Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 24, pp. 1–21.

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

DOI: 10.58997/ejde.2025.24

EXISTENCE OF POSITIVE SOLUTIONS FOR SYSTEMS OF

QUASILINEAR SCHRÖDINGER EQUATIONS

AYESHA BAIG, ZHOUXIN LI

Abstract. In this article, we study the existence of standing wave solutions

for the quasilinear Schrödinger system

−ε2∆u+W (x)u− κε2∆(u2)u = Qu(u, v) in RN ,

−ε2∆v + V (x)v − κε2∆(v2)v = Qv(u, v) in RN ,

u, v > 0 in RN , u, v ∈ H1(RN ).

where N ≥ 3, κ > 0, ε > 0, W,V : RN → R are continuous functions that fall

into two classes of potentials. To overcome the lack of differentiability, we use
the dual approach developed by Colin–Jeanjean. The existence of solutions is

obtained using Del Pino–Felmer’s penalization technique with an adaptation

of Alves’ arguments [1].

1. Introduction

In this article, we examine the existence of solutions to the system of quasilinear
Schrödinger equations (QLSE):

−ε2∆u+W (x)u− κε2∆(u2)u = Qu(u, v) in RN ,

−ε2∆v + V (x)v − κε2∆(v2)v = Qv(u, v) in RN ,

u, v > 0 in RN , u, v ∈ H1(RN ).

(1.1)

where N ≥ 3, κ > 0, ε > 0, W,V : RN → R are continuous functions that fall
into two classes of potentials introduced in [1]. The functions Qu, Qv : R+

2 → R
are continuous functions denoting partial derivatives of the function Q : R+2 → R,
which belongs to the class of C1 and is p-homogeneous.

Systems of type (1.1) are related to various applications in hydrodynamics, Hei-
delberg ferromagnetism, Magnus theory, condensed matter theory, and dissipative
quantum mechanics.

2020 Mathematics Subject Classification. 35Q55, 58E05, 58E30.
Key words and phrases. Quasilinear Schrödinger system; Cerami sequence; dual approach;

positive solution.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted April 17, 2024. Published March 3, 2025.

1



2 A. BAIG, Z. LI EJDE-2025/24

By a simple change of variables, system (1.1) is equivalent to the system

−∆u+W (εx)u− κ∆(u2)u = Qu(u, v) in RN ,

−∆v + V (εx)v − κ∆(v2)v = Qv(u, v) in RN ,

u, v > 0 in RN

(1.2)

Considering the potential values of κ, assumptions regarding potentials, and
various nonlinearity types, numerous studies have explored the existence of solutions
for system (1.1), particularly when κ ̸= 0, as seen in [5, 6, 8, 10, 11, 19].

Recently, numerous articles have examined the scalar equation:

−ε2∆u+ V (x)u− κε2∆(u2)u = g(u), in RN , (1.3)

where N ≥ 3, κ ∈ R, ε > 0 are real parameters, and V : RN → R satisfies certain
geometries. The function g : R → R is continuous. This kind of equation frequently
appears in various models, notably in connection with standing wave phenomena
within the quasilinear Schrödinger equation.

−iε
∂z

∂t
= −ε2∆z+F (x)z−κε2∆ρ(|z|2)ρ′(|z|2)z−f(|z|2)z, for all x ∈ RN . (1.4)

where z : R × RN → C, F represents the potential, κ is a real constant, and f, ρ
are real-valued functions. When ρ(s) = s, this equation arises in fields such as fluid
mechanics, plasma physics, dissipative mechanics, and condensed matter theory.
The stationary solutions of (1.4) take the form

z(t, x) = exp
(
− iEt

ε

)
u(x), E ∈ R, (1.5)

where u represents the solution to equation (1.3) with V (x) = F (x) − E and
g(u) = f(u2)u. For the physical motivation of equation (1.4), readers are referred
to [13, 12, 14] and references therein.

The semilinear scenario, identified by κ = 0, has undergone thorough examina-
tion in recent years. For instance, del Pino and Felmer [17] studied the problem:

−ε2∆u+ V (x)u = q(u), in RN , u > 0 in RN , u ∈ H1(RN ). (1.6)

where ε > 0, N ≥ 3, and q : R → R is a subcritical nonlinearity. The function V is
a locally Hölder continuous potential satisfying

0 < α = inf
x∈RN

V (x) ≤ V0 = inf
Ω

V (x) < min
∂Ω

V (x). (1.7)

In [17], the authors introduced the penalization method and proved that if V
satisfies (1.7), then (1.6) has a solution uε that concentrates at a minimum of V .

Alves, do Ó, and Souto [3] also studied (1.6) and proved the same result as in [17]
for V satisfying (1.7), with the subcritical nonlinearity perturbed by a critical term.

Alves [1] studied problem (1.6) with the nonlinearity q : R → R being continuous
and having subcritical or critical growth. Alves [1] introduced for the first time two
interesting classes of potential V , namely:
Class 1: The potential V satisfies the Palais-Smale (PS) condition, with the fol-
lowing conditions:

(A1) There exists V0 > 0 such that V (x) ≥ V0, for all x ∈ RN , where V0 =
infRN V (x).

(A2) V ∈ C2(RN ) and V, ∂V
∂xi

, ∂2V
∂xi∂xj

are bounded across RN , for all i, j ∈
{1, 2, 3, . . . , N}.



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 3

(A3) V adheres to the Palais-Smale (PS) condition, that is, if (xn) ⊂ RN , with
(V (xn)) being bounded and ∇V (xn) → 0, then (xn) possesses a convergent
subsequence.

Class 2: The potential V lacks critical points along the boundary of some bounded
domain. Within this class of potentials, V satisfies (A1), (A2) and

(A4) there exists a domain Λ ⊂ RN where ∇V (x) ̸= 0 for all x ∈ ∂Λ.

Given that V falls into either Class 1 or Class 2 and taking into account certain
conditions met by the nonlinearity, the author demonstrated the existence of a
positive solution for ε > 0 sufficiently small.

Alves [2] explored the presence and concentration of solutions for the system
derived from (1.1) with κ = 0:

−ε2∆u+W (x)u = Qu(u, v) in RN ,

−ε2∆v + V (x)v = Qv(u, v) in RN ,

u, v > 0 in RN , u, v ∈ H1(RN ).

(1.8)

where the functions W,V : RN → R are Hölder continuous satisfying W (x), V (x) ≥
α > 0 in RN and the condition:

(5) There exists an open and bounded set Λ ⊂ RN , with x0 ∈ Λ and ρ > 0,
such that W (x), V (x) ≥ ρ, for all x ∈ ∂Λ and W (x0), V (x0) < ρ.

Severo and Silva [19] employed the variational approach within an appropriate
Orlicz space to examine a system of type (1.1) with κ = 1. Recently, Arruda-
Figueiredo and Nascimento [4] considered the two classes of potentials introduced
by Alves in [1] and showed the existence of solutions for the system (1.8).

Motivated by these works, and mainly by [1, 2, 4, 17, 19], we study system (1.1)
for κ = 1. We shall refer to the potential V as belonging to Class 1 when it satisfies
(A6)–(A8), and as belonging to Class 2 when it satisfies (A6), (A7), (A9) where

(A6) There exist V∞,V0 > 0 such that V0 ≤ V(x) ≤ V∞, for all x ∈ RN , where
V0 = infRN V(x).

