Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 32, pp. 1–9.

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

DOI: 10.58997/ejde.2024.32

EXISTENCE OF SEMI-NODAL SOLUTIONS FOR ELLIPTIC

SYSTEMS RELATED TO GROSS-PITAEVSKII EQUATIONS

JOÃO PABLO PINHEIRO DA SILVA, EDCARLOS DOMINGOS DA SILVA

Communicated by Claudianor O. Alves

Abstract. In this work we consider existence of semi-nodal solutions, i.e.,
solutions of the form (u, v) with u > 0 and v± := max{0,±v} ̸≡ 0 for a class

of elliptic systems related to the Gross-Pitaevskii equation.

1. Introduction

This work concerns the elliptic system

−∆u = λ1u+ µ1|u|2p−2u+ β|u|p−2u|v|q, in Ω

−∆v = λ2v + µ2|v|2q−2v + β|u|p|v|q−2v, in Ω

u = v = 0, on ∂Ω .

(1.1)

For p = q = 2 the cubic system (1.1) arises in mathematical models for various
physics problems, especially in nonlinear optics and Bose-Einstein condensation,
see [14, 17]. In those works present information on the physical significance of non-
cubic nonlinearities and on the existence and multiplicity of solutions. Furthermore,
when λi < 0, system (1.1) comes from the study of solitary wave solutions of the
coupled Gross-Pitaevskii equations,

−i
∂

∂t
Φ1 = ∆Φ1 + µ1|Φ1|2Φ2 + βΦ2

2Φ1, x ∈ Ω, t > 0

−i
∂

∂t
Φ2 = ∆Φ2 + µ2|Φ2|2Φ1 + βΦ2

1Φ2, x ∈ Ω, t > 0

Φj = Φj(x, t) ∈ C, j = 1, 2

Φj(x, t) = 0, x ∈ ∂Ω, t > 0, j = 1, 2.

(1.2)

When Φ1(x, t) = e−iλ1tu and Φ2(x, t) = e−iλ2tv, system (1.2) reduces to (1.1). In
the Kerr-like photorefractive media, the solution Φj represents the jth element of
the beam (see [2]). The self-focusing in the jth component of the beam is related
to the positive constant µj , whereas the coupling constant β > 0 signifies the
interaction between the two beam components. When µj = 0, the self-focusing

2020 Mathematics Subject Classification. 35J47, 35J50.
Key words and phrases. Elliptic systems; variational methods; semi-nodal solutions;

Gross-Pitaevskii equation.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted June 10, 2023. Published April 25, 2024.

1



2 J. P. P. D. SILVA, E. D. SILVA EJDE-2024/32

has been suppressed, and this type of situation is also relevant in optics (see for
example [15, 16, 20]). The problem denoted by system (1.2) is also encountered in
the Hartree-Fock theory for a binary mixture of Bose-Einstein condensates in two
different hyperfine states |1⟩ and |2⟩ (see for example [13]). In this context, each
Φj represents the corresponding condensate amplitude, while µj and β denote the
intra and interspecies scattering lengths. The self-interactions of the single state
|j⟩ are represented by the sign of µj , with µj > 0 indicating the focusing case
and µj < 0 corresponding to the defocusing case. When the intraspecies scattering
length µj is zero, it means that the interaction between particles of the same species
is extremely weak or nonexistent (see for example [22]). In addition, the sign of β
plays a crucial role in determining whether the interactions between states |1⟩ and
|2⟩ are attractive or repulsive. Specifically, if β > 0, the interactions are attractive,
while β < 0 implies that the interactions are repulsive. This feature is important in
understanding the competition between different states and can have a significant
impact on the behavior of the system as a whole.

Recently, there has been growing interest in studying systems of the form (1.1)
that are related to the system (1.2) in the cubic case p = q = 2. This is evidenced
by the increasing number of research papers published on the topic, among which
we highlight [1, 3, 8, 9, 10, 18, 19, 21, 25] and references therein. On this subject,
we also refer the interested reader to [23].

