Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 31, pp. 1–13.

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

DOI: 10.58997/ejde.2024.31

NODAL SOLUTIONS FOR NONLINEAR SCHRÖDINGER

SYSTEMS

XUE ZHOU, XIANGQING LIU

Abstract. In this article we consider the nonlinear Schrödinger system

−∆uj + λjuj =
k∑

i=1

βiju
2
i uj , in Ω,

uj(x) = 0, on ∂Ω, j = 1, . . . , k,

where Ω ⊂ RN (N = 2, 3) is a bounded smooth domain, λj > 0, j = 1, . . . , k,

βij are constants satisfying βjj > 0, βij = βji ≤ 0 for 1 ≤ i < j ≤ k. The
existence of sign-changing solutions is proved by the truncation method and

the invariant sets of descending flow method.

1. Introduction

We consider the nonlinear Schödinger system

−∆uj + λjuj =

k∑
i=1

βiju
2
iuj , in Ω,

uj(x) = 0, on ∂Ω, j = 1, . . . , k,

(1.1)

where Ω ⊂ RN (N = 2, 3) is a bounded domain with smooth boundary, and λj > 0,
βjj > 0, 1 ≤ j ≤ k, βij = βji, 1 ≤ i < j ≤ k are constants.

This type of coupled systems, also known as Gross-Pitaevskii equations, have
applications in many physical problems such as nonlinear optics and multispecies
Bose-Einstein condensates [8, 18]. Physically, βjj , βij (i ̸= j) are the intraspecies
and interspecies scattering lengths respectively. In the physics literature, the signs
of the coupling constants βij being positive or negative determine the nature of
the system being attractive or repulsive. In the repulsive case (βij < 0, i ̸= j,
i, j = 1, . . . , k), the components tend to segregate with each other leading to phase
separations. These phenomena have been documented in experiments as well as in
numeric simulations; see [4, 17] and references therein. Mathematical work has been
done extensively in recent years, refer the reader to [1, 3, 7, 9, 14, 15, 16, 19] for the
existence theory and the studies of qualitative property of solutions to attractive
and repulsive systems.

2020 Mathematics Subject Classification. 35A15, 35B20, 35J10.
Key words and phrases. Schrödinger system; sign-changing solutions; truncation method;

method of invariant sets of descending flow.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted December 9, 2023. Published April 24, 2024.

1



2 X. ZHOU, X. LIU EJDE-2024/31

Over the years there have been systematic studies on nodal solutions for scalar
equations by using a combination of minimax methods and the method of invari-
ant sets of gradient flows. We refer the reader to [2, 6, 13]. However, most of
the methods in treating scalar equations are not applicable directly to systems.
In [14, 15] a construction of invariant sets has been developed to locate multiple
nontrivial solutions, but without giving any information about nodal property of
the components of solutions. Compared with scalar equations, there are many new
challenges for coupled equations in dealing with the existence of multiple solutions,
in particular multiple sign-changing solutions. An attempt was made in [10, 11]
for establishing an abstract framework to deal with sign-changing solutions for sys-
tems that share some of the above features. The authors in [10, 11] developed
the method of multiple invariant sets of decreasing flow. In [10] for the subcritical
case infinitely many sign-changing solutions were established. Specially, Chen, Lin
and Zou [5] proved the existence of multiple sign-changing (i.e., both two compo-
nents change sign) and semi-nodal solutions (i.e., one component changes sign and
the other one is positive) for coupled Schrödinger equations for the case of k = 2,
β12 = β21 = β > 0. Motivated by the works we mentioned above, in this paper we
consider the existence of sign-changing solutions for the system (1.1) in the general
case, by using the method of invariant sets of decreasing flow (see [10]) and the
truncation method (see [12]).

We assume that

(A1) Ω ⊂ RN , N = 2, 3, k ≥ 2, λj > 0 for j = 1, . . . , k.
(A2) βjj > 0, βij = βji ≤ 0 for 1 ≤ i < j ≤ k.

