Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 19, pp. 1–29.

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

DOI: 10.58997/ejde.2025.19

MULTIPLICITY RESULTS FOR CRITICAL FRACTIONAL

AMBROSETTI-PRODI TYPE SYSTEM WITH NONLINEARITIES

INTERACTING WITH THE SPECTRUM

EDUARDO H. CAQUI, SANDRA M. DE S. LIMA, FÁBIO R. PEREIRA

Abstract. We study the existence of solutions for Ambrosetti-Prodi type

systems involving the fractional Laplace operator, and having nonlinearities

reaching critical growth and interacting in some sense with the spectrum of
the operator. The resonant case in λk,s for k > 1 is also studied.

1. Introduction

Let s ∈ (0, 1), N > 2s and Ω ⊂ RN be a bounded smooth domain. In this paper
we study the existence of solutions for the critical fractional system

(−∆)su = au+ bv +
α

α+ β
u+

α−1v+
β + ξ1u+

α+β−1 + f in Ω,

(−∆)sv = bu+ cv +
β

α+ β
u+

αv+
β−1 + ξ2v+

α+β−1 + g in Ω,

u = v = 0 in RN \ Ω,

(1.1)

where

(−∆)su(x) := C(N, s) lim
ε↘0

∫
RN\Bε(x)

u(x)− u(y)

|x− y|N+2s
dy, x ∈ RN ,

is the fractional Laplace operator with

C(N, s) =
(∫

RN

1− cos(ζ1)

|ζ|N+2s
dζ

)−1

a positive dimensional constant, α, β > 1 are real constants such that the sum
α + β is the fractional critical Sobolev exponent 2∗s := 2N

N−2s , ξ1, ξ2 ≥ 0, w+ =

max{w(x), 0}, and the forcing terms f and g are of the form f = tϕ1,s + f1 and
g = rϕ1,s + g1, in such a way that the pair (t, r) ∈ R2, f1, g1 ∈ Lq(Ω) for some

q > N
2s and

∫
Ω
f1ϕ1,s dx =

∫
Ω
g1ϕ1,s dx = 0 with ϕ1,s the positive eigenfunction

associated with the first eigenvalue λ1,s of the operator (−∆)s with homogeneous
Dirichlet boundary condition.

2020 Mathematics Subject Classification. 35B06, 35B09, 35J15, 35J20.
Key words and phrases. Ambrosetti-Prodi problem; fractional systems; resonance;

critical growth.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted April 6, 2024. Published February 26, 2025.

1



2 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

With the above decomposition, to state and compare our results to the scalar
case, it is convenient to rewrite system (1.1) as

(−
−→
∆)sU = AU +∇F (U) + Tϕ1,s + F1 in Ω,

U = 0 in RN \ Ω
(1.2)

where

U =

(
u
v

)
, (−

−→
∆)sU =

(
(−∆)su 0

0 (−∆)sv

)
, A =

(
a b
b c

)
∈M2×2(R),

where ∇ is the gradient operator,

F (U) =
1

α+ β

(
u+

αv+
β + ξ1u+

α+β + ξ2v+
α+β

)
, T =

(
t
r

)
, F1 =

(
f1
g1

)
,

Let µ1, µ2 be real eigenvalues of the symmetric matrix A, which will assume
µ1 ≤ µ2. Thus, it is satisfied that µ1|U |2 ≤ (AU,U)R2 ≤ µ2|U |2 for all U :=
(u, v) ∈ R2. The interaction of these eigenvalues with the spectrum of (−∆)s will
play an important role in the study of existence of the solutions.

We recall that Ambrosetti and Prodi [2] in 1972, studied the boundary value
problem

−∆u = f(u) + g(x) in Ω,

u = 0 on ∂Ω,
(1.3)

where g ∈ C0,α(Ω) with α ∈ (0, 1), f ∈ C2(R) such that f(0) = 0, f ′′(t) > 0 for all
t ∈ R and

0 < lim
t→−∞

f ′(t) < λ1 < lim
t→+∞

f ′(t) < λ2,

where 0 < λ1 < λ2 ≤ · · · ≤ λk . . . denote the eigenvalues of (−∆, H1
0 (Ω)). The

authors showed that there exists in C0,α(Ω), a closed connected C1 manifold M1 of
codimension 1 which splits the space into two connected components M0 and M2

such that, if g ∈ M0, the problem (1.3) has no solution; if g ∈ M1, the problem
(1.3) has exactly one solution and if g ∈M2, the problem (1.3) has exactly two solu-
tions. After the pioneering work by Ambrosetti and Prodi [2], many existence and
multiplicity results have been investigated in different directions. In particular, Ruf
and Srikanth [30] established a multiplicity result for the local subcritical problem
−∆u = λu+up++f(x) in Ω, u = 0 on ∂Ω provided that the non-homogeneous term
f has the form f(x) = h(x)+tφ1(x) (h ∈ Lr(Ω) with r > N), λ is not an eigenvalue
of (−∆, H1

0 (Ω)) and t > T , for some sufficiently large number T = T (h). Still in
the local scalar case, but with nonlinearity in the critical growth (p = 2∗ − 1), the
problem above mentioned has been studied by De Figueiredio and Yang [17]. They
proved the existence of two solutions when N > 6. This result was extended by
Calanchi and Ruf [10] using the technique developed in [21]. Works related to this
subject in the local scalar case, we recommend [4] and in the nonlocal operators
situation, [3] and [20] (see references therein). For the critical system in the local
operators situation, problem (1.1) was studied, for instance, in [18] and [27] when
µ2 < λ1 and by [25] in the uncoupled case. For the fractional subcritical system,
(1.1) was studied, for instance in [24].

The purpose of this work is to prove the existence of solutions nonlocal gradient
systems of elliptic equations (1.1) involving critical nonlinearities on the hypothesis
of an interaction of the eigenvalues µ1, µ2 of the matrix A with eigenvalues of the
fractional Laplace operator (−∆)s. When µ2 < λ1,s, this system belongs to the



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 3

class of the so called Ambrosetti-Prodi type problems [2] which have been studied
by several authors in the last decades with different approaches.

Problem (1.1) is an extension to systems involving fractional Laplace operator
of the equation considered in [30], [17] and [10], in which (1.1) was studied in the
local operators case (s = 1) and nonlocal operators (0 < s < 1) in [3] (see [20] also)
and with the particular matrix

A =

(
λ 0
0 λ

)
∈M2×2(R).

In this article, we complement the results achieved in [24], proving that the system
(1.1) (or (1.2)) has at least two solutions for sufficiently large values of parameters
(t, r), the first solution is negative and obtained explicitly depending on the non-
homogeneous terms f and g. The second solution is obtained via the Mountain
Pass Theorem when µ2 < λ1,s, or applying the Linking Theorem in the case λk,s <
µ1 ≤ µ2 < λk+1,s if k ≥ 1. The resonant case λk,s = µ1 for k > 1 is also treated
here. Finally, we should point out that the corresponding local problem governed
by the standard Laplacian operator can be recovered by letting s→ 1.

To show the existence of solutions, difficulties arise when we consider fractional
operators. As we know, in [10], the approximate eigenfunctions technique was
used to facilitate the estimates of the energy functional associated with the lo-
cal scalar problem in the space H1

0 (Ω) (for local critical systems, also see [27]).
However, as noted in [23], in the nonlocal case, it is not possible to employ any
more the same idea as in [10] or [27], since u and v are not orthogonal in the
fractional space Xs

0(Ω) even though they have disjoint supports. On the other
hand, further complications arise due to the presence of the mathematical term
F (u, v) = 1

α+β

[
u+

αv+
β + ξ1u+

α+β + ξ2v+
α+β

]
that includes either an uncoupled

or a coupled nonlinearity.
Because of these obstacles, we develop similar techniques to these known for the

Laplacian operator.
It is important to point out that, with the aid of [19], our results are still valid

for the general case ∇F (u, v) when F is a (α+β)−homogeneous nonlinearity, which
includes a larger class of functions.

The proof of the Theorem below follows arguments as in [24], so we will omit it.

Theorem 1.1 (Existence of a negative solution). Let A ∈M2×2(R) be a symmetric
matrix such that

det(λj,sI −A) ̸= 0, for j = 1, 2, . . . . (1.4)

Assume that F1 = (f1, g1) ∈ Lq(Ω)× Lq(Ω) for some q > N
2s and consider

R =
{
(t, r) ∈ R2 : br + (λ1,s − c)t < η det(λ1,sI −A) and

(λ1,s − a)r + bt < ϑdet(λ1,sI −A)
}
.

Then there exist η, ϑ ≪ 0 such that system (1.2) has a solution (uT , vT ) (with
uT < 0 and vT < 0 in Ω) for every T ∈ R.

Remark 1.2. Suppose that det(λ1,sI −A) > 0 and

λ1,s > max{a, c}. (1.5)

Then the set R is a region between lines satisfying:



4 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

(i) If b = 0,

R = (−∞, η
λ1,s − c

det(λ1,sI −A)
)×

(
−∞, ϑ

λ1,s − a

det(λ1,sI −A)

)
⊂ R2.

(ii) If b > 0,

R =
{
(t, r) ∈ R2 : r < η

det(λ1,sI −A)

b
− (λ1,s − c)

b
t and

r < ϑ
det(λ1,sI −A)

λ1,s − a
− b

λ1,s − a
t
}
.

(iii) If b < 0,

R =
{
(t, r) ∈ R2 : r > η

det(λ1,sI −A)

b
− (λ1,s − c)

b
t and

r < ϑ
det(λ1,sI −A)

λ1,s − a
− b

λ1,s − a
t
}
.

On the other hand, if det(λ1,sI −A) > 0 and

λ1,s < min{a, c}, (1.6)

then the set R satisfies:

(i) If b = 0,

R = (η
λ1,s − c

det(λ1,sI −A)
,+∞)× (ϑ

λ1,s − a

det(λ1,sI −A)
,+∞) ⊂ R2.

(ii) If b > 0,

R =
{
(t, r) ∈ R2 : r < η

det(λ1,sI −A)

b
− (λ1,s − c)

b
t and

r > ϑ
det(λ1,sI −A)

(λ1,s − a)
− b

(λ1,s − a)
t
}
.

(iii) If b < 0,

R =
{
(t, r) ∈ R2 : r > η

det(λ1,sI −A)

b
− (λ1,s − c)

b
t and

r > ϑ
det(λ1,sI −A)

(λ1,s − a)
− b

(λ1,s − a)
t
}
.

Note that, since det(λ1,sI − A) ̸= 0, the lines that define the region R are not
parallel. Moreover, if det(λ1,s I −A) < 0 a similar result can be obtained as in the
Remark 1.2.

The following are the main results of this article.

