Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 56, pp. 1–10.

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

DOI: 10.58997/ejde.2025.56

POSITIVE SOLUTIONS FOR N-DIMENSIONAL FOURTH-ORDER SYSTEMS

UNDER A PARAMETRIC CONDITION

PABLO ÁLVAREZ-CAUDEVILLA, CRISTINA BRÄNDLE, DEVASHISH SONOWAL

Communicated by Giovanni Molica Bisci

Abstract. We establish the existence of positive solutions for a system of coupled fourth-order
partial differential equations on a bounded domain Ω ⊂ Rn,

∆2u1 + β1∆u1 − α1u1 = f1(x, u1, u2),

∆2u2 + β2∆u2 − α2u2 = f2(x, u1, u2),

for x ∈ Ω, subject to homogeneous Navier boundary conditions, where the functions f1, f2 :

Ω× [0,∞)× [0,∞) → [0,∞) are continuous, and α1, α2, β1 and β2 are real parameters satisfying
certain constraints related to the eigenvalues of the associated Laplace operator.

1. Introduction

Fourth-order nonlinear differential equations naturally appear in models concerning physical,
biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of
structures, or soil settlement, see, for example, [12] and [5] for an exposition of several models
involving higher order operators. As it is explained in [12], typically in the literature we find
fourth-order ordinary differential equations of the form

u(iv)(x) = g(x, u(x), u′′(x)), 0 < x < 1,

u(0) = u(1) = u′′(0) = u′′(1),
(1.1)

under different conditions on the function g. Such kind of equations are used to model the
deformations of an elastic beam in equilibrium state, whose two ends are simply supported [7, 8].
Because of its physics applications one normally looks for the existence of positive solutions. Thus,
for the particular one-dimensional case there are several papers where such an existence is analyzed;
see [3, 7, 8, 10] for further details and references therein. Although there are numerous references
dealing with this type of one dimensional problems, little is yet known about the behavior of
the solution in higher dimensions, both for a single equation or a system. Indeed, for one single
equation the work [3] is probably the only one mentioning higher dimension equations. On the
other hand, we would like to mention the work of Wang-Yang [15] where a system of fourth order
differential equations was analyzed obtaining the existence of positive solutions, however in one
dimension.

In this work we consider a bounded smooth domain Ω ⊂ RN , with N ≥ 1 and we generalize
the one dimensional system studied in [15] to a system with two coupled equations of the form

∆2u1 = g1(x, , u1, u2,∆u1),

∆2u2 = g2(x, u1, u2,∆u2),

2020 Mathematics Subject Classification. 35J70, 35J47, 35K57.

Key words and phrases. Coupled system; higher order operator.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted August 14, 2024. Published May 29, 2025.

1



2 P. ÁLVAREZ-CAUDEVILLA, C. BRÄNDLE, D. SONOWAL EJDE-2025/56

for some functions g1 and g2. More precisely, and in relation to the previously analyzed problems
in 1D, we will be interested in discussing the existence of positive solutions (u1, u2) to the system

∆2u1 + β1∆u1 − α1u1 = f1(x, u1, u2), x ∈ Ω.

∆2u2 + β2∆u2 − α2u2 = f2(x, u1, u2), x ∈ Ω,
(1.2)

under the homogeneous Navier boundary conditions

u1 = ∆u1 = u2 = ∆u2 = 0 on ∂Ω. (1.3)

We will assume that the functions f1, f2 : Ω × [0,∞) × [0,∞) → [0,∞) are continuous and
α1, α2, β1, β2 are real parameters. Our aim in this article is to show that under additional condi-
tions on the parameters (the so-called non-resonance condition, see [3]) of the system and on the
growth conditions of f1 and f2, the system has at least one positive solution (see Section 4).
Outline of this paper: In Section 2 we show some crucial and important results and properties for
Green’s functions that will be essential in proving the existence of solutions for problem (1.2)–
(1.3). Those existence results will be obtained through the application of general fixed point theory
showed in Section 3. The final Section 4 is devoted to the proof of the existence result.