In this work, we investigate the existence of semi-nodal solutions for system
(1.1), that is, solutions where u > 0 in Ω and v± := max{0,±v} ̸≡ 0 in Ω, which
has also received attention in recent studies, in particular, we are interested in the
case where µ1 = µ2 = 0 and p + q < 2∗, β > 0 and N ≤ 5. Clapp and Soares
[11] dealt with the case where p = q < 2∗/2 = N/(N − 2), λj = −1, µj = 0 and
Ω = RN with N ≥ 4, among other results, they showed the existence of semi-nodal
solutions subject to the mentioned conditions. Chen, Lin & Zou [5, 6] dealt with
the case p = q = 2, λj < 0, µj > 0, β > 0, and Ω ⊂ RN bounded with N ∈ {1, 2, 3},
they showed existence and multiplicity results of nodal solutions for (1.1). In [7],
the same authors provided the existence of semi-nodal solutions for (1.1) for the
critical case p = q = 2∗/2 with Ω ⊂ RN bounded, N ≥ 6, µj > 0, λj ∈ (0, λ1(Ω))
and β < 0, here λ1(Ω) is the first eigenvalue of (−∆, H1

0 (Ω)). In this work, we
are interested in the case where µj = 0, λj < λ1(Ω), β > 0, p > 1, q > 2 with
p + q < 2∗. In particular, 3 < p + q < 2∗ which implies that 3 < 2∗. Hence our
main result applies only for the cases N ∈ {3, 4, 5} where p+ q < 2∗. Furthermore,
assuming that N ∈ {1, 2}, it suffices that p > 1 and q > 2 because 2∗ = +∞. For
the sake of convenience, we will change the notation of system (1.1) to this case,
more specifically, we will consider the system

−∆u = λu+ ξup−1|v|q, in Ω

−∆v = µv + τup|v|q−2v, in Ω

u = v = 0, on ∂Ω

u > 0, v± ̸≡ 0 in Ω.

(1.3)

Our main result reads as follows.

Theorem 1.1. Assume that Ω ⊂ RN is a smooth bounded domain, p > 1, q > 2
with p + q < 2∗ = 2N/(N − 2) for N ∈ {3, 4, 5}, and p + q < +∞ for N ∈ {1, 2},



EJDE-2024/32 SOLUTIONS TO SEMI-NODAL SOLUTIONS 3

λ, µ < λ1(Ω), where λ1(Ω) is the first eigenvalue of (−∆, H1
0 (Ω)). Then there exists

a pair of solution u, v ∈ C2(Ω) to (1.3).

Our approach is based on minimization arguments presented in [4, 24] with
the necessary technical modifications. The main difficulties in our approach are
to avoid semi-trivial solutions (i.e. solution of the form (u, 0) or (0, v)) and to
construct Palais-Smale sequence that converges to the infimum of the functional
associated with system (2.21) restricted to a certain subset of the Nehari manifold.

2. Main result

To present our main result, we use the following notation: H := H1
0 (Ω)×H1

0 (Ω),

∥u∥2λ := ∥u∥2 − λ|u|22, and ∥v∥2µ := ∥v∥2 − µ|v|22, where ∥f∥ :=
( ∫

Ω
|∇f |2dx

)1/2
is the norm of H1

0 (Ω), we will write |f |s as the norm of Ls(Ω) and f±(x) :=
max{0,±f(x)}. Given the condition λ, µ < λ1(Ω), we see that there exists Cλµ :=
Cλµ(λ1(Ω)) such that

Cλµ∥u∥ ≤ ∥u∥λ ≤ C−1
λµ ∥u∥ and Cλµ∥v∥ ≤ ∥v∥µ ≤ C−1

λµ ∥v∥ for all u, v ∈ H1
0 (Ω)

(2.1)
To obtain solutions for system (1.3), we define the functional Iλµ ∈ C1(H,R)

given by

Iλµ(u, v) =
1

2
∥u∥2λ +

1

2
∥v∥2µ − 1

p+ q

∫
Ω

|u|p|v|q dx

Here we shall follow same ideas from [4] which allows us to minimize the functional
Iλµ over the following subsets of the Nehari manifold

Nλ := {(u, v) ∈ H : I ′λµ(u, v)(u, 0) = 0, u ̸≡ 0, v ̸≡ 0},
N±

µ := {(u, v) ∈ H : I ′λµ(u, v)(0, v
±) = 0, u ̸≡ 0, v± ̸≡ 0},

Mλµ := Nλ ∩N+
µ ∩N−

µ

The following result is of fundamental importance for constructing a Palais-Smale
sequence at the level where we obtain solutions to our problem. This approach is
based on an idea presented in the work [24].

Lemma 2.1. Let (u, v) ∈ Mλµ and z, w ∈ H1
0 (Ω) \ {0} then for all δ > 0 there are

unique positive numbers t = t(δ), r = r(δ), and s = s(δ) such that(
t(u− δz), r(v − δw)+ − s(v − δw)−

)
∈ Mλµ.