Theorem 1.3. Assume that N > 6s, ξ1, ξ2 > 0, α + β = 2∗s and that one of the
following 2 conditions hold:

0 < µ1 ≤ µ2 < λ1,s, (1.7)

λk,s < µ1 ≤ µ2 < λk+1,s, for some integer k ≥ 0. (1.8)

Then, system (1.2) has a second solution.

Remark 1.4. Hypothesis (1.7) implies that the conditions (1.4) and (1.5) are sat-
isfied and the hypothesis (1.8) implies in (1.4) and (1.6). In both cases, det(λ1,sI−
A) > 0.



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 5

Theorem 1.5. Suppose N > 6s and

ξ1, ξ2 > 0 and λk,s = µ1 ≤ µ2 < λk+1,s, for some k > 1.

In addition assume that

F1 = (f1, g1) ∈ (Ker((−
−→
∆)s − λk,sI))

⊥. (1.9)

Then system (1.2) has a second solution.

2. Preliminaries

For each measurable function u : RN → R the Gagliardo seminorm is defined by

[u]s :=
(
C(N, s)

∫
R2N

|u(x)− u(y)|2

|x− y|N+2s
dx dy

)1/2

=
(∫

RN

|(−∆)s/2u|2dx
)1/2

.

The second equality follows from [13, Proposition 3.6] when the above integrals are
finite. Then, we consider the fractional Sobolev space

Hs(RN ) = {u ∈ L2(RN ) : [u]s <∞}, ∥u∥Hs = (∥u∥2L2 + [u]2s)
1/2,

which is a Hilbert space. We use the closed subspace

Xs
0(Ω) := {u ∈ Hs(RN ) : u = 0 a.e. in RN \ Ω}.

By Theorems 6.5 and 7.1 in [13], the imbedding Xs
0(Ω) ↪→ Lr(Ω) is continuous

for r ∈ [1, 2∗s] and compact for r ∈ [1, 2∗s). Fractional Sobolev embeddings with
radial potentials have recently been explored in [16], offering further insights into
the behavior of solutions in these functions spaces. Because the fractional Sobolev
inequality, Xs

0(Ω) is a Hilbert space with inner product

⟨u, v⟩Xs
0
:= C(N, s)

∫
R2N

(u(x)− u(y))(v(x)− v(y))

|x− y|N+2s
dx dy,

which induces the norm ∥ · ∥Xs
0
= [ · ]s. Observe that by [13, Proposition 3.6], we

have the identity

∥u∥2Xs
0
=

2

C(N, s)
∥(−∆)s/2u∥2RN , u ∈ Xs

0(Ω).

Then it is proved that for u, v ∈ Xs
0(Ω),

2

C(N, s)

∫
RN

u(x)(−∆)sv(x) dx =

∫
R2N

(u(x)− u(y))(v(x)− v(y))

|x− y|N+2s
dx dy,

in particular, (−∆)s is self-adjoint in Xs
0(Ω).

Now, we consider the Hilbert space given by the product space

Y (Ω) := Xs
0(Ω)×Xs

0(Ω),

equipped with the inner product

⟨(u, v), (φ,ψ)⟩Y := ⟨u, φ⟩Xs
0
+ ⟨v, ψ⟩Xs

0

and the norm
∥(u, v)∥Y := (∥u∥2Xs

0
+ ∥v∥2Xs

0
)1/2.

The space Lr(Ω)× Lr(Ω) (r > 1) is considered with the standard product norm

∥(u, v)∥Lr×Lr := (∥u∥2Lr + ∥v∥2Lr )1/2.

Also, we recall that

µ1|U |2 ≤ (AU,U)R2 ≤ µ2|U |2 for all U := (u, v) ∈ R2, (2.1)



6 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

where µ1 ≤ µ2 are the eigenvalues of the symmetric matrix A. In this article, we
consider the following notation for product space S × S := S2 and

w+(x) := max{w(x), 0}, w−(x) := max{−w(x), 0}

for positive and negative part of a function w. Consequently we obtain w = w+ −
w−.

Since we want to obtain a solution for problem (1.1) with critical growth, we
defined S as the best constant for the Sobolev-Hardy embedding

Xs
0(Ω) ↪→ L2∗s (Ω).

The constant is

S = Sα+β(Ω) = inf
u∈Xs

0 (Ω)\{0}

{ ∥u∥2Xs
0( ∫

Ω
|u|2∗sdx

)2/2∗s }.
Chen, Li and Ou [11] proved that the best Sobolev constant Sα+β = S is achieved
by w, where w is the unique positive solution (up to translations and dilations) of

(−∆)sw = w2∗s−1, in RN , w ∈ L2∗s (Ω).

For the case of problems involving systems, we need the definition

Ss = Ss(α, β)(Ω) = inf
(u,v)∈Y \{0}

∥(u, v)∥2Y( ∫
Ω
|u|α|v|β + ξ1|u|α+β + ξ2|v|α+β dx

)2/2∗s .
The following result establishes a relationship between S and Ss. In local case,

it was proved in [1], which the proof in our case follows arguing as was done there
combined with the arguments in [14] and [15] for the nonlocal case.

Lemma 2.1. Let Ω be a domain (not necessarily bounded), then there exists a pos-
itive constant m such that Ss = mS. Moreover, if w0 achieves S then (s0w0, t0w0)
achieves Ss for some positive constants s0 and t0.

Remark 2.2. The constant m in the previous lemma is given by m =M−1, where
M = max J(s, t) is attained in some (B,C) (with B,C > 0) of the compact set
{(s, t) ∈ R2 : |s|2 + |t|2 = 1} with

J(s, t) := (|s|α|t|β + ξ1|s|α+β + ξ2|t|α+β)
2

α+β .

Therefore,

B2 + C2

(BαCβ + ξ1Bα+β + ξ2Cα+β)
2

α+β

= m.

2.1. An eigenvalue problem. For λ ∈ R, we consider the problem with homoge-
neous Dirichlet boundary condition

(−∆)su = λu in Ω,

u = 0 in RN \ Ω.
(2.2)

If (2.2) admits a weak solution u ∈ Xs
0(Ω) \ {0}, then λ is called an eigenvalue and

u a λ-eigenfunction. The set of all eigenvalues is referred as the spectrum of (−∆)s

in Xs
0(Ω) and denoted by σ((−∆)s). Since K = [(−∆)s]−1 is a compact operator,

the problem (2.2) can be written as u = λKu with u ∈ L2(Ω), hence the following
results are true (see [33], [35]).



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 7

(i) problem (2.2) admits an eigenvalue λ1,s = minσ((−∆)s) > 0 that can be
characterized as follows

λ1,s = min
u∈Xs

0\{0}

∫
RN |(−∆)s/2u(x)|2 dx∫

RN |u(x)|2 dx
; (2.3)

(ii) there exists a non-negative function φ1,s ∈ Xs
0(Ω), which is an eigenfunction

corresponding to λ1,s, attaining the minimum in (2.3);
(iii) all λ1,s-eigenfunctions are proportional, and if u is a λ1,s-eigenfunction, then

either u(x) > 0 a.e. in Ω or u(x) < 0 a.e. in Ω;
(iv) the set of the eigenvalues of problem (2.2) consists of a sequence {λk,s}

satisfying

0 < λ1,s < λ2,s ≤ λ3,s ≤ · · · ≤ λj,s ≤ λj+1,s ≤ . . . , λk,s → ∞, as k → ∞,

which is characterized by

λk+1,s = min
u∈Pk+1\{0}

∫
R2N

|u(x)−u(y)|2
|x−y|N+2s dx dy∫

RN |u(x)|2 dx
(2.4)

where

Pk+1 = {u ∈ Xs
0(Ω) : ⟨u, φj,s⟩X = 0, j = 1, 2, . . . , k};

(v) if λ ∈ σ((−∆)s) \ {λ1,s} and u is a λ-eigenfunction, then u changes sign in
Ω.

(vi) Denote by φk,s the eigenfunction associated to the eigenvalue λk,s, for each
k ∈ N. The sequence {φk,s} is an orthonormal basis either of L2(Ω) or of Xs

0(Ω).

Remark 2.3. Every eigenfunction of (−∆)s is in C0,σ(Ω) for some σ ∈ (0, 1) (see
[33, Theorem 1] or [31, Proposition 2.4]).

3. Proof of Theorem 1.1

The proof of the Theorem 1.1 needs the following lemma (see details in [24]).

Lemma 3.1. If (1.4) hold and F1 ∈ L2(Ω)× L2(Ω), then the system

(−
−→
∆)sU = AU + F1 in Ω,

U = 0 in RN \ Ω,
(3.1)

has a unique solution U0 = (u0, v0) ∈ Y (Ω).

Remark 3.2. If (1.9) holds, using the Fredholm alternative, we have that (3.1)
has a unique solution.

Remark 3.3. If F1 ∈ Lq(Ω)× Lq(Ω) with q > N
2s , by [6, Theorem 3.13], we know

that the solution U0 = (u0, v0) ∈ C0(Ω)× C0(Ω).
If F1 ∈ L∞(Ω) × L∞(Ω), by [29, Proposition 4.6], the solution U0 = (u0, v0) ∈
C0,s(Ω)× C0,s(Ω).

If s = 1/2 and F1 ∈ C0,σ
0 (Ω) × C0,σ

0 (Ω), with 0 < σ < 1 and N > 2s, then

U0 ∈ C1,σ(Ω) × C1,σ(Ω) and ∥U0∥(C1,σ(Ω))2 ≤ c∥F1∥(C0,σ(Ω))2 (see [9, Proposition

3.1] and
if s > 1/2, arguing as in [5], we have that U0 ∈ C1,2s−1(Ω)×C1,2s−1(Ω). Moreover,
a bootstrap argument ensures that if the function F1 ∈ C0(Ω)×C0(Ω) and N > 2s,
then the solution U0 given by Lemma 3.1 satisfies ∥U0∥(C0,σ(RN ))2 ≤ c∥F1∥(Lq(Ω))2 ,



8 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

where σ = min{s, 2s− N
q }, for some constant depending only on N, s, q and Ω (see

[28, Proposition 1.4].

We are ready to prove the existence of a negative solution for system (1.2).

Proof of Theorem 1.1. We will prove the theorem when the conditions (1.4) and
(1.6) hold (other cases (1.4) and (1.5) or (1.9)) are analogous to this and left for
the reader).