2. Preliminaries: the case of a single equation

To show the existence of solution for system (1.2) under the homogeneous Navier boundary
conditions (1.3) it seems to be convenient to study the behavior of the problem

∆2u+ β∆u− αu = 0 in Ω,

u = ∆u = 0 on ∂Ω,
(2.1)

where α and β are two real parameters. As a first step, observe that this fourth order equation (2.1)
can be rewritten as

Lµ1Lµ2u := (−∆− µ1)(−∆− µ2)u = 0, with β = µ1 + µ2 and α = −µ1µ2. (2.2)

The eigenvalues µ1 and µ2 are also the roots of the polynomial P (µ) = µ2 − βµ− α, so that

µ1 =
β +

√
β2 + 4α

2
, µ2 =

β −
√
β2 + 4α

2
.

We observe that these kinds of algebraic computations, as well as (2.2), are relatively standard
for this type of problems; see several examples in the book of Peletier-Troy [12].

It will be also useful to introduce λk, the eigenvalues of the Laplace operator (−∆) under
homogeneous Dirichlet boundary conditions in Ω. For such an eigenvalue problem we actually
have a family of infinitely many positive ordered eigenvalues, i.e.

0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ . . .

associated with a complete family of eigenfunctions {ϕk}∞k=1. Now, for the eigenvalue problem
(2.1) we can establish the following result, providing us with an existence condition for non-trivial
solutions in terms of the parameters α and β.

Proposition 2.1. The eigenvalue problem (2.1) has (at least) a non-trivial solution if and only
if the pair (α, β) satisfies

α

λ2k
+

β

λk
= 1, for some k ∈ N. (2.3)

Proof. First of all observe that, instead of (2.2) (or (2.1)), we might consider the equivalent pair
of Helmholz equations

(−∆− µ2)u = v,

(−∆− µ1)v = 0.
(2.4)

If (2.1) has a nontrivial solution, then it is clear that µ1 = λk or µ2 = λk, for some k ∈ N. Indeed,
assume, by contradiction, that µ1 ̸= λk and µ2 ̸= λk, for all k ∈ N. In that case, the second
equation in (2.4) has only the trivial solution and consequently, the solution for the first equation
is also only the trivial one. Moreover, if ϕk denotes the eigenfunction for (−∆) in Ω under Dirichlet



EJDE-2025/56 N-DIMENSIONAL FOURTH-ORDER SYSTEMS 3

boundary conditions associated with the eigenvalue λk, then, in any case, µ1 = λk or µ2 = λk,
the function u = ϕk is a solution to (2.2) and hence to (2.1). So that, finally, substituting ϕk into
the fourth order equation (2.1) we arrive at the equality (2.3).

Conversely, if (2.3) holds, it is straightforward to see that u = ϕk is a solution to (2.1). □

Remark 2.2. Observe that, as in [3], without further assumptions on α and β, the coefficients
µ1 and µ2 may be complex.

In addition to the homogeneous problem (2.1), we consider the inhomogeneous problem

∆2u+ β∆u− αu = h(x) in Ω,

u = ∆u = 0 on ∂Ω,
(2.5)

for h positive and continuous in Ω. We will impose that α and β verify

β < 2λ1, and β2 ≥ −4α. (2.6)

These conditions imply that µ1 ≥ µ2 > −λ1. In particular, λk + µ > 0. If we denote by Gi(x, τ),
the Green’s function of the linear boundary problem

(−∆− µi)u = 0 in Ω

together with Dirichlet homogeneous boundary conditions, then, using again the expression (2.2),
we have that the solution of (2.5) is unique and it can be expressed as

u(x) =

∫
Ω

∫
Ω

G1(x, τ)G2(τ, s)h(s) ds dτ, x ∈ Ω. (2.7)

Observe that, due to the Fredholm alternative, the inhomogeneous problem has a unique solution,
which is given by the Green’s function (2.7) if (2.3) fails. This indeed happens if µi ̸= −λk which
implies that the operators (−∆− µi) do not have the eigenvalue 0 and hence they are invertible.

We now prove some important properties of the Green’s function that will play a crucial role
in our analysis. Note that in general, for the problem

Lu = f in Ω, u = 0 on ∂Ω (2.8)

with L a second order differential operator (self-adjoint to have some extra properties) we find a
solution to (2.8) as

u(x) = L−1[f ](x) =

∫
Ω

G(x, τ)f(τ)dτ, (2.9)

in terms of the Green’s function G. Thus, the existence of a Green’s function is equivalent to that
of a unique solution of (2.8).

