the principal eigenvalue
with respect to the potential and the domain, we actually have that the sequence
{(ϕλn , ψλn)T } is a Cauchy sequence in X so that the limit (ϕ∗, ψ∗)

T satisfies that

(ϕ∗, ψ∗) ≥ (0, 0), and

∫
Ω

|ϕ∗|p +

∫
Ω

|ψ∗|q = 1.



EJDE-2022/50 COOPERATIVE SYSTEMS INVOLVING p-LAPLACIAN OPERATORS 21

Step 2. {(ϕ∗, ψ∗)T } belongs to the functional space W 1,p
0 (Ωa0) ×W 1,q

0 (Ωd0). First
thanks to (3.1) we have that the sequence {(ϕλn , ψλn)T } is bounded in X. Then,
as performed above we can extract a subsequence, again labeled {(ϕλn , ψλn)T },
weakly convergent in X and strongly in Lp(Ω)×Lq(Ω) to some function (ϕ∗, ψ∗)

T ∈
Lp(Ω) × Lq(Ω). Actually, we have proved that the sequence converges strongly in

W 1,p
0 (Ω)×W 1,q

0 (Ω).
Furthermore, according to (3.1) it follows that

λn

∫
Ω

aϕpλn ≤ K and λn

∫
Ω

dψqλn ≤ K.

Then, since {(ϕλn , ψλn)T } converges strongly in Lp(Ω) × Lq(Ω) to (ϕ∗, ψ∗)
T ∈

Lp(Ω)× Lq(Ω). Indeed,

lim
n→∞

∫
Ω

aϕpλn = 0 and lim
n→∞

∫
Ω

dψqλn = 0,

from which we easily deduce that ϕ∗ = 0 a.e. in Ωa+ and, ψ∗ = 0 a.e. in Ωd+.
Consequently, we can conclude that

(ϕ∗, ψ∗)
T ∈W 1,p

0 (Ωa0)×W 1,q
0 (Ωd0).

Step 3. (ϕ∗, ψ∗)
T eigenfunction of the limiting problem in W 1,p

0 (Ωa0)×W 1,q
0 (Ωd0).

As a consequence of Step 1 it follows that

τ1(λn)|ϕλn |p−2ϕλn + b|ϕλn |α−1ϕλn |ψλn |βψλn ,

τ1(λn)|ψλn |q−2ψλn + c|ϕλn |αϕλn |ψλn |β−1ψλn ,

converge strongly in Lp(Ω)× Lq(Ω) to

τ1|ϕ∗|p−2ϕ∗ + b|ϕ∗|α−1ϕ∗|ψ∗|βψ∗ and τ1|ψ∗|q−2ψ∗ + c|ϕ∗|αϕ∗|ψ|β−1ψ∗,

respectively. Therefore, from the assumptions on the potentials a and d and Step 2
we find that up to a subsequence, {(ϕλn , ψλn)T } converges strongly in W 1,p

0 (Ω) ×
W 1,q

0 (Ω) and the limit

(ϕ∗, ψ∗)
T ∈W 1,p

0 (Ωa0)×W 1,q
0 (Ωd0),

is a solution of the eigenvalue problem

−∆pϕ∗ − b|ϕ|α−1ϕ∗|Pψ∗|βPψ∗ = τ1ϕ∗

−∆qψ∗ − c|Pϕ∗|αPϕ∗|ψ∗|β−1ψ∗ = τ1ψ∗
(3.6)

for (ϕ∗, ψ∗)
T ∈W 1,p

0 (Ωa0)×W 1,q
0 (Ωd0) and where P stands for the projection on the

subdomain where both potentials vanish at the same time, i.e.

Pφ = χΩa0∩Ωd0
φ, with φ = ϕ∗, ψ∗.

Thus, P is defined to be zero if the intersection is null, Ωa0 ∩Ωd0 = ∅. Furthermore,
by the uniqueness of the principal eigenvalue of a singular p-Laplacian equation the
sequence converges to the eigenfunction associated with the principal eigenvalue τ1
for the uncoupled system (1.13), i.e. the principal eigenfunction whose components
are the corresponding eigenfunctions for each equation in (1.13). Actually, we



22 P. ÁLVAREZ-CAUDEVILLA EJDE-2022/50

observe that if that was the case the limiting eigenvalue would be the infimum
between the corresponding eigenvalues of the following uncoupled system

−∆pϕ∗ = τ1|ϕ∗|p−2ϕ∗ in Ωa0 ,

−∆qψ∗ = τ1|ψ∗|q−2ψ∗ in Ωd0,

under homogeneous boundary conditions and such that

τ1 = inf{τ1[−∆p,Ω
a
0 ], τ1[−∆q,Ω

d
0]}.

Acknowledgements. This research was partially supported by the Ministry of
Economy and Competitiveness of Spain, under research project PID2019-106122GB-
I00.

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Pablo Álvarez-Caudevilla

Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. Universidad 30,
28911, Leganés, Madrid, Spain

Email address: pcaudev@math.uc3m.es


	1. Introduction
	1.1. Models and preliminaries
	1.2. Motivation
	1.3. Main results

	2. p-Laplacian eigenvalue problem
	2.1. Operator properties
	2.2. Principal eigenvalue properties

	3. Proof of the main results
	Acknowledgements

	References