By Lemma 3.1 and Remark 3.3, the system

(−
−→
∆)sU = AU + F1 in Ω,

U = 0 in RN \ Ω,

has a unique solution U0 = (u0, v0) ∈ C0(Ω)× C0(Ω). Also

(w, z) =
( (λ1,s − c)t+ br

det(λ1,sI −A)
ϕ1,s,

bt+ (λ1,s − a)r

det(λ1,sI −A)
ϕ1,s

)
is the unique solution of the system

(−
−→
∆)sU = AU + Tϕ1,s in Ω,

U = 0 in RN \ Ω.
Consequently, if

uT =
(λ1,s − c)t+ br

det(λ1,sI −A)
ϕ1,s + u0,

vT =
bt+ (λ1,s − a)r

det(λ1,sI −A)
ϕ1,s + v0,

then UT = (uT , vT ) is a solution of the system

(−
−→
∆)sU = AU + Tϕ1,s + F1 in Ω,

U = 0 in RN \ Ω.
Clearly if uT and vT are negative in Ω, we deduce also that UT is a solution of

(1.2). Therefore, to conclude the proof under the conditions (1.4) and (1.6) (see
Remark 1.2), it suffices to show the existence of an unbounded region R ⊂ R2

where uT and vT are negative in Ω for every T = (t, r) ∈ R. using (1.2), (1.5),
positivity and regularity of ϕ1,s (see [33]) and regularity of the functions u0 R ⊂ R2

such that (uT , vT ) is a negative solution in Ω.
Indeed, since ϕ1,s ∈ C0,σ(Ω) is strictly positive in Ω (see corollary 4.8 in [22])

and u0, v0 ∈ C0(Ω), there exists η, ϑ≪ 0 such that

ηφ1,s + u0 < 0 in Ω,

ϑφ1,s + v0 < 0 in Ω.

□

4. Proof of Theorem 1.3

Let UT := (uT , vT ) be the negative solution with uT , vT < 0 in Ω given by
Theorem 1.1 for T ∈ R. Notice that if U ̸= (0, 0) is a solution of

(−
−→
∆)sU = AU +∇F (U + UT ) in Ω,

U = 0 in RN \ Ω,
(4.1)



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 9

then U = U + UT is a (second) solution of system (1.2) with U + UT ̸= UT .
Therefore, to prove the Theorem 1.3, we only have to show that the system (4.1)
has a nonzero solution for every T ∈ R.

Observe that the weak solutions of (4.1) are the critical points of the functional
Iλ,s : Y (Ω) → R given by

Iλ,s(U) =
C(N, s)

2

∫
R2N

|u(x)− u(y)|2 + |v(x)− v(y)|2

|x− y|N+2s
dx dy

− 1

2

∫
Ω

(AU,U)R2 dx−
∫
Ω

F (U + UT ) dx,

where

F (U) :=
1

α+ β

[
uα+v

β
+ + ξ1u+

α+β + ξ2v+
α+β

]
,

for every U = (u, v) ∈ R2, and U = 0 is a critical point for Iλ,s with Iλ,s(0) = 0.

Remark 4.1. The nonlinearity F is (α+ β)-homogeneous, i.e.

F (λU) = λα+βF (U), ∀U ∈ R2, ∀λ ≥ 0.

In particular:

(i) (∇F (U), U)R2 = uFu(U) + vFv(U) = (α+ β)F (U) for all U = (u, v) ∈ R2.
(ii) Fu and Fv are (α+ β − 1)-homogeneous.
(iii) There exists K > 0 such that

Fu(U) ≤ K(|u|α+β−1 + |v|α+β−1),

Fv(U) ≤ K(|u|α+β−1 + |v|α+β−1),

for all U = (u, v) ∈ R2.
Since F (U) = F (u+, v+) for all U = (u, v) ∈ R2, we deduce that

|∇F (U)| ≤ K(uα+β−1
+ + vα+β−1

+ )

for some constant K > 0.

4.1. Geometry of the functional Iλ,s. In this subsection, we demonstrate that
the functional Iλ,s satisfies the geometric structure required by the Linking The-
orem (see [26, Theorem 5.3]) when λk,s ≤ µ1 ≤ µ2 < λk+1,s, for some k ≥ 1. In
particular, if µ2 < λ1,s holds, then the functional satisfies the conditions of the
Mountain Pass Theorem.

Since Y (Ω) is a Hilbert space, we consider the orthogonal decomposition Y (Ω) =
E−

k ⊕ E+
k , where

E−
k = span{(0, φ1,s), (φ1,s, 0), (0, φ2,s), (φ2,s, 0), . . . , (0, φk,s), (φk,s, 0)}

and E+
k = (E−

k )⊥, for 1 ≤ k ∈ N. Note that E+
k = (P)2 and U ∈ Y (Ω), then

U = U− + U+ with U− ∈ E−
k and U+ ∈ E+

k .
Therefore, from the variational characterization (2.4), we have the following

estimates:

∥U∥2Y ≥ λk+1,s∥U∥2L2×L2 , for all U ∈ E+
k ,

∥U∥2Y ≤ λk,s∥U∥2L2×L2 , for all U ∈ E−
k .

Let

Sρ := ∂Bρ ∩ E+
k ,



10 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

Q := {U ∈ Y (Ω) : U =W + ζE, W ∈ E−
k , ∥W∥Y ≤ r, 0 ≤ ζ ≤ R},

where E ∈ E+
k , 0 < ρ < R and r > 0 will be chosen later so that the following

conditions hold:

inf
U∈Sρ

Iλ,s(U) ≥ σ > 0,

max
U∈∂Q

Iλ,s(U) ≤ α0, with α0 < σ,

max
U∈Q

Iλ,s(U) ≤ s

N
S

N
2s .

Proposition 4.2. Suppose Ω is a smooth bounded domain of RN , α+ β = 2∗s and
λk,s < µ1 ≤ µ2 < λk+1,s, for some k ∈ N. Then there exists ρ0 > 0 and a function
α : [0, ρ0] → R+ such that

Iλ,s(U) ≥ α(ρ) for all U ∈ Sρ := ∂Bρ(0) ∩ E+
k .

Explicitly the maximum value of α(ρ) is

α̂ =
s

N
SN/2s

(
1− µ2

λk+1,s

) N
2s

1

(1 + ξ)
N−2s

2s

(4.2)

and it is assumed that

ρ̂ = S
N
4s (1− µ2

λk+1,s
)

N−2s
4s

1

(1 + ξ)
N−2s

4s

,

where S is the best constant for the embedding of Xs
0 in L2∗s and ξ =: max{ξ1, ξ2}.

Proof. Using that (A(U), U)R2 ≤ µ2|u|2, we obtain

Iλ,s(U) ≥ 1

2
∥U∥2Y − µ2

2

∫
Ω

|U |2 dx− 1

α+ β

∫
Ω

[
ξ1(u+ ur,t)

2∗s
+

+ ξ2(v + vr,t)
2∗s
+ + (u+ ur,t)

α
+(v + vr,t)

β
+

]
dx.

Note that

sαtβ ≤ sα+β + tα+β for all s, t ≥ 0, (4.3)∫
Ω

(u+ ur,t)
2∗s
+ dx ≤

∫
Ω

|u|2
∗
s dx ≤ S−2∗s/2∥u∥2

∗
s

Xs
0
= S− N

N−2s ∥u∥2
∗
s

Xs
0
. (4.4)

Similarly, ∫
Ω

(v + vr,t)
2∗s
+ dx ≤ S− N

N−2s ∥v∥2
∗
s

Xs
0
. (4.5)

Then, by (4.3), (4.4) and (4.5), we have

Iλ,s(U) ≥ 1

2

(
1− µ2

λk+1,s

)
∥U∥2Y

−
( (1 + ξ1)

α+ β
S− N

N−2s ∥u∥2
∗
s

Xs
0
+

(1 + ξ2)

α+ β
S− N

N−2s ∥v∥2
∗
s

Xs
0

)
.

Since ξ =: max{ξ1, ξ2} ≥ ξ1, ξ2, we obtain

Iλ,s(U) ≥ 1

2

(
1− µ2

λk+1,s

)
ρ2 − (1 + ξ)

α+ β
S− N

N−2s ρ2
∗
s =: α(ρ),



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 11

where ρ = ∥U∥Y . Using a standart calculus argument, we obtain that the maximum
of α(ρ) is attained at

ρ0 =
1

(1 + ξ)
N−2s

4s

SN/4s
(
1− µ2

λk+1,s

)N−2s
4s

.

So, the function α : [0, ρ0] → R+ is such that Iλ,s(U) ≥ α(ρ) for all U ∈ Sp and
the maximum value is

α(ρ0) =
s

N
SN/2s

(
1− µ2

λk+1,s

) N
2s 1

(1 + ξ)
N−2s

2s

. (4.6)

Therefore, Iλ,s(U) ≥ α(ρ) for all U ∈ Sρ. The proof of the proposition is complete.
□

It is well know (see [12, Theorem 1.1]) that S = Sα+β is achieved by

ũ(x) := k(µ2 + |x− x0|2)−
N−2s

2 , (4.7)

with k ∈ R \ {0}, µ > 0 and x0 ∈ RN fixed constants.
Equivalently, we see that

S = inf
u∈Xs

0\{0}, ∥u∥L
2∗s =1

∫
R2N

|u(x)− u(y)|2

|x− y|N+2s
dx dy =

∫
R2N

|u(x)− u(y)|2

|x− y|N+2s
dx dy

where u(x) = ũ(x)/∥ũ∥L2∗s . By translation, suppose x0 = 0 in (4.7). Then, the

function u∗(x) = u
(

x

S
1
2s

)
, x ∈ RN , is a solution for the problem

(−∆)su = |u|2
∗
s−2, in RN (4.8)

satisfying

∥u∗∥2
∗
s

L2∗s (RN )
= S

N
2s .

As in [32], for every ϵ > 0 we define the family of functions

Uϵ(x) := ϵ−
N−2s

2 u∗
(x
ϵ

)
, x ∈ RN ,

then Uϵ is a solution of (4.8) and satisfies for all ϵ > 0,∫
R2N

|Uϵ(x)− Uϵ(y)|2

|x− y|N+2s
dx dy =

∫
R2N

|Uϵ(x)|2
∗
s dx dy = S

N
2s .

Now, take a fixed δ > 0 such that B4δ ⊂ Ω. Let η ∈ C∞
c (RN ) be a cut-off function

such that 0 ≤ η ≤ 1 in RN , η = 1 in Bδ and η = 0 in RN \B2δ, where Br = Br(0)
is the ball centered at the origin and with radius r > 0.

We define the family of nonnegative truncated functions

uϵ(x) := η(x)Uϵ(x) x ∈ RN , (4.9)

and note that uϵ ∈ Xs
0 .

The following Brezis-Nirenberg estimates for nonlocal setting was proved in [32]
(also see [34]), which are similar to those proved for the local case in [8].