Solutions of (1.1) correspond to critical points of the functional

I(U) =
1

2

∫
Ω

k∑
j=1

(|∇uj |2 + λju
2
j ) dx− 1

4

∫
Ω

k∑
i,j=1

βiju
2
iu

2
j dx

for U = (u1, . . . , uk) ∈ X = H1
0 (Ω)× · · · ×H1

0 (Ω), the k-fold product of (H1
0 (Ω))

k.
We shall use the equivalent inner products

(u, v)j =

∫
Ω

(∇u∇v + λjuv)dx, j = 1, . . . , k

and the induced norm ∥ · ∥j . The inner product

(U, V ) =

k∑
j=1

(u, v)j , U = (u1, . . . , uk), V = (v1, . . . , vk),

gives rise to a norm ∥ · ∥ on X.
Firstly, we introduce the following perturbation problem. We assume U =

(u1, . . . , uk), ε ∈ R is a small parameter, F (U, ε), ∂F
∂uj

(U, ε) are continuous functions,

and F (U, ε) = F (−U, ε). For ε = 0, we understand

F (U, 0) = 0,
∂F

∂uj
(U, 0) = 0.

Then we consider the perturbed problem

−∆uj + λjuj =

k∑
i=1

βiju
2
iuj +

∂F

∂uj
(U, ε), in Ω,

uj(x) = 0, on ∂Ω, j = 1, . . . , k.

(1.2)



EJDE-2024/31 SCHRÖDINGER SYSTEMS 3

Here are our main results.

Theorem 1.1. Assume (A1), (A2) hold. Then system (1.1) has infinitely many
solutions with each component being sign-changing.

Theorem 1.2. Assume (A1), (A2) hold and let l ∈ N+. Then there exists εl > 0
such that for |ε| ≤ εl, the system (1.2) has l pairs of sign-changing solutions.

Corollary 1.3. For each l ∈ N+, there exists βl > 0 such that for βij = βji ≤ βl

with 1 ≤ i < j ≤ k, system (1.1) has at least l pairs of sign-changing solutions.

Note that we do not assume any growth conditions for the perturbation function
F . To apply critical point theorem [10, 11], we firstly have the following truncated
function; the idea comes from [12]. For M > 0, we define

FM (U, ε) = F
(
fM (|U |) U

|U |
)
,

where fM is a monotonic smooth function, satisfying fM (t) = t if t ≤ M , fM (t) =
M + 1

2 if t ≥ M. Then we consider the truncated system

−∆uj + λjuj =

k∑
i=1

βiju
2
iuj +

∂FM

∂uj
(U, ε), in Ω,

uj(x) = 0, on ∂Ω, j = 1, . . . , k.

(1.3)

If U = (u1, . . . , uk) is a solution of (1.3), and there exists M > 0 such that |U(x)| <
M for all x ∈ Ω, then U is also a solution of the perturbed problem (1.2). System
(1.3) has a variational structure given by the functional

IM (U) = I(U)−
∫
Ω

FM (U, ε) dx

=
1

2

∫
Ω

k∑
j=1

(|∇uj |2 + λju
2
j ) dx− 1

4

∫
Ω

k∑
i,j=1

βiju
2
iu

2
j dx−

∫
Ω

FM (U, ε) dx .

(1.4)
This article organized as follows. In Section 2, we study the truncated functional

IM , and construct a sequence of critical values for IM by using the method of
multiple invariant sets of descending flow. In Section 3, we obtain the sign-changing
solutions of the perturbed problem (1.2), then we obtain the main result.

Throughout this article, we use ∥ · ∥Lp and ∥ · ∥ to denote the norms of Lp and
X, respectively. c, c1, . . . denote constants that are independent of the sequences
in the arguments but maybe different from line to line, and c(·) will be used to
indicate the dependency of the constant c on the relevant quantity.

2. Critical points of the truncated functional IM

To obtain sign-changing critical points of IM , we apply an abstract critical point
theorem (Theorem 2.1) to the truncated functional IM .

Theorem 2.1. Let X be a Banach space, f be an even C1-functional on X, A be
an odd, continuous mapping from X to X, and Pj , Qj, j = 1, . . . , k be open convex
subsets of X with Qj = −Pj. Denote W = ∪k

j=1(Pj ∪Qj), Σ = ∩k
j=1(∂Pj ∩ ∂Qj).

Assume

(A3) f satisfies the Palais-Smale condition.