(A7) V ∈ C2(RN ) and V, ∂V
∂xi

, ∂2V
∂xi∂xj

are bounded in RN , for i, j ∈ {1, 2, . . . , N}.
(A8) V satisfies the PS-condition, that is, if (xn) ⊂ RN , such that V(xn) is

bounded and ∇V(xn) → 0, then (xn) possesses a convergent subsequence.
(A9) There exists a domain Λ ⊂ RN exists where ∇V(x) ̸= 0 for every x ∈ ∂Λ.

Notation Let H1(RN ) be the Sobolev space with norm

∥u∥H1(RN ) :=
(
∥u∥2L2(RN ) + ∥∇u∥2L2(RN )

)1/2

.

Let 2∗ be the Sobolev critical exponent

2∗ =
2N

N − 2
for N > 2.

Now, we present the assumptions on the function Q. Let R2
+ := [0,+∞) ×

[0,+∞), we assume that the nonlinearity Q ∈ C1(R2
+,R) is p-homogeneous with

subcritical growth. More precisely, our hypotheses on Q are:

(A10) There exists p ∈ (4, 2.2∗), such that Q(tu, tv) = tpQ(u, v) for all t > 0,
(u, v) ∈ R2

+, where 2∗ = 2N
N−2 and N ≥ 3.



4 A. BAIG, Z. LI EJDE-2025/24

(A11) There exists C > 0 such that

|Qu(u, v)|+ |Qv(u, v)| ≤ C(|u|p−1 + |v|p−1)

for all (u, v) ∈ R2
+.

(A12) Qu(0, 1) = 0, Qv(1, 0) = 0.
(A13) Qu(1, 0) = 0, Qv(0, 1) = 0.
(A14) Q(u, v) > 0 for each u, v > 0.
(A15) Qu(u, v), Qv(u, v) ≥ 0 for each (u, v) ∈ R2

+.

Since Q is a homogeneous function of degree p > 4, it follows that

pQ(u, v) = uQu(u, v) + vQv(u, v).

Moreover, ∇Q is a homogeneous function of degree p− 1.
A prototype of function Q that satisfies (A11)–(A15) is

H(u, v) := a|u|p +
∑

αi+βi=p

bi|u|αi |v|βi + c|v|p,

where a, bi, c ∈ R, αi + βi = p, αi, βi ≥ 1, i ∈ I, with I denoting a finite subset of
N.

Definition 1.1. We say that the pair (u, v) ∈ H1(RN ) ∩L∞
loc(RN ) is a solution to

(1.1) if u, v > 0 almost everywhere in RN and satisfies

ε2
∫
RN

(1 + 2u2)∇u∇φ+ 2

∫
RN

|∇u|2uφ+

∫
RN

W (εx)uφ =

∫
RN

Qu(u, v)φ,

for all φ ∈ C∞
0 (RN ), and

ε2
∫
RN

(1 + 2v2)∇v∇ϕ+ 2

∫
RN

|∇v|2vϕ+

∫
RN

V (εx)vϕ =

∫
RN

Qv(u, v)ϕ,

for all ϕ ∈ C∞
0 (RN ).

Theorem 1.2. Assume that W and V satisfy (A6) and that either W or V falls into
Class 1 or class 2. Furthermore, suppose that Q satisfies (A10)–(A15). Then system

(1.1) has a solution for each ε ∈ (0, ε0). Moreover, uε, vε ∈ C1,α
loc (RN ) ∩ L∞(RN ),

and there exist constants C1, C2, C3, C4 > 0 satisfying

uε(x) ≤ C1 exp(−C2|x/ε|), vε(x) ≤ C3 exp(−C4|x/ε|), ∀x ∈ RN .

Remark 1.3. Theorem 1.2 extends the findings of Arruda-Figueiredo and Nasci-
mento [4, Theorem 1.1] in at least two ways:

(1) The first is that we consider (κ ̸= 0), which leads to entirely different esti-
mates (regarding the functional and the solution to the auxiliary problem)
when compared to the case of (κ = 0).

(2) The second difference is that, unlike [4, Theorem 1.1], we do not require
both potentials V and W to belong to the same Class 1 or Class2. We only
require that one of the potentials belongs to one of these classes and that
the other satisfies condition (A6).

We recall that J ∈ C1(E,R) satisfies the Cerami condition on level b, denoted
by the (Ce)b condition, if any sequence (un) ⊂ E for which

(i) J(un) → b,
(ii) ∥J ′(un)∥E′(∥un∥+ 1) → 0 as n → ∞,



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 5

possesses a convergent subsequence.
J satisfies the Cerami condition, denoted by (Ce), if it satisfies (Ce)b for every

b ∈ R. We say that (un) ⊂ E is a (Ce)b sequence if it satisfies (i) and (ii). We also
say that (un) ⊂ E is a (Ce) sequence if it is a (Ce)b sequence for some b ∈ R.

To demonstrate our primary finding, we will use the following variant of the
Mountain Pass Theorem.

Theorem 1.4 ([20]). Let E be a real Banach space and J : E → R be a functional
of class C1. Let S be a closed subset of E, which disconnects (arcwise) E into
distinct connected components E1 and E2. Suppose further that J(0) = 0 and

(1) 0 ∈ E1 and there exists α > 0 such that J(v) ≥ α for all v ∈ S.
(2) There exists e ∈ E2 such that J(e) < 0.

Then, J possesses a (Ce)c sequence with c ≥ α given by

c := inf
γ∈Γ

max
t∈[0,1]

J(γ(t)) ≥ α,

where

Γ := {γ ∈ C([0, 1], E) : γ(0) = 0, γ(1) ∈ J−1((−∞, 0]) ∩ E2}.

This article is structured as follows: In Section 2, we reframe the system and
introduce an auxiliary system. Initially, we propose an equivalent system through
an appropriate variable transformation as discussed in [7, 15]. To address the
issue of compactness, we then define the auxiliary system (2.10), following the
methodology presented in [8]. Section 3 focuses on the analysis of the positive
solution of the auxiliary system (2.10), employing a variant of the Mountain Pass
Theorem (Theorem 1.4) that does not require the (PS) condition. This is used to
generate a Cerami sequence at the mountain-pass level. Subsequently, we adapt
Del Pino’s strategies to identify a solution for the auxiliary problem (2.10) and
examine certain solution properties of the auxiliary system. Finally, Section 4 is
dedicated to the proof of Theorem 1.2.

2. Reformulation of the system and the auxiliary system

Assuming (A6), we consider the closed subspace of H1(RN )×H1(RN ),

X =
{
(w, z) ∈ H1(RN )×H1(RN ) :

∫
RN

[W (εx)w2 + V (εx)z2] < ∞
}

which is a Hilbert space when endowed with the norm

∥(w, z)∥2 =

∫
RN

[|∇w|2 + |∇z|2 +W (εx)w2 + V (εx)z2].

The natural functional associated with (1.2) is

Jε(u, v) =
1

2

∫
RN

[(1 + 2u2)|∇u|2 + (1 + 2v2)|∇v|2 +W (εx)u2 + V (εx)v2]

−
∫
RN

Q(εx, u, v),

which is not well-defined in X. To address this challenge, we adopt the variable
transformation proposed by Colin and Jeanjean [7], and by Liu, Wang, and Wang
[15].