Lemma 4.3. Suppose s ∈ (0, 1) and N > 2s, then for ϵ > 0 small enough, the
following estimates hold:∫

R2N

|uϵ(x)− uϵ(y)|2

|x− y|N+2s
dx dy ≤ SN/2s +O(ϵN−2s),



12 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

∫
RN

|uϵ(x)|2 dx ≥


Csϵ

2s +O(ϵN−2s) if N > 4s,

Csϵ
2s|logϵ|+O(ϵ2s) if N = 4s,

Csϵ
N−2s +O(ϵ2s) if 2s < N ≤ 4s,∫

RN

|uϵ(x)|2
∗
s dx = SN/2s +O(ϵN ),

∥uϵ∥L1(RN ) = O(ϵ
N−2s

2 ),

∥uϵ∥
2∗s−1

L2∗s−1(RN )
= O(ϵ

N−2s
2 ).

We denote by P− the orthogonal projection of Xs
0 in B−

k = span{ϕ1, ϕ2, . . . , ϕk}
and P+ the orthogonal projection of Xs

0 in A+
k := (B−

k )⊥.
Depending on ϵ > 0 we choose the vetorial function

e = e⃗ϵ = (B(P+uϵ), C(P+uϵ)) ∈ E+
k ,

where uϵ is given in (4.9) and B and C are given by Remark 2.2. We will denote
P+uϵ by eϵ and consequently e⃗ϵ = (Beϵ, Ceϵ).

Remark 4.4. (i) eϵ ∈ A+
k ;

(ii) ⟨(Beϵ, Ceϵ), (0, ϕj)⟩L2×L2 = 0 = ⟨(Beϵ, Ceϵ), (ϕj , 0)⟩L2×L2 , for all j = 1, . . . , k.
Then e = e⃗ϵ ∈ E+

k .

The following results was proved in [3], which are similar to those proved for the
local case in [17].

Lemma 4.5. For s ∈ (0, 1), N > 2s, and ϵ > 0 small enough, the following
estimates hold:

∥P+uϵ∥2Xs
0
≤ [uϵ]

2
s ≤ SN/2s +O(ϵN−2s),∣∣∣∥P+uϵ∥

2∗s
L2∗s (Ω)

− ∥uϵ∥
2∗s
L2∗s (Ω)

∣∣∣ ≤ CϵN−2s,

∥P+uϵ∥L1(Ω) ≤ Cϵ
N−2s

2 ,

∥P+uϵ∥
2∗s−1

L2∗s−1(RN )
≤ Cϵ

N−2s
2 ,

|P−uϵ(x)| ≤ Cϵ
N−2s

2 , for x ∈ Ω.

(4.10)

Fix K > 0 and define Ωϵ,K = {x ∈ Ω : eϵ(x) = (P+uϵ)(x) > K}. By (4.10) we
deduce that

eϵ(0) = (P+uϵ)(0) = uϵ(0)− P−uϵ(0) ≥
C0

∥ũ∥L2∗s (RN )

ϵ−
(N−2s)

2 − Cϵ
N−2s

2 ,

which implies that P+uϵ(0) → ∞ as ϵ→ 0. By the continuity of P+uϵ, there exists
ν > 0 such that Bν ⊂ Ωϵ,K . Therefore, we have the result below.

Lemma 4.6. For s ∈ (0, 1) and N > 2s, we have

∥P+uϵ∥
2∗s
L2∗s (Ωϵ,K)

= ∥uϵ∥
2∗s
L2∗s (Ω)

+O(ϵN−2s).

∥P+uϵ∥
2∗s−1

L2∗s−1(Ωϵ,K)
= ∥uϵ∥

2∗s−1

L2∗s−1(Ω)
+O(ϵ

N+2s
2 ).

∥P+uϵ∥L1(Ωϵ,K) = ∥uϵ∥L1(Ω) +O(ϵN ).

To prove the geometric conditions of the Linking Theorem, we need two results
that can be found in [17] and [18] for the case when s = 1. The proof is similar for
s ∈ (0, 1).



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 13

Lemma 4.7. Given u, v ∈ Lp(Ω) with 2 ≤ p ≤ 2∗s and u + v > 0 a.e. on a
measurable subset Σ ⊂ Ω, it holds∣∣ ∫

Σ

(u+ v)p dx−
∫
Σ

|u|p dx−
∫
Σ

|v|p dx
∣∣ ≤ C

∫
Σ

(|u|p−1|v|+ |u∥v|p−1) dx,

with a constant C > 0 depending only on p.

Lemma 4.8. Given (a, b), (u, v) ∈ Lp(Ω)× Lq(Ω) with p, q ≥ 2 and p+ q ≤ 2∗s. If
a+ b, u+ v > 0 a.e. on a measurable subset Σ ⊂ Ω and H(x, y) = |x|p|y|q, then∣∣ ∫

Σ

H(a+ u, b+ v) dx−
∫
Σ

H(u, v) dx−
∫
Σ

H(a, b) dx
∣∣

≤ C
[ ∫

Σ

(|a|p−1|b|q|u|+ |a|p−1|v|+ q|u|+ |u|p−1|b|q|a|+ |u|p−1|v|q|a|) dx

+

∫
Σ

(|a|p−1|v|q−1|b||u|+ |u|p|b|+ q + |u|p|v|q−1|b|) dx

+

∫
Σ

(|a|p|b|q−1|v|+ |a|p|v|+ q + |u|p−1|b|q−1|a∥v|) dx

+

∫
Σ

(|b|q−1|u|p|v|+ |v|q−1|a|p|b|) dx
]
,

(4.11)

where the constant C > 0 depends only on p+ q.

Proof. Let us define

h(ζ) :=

∫
Σ

[H(a+ ζu, b+ ζv)−H(ζu, ζv)] dx.

Using the Fundamental Theorem of the Calculus, |h(1) − h(0)| =
∫ 1

0
h′(ζ) dζ, and

consequently∫
Σ

[H(a+ u, b+ v)−H(a, b)] dx

≤
∫ 1

0

∫
Σ

|((∇H(a+ ζu, b+ ζv)−∇H(ζu, ζv)), (u, v))R2 | dx dζ.
(4.12)

Applying the Mean Value Theorem to the function ∇H(x, y), there exist θ1, θ2 ∈
(0, 1) such that

∇H(a+ ζu, b+ ζv)−∇H(ζu, ζv)

=
(
p|a+ ζu|p−2(a+ ζu)|b+ ζv|q − p|ζu|p−2(ζu)|ζv|q,

q|a+ ζu|p|b+ ζv|q−2(b+ ζv)− q|ζu|p|ζv|q−2(ζv)
)

=
(
p(p− 1)|(1− θ1)a+ ζu|p−2|(1− θ1)b+ ζv|+ qa

+ pq|(1− θ1)a+ ζu)|p−2((1− θ1)a+ ζu)|(1− θ1)b+ ζv|q−2(1− θ1)b+ ζv)b,

pq|(1− θ2)a+ ζu)|p−2((1− θ2)a+ ζu)|(1− θ2)b+ ζv|q−2(1− θ2)b+ ζv)a

+ q(q − 1)|(1− θ2)b+ ζv|q−2|(1− θ2)a+ ζu|pb
)
.

(4.13)
Inequality (4.11) follows by substituting (4.13) in (4.12) and making some additional
estimations. □



14 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

The following inequality which is a direct consequence of Young Inequality, is
essential for the proof of Theorem 1.3.

Lemma 4.9. If α, β > 1, α+ β = 2∗s and α >
2∗s−1

2 , then there is p > 2 such that,
for each ϵ > 0, the following inequality holds

|s|α|t|β ≤ Cϵ|s|2
∗
s−1 + Cϵp|t|βp

where Cϵ and C are positive constants.

Lemma 4.10. Suppose A,B,C and θ positive numbers. Consider the function

Φϵ(s) = 1
2s

2A − 1
2∗s
s2

∗
sB + s2

∗
s ϵθC with s > 0. Then sϵ =

(
A

B−2∗sϵ
θC

) 1
2∗s−2 is the

maximum point of Φϵ and

Φϵ(s) ≤ Φϵ(sϵ) =
s

N

( AN

BN−2s

) 1
2s

+O(ϵθ).

Lemma 4.11. If λk,s < µ1 ≤ µ2 < λk+1,s, there are constants r0, R0 > 0 and
ϵ0 > 0 such that, for r > r0, R > R0 and 0 < ϵ ≤ ϵ0, we have

Iλ,s|∂Q < α̂,

with α̂ > 0 as in Proposition 4.2.

Proof. Let ∂Q = Γ1 ∪ Γ2 ∪ Γ3, where

Γ1 = BR ∩ E−
k ,

Γ2 = {U ∈ Y : U =W + se⃗ϵ with W ∈ E−
k , ∥W∥Y = r, 0 ≤ s ≤ R},

Γ3 = {U ∈ Y : U =W +Re⃗ϵ with W ∈ E−
k ∩Br(0)}.

We will show that for each Γi we have Iλ,s|Γi
< α̂, for all i = 1, 2, 3.

(i) For all U ∈ Γ1(⊂ E−
K), using (2.1), we infer that

Iλ,s(U) ≤ 1

2
∥U∥2Y − µ1

2

1

λk,s
∥U∥2Y =

1

2

(
1− µ1

λk,s

)
∥U∥2Y ≤ 0.

(ii) Let U ∈ Γ2, then U = W + se⃗ε with W = (w1, w2) ∈ E−
k and e⃗ :=

(B(P+uε), C(P+uε)) = (Beε, Ceε), where the positive constants B and C are cho-
sen as in Remark 2.2.

Therefore,

Iλ,s(U)

≤ 1

2

(
1− µ1

λk,s

)
∥W∥2Y +

s2

2
(B2 + C2)∥eε∥2Xs

0

− 1

α+ β

∫
Ω

(w1 + sBeε + urt)
α
+(w2 + sCeε + vrt)

β
+ dx

− ξ1
α+ β

∫
Ω

(w1 + sBeε + urt)
α+β
+ dx− ξ2

α+ β

∫
Ω

(w2 + sCeε + vrt)
α+β
+ dx.

Consider the maximum value α̂ of the function α(ρ) like in (4.6), and define

s0 :=

√
2 s
N S

N/2s(1− µ2

λk+1,s
)

N
2s

1

(1+ξ)
N−2
2s√

sup0<ε≤1 ∥e⃗ε∥2Y
=

√
2α̂√

sup0<ε≤1 ∥e⃗ε∥2Y
. (4.14)

To comply the condition Iλ,s|Γ2
< α̂, we distinguish two cases.