Lemma 2.3. For all x, τ ∈ Ω, the Green’s function G associated with the differential operator
(−∆− µ) with Dirichlet boundary condition satisfies the following properties:

(i) G(x, τ) = G(τ, x) and G(x, τ) > 0;

(ii) G(x, τ) ≤ C
√
G(τ, τ), where C > 0 is a constant;

(iii) G(x, τ) ≥ δψ2(x)
√
G(τ, τ), where δ > 0 is a constant and ψ is the L2-normalized eigen-

function associated with the first eigenvalue, λ1, of the linear operator (−∆ − µ) with
Dirichlet boundary condition.

Remark 2.4. Since G(x, τ) = 0 for x ∈ ∂Ω, we obtain that (ii) and (iii) above, are trivially
satisfied for x ∈ ∂Ω.

Proof of Lemma 2.3. First, observe that G is the Green’s function associated with the self-adjoint
operator (−∆− µ), whose eigenvalues and eigenfunctions are just λk + µ and ϕk. Moreover, (2.6)
implies that λk + µi > 0. Therefore, the symmetry property in (i) and the positivity of G are
straightforward, see for instance [9].



4 P. ÁLVAREZ-CAUDEVILLA, C. BRÄNDLE, D. SONOWAL EJDE-2025/56

To prove the following items we use the so-called bilinear representation for the Green’s functions
of Helmholtz type equations (see [4] for further details on the original paper of Sommerfield [14]).
In particular, we have

G(x, τ) =

∞∑
k=1

ϕk(x)ϕk(τ)

λk + µ
. (2.10)

The boundedness of G and the Cauchy-Schwartz inequality applied to (2.10) yields estimate (ii).
Finally, to obtain estimate (iii), it suffices to show that, there exists a constant δ such that

inf
x,τ∈Ω

G(x, τ)

ψ2(x)
√
G(τ, τ)

≥ δ > 0. (2.11)

To this aim, we observe first that, from (i) we have that G is positive in Ω and moreover, G is

bounded in Ω by continuity. Therefore, ψ2(x)
√
G(τ, τ) is uniformly bounded in Ω. Hence, the

infimum in (2.11) could only be zero if the numerator approaches zero. However, we will show
that this infimum cannot be achieved when the numerator G(x, τ) approaches zero.

Indeed, for each ϵ > 0, consider the set Aϵ = {x ∈ Ω | 0 < ϵ
2 < G(x, τ) < ϵ, for all τ ∈ Ω}.

Since (−∆ − µ)ψ = λ1ψ, with homogeneous Dirichlet boundary data, using the representation
formula, we obtain that for each x ∈ Aϵ,

ψ(x) = λ1

∫
Ω

G(x, y)ψ(y)dy ≤ ϵλ
1+n

4
1 |Ω|, (2.12)

where the inequality follows from the estimate |ψ(y)| ≤ λ
n
4
1 , for all y ∈ Ω [2]. Finally, using (2.12),

for any x ∈ Aϵ we obtain

G(x, τ)

ψ2(x)
√
G(τ, τ)

≥
(
2ϵλ

2+n
2

1 max
Ω

√
G(τ, τ)

)−1

.

Hence, as ϵ tends to zero,
G(x, τ)

ψ2(x)
√
G(τ, τ)

→ ∞.

Therefore, the infimum (2.11) is attained at a point (x, τ) such that G(x, τ) is positive, and due
to the continuity and boundedness of G, (2.11), and hence (iii), follows. □

Finally, we include an estimation for the solution of equation (2.5). Its proof is similar to the
one performed in [10] and we omit it here. In the sequel we will denote the maximum norm in
C(Ω) by ∥u∥ = maxΩ u(x).