4 X. ZHOU, X. LIU EJDE-2024/31

(A4) c∗ = infx∈Σ f(x) > 0.
(A5) For each b0 > 0 and c0 > 0, there exists b = b(b0, c0), such that if |f(x)| ≤

c0, ∥Df(x)∥ ≥ b0, then

⟨Df(x), x−Ax⟩ ≥ b∥x−Ax∥ > 0.

(A6) A(∂Pj) ⊂ Pj , A(∂Qj) ⊂ Qj, j = 1, . . . , k.

We define

Γj = {E ⊂ X : E is compact, −E = E, γ(E ∩ σ−1(Σ)) ≥ j for σ ∈ Λ},
Λ =

{
σ ∈ C(X,X) : σ is odd, σ(Pj) ⊂ Pj , σ(Qj) ⊂ Qj , j = 1, . . . , k,

σ(x) = x if f(x) < 0
}

where γ = γ(E) denotes the genus of a symmetric set E

γ = min{n : there is an odd map φ(j) : E → Rn \ {0}}.

We ssume that

(A7) Γj is nonempty for j = 1, 2, . . . .

Then we define

cj = inf
E∈Γj

sup
x∈E\W

f(x), j = 1, 2, . . . ,

Kc = {x ∈ X : Df(x) = 0, f(x) = c}, K∗
c = Kc \W.

Then

(1) cj ≥ c∗, K
∗
cj ̸= ∅ for j = 1, 2, . . . .

(2) cj → +∞, as j → ∞.
(3) If cj = cj+1 = · · · = cj+l−1 = c, then γ(K∗

c ) ≥ l.

Lemma 2.2. IM is a C1-functional on X, and satisfies the Palais-Smale condition.

Proof. It is easy to verify that IM is a C1-functional. Also, for Φ = (φ1, . . . , φk) ∈
X, we have

⟨DIM (U), Φ⟩ =
∫
Ω

k∑
j=1

(∇uj∇φj + λjujφj) dx−
∫
Ω

k∑
i,j=1

βiju
2
iujφj dx

−
∫
Ω

k∑
j=1

∂FM

∂uj
(U, ε)φj dx,

(2.1)

there exists an arbitrary small constant εM , such that for |ε| ≤ εM , we have

IM (U)− 1

4
⟨DIM (U), U⟩

=
1

4

∫
Ω

k∑
j=1

(|∇uj |2 + λju
2
j ) dx−

∫
Ω

(
FM (U, ε)− 1

4

k∑
j=1

∂FM

∂uj
(U, ε)uj

)
dx

≥ 1

4
∥U∥2 − c.

(2.2)

Then any Palais-Smale sequence of IM is bounded inX. Let Un = (un,1, . . . , un,k) ∈
X be a Palais-Smale sequence of the functional IM . Notice that the imbedding



EJDE-2024/31 SCHRÖDINGER SYSTEMS 5

H1
0 (Ω) ↪→ L4(Ω) is compact and we can assume that Un → U in L4(Ω). Then we

have∫
Ω

k∑
j=1

(|∇(un,j − um,j)|2 + λj(un,j − um,j)
2) dx

= ⟨DIM (Un)−DIM (Um), Un − Um⟩+
∫
Ω

k∑
i,j=1

βiju
2
n,iun,j(un,j − um,j) dx

−
∫
Ω

k∑
i,j=1

βiju
2
m,ium,j(un,j − um,j) dx

+

∫
Ω

k∑
j=1

(∂FM

∂uj
(Un, ε)−

∂FM

∂uj
(Um, ε)

)
(un,j − um,j) dx

≤ o(1) + c∥Un∥3L4(Ω)

(∫
Ω

k∑
j=1

(un,j − um,j)
4 dx

)1/4

+ c∥Um∥3L4(Ω)

(∫
Ω

k∑
j=1

(un,j − um,j)
4 dx

)1/4

+

∫
Ω

k∑
j=1

∣∣∂FM

∂uj
(Un, ε)−

∂FM

∂uj
(Um, ε)

∣∣ |un,j − um,j | dx

≤ o(1) + c∥Un − Um∥L4(Ω) → 0, as n,m → ∞.