6 A. BAIG, Z. LI EJDE-2025/24

For this we consider w = f−1(u) and z = f−1(v), where f : R → R is defined by

f ′(t) =

{
1√

1+2f2(t)
, in [0,+∞),

−f ′(−t), in (−∞, 0].
(2.1)

Lemma 2.1. The function f satisfies the following properties:

(1) f is uniquely defined, C∞, and invertible.
(2) |f ′(t)| ≤ 1 and |f(t)| ≤ |t|, for t ∈ R.
(3) f(t)

t → 1 as t → 0.

(4) f(t)√
t
→ 4

√
2 as t → ∞.

(5) |f(t)|
2 ≤ |t|f ′(t) ≤ |f(t)|.

(6) |f(t)| ≤ 21/4|t|1/2, for all t ∈ R.
(7) f2(t)

2 ≤ tf(t)f ′(t) ≤ f2(t), for all t ∈ R.
(8) There exist constants C1, C2 > 0, such that:

|f(t)| ≥ C1|t|, if |t| ≤ 1;

|f(t)| ≥ C2|t|1/2, if |t| ≥ 1.

(9) |f(t)f ′(t)| ≤ 1√
2
, for all t ∈ R.

(10) The function t → fq(s)f ′(s) is increasing on (0,∞) for each q > 1.

With the exception of property (10), all other properties are derived from [9,
Lemma 2.1] (see also [7, 16, 15]). For property (10), refer to Reference [8, Remark
3.1].

Following the variable transformation

Iε(w, z) := Jε(f(w), f(z)),

we obtain the functional

Iε(w, z) =
1

2

∫
RN

[|∇w|2 + |∇z|2 +W (εx)f(w)2 + V (εx)f(z)2]

−
∫
RN

Q(εx, f(w), f(z)),

which is well-defined in X. More precisely, Iε is of class C1(X,R) (because of (A6),
(A11), and the properties of f). The Gateaux derivative is

I ′ε(w, z)(ϕ, φ)

=

∫
RN

[∇w∇ϕ+∇z∇φ] +

∫
RN

[W (εx)f(w)f ′(w)ϕ+ V (εx)f(z)f ′(z)φ]

−
∫
RN

[Qw(εx, f(w), f(z))f
′(w)ϕ+Qz(εx, f(w), f(z))f

′(z)φ] .

(2.2)

for all (w, z), (ϕ, φ) ∈ X.
Let (w, z) ∈ X be a critical point of the functional Iε. Then (w, z) constitutes a

weak solution to the reformulated system below:

−∆w +W (εx)f(w)f ′(w) = Qw(εx, f(w), f(z))f
′(w), in RN ,

−∆z + V (εx)f(z)f ′(z) = Qz(εx, f(w), f(z))f
′(z), in RN ,

w, z > 0, w, z ∈ H1(RN ).

(2.3)



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 7

Proposition 2.2. If (w, z) ∈ X∩ [L∞
loc(RN )]2 is a critical point of Iε, then (u, v) =

(f(w), f(z)) is a solution for (1.2).

For a poof of the above propostion, see [19, Proposition 2.5].
To apply the variational method and find a solution to (1.2), we will use the

Penalization Method developed by del Pino and Felmer [17], following the ideas of
Alves [2]. Given our interest in securing a positive solution for (1.2), we assume
that

Q(u, v) = 0 if u ≤ 0 or v ≤ 0. (2.4)

Let us fix a > 0 and let η : R → R be a non-increasing C1 function satisfying

η ≡ 1 in (−∞, a], η ≡ 0 in [5a,+∞), η′ ≤ 0, and |η′| ≤ C

a
, (2.5)

where the constant C is independent of a.

Using the function η, we define Q̂ : R2 → R by

Q̂(s, t) = η(|(s, t)|)Q(s, t) + [1− η(|(s, t)|)]A(s2 + t2), (2.6)

where

A := max
{Q(s, t)

s2 + t2
: (s, t) ∈ R2, a ≤ |(s, t)| ≤ 5a

}
. (2.7)

Note that A > 0 and A → 0 as a → 0+. Thus, we can assume that:

A <
1

4
min{W0, V0}, W (x) ≥ W0 > 0, V (x) ≥ V0 > 0, (2.8)

where W0, V0 are obtained from (A6).
Now, fixing a bounded domain Ω ⊂ RN , we define the function H : RN×R2 → R

by:

H(x, s, t) = χΩ(x)Q(s, t) + [1− χΩ(x)]Q̂(s, t), (2.9)

where χΩ denotes the characteristic function of Ω.

Lemma 2.3. The function H and its derivatives Hs and Ht satisfy the following
properties:

(A16) pH(x, s, t) = sHs(x, s, t) + tHt(x, s, t) for each x ∈ Ω.
(A17) 2H(x, s, t) ≤ sHs(x, s, t) + tHt(x, s, t) for each x ∈ RN \ Ω.
(A18) Fixing k = 4p

p−2 , we can choose a > 0 sufficiently small such that

sHs(x, s, t) + tHt(x, s, t) ≤
1

k
[W (x)s2 + V (x)t2], in RN \ Ω,

|Hs(x, s, t)|
a

,
|Ht(x, s, t)|

a
≤ 1

4
min{W0, V0}, in RN \ Ω.

For a proof of the above lemma see [2, Lemma 2.2]. Now, our objective is to
study the existence of solutions for the auxiliary system

−∆w +W (εx)f(w)f ′(w) = Hw(εx, f(w), f(z))f
′(w), in RN ,

−∆z + V (εx)f(z)f ′(z) = Hz(εx, f(w), f(z))f
′(z), in RN ,

w, z > 0, w, z ∈ H1(RN ).

(2.10)

Remark 2.4. If (w, z) is a solution of (2.10) satisfying

|(f(w(x)), f(z(x)))| ≤ a, ∀x ∈ RN \ Ωε,

then (w, z) will be a solution of (2.3), where Ωε := {x ∈ RN : εx ∈ Ω}. Thus, our
goal is to obtain solutions of (2.10) with this property.



8 A. BAIG, Z. LI EJDE-2025/24

Associated with the system (2.10), we define on X the functional

Φε(w, z) =
1

2

∫
RN

[|∇w|2 + |∇z|2 +W (εx)f2(w) + V (εx)f2(z)]

−
∫
RN

H(εx, f(w), f(z)).

(2.11)

Under conditions (A7) and (A11), it is possible to show that the functional Φε

is of class C1 with Gateaux derivative

Φ′
ε(w, z)(ϕ, φ)

=

∫
RN

[∇w∇ϕ+∇z∇φ] +

∫
RN

[W (εx)f(w)f ′(w)ϕ+ V (εx)f(z)f ′(z)φ]

−
∫
RN

[Hw(εx, f(w), f(z))f
′(w)ϕ+Hz(εx, f(w), f(z))f

′(z)φ],

(2.12)

for any (w, z), (ϕ, φ) ∈ X. Therefore, the critical points of Φε are precisely the
weak solutions of (2.10).

For each ρ > 0, consider the set

Σρ =
{
(w, z) ∈ X : Ψ(w, z) = ρ2

}
, (2.13)

where

Ψ(w, z) =

∫
RN

[|∇w|2 + |∇z|2 +W (εx)f2(w) + V (εx)f2(z)]. (2.14)

Since Ψ is continuous, Σρ is a closed subset in X that disconnects X into

X1 := {(w, z) ∈ X : Ψ(w, z) > ρ2}, X2 := {(w, z) ∈ X : Ψ(w, z) < ρ2}.