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 15

Case 1: 0 ≤ s ≤ s0. The expression of Iλ,s provides the estimate

Iλ,s(U) ≤ s2

2
∥e⃗ε∥2Y ≤ s20

2
sup

0<ε≤1
∥e⃗ε∥2Y = α̂,

which concludes this case.
Case 2: s > s0. We define

K := sup
{∥∥W + (ur,t, vr,t)

s

∥∥
L∞×L∞ : s0 ≤ s ≤ R, ∥W∥Y = r, W ∈ E−

k

}
,

with K > 0 independent of R. Then, by (4.9) and (4.10) we have

eϵ(0) = (P+uϵ)(0) = uϵ(0)− P−uϵ(0)

≥ C0

∥ũ∥L2∗s (RN )

ϵ−
(N−2s)

2 − cϵ
N−2s

2 → +∞,

as ϵ→ 0 because N > 2s. By the continuity of eϵ, we have

Ωϵ = {x ∈ Ω : eϵ(x) = (P+uϵ)(x) > K} ≠ ∅

for ϵ > 0 small enough. Therefore, by Lemmas 4.7 and 4.8, for j = 1 and zr,t = ur,t
or j = 2 and zr,t = vr,t, we have∫

Ωϵ

(
Beϵ +

wj + zr,t
s

)α+β

dx

≥
∫
Ωϵ

|Beϵ|2
∗
s dx+

∫
Ωϵ

∣∣wj + zr,t
s

∣∣α+β
dx

− C

∫
Ωϵ

(
|Beϵ|2

∗
s−1

∣∣wj + zr,t
s

∣∣+ |Beϵ|
∣∣wj + zr,t

s

∣∣2∗s−1
)
dx

(4.15)

and∫
Ωϵ

(
Beϵ +

w1 + ur,t
s

)α

+

(
Ceϵ +

w2 + vr,t
s

)β

+
dx

≥
∫
Ωϵ

BαCβ |eϵ|α+β dx+

∫
Ωϵ

∣∣∣∣w1 + ur,t
s

∣∣∣∣α ∣∣w2 + vr,t
s

∣∣βdx
−K

∫
Ωϵ

(∣∣w1 + ur,t
s

∣∣α−1∣∣w2 + vr,t
s

∣∣β |Beϵ|+ ∣∣w1 + ur,t
s

∣∣α−1|Ceϵ|β |Beϵ|

+ |Beϵ|α−1
∣∣w2 + vr,t

s

∣∣β∣∣w1 + ur,t
s

∣∣+ |Beϵ|α−1|Ceϵ|β
∣∣w1 + ur,t

s

∣∣) dx
−K

∫
Ωϵ

(∣∣w1 + ur,t
s

∣∣α−1∣∣w2 + vr,t
s

∣∣|Ceϵ|β−1|Beϵ|+ |Beϵ|α
∣∣w2 + vr,t

s

∣∣β
+ |Beϵ|α|Ceϵ|β−1

∣∣w2 + vr,t
s

∣∣) dx
−K

∫
Ωϵ

(∣∣w1 + ur,t
s

∣∣α∣∣w2 + vr,t
s

∣∣β−1|Ceϵ|+
∣∣w1 + ur,t

s

∣∣α|Ceϵ|β
+ |Beϵ|α−1

∣∣w2 + vr,t
s

∣∣β−1∣∣w1 + ur,t
s

∣∣|Ceϵ|) dx
−K

∫
Ωϵ

(∣∣w2 + vr,t
s

∣∣β−1|Beϵ|α|Ceϵ|

+ |Ceϵ|β−1
∣∣w1 + ur,t

s

∣∣α∣∣w2 + vr,t
s

∣∣) dx.

(4.16)



16 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

Then using the estimates (4.15) and (4.16), we can see that, for ϵ > 0 small
enough,

Iλ,s(U) ≤ 1

2

(
1− µ1

λk,s

)
∥W∥2Y +

s2

2
(B2 + C2)∥eϵ∥2Xs

0

− s2
∗
s

2∗s
(BαCβ + ξ1B

2∗s + ξ2C
2∗s )∥eϵ∥

2∗s
L2∗s (Ωϵ)

+K
s2

∗
s

2∗s

(
∥eϵ∥

2∗s−1

L2∗s−1(Ωϵ)
+ ∥eϵ∥L1(Ωϵ) + ∥eϵ∥α+1

Lα+1(Ωϵ)
+ ∥eϵ∥β+1

Lβ+1(Ωϵ)

+ ∥eϵ∥α−1
Lα−1(Ωϵ)

+ ∥eϵ∥β−1
Lβ−1(Ωϵ)

+ ∥eϵ∥βLβ(Ωϵ)
+ ∥eϵ∥αLα(Ωϵ)

)
.

Now, for each j ∈ {α, β, α− 1, β − 1}, there exists Cj > 0 such that

∥eϵ∥jLj(Ωϵ)
≤ Cj∥eϵ∥

2∗s−1

L2∗s−1(Ωϵ)

and by lemma 4.9, for each j ∈ {α+ 1, β + 1}, there exists Kj > 0 such that

∥eϵ∥jLj(Ωϵ)
≤ Kj(∥eϵ∥

2∗s−1

L2∗s−1(Ωϵ)
+ ϵp),

with p > 2. Therefore, using the above estimate and the Lemmas 4.3, 4.5 and 4.6,
we obtain

Iλ,s(U) ≤ 1

2

(
1− µ1

λk,s

)
∥W∥2Y +Φϵ(s),

where

Φϵ(s) :=
s2

2
(B2 + C2)S

N
2s − s2

∗
s

2∗s
(BαCβ + ξ1B

2∗s + ξ2C
2∗s )S

N
2s +Ks2

∗
sO(ϵq)

with q = min{N−2s
2 , p}. Then, applying Lemma 4.10, we obtain

Iλ,s(U) ≤ 1

2

(
1− µ1

λk,s

)
r2 +

s

N

( [(B2 + C2)S
N
2s ]N

[(BαCβ + ξ1B2∗s + ξ2C2∗s )S
N
2s ]N−2s

) 1
2s

+O(ϵq)

=
1

2

(
1− µ1

λk,s

)
r2 +

s

N

( (B2 + C2)
N
2s

(BαCβ + ξ1B2∗s + ξ2C2∗s )
N−2s

2s

)
S

N
2s +O(ϵq).

Since λk,s < µ1 and ε > 0 can be made arbitrarily small, we can choose r > 0 to be
arbitrarily large in the inequality above such that Iλ,s(U) < 0. This leads to the
conclusion stated in the proposition for U ∈ Γ2.
(iii). Let U ∈ Γ3. It can be expressed by Γ3 definition as U = W + Re⃗ε with
W ∈ E−

k ∩Br(0). Analogously to the case (ii), we obtain

Iλ,s(U) ≤ 1

2

(
1− µ1

λk,s

)
∥W∥2Y + (B2 + C2)

R2

2
∥eϵ∥2Xs

0

− BαCβ

2∗s
R2∗s

∫
Ω

(
eϵ +

w1 + ur,t
BR

)α

+

(
eϵ +

w2 + vr,t
CR

)β

+
dx

− ξ1B
2∗s

2∗s
R2∗s

∫
Ω

(
eϵ +

w1 + ur,t
BR

)2∗s

+
dx

− ξ2C
2∗s

2∗s
R2∗s

∫
Ω

(
eϵ +

w2 + vr,t
CR

)2∗s

+
dx.



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 17

Because of the boundedness of the functions W ∈ E−
k ∩ Br(0), ur,t and vr,t, there

exists k > 0 such that ∥w1 + ur,t∥L∞ ≤ k and ∥w2 + vr,t∥L∞ ≤ k. Again, since
eϵ(0) = P+uϵ(0) → ∞ as ϵ → 0, there exists ϵ0 > 0 such that for all 0 < ϵ < ϵ0,
we have eϵ(0) > 2k. Then, by the continuity of eϵ we can find R1 = R1(ϵ) > 0 and
η = η(ϵ) > 0 such that |χ| ≥ η for all R > R1, where

χ :=
{
x ∈ Ω : eϵ(x) +

w1(x) + ur,t(x)

BR
> 1 and eϵ(x) +

w2(x) + vr,t(x)

CR
> 1

}
.

Then, we find ϵ0, R0 > 0 such that for 0 < ϵ < ϵ0 and R > R0, we have

Iλ,s(U) ≤ 0, for all U ∈ Γ3.

Let R0 > max{R1, R2}, where R0 is such that αR2
0 −R

2∗s
0 < 0, with

α =
(B2 + C2)2∗s

2(BαCβ + ξ1B2∗s + ξ2C2∗s )
(η−1∥eϵ∥2Xs

0
).

Then, for ϵ > 0 above, and R > R0 we find that

Iλ,s(U) ≤ 1

2

(
1− µ1

λk,s

)
∥W∥2Y + (B2 + C2)

R2

2
∥eϵ∥2Xs

0 (Ω)

− R2∗s

2∗s
BαCβ |χ| − ξ1

R2∗s

2∗s
B2∗s |χ| − ξ2

R2∗s

2∗s
C2∗s |χ|

≤ (B2 + C2)
R2

2
∥eϵ∥2Xs

0 (Ω) − (BαCβ + ξ1B
2∗s + ξ2C

2∗s )
R2∗s

2∗s
η < 0.

This completes the proof. □

Lemma 4.12. Let s ∈ (0, 1), λk,s ≤ µ1 ≤ µ2 < λk+1,s and N > 6s. Then we have
the estimate

max
Q

Iλ,s <
s

N
S

N
2s

Proof. Let ϵ < ϵ0 fixed that the linking theorem geometry holds. For W + se⃗ϵ ∈ Q,
we have

Iλ,s(W + se⃗ϵ) ≤
1

2

(
1− µ1

λk,s

)
∥W∥2Y +

s2

2
∥e⃗ϵ∥2Y − µ1

2
s2∥e⃗ϵ∥2L2×L2

−
∫
Ω

F (w + se⃗+ UT ) dx.

Let s0 be defined as in (4.14).
Case 1: 0 < s ≤ s0. Arguing as in the proof of Lemma 4.11 and bearing in mind
(4.2), we can see that

Iλ,s(W + se⃗ϵ) ≤
s2

2
∥e⃗ϵ∥2Y ≤ s20

2
sup

0<ε≤1
∥e⃗ε∥2Y = α̂ <

s

N

1

(1 + ξ)
N−2s

2s

S
N
2s . (4.17)

Now, by Lemma 2.1 and Remark 2.2, we obtain

S
N
2s =

(BαCβ + ξ1B
2∗s + ξ2C

2∗s )
N−2s

2s

(B2 + C2)
N
2s

S
N
2s
s

≤ (1 + ξ)
N−2s

2s
[(B2 + C2)

2∗s
2 ]

N−2s
2s

(B2 + C2)
N
2s

S
N
2s
s

= (1 + ξ)
N−2s

2s S
N
2s
s ,



18 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

and consequently by the estimate (4.17), we conclude that

Iλ,s(W + se⃗ϵ) <
s

N
S

N
2s
s .