Therefore, we conclude that up to a subsequence a Palais-Smale sequence Un is a
Cauchy sequence in X, hence a convergent sequence. □

Definition 2.3. An odd and continuous operator A : U = (u1, . . . , uk) ∈ X 7→
V = (v1, . . . , vk) = AU ∈ X is defined by the system∫

Ω

(∇vj∇φj + λjvjφj) dx−
∫
Ω

k∑
i=1,i̸=j

βiju
2
i vjφj dx

=

∫
Ω

βjju
3
jφj dx+

∫
Ω

∂FM

∂uj
(U, ε)φj dx,

(2.3)

For j = 1, . . . , k and Φ = (φ1, . . . , φk) ∈ X.

Lemma 2.4. The operator A is well-defined and continuous.

Proof. Note that V = AU can be obtained by solving the minimization problem

inf{G(V ) : V ∈ X}

where

G(V ) =
1

2

∫
Ω

k∑
j=1

(|∇vj |2 + λjv
2
j ) dx− 1

2

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i v

2
j dx

−
∫
Ω

k∑
j=1

βjju
3
jvj dx−

∫
Ω

k∑
j=1

∂FM

∂uj
(U, ε)vj dx.



6 X. ZHOU, X. LIU EJDE-2024/31

Let V = AU , V̄ = AŪ , V̄ = (v̄1, . . . , v̄k), Ū = (ū1, . . . , ūk). By (2.3), we have

∥V − V̄ ∥2

=

∫
Ω

k∑
j=1

(|∇(vj − v̄j)|2 + λj(vj − v̄j)
2) dx

=

∫
Ω

k∑
i,j=1,i̸=j

βij(u
2
i vj − ū2

i v̄j)(vj − v̄j) dx+

∫
Ω

k∑
j=1

βjj(u
3
j − ū3

j )(vj − v̄j) dx

+

∫
Ω

k∑
j=1

(∂FM

∂uj
(U, ε)− ∂FM

∂uj
(Ū , ε)

)
(vj − v̄j) dx

≤ c

∫
Ω

k∑
i,j=1,i̸=j

|u2
i − ū2

i | |vj | |vj − v̄j | dx+ c

∫
Ω

k∑
j=1

|u3
j − ū3

j | |vj − v̄j | dx

+

∫
Ω

k∑
j=1

∣∣∂FM

∂uj
(U, ε)− ∂FM

∂uj
(Ū , ε)

∣∣ |vj − v̄j | dx

≤ c(∥U − Ū∥ ∥V − V̄ ∥+ ∥∂FM

∂uj
(U, ε)− ∂FM

∂uj
(Ū , ε)∥ ∥V − V̄ ∥),

hence AU −AŪ = V − V̄ → 0 as U → Ū in X. □

Lemma 2.5. For each b0, c0 > 0, then the following property holds: if |IM (U)| ≤ c0
and ∥DIM (U)∥ ≥ b0, then there exists b = b(b0, c0) such that

⟨DIM (U), U −AU⟩ ≥ b∥U −AU∥ > 0.

Proof. We have

⟨DIM (U),Φ⟩

=

∫
Ω

k∑
j=1

(∇(uj − vj)∇φj + λj(uj − vj)φj) dx−
∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)φj dx

= ⟨U − V, Φ⟩ −
∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)φj dx

(2.4)
for Φ = (φ1, . . . , φk) ∈ X. By using Φ = U − V in (2.4), we obtain

⟨DIM (U), U − V ⟩ = ∥U − V ∥2 −
∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)

2 dx.

Notice that if βij = βji ≤ 0 for 1 ≤ i < j ≤ k, then

⟨DIM (U), U − V ⟩ ≥ ∥U − V ∥2 (2.5)

and

⟨DIM (U), U − V ⟩ ≥ −
∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)

2 dx. (2.6)



EJDE-2024/31 SCHRÖDINGER SYSTEMS 7

It follows from (2.4) and (2.6) that

|⟨DIM (U),Φ⟩| =
∣∣⟨U − V, Φ⟩ −

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)φj dx

∣∣
≤ ∥U − V ∥ ∥Φ∥+

(
−

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
i (uj − vj)

2 dx
)1/2

×
(
−
∫
Ω

k∑
i,j=1,i̸=j

βiju
2
iφ

2
j dx

)1/2

≤ ∥U − V ∥ ∥Φ∥+ c∥U∥L4(Ω)∥Φ∥L4(Ω)⟨DIM (U), U − V ⟩1/2

which implies that

∥DIM (U)∥ ≤ ∥U − V ∥+ c∥U∥⟨DIM (U), U − V ⟩1/2. (2.7)