The following lemma guarantees that the functional Φε meets the geometric condi-
tions required by the Mountain Pass Theorem.

Lemma 2.5. The functional Φε satisfies the following conditions:

(i) There exist constants ρ, α > 0 such that Φε(w, z) ≥ α, for all (w, z) ∈ Σρ.
(ii) For every ε ∈ (0, 1], there exist (e1, e2) ∈ X2 such that Φε(e1, e2) ≤ 0.

Proof. Using (A16)–(A18), we obtain∫
RN

H(εx, f(w), f(z)) ≤
∫
Ωε

H(εx, f(w), f(z))

+
1

2k

∫
RN/sΩε

[W (εx)f(w)2 + V (εx)f(z)2].

By (2.9) and (A12), we have∫
RN

H(εx, f(w), f(z)) ≤ C

∫
RN

(|f2(w)|p/2 + |f2(z)|p/2)

+
1

2k

∫
RN\Ωε

[W (εx)f(w)2 + V (εx)f(z)2].
(2.15)



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 9

Applying Hölder’s inequality and the embedding of D1,2(RN ) into L2∗(RN ), we
obtain ∫

RN

H(εx, f(w), f(z)) dx

≤ C
(∫

RN

|f2(w)|2 dx
)σp/2(∫

RN

|f2(w)|2
∗
dx

)1−σp
2

+ C
(∫

RN

|f2(z)|2 dx
)σp/2(∫

RN

|f2(z)|2
∗
dx

)1−σp
2

+
1

2k

∫
RN\Ωε

[
W (εx)f(w)2 + V (εx)f(z)2

]
dx

≤ Cρσp
(∫

RN

|∇(f2(w))|2 dx
) (1−σp

2
)2∗

2

+ Cρσp
(∫

RN

|∇(f2(z))|2 dx
) (1−σp

2
)2∗

2

+
1

2k

∫
RN\Ωε

[
W (εx)f(w)2 + V (εx)f(z)2

]
dx

(2.16)

where σ := 2.2∗−p
(2∗−1)p and C > 0.

Noting that for each (w, z) ∈ Σρ,∫
RN

|∇(f2(w))|2 ≤ 2

∫
RN

|∇w|2 ≤ 2ρ2,∫
RN

|∇(f2(z))|2 ≤ 2

∫
RN

|∇z|2 ≤ 2ρ2.

(2.17)

From (2.11), (2.17), and (2.16), we obtain

Φε(w, z) ≥
k − 1

2k

∫
RN

[
|∇w|2 + |∇z|2 +W (εx)f(w)2 + V (εx)f(z)2

]
dx

− C

V
σp/2
0

ρσpCρ(1−
σp
2 ) 2∗

2 − C

W
σp/2
0

ρσpCρ(1−
σp
2 ) 2∗

2 .

Thus,

Φε(w, z) ≥
k − 1

2k
ρ2Cρ(2N+2p)/(N+2), ∀(w, z) ∈ Σρ.

Since 2N+2p
N+2 > 2 because p > 4, we can choose α > 0 sufficiently small, such

that

Φε(w, z) ≥ α > 0, ∀(w, z) ∈ Σρ,

which proves (i). Now, note that, by condition (H1), there exist constants C3, C4 >
0 such that

H(εx, s, t) ≥ C3|(s, t)|p − C4, ∀(x, s, t) ∈ Ω× R2. (2.18)

Assume without loss of generality that 0 ∈ Ω. Let ϕ ∈ C∞
0 (RN , [0, 1]) be such

that suppϕ ⊂ Br(0) ⊂ Ω, for some r > 0. Since Br(0) ⊂ Ωε for all ε ∈ (0, 1],
f(tϕ) ≥ 0 for all t ≥ 0, and W (εx) ≤ W∞, V (εx) ≤ V∞ in RN , by (2.18), for all
t ≥ 0, we have

Φε((tϕ, tϕ)) ≤
t2

2

∫
Br(0)

[
2|∇ϕ|2 + 1

2

∫
Br(0)

[W∞ + V∞]f2(tϕ)



10 A. BAIG, Z. LI EJDE-2025/24

− C3

∫
Br(0)

|(f(tϕ), f(tϕ))|p + C4|Br(0)|
]
.

Thus,

Φε((tϕ, tϕ)) ≤ t2
[1
2

∫
Br(0)

(
2|∇ϕ|2 + [W∞ + V∞]ϕ2

)
− C3

∫
Br(0)

|f(tϕ)|p
]

+ C4|Br(0)|.

From property (6) of Lemma 2.1, it follows that the function f(t)
t is decreasing

for t > 0. Since 0 ≤ tφ ≤ t, for all x ∈ Ωε and t > 0, we obtain f(t)φ(x) ≤ f(tφ(x)).
Hence, for all ε ∈ (0, 1] and t ≥ 0, we have

Φε((tϕ, tϕ))

≤ t2
[1
2

∫
Br(0)

(
2|∇ϕ|2 + [W∞ + V∞]ϕ2

)
− C3

fp(t)

t2

∫
Br(0)

|ϕ|p
]
+ C4|Br(0)|.

(2.19)
From property (5) of Lemma 2.1 and since p > 4, we conclude that

lim
t→+∞

fp(t)

t2
= lim

t→+∞

(f(t)√
t

)p

t
p
2−2 = +∞. (2.20)

Hence, by (2.19) and (2.20), the proof of (ii) is complete. □

3. Existence of positive solutions for the auxiliary system

By Theorem 1.4 and Lemma 2.5, there exists a Cerami sequence for Φε at the
level

cε = inf
γ∈Γε

max
t∈[0,1]

Φε(γ(t)), (3.1)

where

Γε =
{
γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) ∈ Φ−1

ε ((−∞, 0]) ∩X2

}
,

X2 =
{
(w, z) ∈ X : Ψ(w, z) < ρ2

}
.

That is, there exists (wn, zn) ⊂ X, such that

Φε(wn, zn) = cε + on(1), and (1 + ∥(wn, zn)∥)∥Φ′
ε(wn, zn)∥∗ = on(1). (3.2)

Lemma 3.1. Every Cerami sequence (wn, zn) for Φε is bounded in X.

Proof. Indeed, let

(ϕn, φn) :=
( f(wn)

f ′(wn)
,
f(zn)

f ′(zn)

)
.

As

∇ϕn =
(
1 +

2f2(wn)

1 + 2f2(wn)

)
∇wn, ∇φn =

(
1 +

2f2(zn)

1 + 2f2(zn)

)
∇zn,

we have

∥ϕn∥ ≤ 2∥wn∥, ∥φn∥ ≤ 2∥zn∥.
Hence, by (3.2), Φ′

ε(wn, zn)(ϕn, φn) = on(1); thus,

cε + on(1) = Φε(wn, zn)−
1

p
Φ′

ε(wn, zn)(ϕn, φn)

=

∫
RN

[1
2
− 1

p

(
1 +

2f2(wn)

1 + 2f2(wn)

)]
|∇wn|2



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 11

+

∫
RN

[1
2
− 1

p

(
1 +

2f2(zn)

1 + 2f2(zn)

)]
|∇zn|2

+
(1
2
− 1

p

) ∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
+

1

p

∫
RN

[f(wn)Hw(εx, f(wn), f(zn)) + f(zn)Hz(εx, f(wn), f(zn))]

− 1

p

∫
RN

pH(εx, f(wn), f(zn)).