Lemma 2.5. The solution u of the boundary value problem (2.5) satisfies for all x ∈ Ω,

u(x) ≥ δ1δ2C0ψ
2
1(x)

C1C2|Ω|
√
M1

∥u∥, (2.13)

with

C0 =

∫
Ω

√
G1(x, x)ψ

2
2(x) dx and M1 = max

Ω
G1(x, x),

where δi and Ci, with i = 1, 2 are the quantities δ and C (respectively) appearing in Lemma 2.3
relatively to Green’s function Gi, ψi are the L2-normalized eigenfunctions associated with the first
eigenvalue of the linear operator (−∆− µi) with Dirichlet boundary condition.

3. Fixed point theory

From the previous section it can be seen that we actually can work with integral equations to
prove the existence of solutions for problem (1.2) (based on the Green’s functions). In fact, first
of all we observe that we can adapt the expression (2.7) to the solution of (1.2) and write

uj(x) =

∫
Ω

∫
Ω

G1,j(x, τ)G2,j(τ, s)fj(s, u1(s), u2(s))ds dτ, x ∈ Ω,



EJDE-2025/56 N-DIMENSIONAL FOURTH-ORDER SYSTEMS 5

with j = 1, 2. In view of this expression, we shall mainly discuss the existence results for (1.2) by
using the fixed point index theory. Thus, we define the following mappings:

Tj(u1, u2)(x) =

∫
Ω

∫
Ω

G1,j(x, τ)G2,j(τ, s)fj(s, u1(s), u2(s))ds dτ,

T (u1, u2)(x) = (T1(u1, u2)(x), T2(u1, u2)(x)) ,

(3.1)

for all x ∈ Ω, (u1, u2) ∈ C(Ω) × C(Ω). Observe that the existence of non-trivial solutions for
system (1.2) is equivalent to the existence of a nontrivial fixed point of T . Therefore, we just need
to find the nontrivial fixed point of T to establish the existence of non-trivial solutions for (1.2).

We introduce some results of fixed point index theory which will be play a key role in the
subsequent analysis, see [1], [6] and [11].

Definition 3.1 ([6, Chapter 1]). Let (E, ∥ ∥) be a real Banach space. A non-empty, closed, convex
set K ⊂ E is called a cone if the following conditions are met:

(i) If v ∈ K, and a ≥ 0 then av ∈ K;
(ii) If v ∈ K and −v ∈ K, then v = 0.

In our setting we consider the Banach space E := C(Ω)× C(Ω) endowed with the norm

∥(u1, u2)∥ = ∥u1∥+ ∥u2∥.

Recall that, by abuse of notation, we have denoted the maximum norm of C(Ω) also by ∥ · ∥.
Having this in mind we define for the closed subset Ω0 of Ω, the set

P = {(u1, u2) ∈ E : (u1(x) + u2(x)) ≥ σ∥(u1, u2)∥, for all x ∈ Ω0}, with σ = min{σ1, σ2}
where

σj =
δ1,jδ2,jm1,jC0,j

C1,jC2,j

√
M1,j |Ω|

,

with δi,j , C0,j , Ci,j , and M1,j as the corresponding quantities coming from Lemma 2.3 for uj , and
m1,j = minΩ0

ψ2
1,j(x), i, j = 1, 2. Note that 0 < σj <∞ for all x ∈ Ω0.

Lemma 3.2. The set P is a nonempty, convex, and closed subset of E. Moreover, P is a cone
of E.

Proof. It is clear that P is nonempty as 0 ∈ P. Also, thanks to the continuity of u1 and u2 it
follows that P is closed. To see that it is convex, observe that if (u1, u2), (v1, v2) ∈ P for t ∈ (0, 1)

t(u1 + u2) + (1− t)(v1 + v2) ≥ σ(t(∥u1∥+ ∥u2∥) + (1− t)(∥v1∥+ ∥v2∥))
≥ σ(∥tu1 + (1− t)v1∥+ ∥tu2 + (1− t)v2∥),

which implies t(u1, u2) + (1− t)(v1, v2) ∈ P.
Finally we prove that P is a cone of the Banach space E. Indeed, it is straightforward to

check that if (u1, u2) ∈ P and a ≥ 0, then (au1, au2) ∈ P. Moreover, assume that (u1, u2) and
(−u1,−u2) belong to P. Then, 0 ≥ 2σ(∥u1∥ + ∥u2∥). Since σ > 0 inside P we conclude that
(u1, u2) = (0, 0). □

We now present a lemma that outlines some properties of the mapping T .