There exists a small constant εM , so that for |ε| ≤ εM , by (1.4) and (2.4), we have

IM (U)− 1

4
⟨U − V, U⟩

= IM (U)− 1

4
⟨DIM (U), U⟩ − 1

4

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
iuj(uj − vj) dx

=
1

4
∥U∥2 +

∫
Ω

(1
4

k∑
j=1

∂FM

∂uj
(U, ε)uj − FM (U, ε)

)
dx

− 1

4

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
iuj(uj − vj) dx

≥ 1

4
∥U∥2 − 1

4

∫
Ω

k∑
i,j=1,i̸=j

βiju
2
iuj(uj − vj) dx− c.

(2.8)

So by (2.8), we obtain

∥U∥2

≤ c(1 + |IM (U)|) + c|⟨U − V,U⟩|+ c
∣∣∣ ∫

Ω

k∑
i,j=1,i̸=j

βiju
2
iuj(uj − vj) dx

∣∣∣
≤ c(1 + |IM (U)|) + c∥U − V ∥2 + 1

4
∥U∥2 + c∥U∥2L4(Ω)⟨DIM (U), U − V ⟩1/2.

(2.9)

Given a positive constant a, if

⟨DIM (U), U − V ⟩ ≥ a2,

then by (2.5) we can easily obtain

⟨DIM (U), U − V ⟩ ≥ a∥U − V ∥ > 0.

The conclusion holds; if not, let

⟨DIM (U), U − V ⟩ ≤ a2, (2.10)

by (2.9) and (2.10), we have

∥U∥2 ≤ c(1 + |IM (U)|+ ∥U − V ∥2) + c0a∥U∥2. (2.11)



8 X. ZHOU, X. LIU EJDE-2024/31

Hence, taking a such that c0a ≤ 1/2, then we have

∥U∥2 ≤ c(1 + |IM (U)|+ ∥U − V ∥2). (2.12)

Substituting (2.12) into (2.7), we obtain

∥DIM (U)∥

≤ ∥U − V ∥+ c(1 + |IM (U)|+ ∥U − V ∥2)1/2⟨DIM (U), U − V ⟩1/2

≤ ∥U − V ∥+ 1

2
∥DIM (U)∥+ c(1 + |IM (U)|+ ∥U − V ∥2)∥U − V ∥.

(2.13)

Therefore

∥DIM (U)∥ ≤ c(1 + |IM (U)|+ ∥U − V ∥2)∥U − V ∥.
If |IM (U)| ≤ c0 and ∥DIM (U)∥ ≥ b0 > 0, we deduce that there exists b =

b(b0, c0) such that ∥U − V ∥ > b. So it follows from (2.5) that

⟨DIM (U), U −AU⟩ ≥ b∥U −AU∥ > 0. □

Let Pj , Qj for j = 1, . . . , k be open convex subsets of X, defined by

Pj = Pj(δ) = {U = (u1, . . . , uk) ∈ X : ∥u−
j ∥L4(Ω) < δ},

Qj = Qj(δ) = {U = (u1, . . . , uk) ∈ X : ∥u+
j ∥L4(Ω) < δ}.

Lemma 2.6. There exist δ > 0 and εM > 0 such that for |ε| ≤ εM , it holds that

A(∂Pj) ⊂ Pj , A(∂Qj) ⊂ Qj , for j = 1, . . . , k.

Proof. Choose Φ = V + = (v+1 , . . . , v
+
k ) as test function in (2.3), we have∫

Ω

(
|∇v+j |

2 + λj(v
+
j )

2
)
dx−

∫
Ω

k∑
i=1,i̸=j

βiju
2
i (v

+
j )

2 dx

=

∫
Ω

βjju
3
jv

+
j dx+

∫
Ω

∂FM

∂uj
(U, ε)v+j dx

≤ c
(∫

Ω

(u+
j )

3v+j dx+

∫
Ω

∣∣∂FM

∂uj
(U, ε)

∣∣v+j dx
)
.