Case 2: s > s0. As in the proof of Lemma 4.11, from (4.15), Lemma 4.3 and
Lemma 4.6, we obtain

Iλ,s(W + se⃗ϵ) ≤
1

2
s2

(
∥e⃗ϵ∥2Y − µ1∥e⃗ϵ∥2L2×L2

)
−
∫
Ω

F (w + se⃗+ UT ) dx.

On the other hand,

F (w + se⃗+ UT ) =
1

2∗s

[
(sB)α

(
eϵ +

w1 + ur,t
sB

)α

+
(sC)β

(
eϵ +

w2 + vr,t
sC

)β

+

+ ξ1(sB)2
∗
s

(
eϵ +

w1 + ur,t
sB

)2∗s

+
+ ξ2(sC)

2∗s

(
eϵ +

w2 + vr,t
sC

)2∗s

+

]
.

Using the previous arguments, we obtain

Iλ,s(W + se⃗ϵ) ≤ Φϵ(s),

where

Φϵ(s) :=
1

2
s2

(
∥e⃗ϵ∥2Y − µ1∥e⃗ϵ∥2L2×L2

)
− s2

∗
s

2∗s
(BαCβ + ξ1B

2∗s + ξ2C
2∗s )∥eϵ∥

2∗s
L2∗s (Ωϵ)

+Ks2
∗
sO(ϵq).

Applying Lemma 4.10 to the function Φϵ, by Lemmas 4.3, 4.5 and 4.6 and by the
choice of B and C, we have

Φϵ(s) ≤ Φs(sϵ)

≤ s

N

[
(B2 + C2)S

N
2s +O(ϵN−2s)− µ1Ce

2s +O(ϵN−2s)
] N

2s[
(BαCβ + ξ1B2∗s + ξ2C2∗s )S

N
2s +O(ϵN ) +O(ϵN−2s)

]N−2s
2s

+O(ϵq)

≤ s

N

[ (B2 + C2)

(BαCβ + ξ1B2∗s + ξ2C2∗s )
N−2s

N

S
] N

2s − µ1

N
O(ϵ2s) +O(ϵq).

Since p > 2 and N > 6s we garantee q := min{N−2s
2 , p} > 2s. Than, taking ϵ > 0

sufficiently small, we obtain

Iλ,s(W + se⃗ϵ) ≤
s

N
S

N
2s
s . □

4.2. Palais-Smale condition for the functional Iλ,s. In this subsection we
discuss a compactness property for the functional Iλ,s, given by the Palais-Smale
condition.

Lemma 4.13. If k ≥ 0 and λk,s < µ1 ≤ µ2 < λk+1,s. Then every (PS)c sequence
of Iλ,s is bounded.

Proof. The Fréchet derivative of the functional Iλ,s is

I ′
λ,s(u, v)(ϕ, ψ) = ⟨(u, v), (φ,ψ)⟩Y −

∫
Ω

(A(u, v), (ϕ, ψ))R2 dx

−
∫
Ω

(∇F (u+ uT , v + vT ), (ϕ, ψ))R2 dx,

for every (u, v), (ϕ, ψ) ∈ Y (Ω).



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 19

Let (Un) ⊂ Y (Ω) be a (PS)c-sequence, i.e. satisfying Iλ,s(Un) = c + o(1) and
⟨I ′

λ,s(Un),Ψ⟩ = o(1)∥Ψ∥Y for all Ψ = (ψ, ξ) ∈ Y (Ω). Therefore,

Iλ,s(Un)−
1

2
I ′
λ,s(Un)Un

=
1

2

∫
Ω

(∇F (Un + UT ), Un)R2 dx−
∫
Ω

F (Un + UT ) dx

≤ c+ o(1) + o(1)∥Un∥Y .

(4.18)

Then

1

2

∫
Ω

(∇F (Un + UT ), Un)R2 dx−
∫
Ω

F (Un + UT ) dx

=
1

2

∫
Ω

( α

α+ β
(un + ur,t)

α−1
+ (vn + vr,t)

β
+un + ξ1(un + ur,t)

α+β−1
+ un

+
β

α+ β
(un + ur,t)

α
+(vn + vr,t)

β−1
+ vn + ξ2(vn + vr,t)

α+β−1
+ vn

)
dx

− 1

α+ β

∫
Ω

(
(un + ur,t)

α
+(vn + vr,t)

β
+ + ξ1(un + ur,t)

α+β
+

+ ξ2(vn + vr,t)
α+β
+

)
dx

≤ c+ o(1) + o(1)∥Un∥Y .

(4.19)

Now note that

∫
Ω

(
(un + ur,t)

α−1
+ (vn + vr,t)

β
+un

)
dx

=

∫
Ω

(
(un + ur,t)

α−1
+ (un + ur,t)+(vn + vr,t)

β
+

)
dx

−
∫
Ω

(
(un + ur,t)

α−1
+ (vn + vr,t)

β
+ur,t

)
dx

(4.20)

and ∫
Ω

(
(un + ur,t)

α+β−1
+ un

)
dx

=

∫
Ω

(un + ur,t)
α+β
+ dx−

∫
Ω

(un + ur,t)
α+β−1
+ ur,t dx.

(4.21)

Substituting (4.20), (4.21) and expressions similar to these in (4.19), yields

∫
Ω
(un + ur,t)

α
+(vn + vr,t)

β
+dx,∫

Ω
(un + ur,t)

α+β
+ dx,∫

Ω
(vn + vr,t)

α+β
+ dx

 ≤ c+ o(1) + o(1)∥Un∥Y . (4.22)



20 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

Now, using (2.1) and that Ψ = U+
n = (u+n , v

+
n ) ∈ E+

k , we obtain(
1− µ2

λk+1,s

)
∥U+

n ∥2Y

≤ ∥U+
n ∥2Y −

∫
Ω

(AU+
n , U

+
n )R2 dx

=

∫
Ω

(∇F (Un + UT ), U
+
n )R2 dx− ⟨I ′

λ,s(Un), (U
+
n )⟩

≤
∫
Ω

Fu(Un + UT )|u+n | dx+

∫
Ω

Fv(Un + UT )|v+n | dx+ C∥U+
n ∥Y .

(4.23)

Hence, by Remark 4.1 (iii), there exists a constant K > 0 such that

Fu(U) ≤ K
(
(u)α+β−1

+ + (v)α+β−1
+

)
Fv(U) ≤ K

(
(u)α+β−1

+ + (v)α+β−1
+

)
.

Then ∫
Ω

Fu(Un + UT )|u+n | dx+

∫
Ω

Fv(Un + UT )|v+n | dx

≤ K

∫
Ω

(
(un + uT )

α+β−1
+ + (vn + vT )

α+β−1
+

)
|u+n |dx

+K

∫
Ω

(
(un + uT )

α+β−1
+ + (vn + vT )

α+β−1
+

)
|v+n |dx.

and using Hölder’s inequality with p = α+β
α+β−1 and q = α + β, Young’s inequality,

we deduce that∫
Ω

Fu(Un + UT )|u+n | dx+

∫
Ω

Fv(Un + UT )|v+n | dx

≤ K
{
ϵ∥u+n ∥2L2∗s + Cϵ

[
∥(un + uT )+∥

2(2∗s−1)

L2∗s
+ ∥(vn + vT )+∥

2(2∗s−1)

L2∗s

]}
+K

{
ϵ∥v+n ∥2L2∗s + Cϵ

[
∥(un + uT )+∥

2(2∗s−1)

L2∗s
+ ∥(vn + vT )+∥

2(2∗s−1)

L2∗s

]}
.

Using (4.22), in view of the embedding X(Ω) ↪→ Lr(Ω) for r ≤ 2∗s, we obtain∫
Ω

Fu(Un + UT )|u+n | dx+

∫
Ω

Fv(Un + UT )|v+n | dx

≤ ϵC1∥U+
n ∥2Y + C2Cϵ + 4ϵn∥Un∥

N+2s
N

Y .

By (4.23), taking ϵ > 0 small enough, we conclude that

∥U+
n ∥2Y ≤ C3 + C4∥Un∥

N+2s
N

Y + C5∥U+
n ∥Y . (4.24)

Analogously, the following estimate is valid

∥U−
n ∥2Y ≤ C6 + C7∥Un∥

N+2s
N

Y + C8∥U+
n ∥Y . (4.25)

Using the estimates (4.24) and (4.25), we obtain

∥Un∥2Y ≤ C + C∥Un∥
N+2s

N

Y + C∥Un∥Y .

Since N+2s
N < 2, we conclude that (Un) is bounded in Y (Ω). □



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 21

Lemma 4.14. If k ≥ 0 and λk,s < µ1 ≤ µ2 < λk+1,s, then the functional Iλ,s
satisfies the (PS) condition at level c with c < s

N S
N
2s
s .

Proof. Let (Un) ⊂ Y (Ω) be a sequence satisfying

Iλ,s(Un) → c and I ′
λ,s(Un) → 0 in the dual space Y (Ω)′,

as n → ∞. By Lemma 4.13 we have that (Un) is bounded. Hence passing to a
subsequence, we may suppose that

Un ⇀ U in Y (Ω),

Un → U in Lp(Ω)× Lp(Ω), for all p ∈ [1, 2∗s),

Un → U a.e in RN .

(4.26)

Hence, U is s weak solution to

(−
−→
∆)sU = AU +∇F (U + UT ) in Ω,

U = 0 in RN \ Ω,
(4.27)

that is, for any Ψ ∈ Y (Ω) it holds

⟨U,Ψ⟩Y −
∫
Ω

(AU,Ψ)R2 =

∫
Ω

(∇F (U + UT ),Ψ)R2 dx. (4.28)

In particular, taking Ψ = U in (4.28), we obtain

∥U∥2Y −
∫
Ω

(AU,U)R2 dx =

∫
Ω

(∇F (U + UT ), U)R2 dx (4.29)

Note that by (4.27)
(
⟨I ′

λ,s(U), U⟩ = 0
)
and (4.29) we obtain

Iλ,s(U) =
1

2

∫
Ω

(∇F (U + UT ), U)R2dx−
∫
Ω

F (U + UT ) dx ≥ 0. (4.30)

By applying the Brezis-Lieb Lemma [7], it follows that

∥(Un + UT )+∥
2∗s
L2∗s×L2∗s

= ∥(Un − U)+∥
2∗s
L2∗s×L2∗s

+ ∥(U + UT )+∥
2∗s
L2∗s×L2∗s

+ o(1)

∥Un − U∥2Y = ∥Un∥2Y − ∥U∥2Y + o(1)

(4.31)
and by applying the Brezis-Lieb Lemma for homogeneous functions [19], we con-
clude that∫

Ω

F (Un + UT ) dx =

∫
Ω

F (U + UT ) dx+

∫
Ω

F (Un − U) dx+ o(1). (4.32)

Also, we have∫
Ω

(∇F (Un + UT ), Un + UT )R2 dx−
∫
Ω

(∇F (U + UT ), U + UT )R2 dx

= (α+ β)

∫
Ω

F (Un − U) dx.