Lemma 3.3. The mapping T : P → P, defined by (3.1), is completely continuous and T (P) ⊂ P.

Proof. Let (u1, u2) ∈ P. For j = 1, 2, using Lemma 2.3 we obtain

Tj(u1, u2)(x) =

∫
Ω

∫
Ω

G1,j(x, τ)G2,j(τ, s)fj(s, u1(s), u2(s))ds dτ

≤ C1,jC2,j

√
M1,j |Ω|

∫
Ω

√
G2,j(s, s)fj(s, u1(s), u2(s))ds

(3.2)

for all x ∈ Ω, and hence

∥Tj(u1, u2)∥ ≤ C1,jC2,j

√
M1,j |Ω|

∫
Ω

√
G2,j(s, s)fj(s, u1(s), u2(s))ds. (3.3)



6 P. ÁLVAREZ-CAUDEVILLA, C. BRÄNDLE, D. SONOWAL EJDE-2025/56

On the other hand, using the estimate (3.3) and Lemma 2.3, we observe for all x ∈ Ω that

Tj(u1, u2)(x) =

∫
Ω

∫
Ω

G1,j(x, τ)G2,j(τ, s)fj(s, u1(s), u2(s))ds dτ

≥ δ1,jδ2,jψ
2
1,j(x)

∫
Ω

∫
Ω

√
G1,j(τ, τ)ψ

2
2,j(τ)

√
G2,j(s, s)fj(s, u1(s), u2(s))ds dτ

≥ δ1,jδ2,jψ
2
1,j(x)

∫
Ω

√
G1,j(τ, τ)ψ

2
2,j(τ)dτ

∫
Ω

√
G2,j(s, s)fj(s, u1(s), u2(s))ds

= δ1,jδ2,jψ
2
1,j(x)C0,j

∫
Ω

√
G2,j(s, s)fj(s, u1(s), u2(s))ds

≥
δ1,jδ2,jψ

2
1,j(x)C0,j

C1,jC2,j

√
M1,j |Ω|

∥Tj(u1, u2)∥, j = 1, 2.

If x ∈ Ω0, we obtain that Tj(u1, u2)(x) ≥ σj∥Tj(u1, u2)∥, which yields

T1(u1, u2)(x) + T2(u1, u2)(x) ≥ σ1∥T1(u1, u2)∥+ σ2∥T2(u1, u2)∥ ≥ σ∥T (u1, u2)∥

and hence T (u1, u2) = (T1(u1, u2), T2(u1, u2)) ∈ P, that is, T (P) ⊂ P.
In addition, note that f1, f2, and Gi,j are continuous. Therefore, we can deduce that T is

completely continuous just applying Ascoli-Arzela Theorem. □

Finally, let us denote

Pr := {(u1, u2) ∈ P : ∥(u1, u2)∥ < r}.
Clearly, for each r > 0,Pr is a relatively open and bounded set of P. Then, by the definition of
cone P and the norm ∥(u1, u2)∥, one can see that

∂Pr := {(u1, u2) ∈ P : ∥(u1, u2)∥ = r}, Pr := {(u1, u2) ∈ P : ∥(u1, u2)∥ ≤ r}.

Since Pr ̸= ∅ and T : Pr → P is a completely continuous mapping, see Lemma 3.2, we obtain that
the fixed point index, i (T,Pr,P) is defined if T (u1, u2) ̸= (u1, u2) for ever (u1, u2) ∈ Pr. Moreover,
if i (T,Pr,P) ̸= 0, this actually implies that the mapping T possesses a fixed point in Pr. Recall
that the fixed point index i is a counter of the number of zeros for a continuous differential
operator. It might be obtained using the Leray-Schauder formula of the topological degree for
compact perturbations of the identity in a Banach space and it is related to the topological degree
of Brouwer. Through these abstract algebraic concepts one can obtain the number of solutions of
a differential equation; see [1], [6] and [11] for further details on fixed point theory an extensive
analysis of such concepts.

The following lemma states how to calculate the fixed point index of T in Pr, i (T,Pr,P). We
include it here omitting the proof, which can be checked in [6].