Since

1 +
2f2(t)

1 + 2f2(t)
≤ 2,

by (A16), we have

cε + on(1) ≥
(1
2
− 2

p

) ∫
RN

(|∇wn|2 + |∇zn|2)

+
(1
2
− 1

p

) ∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
+

1

p

∫
RN\Ωε

[f(wn)Hw(εx, f(wn), f(zn)) + f(zn)Hz(εx, f(wn), f(zn))]

− 1

p

∫
RN

pH(εx, f(wn), f(zn)).

By (A17) and (A18), it follows that:

cε + on(1) ≥
p− 4

2p

∫
RN

(
|∇wn|2 + |∇zn|2

)
+
(1
2
− 1

p

) ∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
+

∫
RN\Ωε

H(εx, f(wn), f(zn))

≥ p− 4

2p

∫
RN

(
|∇wn|2 + |∇zn|2

)
+
(1
2
− 1

p
− 1

2k

)∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
.

Using k = 4p
p−2 , we obtain

cε + on(1) ≥ cp

∫
RN

[
(|∇wn|2 + |∇zn|2) +W (εx)f2(wn) + V (εx)f2(zn)

]
, (3.3)

where

cp := min
{p− 4

2p
,
p− 2

4p

}
> 0, since 4 < p < 2.2∗.

Thus, to show that (wn, zn) is bounded, it suffices to demonstrate the existence of
a constant C > 0 such that∫

RN

(W (εx)w2
n + V (εx)z2n) ≤ C, ∀n ∈ N.



12 A. BAIG, Z. LI EJDE-2025/24

Using Lemma 2.1 and (3.3), we obtain∫
|wn|≤1

W (εx)w2
n ≤ 1

C

∫
|wn|≤1

W (εx)f(wn)
2 ≤ c+ on(1).

Then ∫
{|wn|>1}

W (εx)w2
n ≤ W∞

∫
RN

w2∗

n

≤ W∞S
(∫

RN

|∇wn|2
)2∗/2

≤ W∞S(cε + on(1))
2∗/2.

Similarly, we conclude that (V (εx)z2n) is bounded in L1(RN ). Therefore, (wn, zn)
is bounded in X. □

Lemma 3.2. Suppose that (wn, zn) ⇀ (w, z) in X. Then:

(i) Given ξ > 0, there exists R > 0 such that

lim sup
n→∞

∫
RN\BR(0)

[
|∇wn|2 + |∇zn|2 +W (εx)f2(wn) + V (εx)f2(zn)

]
< ξ.

(ii)

lim
n→∞

∫
RN

[W (εx)f(wn)f
′(wn)wn + V (εx)f(zn)f

′(zn)zn]

=

∫
RN

[W (εx)f(w)f ′(w)w + V (εx)f(z)f ′(z)z] .

(iii)

lim
n→∞

∫
RN

[Hw(εx, f(wn), f(zn))f
′(wn)wn +Hz(εx, f(wn), f(zn))f

′(zn)zn]

=

∫
RN

[Hw(εx, f(w), f(z))f
′(w)w +Hz(εx, f(w), f(z))f

′(z)z] .

(iv) If (wn(x), zn(x)) → (w(x), z(x)) a.e. in RN and

lim
n→∞

∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
=

∫
RN

[
W (εx)f2(w) + V (εx)f2(z)

]
,

then

lim
n→∞

∫
RN

[
W (εx)f2(wn − w) + V (εx)f2(zn − z)

]
= 0.

Proof. Consider the cutoff function ϕR ∈ C∞(RN ), such that ϕR = 0 in BR/2(0),

ϕR = 1 in RN \ BR(0), 0 ≤ ϕR ≤ 1, and |∇ϕR| ≤ C/R, where the constant C > 0
is independent of R.

Since (wn, zn) is bounded, we have

Φ′
ε(wn, zn)

(
ϕR

f(wn)

f ′(wn)
, ϕR

f(zn)

f ′(zn)

)
= on(1).

Thus,∫
RN

(
1 +

2f2(wn)

1 + 2f2(wn)

)
|∇wn|2ϕR +

∫
RN

(
1 +

2f2(zn)

1 + 2f2(zn)

)
|∇zn|2ϕR



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 13

+

∫
RN

( f(wn)

f ′(wn)
∇wn +

f(zn)

f ′(zn)
∇zn

)
∇ϕR

+

∫
RN

W (εx)f2(wn)ϕR +

∫
RN

V (εx)f2(zn)ϕR

=

∫
RN

[Hw(εx, f(wn), f(zn))f(wn)ϕR +Hz(εx, f(wn), f(zn))f(zn)ϕR] + on(1).

Choosing R > 0 such that Ωε ⊂ BR/2(0), by (A17) and (A18), we have∫
RN

(|∇wn|2 + |∇zn|2)ϕR + (1− 1

k
)

∫
RN

[
W (εx)f2(wn) + V (εx)f2(zn)

]
ϕR

≤ −
∫
RN

( f(wn)

f ′(wn)
∇wn +

f(zn)

f ′(zn)
∇zn

)
∇ϕR + on(1).

Using Lemma 2.1, the Cauchy-Schwarz inequality, the boundedness of (wn, zn),
and |∇ϕR| ≤ C/R, we obtain:

(1− 1

k
)

∫
RN\BR(0)

[
|∇wn|2 + |∇zn|2 +W (εx)f2(wn) + V (εx)f2(zn)

]
≤ 2

(
∥wn∥L2(RN )∥∇wn∇ϕR∥+ ∥zn∥L2(RN )∥∇zn∇ϕR∥

)
≤ Mc

R
+ on(1),

which completes the proof of (i).

(ii) The proof of (ii) follows from (i) and the property f(t)f ′(t) ≤ f2(t), we have

lim sup
n→∞

∫
RN\BR(0)

W (εx)f(wn)f
′(wn)wn + V (εx)f(zn)f

′(zn)zn = oR(1). (3.4)

Since wn → w and zn → z in Ls
loc(RN ), we have

wn(x) → w(x), zn(x) → z(x) a.e. in RN ,

|wn(x)| ≤ g1(x), |zn(x)| ≤ g2(x), g1, g2 ∈ Ls(BR(0)), s ∈ [1, 2∗).

By the Lebesgue Dominated Convergence Theorem, we have

lim
n→∞

∫
BR(0)

[W (εx)f(wn)f
′(wn)wn + V (εx)f(zn)f

′(zn)zn]

=

∫
BR(0)

[W (εx)f(w)f ′(w)w + V (εx)f(z)f ′(z)z] .

(3.5)

From (3.4) and (3.5), we obtain item (ii).

(iii) Now, using item (i) and that Φ′
ε(wn, zn)(wn, zn) = on(1), we have

lim sup
n→∞

∫
RN\BR(0)

[
Hw(εx, f(wn), f(zn))f

′(wn)wn

+Hz(εx, f(wn), f(zn))f
′(zn)zn

]
= oR(1).

(3.6)



14 A. BAIG, Z. LI EJDE-2025/24

Using that Ωε ⊂ BR(0), hypothesis (A18), the convergence wn → w, zn → z in
Ls
loc(RN ), and applying the Lebesgue Dominated Convergence Theorem, we obtain

lim
n→∞

∫
BR(0)

[Hw(εx, f(wn), f(zn))f
′(wn)wn +Hz(εx, f(wn), f(zn))f

′(zn)zn]

=

∫
BR(0)

[Hw(εx, f(w), f(z))f
′(w)w +Hz(εx, f(w), f(z))f

′(z)z] .