(4.33)

Then, by using (4.26), (4.31) and (4.33), we deduce that

Iλ,s(Un) =
1

2
∥Un − U∥2Y + Iλ,s(U)−

∫
Ω

F (Un − U) dx+ o(1). (4.34)

On the other hand, by using (4.26), (4.29) and (4.31) and (4.32), we have

⟨I ′
λ,s(Un), Un⟩



22 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

= ∥Un∥2Y −
∫
Ω

(AUn, Un)R2 dx−
∫
Ω

(∇F (Un + UT ), Un)R2 dx

=
[
∥Un − U∥2Y + ∥U∥2 + o(1)

]
−

[ ∫
Ω

(AU,U)R2 dx+ o(1)
]

−
∫
Ω

(∇F (Un + UT ), Un + UT )R2 dx+

∫
Ω

(∇F (Un + UT ), UT )R2 dx

= ∥Un − U∥2Y +

[
∥U∥2Y −

∫
Ω

(AU,U)R2 dx

]
−
[
(α+ β)

∫
Ω

F (Un − U) dx

+

∫
Ω

(∇F (U + UT ), U + UT )R2 dx
]
+

∫
Ω

(∇F (Un + UT ), UT )R2 dx+ o(1)

= ∥Un − U∥2Y +
[ ∫

Ω

(∇F (U + UT ), U)R2 dx
]
−
[
(α+ β)

∫
Ω

F (Un − U) dx

+

∫
Ω

(∇F (U + UT ), U)R2 dx+

∫
Ω

(∇F (U + UT ), UT )R2 dx
]

+

∫
Ω

(∇F (Un + UT ), UT )R2 dx+ o(1)

= ∥Un − U∥2Y − (α+ β)

∫
Ω

F (Un − U) dx+

∫
Ω

(∇F (U + UT ), UT )R2 dx

+

∫
Ω

(∇F (Un + UT ), UT )R2 dx+ o(1).

Taking into account that ⟨I ′
λ,s(Un), Un⟩ → 0 and

∫
Ω
(∇F (Un + UT ), UT )R2 dx →∫

Ω
(∇F (U + UT ), UT )R2 dx as n→ ∞, we deduce that

∥Un − U∥2Y = (α+ β)

∫
Ω

F (Un − U) dx+ o(1). (4.35)

Let

L := lim
n→∞

∥Un − U∥2Y ≥ 0.

If L = 0, then Un → U in Y (Ω) as n→ ∞.
Let L > 0. Then, by the definition of Ss,

Ss ≤
∥U∥2Y( ∫

Ω
|u|α|v|β + ξ1|u|α+β + ξ2|v|α+βdx

) 2
α+β

for all U = (u, v) ̸= (0, 0).

and (4.35), we can infer

∥Un − U∥2Y ≥ Ss

(∫
Ω

(un − u)α+(vn − v)β+ + ξ1(un − u)α+β
+ + ξ2(vn − v)α+β

+ dx
) 2

α+β

= Ss

(
(α+ β)

∫
Ω

F (Un − U) dx
) 2

α+β

which gives

L ≥ SsL
N−2s

N , i.e. L ≥ S
N
2s
s . (4.36)

Now, from (4.30), (4.34), (4.35), (4.36) we obtain

s

N
S

N
2s
s ≤

( 2s

N − 2s

) L
2∗s

≤ c <
s

N
S

N
2s
s ,

which is a contradiction. □



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 23

Proof of Theorem 1.3. In the case where λk,s < µ1 ≤ µ2 < λk+1,s occurs, Proposi-
tion 4.2 and Lemma 4.11 with ε > 0 small enough, ensure that the functional Iλ,s
satisfies the geometric structure required by the Linking Theorem. Therefore, from
the Linking Theorem without the Palais-Smale condition, there exists a sequence
(Un) ⊂ Y (Ω) satisfying Iλ,s(Un) → c and I ′

λ,s(Un) → 0 in Y (Ω)′. By Lemma 4.12,
the critical level satisfies

0 < c := inf
γ∈Γ

sup
U∈Q

Iλ,s(γ(U)) ≤ s

N
S

N
2s
s ,

where Γ := {γ ∈ C0(Q,Y (Ω)) : γ = Id on ∂Q}. By Lemma 4.13, (Un) is
bounded in Y (Ω) and consequently Lemma 4.14 ensures that Un → U in Y (Ω).
If 0 = λ0,µ < µ1 ≤ µ2 < λ1,µ, to show that the functional Iλ,s satisfies the ge-
ometrical conditions of the Mountain Pass Theorem, it is sufficient to take the
finite dimensional subspace E− = {(0, 0)} and to apply the Proposition 4.2 with
E+

k = Y (Ω) such that R∥e⃗ϵ∥Y > ρ with R > 0 sufficient large to ensure that
Iλ,s(Re⃗ϵ) < 0. The (PS)c condition is guaranteed by making k = 0 in Lemmas

4.13 and 4.14. Thus, in both cases, there exists a non-trivial solution U for problem
(4.1). By [24, Remark 4.1], it follows that U+ ̸= 0 and therefore, UT and UT + U
are distinct solutions for problem (1.2). □

5. Resonant case

5.1. Proof of Theorem 1.5. In this subsection we discuss a compactness property
for the functional Iλ,s, given by the Palais-Smale condition for this case.

Lemma 5.1. If N > 6s and λk,s = µ1 ≤ µ2 < λk+1,s for k > 1, the functional Is
satisfies the (PS) condition.

Proof. We follow the notation in the previous proof. Let Un ∈ Y (Ω) such that
Is(Un) → c and I ′

s(Un) → 0 in the dual space Y (Ω)′. Writing Y (Ω) = E−
k−1 ⊕

E+
k ⊕ Zk, consequently we have

Un = U−
n + U+

n + βnYn :=Wn + βnYn,

where U−
n ∈ E−

k−1, U
+
n ∈ E+

k = (E−
k )⊥ and Yn ∈ Zk = span{(φk,s, 0), (0, φk,s)}

with ∥Yn∥Y = 1. Using similar arguments as in (4.24) and (4.25), we obtain

∥Wn∥2Y ≤ C + C∥Un∥τY + C∥Wn∥Y , (5.1)

where τ = N+2s
N . We can assume ∥Un∥Y ≥ 1 (if ∥Un∥Y ≤ 1, the sequence (Un) is

bounded in Y (Ω)). Then, since ∥Un∥Y ≤ ∥Wn∥Y + |βn|, from (5.1), we have

∥Wn∥2Y ≤ C1(∥Wn∥Y + |βn|)τ + C∥Wn∥Y . (5.2)

If βn is bounded, since τ < 2, by (5.2) we conclude that (Un) is bounded in Y (Ω).
Otherwise, we may assume βn → +∞, therefore, from (5.2), it follows that

∥Wn

βn
∥2Y ≤ C1

{ (∥Wn∥Y + |βn|)τ/2

|βn|

}2

+ C
1

βn
∥Wn

βn
∥Y

≤ C1

{ 1

|βn|1−τ/2
∥Wn

βn
∥τ/2Y +

1

|βn|1−τ/2

}2

+ C
1

βn
∥Wn

βn
∥Y .

(5.3)

Using again that τ/2 < 1, the above estimate yields

∥Wn

βn
∥2Y ≤ C2∥

Wn

βn
∥τY + C3∥

Wn

βn
∥Y + C4 (5.4)



24 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

and consequently the sequence {Wn

βn
} is bounded in Y (Ω) and by (5.3), ∥Wn

βn
∥Y → 0.

Therefore, possibly up to a subsequence, Wn/βn → 0 a.e. in Ω and strongly in
Lq(Ω) × Lq(Ω), 1 ≤ q < 2∗s; Yn → Y0 ∈ Zk a.e. in Ω and strongly in Y (Ω) and
Lq(Ω)× Lq(Ω), 1 ≤ q < 2∗s.

Now, taking βnYn ∈ Zk as test function, we obtain

I ′
s(Un)Yn = βn

(
∥Yn∥2Y −

∫
Ω

(AYn, Yn)R2 dx
)
−

∫
Ω

(∇F (Un + UT ), Yn)R2 dx.

Since (Un) is a (PS)-sequence and

1

(βn)
4s

N−2s

(
∥Yn∥2Y −

∫
Ω

(AYn, Yn)R2 dx
)
→ 0,

as n→ ∞, we obtain that

o(1) =
1

(βn)
N+2s
N−2s

I ′
s(Un)(Yn) = − 1

(βn)
N+2s
N−2s

∫
Ω

(∇F (Un + UT ), Yn)R2 dx.

Now, from Remark 4.1 (ii),∫
Ω

(∇F
(Un + UT

βn

)
, Yn)R2 dx =

1

(βn)
N+2s
N−2s

∫
Ω

(∇F (Un + UT ), Yn)R2 dx→ 0. (5.5)

On the other hand, since Un =Wn+βnYn, we have that
Un

βn
→ Y0 in Lq(Ω)×Lq(Ω)

for all 1 ≤ q < 2∗s and a.e. in Ω. So, by the Dominated Convergence Theorem and
by (5.5), it follows that∫

Ω

(∇F
(Un + UT

βn

)
, Yn)R2 dx→

∫
Ω

(∇F (Y0), Y0)R2 dx = 0

and from Remark 4.1 (i), we concluded that
∫
Ω
F (Y0) dx = 0.

Finally, using the notation Y0 = (y01 , y
0
2), it follows that (y01)+ = 0 = (y02)+,

contradicting ∥Y0∥Y = 1 and Y0 ∈ Zk with k > 1, which ensures that at least one
of the functions is not null and changes sign. Thus (Un) is bounded and using the
fact that N > 6s, as in the proof of Lemmas 4.12 and 4.14, we have that (Un)
admits a convergent subsequence. □

5.2. Geometry in the resonant case. In this subsection, we demonstrate that
the functional Iλ,s satisfies the geometric structure required by the Linking Theo-
rem in resonant case, that is, we obtain the following result.

Proposition 5.2. Suppose Ω is a smooth bounded domain of RN , α+ β = 2∗s and
λk,s = µ1 ≤ µ2 < λk+1,s for some k > 1. Then

(i) there exist σ, ρ > 0 such that Is(U) ≥ σ for all U ∈ E+
k with ∥U∥Y = ρ,

(ii) there exists E ∈ E+
k and R > 0 such that R∥E∥Y > ρ and Is(U) ≤ 0, for

all U ∈ ∂Q, where Q = (BR ∩ E−
k )⊕ [0, R]E.