Moreover if ∥z∥, ∥w∥ ≤ 1 and ∥u∥, ∥v∥ ≤ M1, then there are constants M0 =
M0(p, q, λ, µ, λ1(Ω),Ω,M1, N) > 0, and Ci = Ci(p, q, λ, µ, λ1(Ω),Ω,M1, N) > 0
such that

|t′(0)|, |r′(0)|, |s′(0)| ∈ [0, C2] and |t(0)|, |r(0)|, |s(0)| ∈ [C1, C2], (2.2)

∥u∥, ∥v±∥ ≥ M0 (2.3)

Proof. Firstly, we mention that (ϕ, φ) ∈ Mλµ if and only if

∥ϕ∥2λ =
p

p+ q

∫
|ϕ|p|φ|qdx and ∥φ±∥2µ =

q

p+ q

∫
|ϕ|p|φ±|dx.



4 J. P. P. D. SILVA, E. D. SILVA EJDE-2024/32

Therefore, for each (t(u− δz), r(v − δw)+ − s(v − δw)−) ∈ Mλµ, we obtain that

∥t(u− δz)∥2λ =
p

p+ q

∫
|t(u− δz)|p|r(v − δw)+ − s(v − δw)−|q dx

∥r(v − δw)+∥2µ =
q

p+ q

∫
|t(u− δz)|p|r(v − δw)+| dx

∥s(v − δw)−∥2µ =
q

p+ q

∫
|t(u− δz)|p|s(v − δw)−| dx

(2.4)

To make the presentation clear, we define the following functions:

f1(δ) = ∥u− δz∥2λ, f2(δ) =

∫
Ω

|u− δz|p[(v − δw)+]q,

f3(δ) =

∫
Ω

|u− δz|p[(v − δw)−]q, f4(δ) = ∥(v − δw)+∥2µ,

f5(δ) = ∥(v − δw)−∥2µ.
It follows from (2.4) that t(δ), r(δ), and s(δ) are precisely the solutions for the
system

t2f1(δ) =
p

p+ q
tprqf2(δ) +

p

p+ q
tpsqf3(δ), (2.5)

r2f4(δ) =
q

p+ q
tprqf2(δ), (2.6)

s2f5(δ) =
q

p+ q
tpsqf3(δ), (2.7)

Here we observe that the solution t(δ) is given explicitly by

t(δ) =
(
1 +

p

q

) 1
p+q−2 [f1(δ)]

− q−2
2(p+q−2)

{ [f4(δ)]
q

q−2

[f2(δ)]
2

q−2

+
[f5(δ)]

q
q−2

[f3(δ)]
2

q−2

} q−2
2(p+q−2)

. (2.8)

Recall that

f1(0) = ∥u∥2λ, f2(0) =

∫
Ω

|u|p|v+|q =
p+ q

q
∥v+∥2µ,

f3(0) =

∫
Ω

|u|p|v−|q =
p+ q

q
∥v−∥2µ, f4(0) = ∥v+∥2µ,

f5(0) = ∥v−∥2µ
Here we used that (u, v) ∈ Mλµ to determine the values of f2(0) and f3(0). Since
0 < µ < λ1(Ω), H

1
0 (Ω) ↪→ Lp(Ω), and H1

0 (Ω) ↪→ Lq(Ω) by Hölder inequality’s there
exists Cpq = Cpq(Ω) > 0 such that(

1− µ

λ1(Ω)

)
∥v±∥2 ≤ ∥v±∥2µ =

q

p+ q

∫
Ω

|u|p|v±|q ≤ Cpq∥u∥p∥v±∥q. (2.9)

Since q > 2 and ∥u∥, ∥v∥ ≤ M1, expression (2.9) yields a constant M0 > 0 such
that ∥u∥, ∥v±∥2 ≥ M0, hence (2.3) is proved. As ∥u∥, ∥v±∥ ∈ [M0,M1], we easily see
from the expressions of fi(0) that there existK1 = K1(a, b, p, q, λ, µ, λ1(Ω),M1, N) >
0 and K2 = K2(a, b, p, q, λ, µ, λ1(Ω),M1, N) > 0 such that

K1 ≤ fi(0) ≤ K2, 1 ≤ i ≤ 5. (2.10)

A standard calculation shows that

f ′
1(0) = −

(∫
Ω

∇u∇z − λuz
)
, f ′

2(0) = −p

∫
Ω

|u|p−2uz|v+|q − q

∫
|u|p|v+|q−1w,



EJDE-2024/32 SOLUTIONS TO SEMI-NODAL SOLUTIONS 5

f ′
3(0) = −p

∫
Ω

|u|p−2uz|v−|q − q

∫
|u|p|v−|q−1w,

f ′
4(0) = −

∫
Ω

∇v+∇w − µ

∫
v+w, f ′

5(0) = −
∫
Ω

∇v−∇w − µ

∫
v−w.