Lemma 3.4. Let T : P → P be a completely continuous mapping.

(1) If ηT (u1, u2) ̸= (u1, u2) for every (u1, u2) ∈ ∂Pr and 0 < η ≤ 1, then i (T,Pr,P) = 1.
(2) If ηT (u1, u2) ̸= (u1, u2) for every (u1, u2) ∈ ∂Pr, η ≥ 1 and inf(u1,u2)∈∂Pr

∥T (u1, u2)∥ > 0
then i (T,Pr,P) = 0.

4. Existence of positive solutions

In this section we proof the main theorem of the paper. As mentioned, we will establish
conditions on the functions fi and on the parameters of the system (1.2) so that (1.2) has positive
solution (u1, u2). Recall that the functions f1, f2 : Ω× [0,∞)× [0,∞) → [0,∞) are continuous.

More precisely, we will assume that the parameters of the system verify

βi < 2λ1, β2
i ≥ −4αi,

αi

λ2k
+
βi
λk

< 1. (4.1)

Recall, see Section 2, that the first two conditions are related to the existence and properties of
the Green’s functions and the third one to the existence of solutions. As for the functions fi, let



EJDE-2025/56 N-DIMENSIONAL FOURTH-ORDER SYSTEMS 7

us introduce the following notation for convenience:

f0 = lim inf
u+u2→0+

min
x∈Ω

F [x, u1, u2], f∞ = lim inf
u1+u2→+∞

min
x∈Ω

F [x, u1, u2],

f0 = lim sup
u1+u2→0+

max
x∈Ω

F [x, u1, u2], f∞ = lim sup
u1+u2→+∞

max
x∈Ω

F [x, u1, u2],

where

F [x, u1, u2] :=
f1(x, u1, u2) + f2(x, u1, u2)

L1u1 + L2u2
and Li = λ21 − λ1βi − αi for i = 1, 2. Observe that, due to the third condition in (4.1), Li > 0.

Theorem 4.1. If f0 < 1 < f∞, then the system (1.2)–(1.3) has at least one positive solution.

Proof. First, since f0 < 1, there exists ε ∈ (0, 1) and R0 > 0, small, so that

f1(x, u1, u2) + f2(x, u1, u2) ≤ (1− ε)(L1u1 + L2u2), (4.2)

for all x ∈ Ω and u1, u2 ≥ 0, u1 + u2 ≤ R0. We claim now that for every (u1, u2) ∈ ∂PR0
and

0 < η ≤ 1

ηT (u1, u2) ̸= (u1, u2).

Then, following Lemma 3.4, we conclude that

i (T,PR0
,P) = 1. (4.3)

To proof the claim, we argue by contradiction by assuming that there exist (u01, u
0
2) ∈ ∂PR0

and 0 < η0 ≤ 1 such that η0T (u
0
1, u

0
2) = (u01, u

0
2). Then, by definition of T , we have that (u01, u

0
2)

satisfies differential equations

∆2u01 + β1∆u
0
1 − α1u

0
1 = η0f1(x, u

0
1, u

0
2), ∆2u02 + β2∆u

0
2 − α2u

0
2 = η0f2(x, u

0
1, u

0
2) (4.4)

and boundary condition (1.3). Adding these equations and using (4.2) we obtain

∆2u01 + β1∆u
0
1 − α1u

0
1 +∆2u02 + β2∆u

0
2 − α2u

0
2 ≤ (1− ε)(L1u1 + L2u2). (4.5)

Multiplying this expression by ϕ1(x) and integrating by parts we have∫
Ω

(L1u1 + L2u2)ϕ1 ≤ (1− ε)

∫
Ω

(L1uu + L2u2)ϕ1,

which is a contradiction, since (L1u1 + L2u2)ϕ1 ≥ 0 in Ω, and hence (4.3) follows.
On the other hand, we have that due to f∞ > 1, there exists ε ∈ (0, 1) and k > 0, so that

f1(x, u1, u2) + f2(x, u1, u2) ≥ (1 + ε)(L1u1 + L2u2), (4.6)

for all x ∈ Ω and u1, u2 ≥ 0, u1 + u2 ≥ k. Moreover if we take C = k(1 + ε)(L1 + L2) then for all
x ∈ Ω and u1 + u2 ≥ 0

f1(x, u1, u2) + f2(x, u1, u2) ≥ (1 + ε)(L1u1 + L2u2)− C. (4.7)