(3.7)
Thus, from (3.6) and (3.7), we obtain item (iii). □

Proposition 3.3. The functional Φε satisfies the Cerami condition for each level
cε.

Proof. Let (wn, zn) ⊂ X be such that

Φε(wn, zn) = cε + on(1) and (1 + ∥(wn, zn)∥)∥Φ′
ε(wn, zn)∥∗ = on(1).

By Lemma 3.1, (wn, zn) is bounded in X. Thus, up to a subsequence, (wn, zn) ⇀
(w, z) in X. Furthermore, since

Φ′
ε(wn, zn)(wn, zn) = on(1),

using (iii) from Lemma 3.2, we obtain

lim
n→∞

∫
RN

[
|∇wn|2 + |∇zn|2 +W (εx)f(wn)f

′(wn)wn + V (εx)f(zn)f
′(zn)zn

]
= lim

n→∞

∫
RN

[Hw(εx, f(w), f(z))f
′(w)w +Hz(εx, f(w), f(z))f

′(z)z] .

(3.8)
Now, using that Φ′

ε(wn, zn)(ϕ, φ) = on(1) for every ϕ, φ ∈ C∞
0 (RN ), by passing

to the limit combined with the Lebesgue Dominated Convergence Theorem, we
obtain Φ′

ε(w, z)(ϕ, φ) = 0. It follows that Φ′
ε(w, z)(ϕ, φ) = 0 for every ϕ, φ ∈ X. In

particular, Φ′
ε(w, z)(w, z) = 0, i.e.,∫

RN

[
|∇w|2 + |∇z|2 +W (εx)f(w)f ′(w)w + V (εx)f(z)f ′(z)z

]
=

∫
RN

[Hw(εx, f(w), f(z))f
′(w)w +Hz(εx, f(w), f(z))f

′(z)z] .

(3.9)

Combining (3.8) and (3.9), we have

lim
n→∞

∫
RN

[
|∇wn|2 + |∇zn|2 +W (εx)f(wn)f

′(wn)wn + V (εx)f(zn)f
′(zn)zn

]
= lim

n→∞

∫
RN

[
|∇w|2 + |∇z|2 +W (εx)f(w)f ′(w)w + V (εx)f(z)f ′(z)z

]
.

(3.10)
Using (ii) from Lemma 3.2 and (3.10), it follows that

lim
n→∞

∫
RN

[
|∇wn|2 + |∇zn|2

]
=

∫
RN

[
|∇w|2 + |∇z|2

]
.

Since (wn, zn) ⇀ (w, z) in X, we have

lim
n→∞

∫
RN

[∇wn · ∇w +∇zn · ∇z] =

∫
RN

[
|∇w|2 + |∇z|2

]
.



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 15

Thus,

lim
n→∞

∫
RN

[
|∇(wn − w)|2 + |∇(zn − z)|2

]
= 0. (3.11)

Observing that

∥wn − w∥ ≤ C
[
ΨW (wn − w) + ΨW (wn − w)2

∗/2
]
,

∥zn − z∥ ≤ C
[
ΨV (zn − z) + ΨV (zn − z)2

∗/2
]
,

where

ΨV(wn − w) :=

∫
RN

|∇(wn − w)|2 +
∫
RN

V(εx)f2(wn − w),

we conclude that

∥(wn, zn)− (w, z)∥2

≤ C
[
ΨW (wn − w) + ΨW (wn − w)2

∗/2 +ΨV (zn − z) + ΨV (zn − z)2
∗/2

]
≤ C

[
ΨW (wn − w) + ΨV (zn − z) + (ΨW (wn − w) + ΨV (zn − z))2

∗/2
]
.

Using (iv) from Lemma 3.2 and (3.11), up to a subsequence, we have (wn, zn) →
(w, z) in X. This completes the proof of the proposition. □

Theorem 3.4. Suppose (A6), (A10)-(A15) are satisfied. Then, for every ε ∈ (0, 1],
the auxiliary system (2.10) has a weak solution (wε, zε) ∈ X, such that

Φε(wε, zε) = cε and ∥(wε, zε)∥2 ≤ C(cε + c2
∗/2

ε ), (3.12)

where C > 0 is a constant independent of ε, and cε is defined by (3.1).

Proof. Using Lemma 2.5, Proposition 3.3, and Theorem 1.4, we conclude that the
functional Φε has a critical point at

cε := inf
γ∈Γ

max
t∈[0,1]

Φε(γ(t)) ≥ α,

where

Γ :=
{
γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) ∈ Φ−1((−∞, 0]) ∩X2

}
,

and α is given in Lemma 2.5. Thus, there exists (wε, zε) ∈ X, such that

Φε(wε, zε) = cε, and Φ′
ε(wε, zε) = 0.

Therefore, (wε, zε) is a solution of (2.10).
Consider (w̄ε, z̄ε) as a test function and note that H(εx, s, t) = 0 for all s, t ≤ 0

and

W (εx)f(wε)f
′(wε)w̄ε, V (εx)f(zε)f

′(zε)z̄ε ≥ 0.

Then we obtain

∥w̄ε, z̄ε∥2D1,2(RN ) =

∫
RN

[
|∇w̄ε|2 + |∇z̄ε|2

]
≤

∫
RN

∇wε∇w̄ε +

∫
RN

W (εx)f(wε)f
′(wε)w̄ε

+

∫
RN

∇zε∇z̄ε +

∫
RN

V (εx)f(zε)f
′(zε)z̄ε

=

∫
RN

H(εx, f(wε))f
′(wε)w̄ε +

∫
RN

H(εx, f(zε))f
′(zε)z̄ε ≤ 0.



16 A. BAIG, Z. LI EJDE-2025/24

Therefore, ∥w̄ε, z̄ε∥2D1,2(RN ) = 0. Hence, (w̄ε, z̄ε) = 0, and consequently, (wε, zε) =

(w+
ε , z

+
ε ) ≥ 0 a.e. in RN . By elliptic regularity, we have (wε, zε) ∈ C1,α(RN ) (see

the proof of Lemma ??). Thus, wε > 0 in RN .I did not find Lemma
3.4 Proof of (3.12). Let

(w̃ε, z̃ε) =
( f(wε)

f ′(wε)
,
f(zε)

f ′(zε)

)
.

Since

Φε(wε, zε)−
1

p
Φ′

ε(wε, zε)(w̃ε, z̃ε) = cε,

Using (A16), we obtain

cε ≥
∫
RN

[1
2
− 1

p

(
1 +

2f2(wε)

1 + 2f2(wε)

)]
|∇wε|2

+

∫
RN

[1
2
− 1

p

(
1 +

2f2(zε)

1 + 2f2(zε)

)]
|∇zε|2

+
(1
2
− 1

p

) ∫
RN

[
W (εx)f2(wε) + V (εx)f2(zε)

]
+

1

p

∫
RN\Ωε

[f(wε)Hw(εx, f(wε), f(zε)) + f(zε)Hz(εx, f(wε), f(zε))]

− 1

p

∫
RN

pH(εx, f(wε), f(zε)).