Then

∥v+j ∥
2
L4(Ω) ≤ c1∥u+

j ∥
3
L4(Ω)∥v

+
j ∥L4(Ω) + c2∥

∂FM

∂uj
(U, ε)∥L∞(Ω)∥v+j ∥L4(Ω). (2.14)

Take δ > 0 such that c1δ
2 ≤ 1/4 and choose εM > 0, such that for |ε| ≤ εM ,

c2∥∂FM

∂uj
(U, ε)∥L∞(Ω) ≤ δ/4. Then for U ∈ ∂Qj , ∥u+

j ∥L4(Ω) = δ, we have

∥v+j ∥
2
L4(Ω) ≤

1

4
δ∥v+j ∥L4(Ω) +

1

4
δ∥v+j ∥L4(Ω),

hence

∥v+j ∥L4(Ω) ≤
1

2
δ.

That is for U ∈ ∂Qj , we have V = AU ∈ Qj and A(∂Qj) ⊂ Qj , j = 1, . . . , k.
Similarly, A(∂Pj) ⊂ Pj , j = 1, . . . , k. □

Lemma 2.7. There exist δ > 0 and c∗ > 0, such that if U ∈ Σ and |ε| ≤ εM , then
IM (U) ≥ c∗.



EJDE-2024/31 SCHRÖDINGER SYSTEMS 9

Proof. Note that

IM (U) =
1

2

∫
Ω

k∑
j=1

(|∇uj |2 + λju
2
j ) dx− 1

4

∫
Ω

k∑
i,j=1

βiju
2
iu

2
j dx−

∫
Ω

FM (U, ε) dx

≥ 1

2
∥U∥2 − 1

4

∫
Ω

k∑
j=1

βjju
4
j dx−

∫
Ω

FM (U, ε) dx

≥ c1∥U∥2L4(Ω) − c2∥U∥4L4(Ω) − ∥FM (U, ε)∥L∞(Ω).

For U ∈ Σ = ∩k
j=1(∂Pj ∩ ∂Qj), we have

∥U∥4L4(Ω) =

∫
Ω

k∑
j=1

(
(u+

j )
4 + (u−

j )
4
)
dx = 2k∥u+

j ∥
4
L4(Ω) = 2kδ4.

By Lemma 2.6, taking δ > 0 such that c2δ
2 ≤ 1

4c1, and choosing εM such that for

|ε| ≤ εM , we have ∥FM (U, ε)∥L∞(Ω) ≤ 1
4c1δ

2. Therefore,

IM (U) ≥ c1δ
2 − c2δ

4 − 1

4
c1δ

2 ≥ 1

2
c1δ

2 := c∗ > 0 . □

Let

Γj = {E ⊂ X : E is compact, −E = E, γ(E ∩ σ−1(Σ)) ≥ j for σ ∈ Λ},
Λ =

{
σ ∈ C(X,X) : σ odd, σ(Pj) ⊂ Pj , σ(Qj) ⊂ Qj , j = 1, . . . , k,

σ(U) = U if IM (U) < 0
}
,

and γ = γ(E) is the genus of E,

γ = min{n : there is an odd map φ(j) : E → Rn \ {0}}.

Now we define a sequence of critical values of the truncated functional IM ,

cj(M, ε) = inf
E∈Γj

sup
U∈E\W

IM (U), j = 1, 2, . . .

where W = ∪k
j=1(Pj ∪Qj).

Lemma 2.8. The set Γj is nonempty, and there exist dj > 0 independent of M , ε

and ε
(j)
M > 0, such that if |ε| ≤ ε

(j)
M , then cj(M, ε) ≤ dj.

Proof. Let Bnk be the unit closed ball of Rnk. Assume n = j + k. Denote t ∈ Rnk

by t = (t1, . . . , tk) and tm = (t1m, t2m, . . . , tnm) ∈ Rn for m = 1, . . . , k. Let
vim ∈ C∞

0 (Ω), i = 1, . . . , n, m = 1, . . . , k be nk functions in X with disjoint
supports. Define φ(j) : Bnk → X by

φ(j)(t) = R
( n∑
i=1

ti1vi1, . . . ,

n∑
i=1

tikvik
)
∈ X

where R is large enough such that I(φ(j)(t)) < −10 for t ∈ ∂Bnk. Then there exists
εM > 0, so that if |ε| ≤ εM , then we have

IM (φ(j)(t)) ≤ I(φ(j)(t)) + 1 < 0

for t ∈ ∂Bnk. By [11, Lemma 5.6], we have Ej := φ(j)(Bnk) ∈ Γj . Then Γj is
nonempty.