We want to show now that there exists an R > R0, to be chosen later, so that

inf
(u1,u2)∈PR

∥T (u1, u2)∥ > 0, and ηT (u1, u2) ̸= (u1, u2)

for every (u1, u2) ∈ ∂PR and η ≥ 1. As before, this implies, following Lemma 3.4, that

i (T,PR,P) = 0. (4.8)

We argue again by contradiction. Let (u01, u
0
2) ∈ ∂PR and η0 ≥ 1 such that η0T (u

0
1, u

0
2) =

(u01, u
0
2). Then, from (4.4) and following the same steps as in the first part of the proof, using (4.7),

we obtain ∫
Ω

(L1u1 + L2u2)ϕ1 ≥ (1 + ε)

∫
Ω

(L1uu + L2u2)ϕ1 − C

∫
Ω

ϕ1.

Consequently, we obtain that

ε

∫
Ω

(L1u
0
1 + L2u

0
2)ϕ1 ≤ C

∫
Ω

ϕ1. (4.9)



8 P. ÁLVAREZ-CAUDEVILLA, C. BRÄNDLE, D. SONOWAL EJDE-2025/56

On the other hand, since (u01, u
0
2) ∈ PR, we have∫

Ω

(L1u
0
1 + L2u

0
2) ≥ min{L1, L2}σ∥(u01, u02)∥

∫
Ω

ϕ1,

so that we conclude that

∥(u01, u02)∥ ≤ C

σεmin{L1, L2}
:= R1.

If R > R1 this last inequality provides a contradiction with the fact that (u01, u
0
2) ∈ ∂PR.

Next, we show that inf(u1,u2)∈PR
∥T (u1, u2)∥ > 0. To this aim, let R2 = k/σ and for R > R2

take (u1, u2) ∈ ∂PR. Then, by definition of the cone P, for x ∈ Ω0, we have that (u1 + u2)(x) ≥
σ∥(u1, u2)∥ > k . So that, because of (4.6) we find that

f1(x, u1, u2) + f2(x, u1, u2) ≥ (1 + ε)(L1u1 + L2u2)

≥ (1 + ε)min{L1, L2}σ∥(u1, u2)∥
≥ (1 + ε)min{L1, L2}k,

(4.10)

for all x ∈ Ω0. On the other hand, let x0 ∈ Ω0 fixed, as in the proof of Lemma 3.3, using
Lemma 2.3, we obtain

(T1(u1, u2) + T2(u1, u2))(x0)

≥ δ1,1δ2,1ψ
2
1,1C0,1

∫
Ω

√
G2,1(s, s)f1(s, u1(s), u2(s))ds

+ δ1,2δ2,2ψ
2
1,2C0,2

∫
Ω

√
G2,2(s, s)f2(s, u1(s), u2(s))ds

≥ min
j=1,2

{δ1,jδ2,jm1,jC0,j}
∫
Ω0

(√
G2,1(s, s)f1(s, u1(s), u2(s))

+
√
G2,2(s, s)f2(s, u1(s), u2(s))

)
ds

≥ min
j=1,2

{δ1,jδ2,jm1,jm2,jC0,j}
∫
Ω0

f1(s, u1(s), u2(s)) + f2(s, u1(s), u2(s))ds,

where m2,j = minΩ0
G2,j(s, s). From (4.10) we have that∫

Ω0

f1(s, u1(s), u2(s)) + f2(s, u1(s), u2(s))ds ≥ (1 + ε)min{L1, L2}k|Ω0|, (4.11)

so that we conclude that

(T1(u1, u2) + T2(u1, u2))(x0) ≥ (1 + ε) min
j=1,2

{δ1,jδ2,jm1,jm2,jC0,j}min{L1, L2}k|Ω0|.

Therefore,

∥T (u1, u2)∥ ≥ (T1(u1, u2) + T2(u1, u2))(x0) ≥ (1 + ε) min
j=1,2

{δ1,jδ2,jm1,jm2,jC0,j}min{L1, L2}k|Ω0|.