It is important to observe that ∥z∥, ∥w∥ ≤ 1 and ∥u∥, ∥v∥ ≤ M1. By the Sobolev
embedding theorems, there exists a constant K3 = K3(p, q, λ, µ, λ1(Ω),M1, N) > 0
such that

|f ′
i(0)| ≤ K3, 1 ≤ i ≤ 5. (2.11)

From the explicit expression of t(δ) given by in (2.8) it follows that for a certain

Ψ ∈ C1(R5
+) (and Ψ /∈ C1(R5

+)) we can write t(δ) = Ψ(f1(δ), . . . , f5(δ)). Therefore
t′(0) =

∑
f ′
i(0)Ψxi(f1(0), . . . , f5(0)), and from this equality, together with (2.10)

and (2.11) we conclude that there exist Ci = Ci(p, q, λ, µ, λ1(Ω),Ω,M1, N) > 0
such that |t′(0)| ≤ C2 and C1 ≤ t(0) ≤ C2. The other inequalities can be derived
from combining the last estimates with (2.6) and (2.7). □

The following proposition shows the existence of a Palais-Smale sequence that
converges to the infimum of Iλµ over Mλµ. Notice also that p+ q > 2 and

Iλµ(u, v) =
(1
2
− 1

p+ q

) (
∥u∥2λ + ∥v∥2µ

)
for all (u, v) ∈ Mλµ. (2.12)

Hence, inf(u,v)∈Mλµ
Iλµ(u, v) > −∞. In what follows, we will use the notation

∥(φ, ϕ)∥ := ∥φ∥+ ∥ϕ∥ for all φ, ϕ ∈ H1
0 (Ω).

Proposition 2.2. Let cλµ := inf(u,v)∈Mλµ
Iλµ(u, v). Then there exists a sequence

(un, vn) ∈ Mλµ such that

Iλµ(un, vn) → cλµ and I ′λµ(un, vn) → 0.

Moreover, there exist M0,M1 > 0 such that ∥un∥, ∥v±n ∥ ∈ [M0,M1] for all n ∈ N.

Proof. By applying the Ekeland’s variational principle [12], we construct a sequence
(un, vn) ∈ Mλµ such that

Iλµ(un, vn) → cλµ,

Iλµ(un, vn) < Iλµ(φ, ϕ) +
1

n
∥(un − u, vn − v)∥, ∀(φ, ϕ) ∈ Mλµ.

(2.13)

As (un, vn) ∈ Mλµ, then ∥un∥2λ + ∥vn∥2µ =
∫
Ω
|un|p|vn|q which leads us to

p+ q − 1

2(p+ q)

(
∥un∥2λ + ∥vn∥2µ

)
=

1

2

(
∥un∥2λ + ∥vn∥2µ

)
− 1

p+ q

∫
Ω

|un|p|vn|q

= Iλµ(un, vn) = cλµ + on(1).

The above expression and (2.1) gives us

2Cλµ

(
∥un∥2 + ∥vn∥2

)
≤

(
∥un∥2λ + ∥vn∥2µ

)
=

2(p+ q)cλµ
p+ q − 1

+ on(1);

therefore (un, vn) is a bounded sequence, i.e, there exists M1 > 0 such that

∥(un, vn)∥ ≤ M1 for all n ∈ N (2.14)

By the Riesz Representation Theorem, it follows that for every fixed n ∈ N there
exist zn, wn ∈ H1

0 (Ω) such that (zn, wn) ∼= I ′λµ(un, vn)/∥I ′λµ(un, vn)∥; moreover

∥(zn, wn)∥ = 1 and I ′λµ(un, vn)(zn, wn) = ∥I ′λµ(un, vn)∥. (2.15)



6 J. P. P. D. SILVA, E. D. SILVA EJDE-2024/32

From now on, we will assume that n ∈ N is fixed. Let t(δ) := tn(δ), r(δ) :=
rn(δ), s(δ) := sn(δ) be given as stated in Lemma 2.1 and u := un, v := vn, z := zn,
w := wn we will define (u(δ), v(δ)) := (un(δ), vn(δ)) by

(u(δ), v(δ)) :=
(
t(δ)[u− δz], r(δ)[v − δw]+ − s(δ)[v − δw]−

)
∈ Mλµ.