Using the inequality 1 + 2f2(wε)
1+2f2(wε)

≤ 2 and (A18), we obtain:

cε ≥
(1
2
− 2

p

) ∫
RN

[
|∇wε|2 + |∇zε|2

]
+
(1
2
− 1

p

) ∫
RN

[
W (εx)f2(wε) + V (εx)f2(zε)

]
+

1

p

∫
RN\Ωε

[f(wε)Hw(εx, f(wε), f(zε)) + f(zε)Hz(εx, f(wε), f(zε))]

− 1

p

∫
RN

pH(εx, f(wε), f(zε)).

which gives

cε ≥
(p− 4

2p

)[ ∫
RN

[
|∇wε|2 + |∇zε|2 +W (εx)f2(wε) + V (εx)f2(zε)

] ]
−
∫
RN\Ωε

H(εx, f(wε), f(zε)).

Since k = 4p
p−2 , by (A18), Lemma 2.1, and the above inequality, we have

cε ≥
1

2k

[ ∫
RN

|∇wε|2 + |∇zε|2 + C

∫
|wε|,|zε|≤1

W (εx)w2
ε + V (εx)z2ε

+

∫
|wε|,|zε|>1

W (εx)f2(wε) + V (εx)f2(zε)
]
,

(3.13)

for some constant C > 0, independent of ε.



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 17

Using (3.13) and the embedding H1(RN ) ⊂ D1,2(RN ), we obtain∫
|wε|>1

W (εx)|wε|2 ≤ W∞

∫
|wε|>1

|wε|2
∗
≤ W∞S

(∫
RN

|∇wε|2
)2∗/2

. (3.14)

Similarly, the conclusion holds for (V (εx)z2ε).
Combining (3.13) and (3.14), we obtain (3.12), wich completes the proof. □

Lemma 3.5. Let (wε, zε) be the solution of (2.10) obtained in Theorem 3.4, and
let the sequences εn ∈ (0, 1) and (xn) ⊂ RN be such that εn → 0 as n → ∞. The
sequences (θn) and (ϑn) defined by

θn(x) := wεn(x+ xn), ϑn(x) := zεn(x+ xn)

belong to L∞(RN ) ∩ C(RN ) and have subsequences that converge uniformly over
compact sets of RN to θ, ϑ ∈ L∞(RN )∩C(RN ), respectively. Moreover, there exist
constants C1, C2, C3, C4 > 0 such that

θ(x) ≤ C1 exp(−C2|x|), ϑ(x) ≤ C3 exp(−C4|x|), ∀x ∈ RN .

The proof of the above lemma follows from adapting the arguments used in the
proof of [18, Lemma (4.1)], combined with [4, Corollary 4.3].

Lemma 3.6. Suppose that W and V satisfy (A6), and that either W or V belongs
to Class 1 or class 2. Furthermore, suppose that Q satisfies (A10)–(A15). Then

mε := max
x∈∂Ωε

|(wε(x), zε(x))| → 0 as ε → 0+,

where we define Ωε := BRε/ε.

Proof. Assume, by contradiction, that the lemma is not true. Then, there exist
δ > 0 and a sequence εn → 0+, such that

mεn ≥ δ, ∀n ∈ N.
Since wεn , zεn ∈ C1,α(RN ), there exist xn ∈ ∂BRεn/εn such that

w2
εn(xn) + z2εn(xn) ≥ δ2, ∀n ∈ N. (3.15)

We define the functions θn, ϑn : RN → R by

θn(x) := wεn(x+ xn), ϑn(x) := zεn(x+ xn).

By Lemma 3.1, the sequence (wεn , zεn) is bounded in X. Thus, by the invariance of
RN by translation, (θn, ϑn) is also bounded in X. Moreover, (θn, ϑn) is a solution
of the following system in RN :

−∆θn +W (εnx+ εnxn)f(θn)f
′(θn) = Hw(εnx+ εnxn, f(θn), f(ϑn))f

′(θn),

−∆ϑn + V (εnx+ εnxn)f(ϑn)f
′(ϑn) = Hz(εnx+ εnxn, f(θn), f(ϑn))f

′(ϑn),

θn, ϑn > 0, θn, ϑn ∈ H1(RN ).

(3.16)
Note that, up to a subsequence, (θn, ϑn) ⇀ (θ, ϑ) in X, for some (θ, ϑ) ∈ X. By

Lemma 3.5, (θn) and (ϑn) converge uniformly over compact sets of RN to θ and
ϑ, respectively. Moreover, θ, ϑ ∈ C(RN ). Thus, from this fact and the condition
above, it follows that:

θ2(0) + ϑ2(0) ≥ δ2.

Hence,
θ ̸≡ 0 or ϑ ̸≡ 0. (3.17)



18 A. BAIG, Z. LI EJDE-2025/24

Since (W (εnxn)) and (V (εnxn)) are bounded, there exist αW , αV > 0 such that

W (εnxn) → αW , V (εnxn) → αV . (3.18)

It follows from (3.16) that∫
RN

∇θn∇ϕ+

∫
RN

W (εnx+ εnxn)f(θn)f
′(θn)ϕ

=

∫
RN

Hw(εnx+ εnxn, f(θn), f(ϑn))f
′(θn)ϕ+ on(1),

(3.19)

and similarly, ∫
RN

∇ϑn∇φ+

∫
RN

V (εnx+ εnxn)f(ϑn)f
′(ϑn)φ

=

∫
RN

Hz(εnx+ εnxn, f(θn), f(ϑn))f
′(ϑn)φ+ on(1).

(3.20)

Using (3.19) and (3.20), passing to the limit, and the density of C∞
0 (RN ) inH1(RN ),

we obtain∫
RN

[∇θ∇ϕ+ αW f(θ)f ′(θ)ϕ] =

∫
RN

g1(x, f(θ), f(ϑ))f
′(θ)ϕ, (3.21)∫

RN

[∇ϑ∇φ+ αV f(ϑ)f
′(ϑ)φ] =

∫
RN

g2(x, f(θ), f(ϑ))f
′(ϑ)φ, (3.22)

for all (ϕ, φ) ∈ X, where

g1(x, f(θ), f(ϑ)) := Ĩ(x)Qu(f(θ), f(ϑ)) + (1− Ĩ(x))Q̂w(f(θ), f(ϑ)),

g2(x, f(θ), f(ϑ)) := Ĩ(x)Qv(f(θ), f(ϑ)) + (1− Ĩ(x))Q̂z(f(θ), f(ϑ)),

for some function Ĩ ∈ L∞(RN ).

Noting that (θ, ϑ) ∈ W 2,2
loc (RN ) ∩ L∞(RN ) and

∇(ff ′)(w) = (f ′)2(w)∇w + f(w)f ′′(w)∇w, f ′′(w) = −2f(w)[f ′(w)]4,

for all w ∈ H1(RN ), by property (2) of Lemma 2.1, we have that (ff ′)(w) ∈
H1(RN ) for all w ∈ H1(RN ). Thus, there exist ϕj , φj ∈ C∞

0 (RN ) such that

∥ϕj − (ff ′)(θ)∥ ≤ 1

j
, ∥φj − (ff ′)(ϑ)∥ ≤ 1

j
, ∀j ∈ N. (3.23)

We assert that (3.17) and (3.21) imply that θ ̸≡ 0 and ϑ ̸≡ 0. Indeed, suppose,
by contradiction, that θ ̸≡ 0 and ϑ ≡ 0. Since f(0) = 0, by (2.4), (3.15), and (3.19),∫

RN

|∇θ|2 + αW

∫
RN

f(θ)f ′(θ)θ =

∫
RN

g1(x, f(θ), 0)f
′(θ)θ = 0.