Taking the infimum on both sides over (u1, u2) ∈ ∂PR we obtain that inf(u1,u2)∈∂PR
∥T (u1, u2)∥ >

0.
Summing up, for any R0 > 0 small and R > max {R0, R1, R2} we conclude the proof by using

the additivity of fixed point index for (4.3) and (4.8):

i
(
T,PR\PR0

,P
)
= i (T,PR,P)− i (T,PR0

, P ) = −1.

Hence, since R0 is as small as desired, T has a fixed point in PR\{0, 0}, which is a positive solution
of the system (1.2)–(1.3). □

Theorem 4.2. If f∞ < 1 < f0, then the system (1.2)–(1.3) has at least one positive solution.

Proof. The proof follows the same arguments as the previous one. We omit the details and sketch
the main differences. Since 1 < f0, there exists ε ∈ (0, 1) and R0 > 0, small, so that

f1(x, u1, u2) + f2(x, u1, u2) ≥ (1 + ε)(L1u1 + L2u2),



EJDE-2025/56 N-DIMENSIONAL FOURTH-ORDER SYSTEMS 9

for all x ∈ Ω and u1, u2 ≥ 0, u1+u2 ≤ R0. Then for every (u1, u2) ∈ ∂PR0
, through the argument

used in (4.11), we have

∥T (u1, u2)∥ ≥ (1 + ε) min
j=1,2

{δ1,jδ2,jm1,jm2,jC0,j}min{L1, L2}R0|Ω0|.

Hence inf(u1,u2)∈∂PR0
∥T (u1, u2)| > 0.

To show that ηT (u1, u2) ̸= (u1, u2) for any (u1, u2) ∈ ∂PR0 and η ≥ 1 we argue by contradiction
and get that, if there exist (u01, u

0
2) ∈ ∂PR0

and η0 ≥ 1 such that η0T (u
0
1, u

0
2) = (u01, u

0
2), then∫

Ω

(L1u1 + L2u2)ϕ1 ≥ (1 + ε)

∫
Ω

(L1uu + L2u2)ϕ1,

which implies 1 ≥ (1 + ϵ). Hence, we conclude that i (T,PR0
,P) = 0.

On the other hand, f∞ < 1 implies that there exist there exists ε ∈ (0, 1) and k > 0, so

f1(x, u1, u2) + f2(x, u1, u2) ≤ (1− ε)(L1u1 + L2u2),

for all x ∈ Ω and u1, u2 ≥ 0, u1 + u2 ≥ k. Moreover, we can find C > 0 so that, for all x ∈ Ω and
u1 + u2 ≥ 0

f1(x, u1, u2) + f2(x, u1, u2) ≤ (1− ε)(L1u1 + L2u2) + C.

We argue as in the proof of Theorem 4.1 assuming that for R > R0 there exist (u01, u
0
2) ∈ ∂PR

and 0 < η0 ≤ 1 such that η0T (u
0
1, u

0
2) = (u01, u

0
2). We get that

∥(u01, u02)∥ ≤ C

σεmin{L1, L2}
:= R1,

which is a contradiction if R > R1. Hence, if R > max{R0, R1} we obtain i (T,PR,P) = 1.
Finally it follows that i

(
T,PR\PR0

,P
)
= i (T,PR,P)− i (T,PR0

,P) = 1. Consequently, T has

a fixed point in PR\PR0 , which is a positive solution of (1.2)–(1.3). □

Acknowledgments. This research was partially supported by Ministry of Economy and Com-
petitiveness of Spain under research project PID2019-106122GB-I00.

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Pablo Álvarez-Caudevilla

Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain
Email address: pacaudev@math.uc3m.es



10 P. ÁLVAREZ-CAUDEVILLA, C. BRÄNDLE, D. SONOWAL EJDE-2025/56

Cristina Brändle

Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain

Email address: cbrandle@math.uc3m.es

Devashish Sonowal

Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain
Email address: devashish.sonowal@iit.comillas.edu


	1. Introduction
	2. Preliminaries: the case of a single equation
	3. Fixed point theory
	4. Existence of positive solutions
	Acknowledgments

	References