Recall also that Iλµ ∈ C1(H,R), where H = H1
0 (Ω)×H1

0 (Ω). Setting R(X−Y ) :=
Iλµ(X) − Iλµ(Y ) − I ′λµ(X)(X − Y ) for any X,Y ∈ H, we have limX→Y (R(X −
Y )/∥X − Y ∥ ) = 0. Since limδ→0+(u(δ), v(δ)) = (u, v), it is not difficult to see that
(∥u(δ)− u∥+ ∥v(δ)− v∥) /δ → ∥t′(0)u− z∥+ ∥r′(0)v+ − s′(0)v− − w∥ as δ → 0+.
Therefore, o(δ) := R(u(δ)− u, v(δ)− v) satisfies o(δ)/δ → 0 as δ → 0+ and

Iλµ(u(δ), v(δ)) = Iλµ(u, v) + I ′λµ(u(δ), v(δ))(u(δ)− u, v(δ)− v) + o(δ) (2.16)

setting Tδ(φ, ϕ) := I ′λµ(u(δ), v(δ))(φ, ϕ), by (2.13) and (2.16) we have

1

n
∥u(δ)− u, v(δ)− v∥

≥ Iλµ(u, v)− Iλµ(u(δ), v(δ))

= Tδ(u− u(δ), v − v(δ)) + o(δ) =

= (1− t(δ))Tδ(u− δz, 0) + Tδ(u, 0)− Tδ(u− δz, 0)

+ (1− r(δ))Tδ(0, [vn − δw]+) + Tδ(0, v
+)− Tδ(0, [v − δw]+)

− (1− s(δ))Tδ(0, [v − δw]−)− Tδ(0, v
−) + Tδ(0, [v − δw]−) + o(δ)

= (1− t(δ))Tδ(u− δz, 0) + (1− r(δ))Tδ(0, [v − δw]+)

− (1− s(δ))Tδ(0, [v − δw]−) + δTδ(z, w) + o(δ).

As a consequence,

1

n
∥u(δ)− u

δ
,
v(δ)− v

δ
∥ ≥

(1− t(δ)

δ

)
Tδ(u− δz, 0) +

(1− r(δ)

δ

)
Tδ(0, [v − δw]+)

−
(1− s(δ)

δ

)
Tδ(0, [v − δw]−) + Tδ(z, w) +

o(δ)

δ
.

Given that limδ→0+ t(δ) = limδ→0+ r(δ) = limδ→0+ s(δ) = 1, taking the limit as
δ → 0+ , the above inequality gives us

1

n
∥t′(0)u− z, r′(0)v+ − s′(0)v− − w∥

≥ −t′(0)T0(u, 0)− r′(0)T0(0, v
+) + s′(0)T0(0, v

−) + T0(z, w)

= −t′(0)T0(u, 0)− r′(0)T0(0, v
+) + s′(0)T0(0, v

−) + T0(z, w).

(2.17)

Since (u, v) ∈ Mλµ, it follows that T0(u, 0) = T0(0, v
+) = T0(0, v

−) = 0; therefore,
from (2.17) and (2.15) we conclude that

|t′(0)|+ |r′(0)|+ |s′(0)|
n

(∥u, v∥+ ∥z, w∥) ≥ T0(z, w)

= I ′λµ(u, v)(z, w)

= ∥I ′λµ(u, v)∥.