Proof. (i) Let U = (u, v) ∈ E+
k , using the fact that uT , vT < 0, estimate |u|α|v|β ≤

|u|α+β + |v|α+β and the fractional imbedding X ↪→ Lα+β , by (2.1), we have

Is(U) ≥ 1

2
∥U∥2Y − µ2

2
∥U∥2(L2)2 − C

∫
Ω

(|u|α+β + |v|α+β) dx

≥ 1

2

(
1− µ2

λk+1,s

)
∥U∥2Y − C∥U∥α+β

Y ,



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 25

where C > 0 is a constant. Since µ2 < λk+1,s and α + β > 2, for ∥U∥Y = ρ small
enough, we obtain Is(U) ≥ σ.

(ii) Now consider the decomposition ∂Q = Γ1 ∪ Γ2 ∪ Γ3, where

Γ1 = {U ∈ Y (Ω); U = U1 + rE, with U1 ∈ E−
k , ∥U1∥Y = R, 0 ≤ r ≤ R},

Γ2 = {U ∈ Y (Ω); U = U1 +RE, with U1 ∈ E−
k , ∥U1∥Y ≤ R},

Γ3 = BR(0) ∩ E−
k .

Let us show that on each set Γi we have Is |Γi
≤ 0, i = 1, 2, 3.

Fixed R0 > ρ, we choose E = (e1, e2) ∈ E+
k = (E−

k )⊥ (with ei ≥ 0, i = 1, 2)
satisfying

(I) ∥E∥2Y <
(

µ1

λk−1,s
− 1

)
δ2, where δ > 0 is a constant to be obtained later.

(II) e1 ≥ 2
(
K +

∥uT ∥C0

R0

)
and e2 ≥ 2

(
K +

∥vT ∥C0

R0

)
a.e. in some C ⊂ Ω with

|C| > 0, where K > 0 satisfies ∥V ∥(C0)2 ≤ K∥V ∥Y , for all V ∈ E−
k .

Note that this choice is possible because (E−
k )⊥ has unbounded functions; E−

k has
finite dimension and

K = sup
∥V ∥Y =1, V ∈E−

k

∥V ∥(C0)2 .

Estimates on Γ1: For U = U1 + rE ∈ Γ1, we consider U1 = RÛ1 ∈ E−
k with

∥Û1∥E = 1 and we set Û1 = c1Y + c2Ek, where Ek ∈ Zk = span{(φk,s, 0), (0, φk,s)}
and Y ∈ E−

k−1 with ∥Y ∥Y = 1. Then

Is(U) ≤ 1

2
∥U1∥2Y +

r2

2
∥E∥2Y − µ1

2
∥U1∥2(L2)2 −

∫
Ω

F (U + UT ) dx

≤ R2

2
∥Û1∥2Y +

R2

2
∥E∥2Y − µ1R

2

2
∥Û1∥2(L2)2 −

∫
Ω

F (U + UT ) dx

=
R2

2
∥c1Y + c2Ek∥2Y +

R2

2
∥E∥2Y − µ1R

2

2
∥c1Y + c2Ek∥2(L2)2

−
∫
Ω

F (U + UT ) dx

=
R2

2
c21(∥Y ∥2Y − µ1∥Y ∥2(L2)2) +

R2

2
c22(∥Ek∥2Y − µ1∥Ek∥2(L2)2)

+
R2

2
∥E∥2Y −

∫
Ω

F (U + UT ) dx.

Consequently

Is(U) ≤ R2

2
c21

(
1− µ1

λk−1,s

)
∥Y ∥2Y +

R2

2
∥E∥2Y −

∫
Ω

F (U + UT ) dx. (5.6)

Now using the notation Û1 = (û1, v̂1) = (c1y1 + c2e
k
1 , c1y2 + c2e

k
2), where Y =

(y1, y2) ∈ E−
k−1 ∩ B1 and Ek = (ek1 , e

k
2) ∈ Zk ∩ B1, we will prove that there exist

δ > 0 and η > 0 such that

max
i=1,2

{
max
Ω

{c1yi + c2e
k
i ; |c1| ≤ δ}

}
≥ η > 0.

Indeed, by contradiction, assume that there exist sequences (cn1 ), (c
n
2 ) ⊂ R and

Yn = (yn1 , y
n
2 ) ⊂ Y (Ω) with ∥Yn∥Y = 1 such that cn1 → 0, |cn2 | =

√
1− (cn1 )

2 → 1



26 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

and

max
i=1,2

{
max
Ω

{cn1yni + cn2 e
k
i }
}
→ 0, as n→ ∞.

Therefore, cn1y
n
i → 0 and cn2 e

k
i → eki and consequently

max
i=1,2

{
max
Ω

eki (x)
}
= 0.

Hence, we conclude that ek1 ≤ 0 and ek2 ≤ 0 in Ω, which is a contradiction, because
k > 1, Ek = (ek1 , e

k
2) ∈ Zk and ∥Ek∥Y = 1 imply that at least one of the coordinate

functions must change sign. So, we conclude that there exist δ > 0, η > 0 such that

max
{
max
Ω

û1; max
Ω

v̂1 : |c1| ≤ δ
}
≥ η > 0

for all Û1 = c1Y + c2Ek ∈ E−
k with ∥Û1∥Y = 1.

Denoting Ω+ =
{
x ∈ Ω : (û1)(x) ≥ η/2 and (v̂1)(x) ≥ η/2

}
. By equicontinuity

of the functions Û1, we have that |Ω+| ≥ ν > 0, for all Û1 ∈ E−
k ∩B1 and |c1| ≤ δ.

Moreover

uT (x)

R
≥ −∥uT ∥C0

R
> −η

4
and

vT (x)

R
≥ −∥vT ∥C0

R
> −η

4
,

for all R ≥ R0 sufficiently large. Then, since e1, e2 ≥ 0 in Ω,∫
Ω

F (U + UT ) dx ≥ ξ1
α+ β

Rα+β

∫
Ω

(
û1 +

uT
R

)α+β

+
dx

+
ξ2

α+ β
Rα+β

∫
Ω

(
v̂1 +

vT
R

)α+β

+
dx

≥ CRα+β
[ ∫

Ω+

(
û1 −

η

4

)α+β

+
dx+

∫
Ω+

(
v̂1 −

η

4

)α+β

+
dx

]
≥ CRα+β

[ ∫
Ω+

(η
4

)α+β

dx+

∫
Ω+

(η
4

)α+β

dx
]

≥ CRα+β
(η
4

)α+β

|Ω+| = C̃Rα+β ,

for all R sufficiently large. Thus, from (5.6) we can conclude that there exists
R1 > 0 such that

Is(U) ≤ R2

2
δ2
(
1− µ1

λk−1,s

)
+
R2

2
∥E∥2Y − C̃Rα+β < 0,

for all R ≥ R1.
On the other hand, if |c1| ≥ δ > 0, by the choice of E, we obtain

Is(U) ≤ −R
2

2
c21

( µ1

λk−1,s
− 1

)
+
R2

2
∥E∥2Y

≤ −R
2

2

[
δ2
( µ1

λk−1,s
− 1

)
− ∥E∥2Y

]
< 0.



EJDE-2025/19 FRACTIONAL AMBROSETTI-PRODI SYSTEMS 27

Estimates on Γ2: For U = U1 +RE ∈ Γ2, we have

Is(U1 +RE) ≤ 1

2
∥U1∥2Y

(
1− µ1

λk,s

)
+
R2

2
∥E∥2Y −

∫
Ω

F (U1 +RE + UT ) dx. (5.7)

Since λk,s = µ1,

Is(U1 +RE) ≤ R2

2
∥E∥2Y −

∫
Ω

F (U1 +RE + UT ) dx. (5.8)

Now, to estimate the last integral, note that, if U1 = (u1, u2),∫
Ω

F (U1 +RE + UT ) dx ≥ 1

α+ β

[
ξ1R

α+β

∫
Ω

(
e1 +

u1 + uT
R

)α+β

+
dx

+ ξ2R
α+β

∫
Ω

(
e2 +

u2 + vT
R

)α+β

+
dx

]
for R ≥ R0, and by (II) each integral on the right can be estimated as follows∫

Ω

(
ei +

ui + wT

R

)α+β

+
dx ≥

∫
Ω

(
ei −

∥ui∥C0 + ∥wT ∥C0

R

)α+β

+
dx

≥
∫
Ω

(
ei −

(
K +

∥wT ∥C0

R0

))α+β

+
dx

≥
∫
C

(
K +

∥wT ∥C0

R0

)α+β

dx =
(
K +

∥wT ∥C0

R0

)α+β

|C|,

for i = 1, 2 and wT ∈ {uT , vT }. Therefore, by (5.8) and by above estimates,

Is(U1 +RE) ≤ R2

2
∥E∥2Y − c1R

α+β

∫
Ω

(
e1 +

u1 + uT
R

)α+β

+
dx

− c2R
α+β

∫
Ω

(
e2 +

u2 + vT
R

)α+β

+
dx

≤ R2

2
∥E∥2Y − CRα+β .

Since α+ β > 2, for R ≥ R0 we have Is(U) < 0, for all U ∈ Γ2.

Estimates on Γ3: For U ∈ Γ3, we have the estimate

Is(U) ≤ 1

2
∥U∥2Y − µ1

2
∥U∥2(L2)2 ≤ 1

2

(
1− µ1

λk,s

)
∥U∥2Y = 0. (5.9)

Therefore, for all R ≥ R0 > 0, follows that Is(U) ≤ 0 for all U ∈ ∂Q, concluding
the desired result. □

Proof of Theorem 1.5. With the previous results, we conclude the proof of Theorem
1.5. We use a direct application of the Linking Theorem and arguing as in the proof
of Theorem 1.3 to obtain two distinct solutions for problem (1.2). □

Acknowledgments. F. R. Pereira was supported partially by FAPEMIG/Brazil
(RED-00133-21) and FAPEMIG/Brazil (CEX APQ 04528/22).



28 E. H. CAQUI, S. M. D. S. LIMA, F. R. PEREIRA EJDE-2025/19

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Eduardo. H. Caqui

Departamento de Ciencias Sede Breña, Universidad Privada del Norte, Av. Tingo Maŕıa

1122, Cercado de Lima, Lima, Peru
Email address: eduardocaqui11@gmail.com

Sandra M. de S. Lima
Departamento de Ciências Exatas, Biológicas e da Terra, INFES-UFF, Santo Antônio

de Pádua - RJ, 28470-000, Brazil

Email address: sandra.msouzalima@gmail.com

Fábio R. Pereira

Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de
Juiz de Fora, 30161-970, Juiz de Fora - MG, Brazil

Email address: fabio.pereira@ufjf.edu.br