10 X. ZHOU, X. LIU EJDE-2024/31

Next we estimate cj(M, ε) for |ε| ≤ εM . We have

cj(M, ε) = inf
E∈Γj

sup
U∈E\W

IM (U) ≤ sup
U∈Ej

IM (U) ≤ sup
U∈Ej

(
I(U) + 1

)
:= dj . □

3. Proof of main results

In this section, we complete the proof of Theorem 1.1 and Theorem 1.2. For
fixed M > 0 and ε = 0, we will obtain the critical point U of I.

Lemma 3.1. Assume DIM (U) = 0, IM (U) ≤ L. Then there exist εM > 0 and
K = K(L) independent of M, ε, such that for |ε| ≤ εM ,

∥U(x)∥L∞(Ω) ≤ K.

Proof. Denote U = (u1, . . . , uk). By (2.2), for |ε| ≤ εM , we have

L ≥ IM (U)− 1

4
⟨DIM (U), U⟩

=
1

4

∫
Ω

k∑
j=1

(|∇uj |2 + λju
2
j ) dx−

∫
Ω

(
FM (U, ε)− 1

4

k∑
j=1

∂FM

∂uj
(U, ε)uj

)
dx

≥ 1

4
∥U∥2 − c.

(3.1)

We know that there exists C(L) > 0, such that ∥U∥ ≤ C(L). Choose ϕ =
ujT |ujT |2r−2 as the test function in ⟨DIM (uj), ϕ⟩ = 0, where r ≥ 1, T > 1, and
ujT (x) = ±T if ±uj(x) ≥ T , ujT (x) = uj(x) if |uj(x)| ≤ T . We have

∫
Ω

(∇uj∇ϕ+ λjujϕ) dx =

∫
Ω

k∑
i=1

βiju
2
iujϕdx+

∫
Ω

∂FM

∂uj
(U, ε)ϕdx. (3.2)

By (3.2), it is easy to obtain the inequality∫
Ω

∇uj∇ϕdx ≤
∫
Ω

βjju
3
jϕdx+

∫
Ω

|∂FM

∂uj
(U, ε)ϕ| dx. (3.3)

Firstly, we estimate the left-hand side of (3.3),∫
Ω

∇uj∇ϕdx ≥ (2r − 1)

∫
Ω

|∇ujT |2|ujT |2r−2 dx

≥ 2r − 1

r2

∫
Ω

|∇|ujT |r|2 dx

≥ c(2r − 1)

r2

(∫
Ω

(
|ujT |r

)2∗
dx

)2/2∗

.

(3.4)



EJDE-2024/31 SCHRÖDINGER SYSTEMS 11

Let M > 0, there exists εM such that for |ε| ≤ εM , we have ∥∂FM

∂uj
(U, ε)∥L∞(Ω) < 1.

Then the right-hand side of (3.3) satisfies∫
Ω

βjju
3
jϕdx+

∫
Ω

∣∣∂FM

∂uj
(U, ε)ϕ

∣∣ dx
≤ c

(∫
Ω

|uj |3|ujT |2r−1 dx+

∫
Ω

1 · |ujT |2r−1 dx
)

≤ c
(∫

Ω

(
1 + |uj |3

)
|uj |2r−1 dx

)
≤ c

(
1 +

∫
Ω

|uj |3|uj |2r−1 dx
)

≤ c
(
1 +

(∫
Ω

|uj |2
∗
dx

) 2
2∗
(∫

Ω

(|uj |r)
2·2∗
2∗−2 dx

) 2∗−2
2∗

)
≤ c

(
1 +

(∫
Ω

(|uj |r)
2·2∗
2∗−2 dx

) 2∗−2
2∗

)
≤ cmax

{
1,
(∫

Ω

(|uj |r)
2·2∗
2∗−2 dx

) 2∗−2
2∗

}
.