Using that θ ≥ 0, we conclude that θ ≡ 0, and thus, (θ, ϑ) = (0, 0), which
contradicts (3.17). A similar conclusion is obtained if we consider θ ≡ 0 and ϑ ̸≡ 0.
Thus, the assertion is true.

Now, suppose W satisfies the (PS) condition, that is, W belongs to Class 1.

Choosing
∂ϕj

∂xi
as a test function in (3.19), we have∫
RN

∇θn∇
(
∂ϕj

∂xi

)
+

∫
RN

W (εnx+ εnxn)f(θn)f
′(θn)

∂ϕj

∂xi

−
∫
RN

Hu(εnx+ εnxn, f(θn), f(ϑn))f
′(θn)

∂ϕj

∂xi
= on(1).



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 19

Thus, exploring the fact that θ ̸≡ 0 and arguing, we can conclude that

∇W (εnxn) → 0, W (εnxn) → αW .

Thus, (εnxn) is a (PS)αW
sequence for W . Therefore, from (A8), (εnxn) should

have a convergent subsequence, but

|εnxn| = Rεn =
1

εn
→ ∞ as n → ∞.

Therefore, the lemma is true for W in Class 1. The case where V belongs to Class
1 is analogous. □

Lemma 3.7. Suppose that W and V satisfy (A6) and that either W or V belongs
to Class 2. Furthermore, suppose that Q satisfies (A10)-(A15). Then

mε := max
x∈∂Ωε

|(uε(x), vε(x))| → 0 as ε → 0+,

where we define Ωε :=
1
εΛ with Λ given in hypothesis (A9).

Proof. Assume, for the sake of contradiction, that the statement does not hold.
Then, there exists a constant δ > 0 and a sequence εn → 0+ such that

max
x∈∂Ωεn

|(uεn(x), vεn(x))| ≥ δ, ∀n ∈ N.

This implies that for each n, there exists a point xn ∈ ∂Ωεn satisfying

|(uεn(xn), vεn(xn))| ≥ δ.

We define the translated functions

θn(x) := uεn(x+ xn), ϑn(x) := vεn(x+ xn).

Since (uεn , vεn) is a solution of the auxiliary system (2.10), the pair (θn, ϑn) satisfies
the system

−∆θn +W (εnx+ εnxn)f(θn)f
′(θn) = Hw(εnx+ εnxn, f(θn), f(ϑn))f

′(θn),

−∆ϑn + V (εnx+ εnxn)f(ϑn)f
′(ϑn) = Hz(εnx+ εnxn, f(θn), f(ϑn))f

′(ϑn).

From the boundedness of (uεn , vεn) in H1(RN ), the sequences (θn, ϑn) are also
bounded in H1(RN ). Specifically, there exists a constant C > 0 such that

∥θn∥H1(RN ) ≤ C, ∥ϑn∥H1(RN ) ≤ C, ∀n ∈ N.

By the Sobolev embedding theorem, up to a subsequence, (θn, ϑn) converges
weakly in H1(RN ) and strongly in Lp

loc(RN ), for p ∈ [2, 2∗], to some (θ, ϑ) ∈
H1(RN ).

The strong convergence in Lp
loc(RN ) implies that for each compact set K ⊂ RN ,

we have

lim
n→∞

∫
K

|θn(x)− θ(x)|pdx = 0, lim
n→∞

∫
K

|ϑn(x)− ϑ(x)|pdx = 0.

In particular, for K = B1(0) (the unit ball centered at the origin), we have

lim
n→∞

∫
B1(0)

|θn(x)− θ(x)|2dx = 0, lim
n→∞

∫
B1(0)

|ϑn(x)− ϑ(x)|2dx = 0.

From the assumption |(uεn(xn), vεn(xn))| ≥ δ, we have

θn(0) = uεn(xn), ϑn(0) = vεn(xn).



20 A. BAIG, Z. LI EJDE-2025/24

Since (θn, ϑn) converges strongly in Lp
loc(RN ), it follows that

θ(0)2 + ϑ(0)2 = lim
n→∞

(θn(0)
2 + ϑn(0)

2) ≥ δ2.

This implies that at least one of θ or ϑ is not identically zero. Without loss of
generality, assume θ ̸≡ 0.

Using the convergence of (θn, ϑn) and the continuity of W,V , and H, we can
pass to the limit in the weak formulation of the system. This yields

−∆θ + αW f(θ)f ′(θ) = g1(x, f(θ), f(ϑ))f
′(θ),

−∆ϑ+ αV f(ϑ)f
′(ϑ) = g2(x, f(θ), f(ϑ))f

′(ϑ),

where

αW = lim
n→∞

W (εnxn), αV = lim
n→∞

V (εnxn),

and g1, g2 are the limits of Hw and Hz, respectively.
Since W or V belongs to Class 2, we have (A9), which implies that ∇V (x) ̸= 0

for all x ∈ ∂Λ. Now, observe that the sequence (εnxn) lies on ∂Λεn , and as εn → 0,
εnxn → x0 ∈ ∂Λ, and by the continuity of ∇V , we have ∇V (x0) = 0. However,
this contradicts (A9), which requires that ∇V (x0) ̸= 0 for all x0 ∈ ∂Λ.

The contradiction implies that our initial assumption must be false. Therefore,
we conclude that

max
x∈∂Ωε

|(uε(x), vε(x))| → 0 as ε → 0+. □

4. Proof of Theorem 1.2

Proof. Suppose, by contradiction, that there exists yε ∈ RN \ Ωε such that

wε(yε) ≥ f−1
(a
2

)
.

Combining the previous lemma with the fact that |(wε(x), zε(x))| → 0 as |x| → +∞,
see Lemma 3.6, we conclude that there exists a maximum point xε ∈ RN \ Ωε for
wε.

Since (wε, zε) ∈ C2,α
loc (RN ) ∩ L∞(RN ) is a solution of (2.10), we have

W (εxε)f(wε(xε))f
′(wε(xε)) = Hw(εxε, f(wε(xε)), f(zε(xε)))f

′(wε(xε)).

Using that f is an increasing function and f ′ > 0 in (0,∞), we obtain

W0
a

2
≤ Hw(εxε, f(wε(xε)), f(zε(xε))),

which contradicts hypothesis (A18).
Thus, we conclude that

wε(x) < f−1
(a
2

)
in RN \ Ωε.

Similarly, we have

zε(x) < f−1
(a
2

)
in RN \ Ωε.

Hence,

|(f(wε), f(zε))| < a in RN \ Ωε.

Therefore, by the definition of H, (wε, zε) is also a solution of (2.3). □



EJDE-2025/24 SYSTEMS OF QUASILINEAR SCHRÖDINGER EQUATIONS 21

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Ayesha Baig

Department of Mathematics and Statistics, Central South University, Changsha 410083,
China

Email address: ayee sha@qq.com

Zhouxin Li (Corresponding Author)

Department of Mathematics and Statistics, Central South University, Changsha 410083,
China

Email address: 1735881486@qq.com


	1. Introduction
	2. Reformulation of the system and the auxiliary system
	3. Existence of positive solutions for the auxiliary system
	4. Proof of Theorem ??
	References