(2.18)

By Lemma 2.1 there exists C2 > 0 (that does not depend on the index n) such
that |t′(0)| + |r′(0)| + |s′(0)| ≤ 3C2. From (2.18), (2.14) and (2.15) we obtain
∥I ′λµ(un, vn)∥ ≤ 3(2M1 +1)C2/n. The existence of the constant M0 > 0 is guaran-
teed by Lemma 2.1. This completes the proof. □



EJDE-2024/32 SOLUTIONS TO SEMI-NODAL SOLUTIONS 7

Proof of Theorem 1.1. Firstly, we deal with the case N ∈ {3, 4, 5}. Let (un, vn) the
sequence obtained in Proposition 2.2, by (2.12), ∥un∥, ∥v±n ∥ ∈ [M0,M1], M0 > 0
and λ, µ < λ1(Ω), we can deduce that cλµ > 0. It follows from the boundedness of
un, vn in H1

0 (Ω) that there exists u0, v0 ∈ H1
0 (Ω) such that, up to a subsequence,

un ⇀ u0 and vn ⇀ v0 weakly in H1
0 (Ω). Since I ′λµ(un, vn) → 0 and (v±n − v±0 ) is

bounded in H1
0 (Ω), it follows that

∥v±n ∥2µ −
∫
Ω

(
∇vn∇v±0 − µvnv

±
0

)
− Γn = I ′λµ(un, vn)(0, v

±
n − v±0 ) → 0, (2.19)

where

Γn :=
q

p+ q

∫
Ω

|un|p||v±n |q−2v±n (v
±
n − v±0 )dx.

Then Hölder’s inequality gives us∣∣ ∫
Ω

|un|p||v±n |q−2v±n (v
±
n − v±0 )dx

∣∣ ≤ |un|pp+q|v±n |
q−1
p+q|v±n − v±0 |p+q. (2.20)

Since 2 < p + q < 2∗, it follows that H1
0 (Ω) ↪→ Lp+q(Ω) is a compact embedding;

therefore vn ⇀ v0 weakly in H1
0 (Ω) imply that |v±n − v±0 |p+q → 0, which combined

with the boundedness of un, vn in H1
0 (Ω) and (2.20) gives us Γn → 0. As∫

Ω

(∇vn∇v±0 − µvnv
±
0 )dx → ∥v±0 ∥2µ

from (2.19) we obtain ∥v±n ∥2µ → ∥v±0 ∥2µ. Since v±n → v±0 strongly in L2(Ω) it follows

that ∥v±n ∥2 → ∥v±0 ∥2 and therefore v±n → v±0 strongly in H1
0 (Ω). In a completely

analogous manner, we can conclude that un → u0 strongly inH1
0 (Ω). It follows from

Proposition 2.2 that Iλµ(u0, v0) = cλµ, I
′
λµ(u0, v0) = 0 and ∥u0∥ > 0 and ∥v±0 ∥ > 0.

Notice also that we can replaced un by |un| and still have Iλµ(|un|, vn) → cλµ.
Without loss of generality we assume that un ≥ 0 which implies that u0 ≥ 0. Since
I ′λµ(u0, v0) = 0 we deduce that u0, v0 are the weak solutions of the system

−∆u0 = λu0 +
p

p+ q
|u0|p−2u0|v0|q, in Ω,

−∆v0 = µv0 +
q

p+ q
|u0|p|v0|q−2v0, in Ω,

u0 = v0 = 0, on ∂Ω,

u0 ≥ 0, v±0 ̸≡ 0 in Ω.

(2.21)

It follows from the standard theory of elliptic regularity and a bootstrap argument
that u0, v0 ∈ C2(Ω). Furthermore, by using that −∆u0 ≥ 0 in Ω and ∥u0∥ > 0,
the Strong Maximum Principle implies that u0 > 0 in Ω. Once again by using that
p+ q > 2, there exist t, s > 0 satisfying( p

p+ q

) 1

tp−2sq
= ξ and

( q

p+ q

) 1

tpsq−2
= τ

It is easy to verify that u = tu0 and v = sv0 satisfy (1.3). The cases N = 1, 2 with
p + q < +∞ follows in a similar way by using that H1

0 (Ω) ↪→ Lq(Ω) is a compact
embedding for each q ≥ 1. □

Acknowledgements. E. D. da Silva was partially supported by CNPq grants
309026/2020-2. The authors would like to express their sincere gratitude to the
referees for their carefully reading the manuscript and valuable suggestions.



8 J. P. P. D. SILVA, E. D. SILVA EJDE-2024/32

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EJDE-2024/32 SOLUTIONS TO SEMI-NODAL SOLUTIONS 9

João Pablo Pinheiro da Silva

Departamento de Matemática, Universidade Federal do Pará, Belem, Brazil

Email address: jpabloufpa@gmail.com

Edcarlos Domingos da Silva

Departamento de Matemática, Universidade de Federal de Goiás, Goiánia, GO, 74690-

900, Brazil
Email address: edcarlos@ufg.br


	1. Introduction
	2. Main result
	Acknowledgements

	References