(3.5)

Let T → ∞ such that ujT (x) → uj(x). By (3.4) and (3.5), we obtain(∫
Ω

(
|ujT |r

)2∗
dx

) 2
2∗ ≤ cr2

2r − 1
max

{
1,
(∫

Ω

(|uj |r)
2·2∗
2∗−2 dx

) 2∗−2
2∗

}
. (3.6)

Denote d = 2∗

2·2∗
2∗−2

= 2
N−2 > 1, q = 2·2∗

2∗−2 = N . By (3.6), we have(∫
Ω

(|uj |r)qd dx
) 1

qdr ≤
( cr2

2r − 1

) 1
2r max

{
1,
(∫

Ω

(|uj |r)q dx
) 1

qr
}
. (3.7)

Choose r0, such that r0q = 2∗ and
∫
Ω
|uj |qr0 dx < ∞. So(∫

Ω

(|uj |r0)qd dx
) 1

qdr0 ≤
( cr20
2r0 − 1

) 1
2r0 max

{
1,
(∫

Ω

(|uj |r0)q dx
) 1

qr0
}
. (3.8)

Using iteration, we note that r0d = r1 in (3.8), then(∫
Ω

|uj |r1q dx
) 1

qr1 ≤
( cr20
2r0 − 1

) 1
2r0 max

{
1,
(∫

Ω

|uj |r0qdx
) 1

qr0
}
. (3.9)

Therefore, by (3.9), we obtain(∫
Ω

|uj |rk+1q dx
) 1

qrk+1 ≤
( cr2k
2rk − 1

) 1
2rk max

{
1,
(∫

Ω

|uj |rkq dx
) 1

qrk
}

≤
k∏

i=0

( cri
2ri − 1

) 1
2ri max

{
1,
(∫

Ω

|uj |r0q dx
) 1

qr0
}
,

where ri = dir0, we denote C0 =
∏k

i=0

(
cri

2ri−1

) 1
2ri , then

∥uj∥Lr0qdk+1
(Ω)

≤ C0(1 + ∥uj∥L2∗ (Ω)). (3.10)

Let k → ∞ in (3.10), by (3.1), we have

∥uj∥L∞(Ω) ≤ C0(1 + ∥uj∥L2∗ (Ω)) ≤ c = c(L) . □



12 X. ZHOU, X. LIU EJDE-2024/31

Proof of Theorem 1.2. By Lemmas 2.2, 2.5-2.8, for a sufficiently small parameter
ε, the functional IM satisfies the conditions (A3), (A4)–(A7) of the abstract critical
point theorem (Theorem 2.1). Then, cj(M, ε) is a critical value of the functional IM ,
and each component of the corresponding critical point Uj(M, ε) is sign-changing.
That is, Uj(M, ε) is a sign-changing solution of the truncated system (1.3). More-
over, given l ∈ N+, L∗ > 0, by Lemma 2.8, there exists ε∗M > 0 such that for

|ε| ≤ ε∗M = min{ε(1)M , . . . , ε
(l)
M },

cj(M, ε) ≤ L∗ = max{d1, . . . , dl}, j = 1, . . . , l.

By Lemma 3.1, there exist the constant K∗ independent of M , ε, and εM > 0, such
that for |ε| ≤ εM ,

∥Uj(M, ε)∥L∞(Ω) ≤ K∗, j = 1, . . . , l.

Now take M ≥ K∗ + 1, then for |ε| ≤ εl, Uj(ε) := Uj(M, ε), j = 1, . . . , l are
sign-changing solutions of the perturbed system (1.2). □

Note that taking ε = 0, we have F (U, 0) = 0 and ∂F
∂uj

(U, 0) = 0, then the

solutions to the perturbed system (1.2) are also solutions to the original system
(1.1).

In Section 2, we have obtained the sign-changing critical points of the truncated
functional IM . Therefore, by Theorem 1.2, we know that system (1.2) has l pairs
of sign-changing solutions. Then, for ε = 0, the system (1.1) has infinitely many
sign-changing solutions, and we have thus proved the main result.

Acknowledgments. This work was supported by the NSFC 12161093, and by the
Yunnan key Laboratory of Modern Analytical Mathematics and Applications.

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Xue Zhou
Department of Mathematics, Yunnan Normal University, Kunming 650221, China

Email address: niuzhoux@163.com

Xiangqing Liu (corresponding author)

Department of Mathematics, Yunnan Normal University, Kunming 650221, China

Email address: lxq8u8@163.com


	1. Introduction
	2. Critical points of the truncated functional IM
	3. Proof of main results
	Acknowledgments

	References