Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 05, pp. 1–21.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LOCALIZED NODAL SOLUTIONS FOR PARAMETER-

DEPENDENT QUASILINEAR SCHRÖDINGER EQUATIONS

RUI HE, XIANGQING LIU

Abstract. In this article, we apply a new variational perturbation method

to study the existence of localized nodal solutions for parameter-dependent
semiclassical quasilinear Schrödinger equations, under a certain parametric

conditions.

1. Introduction

In this article, we study the existence of localized nodal solutions for the param-
eter-dependent semiclassical quasilinear Schrödinger equation

ε2
N∑

i,j=1

(
Dj(bij(v)Div)− 1

2
Dzbij(v)DivDjv

)
− V (x)v + λ|v|q−2v = 0,

v(x)→ 0 as |x| → ∞,

(1.1)

where x ∈ RN , ε > 0 is a small parameter, Div = ∂v
∂xi

, Dzbij(z) = d
dz bij(z),

2 < q < 4, N ≥ 3, λ > 0, and V is the potential function.
We assume the following conditions on bij and V :

(A1) bij ∈ C1,1(R,R), bij = bji, i, j = 1, . . . , N and there exists c0 > 0 such that

|Dzbij(z)−Dzbij(w)| ≤ c0|z − w| for z, w ∈ R;

(A2) there exist c+, c− > 0 such that

c−(1 + z2)|ξ|2 ≤
N∑

i,j=1

bij(z)ξiξj ≤ c+(1 + z2)|ξ|2 for z ∈ R, ξ = (ξi) ∈ RN ;

(A3) there exists δ > 0 such that

δ

N∑
i,j=1

bij(z)ξiξj ≤
N∑
i,j

(
bij(z) +

1

2
zDzbij(z)

)
ξiξj ≤ q

(1

2
− δ
) N∑
i,j=1

bij(z)ξiξj

for z ∈ R, ξ = (ξi) ∈ RN ;
(A4) bij(z) is even in z;

(A5) V ∈ C1(RN ,R) and there exists c0 > 0 such that

c0 ≤ V (x) ≤ c−1
0 , for x ∈ RN ;

2010 Mathematics Subject Classification. 35B05, 35B45.
Key words and phrases. Quasilinear Schrödinger equation; perturbation method;

truncation technique; nodal solution.
c©2021 Texas State University.

Submitted October 2, 2020. Published January 25, 2021.

1



2 R. HE, X. LIU EJDE-2021/05

(A6) there exists a bounded domain M ⊂ RN with smooth boundary ∂M such
that

〈∇V (x), n(x)〉 > 0, for x ∈ ∂M,

where n(x) is the outer normal of ∂M at the point x ∈ ∂M .

Without loss of generality we assume 0 ∈ M . Under assumption (A6), the
critical set A of V contained M is a nonempty closed set:

A = {x ∈M |∇V (x) = 0}.
For a set B ⊂ RN and δ > 0 we denote

Bδ =
{
x ∈ RN : dist(x,B) = inf

y∈B
|x− y| < δ

}
,

Bδ =
{
x ∈ RN : δx ∈ B

}
.

Our main result reads as follows.

Theorem 1.1. Assume 2 < q < 4, (A1)–(A6). Then for any positive integer k
there exist Λk > 0 and εk > 0 such that if λ ≥ Λk, 0 < ε < εk, then (1.1) has k
pairs of sign-changing solutions ±vj,ε, j = 1, . . . , k. Moreover, for any δ > 0 there
exist α > 0, c = ck > 0 and εk(δ) > 0 such that if 0 < ε < εk(δ), then

|vj,ε(x)| ≤ c exp
{
− α

ε
dist(x,Aδ)

}
, for x ∈ RN , j = 1, . . . , k.

For small ε and 4 < q < 2 ·2∗, the authors in [6] established the existence of a se-
quence of localized nodal solutions concentrating near a given local minimum point
of the potential function V , by developing new variational perturbation method to
treat this class of non-smooth variational problems. There are few results for the
case 2 < q < 4. Motivated by their work, we will use the variational perturba-
tion developed in [6] to deal with the existence and multiplicity of localized nodal
solutions of (1.1), for the case 2 < q < 4. Next we outline the approach.

First, we denote u(x) = v(εx). Then equation (1.1) is equivalent to

N∑
i,j=1

(
Dj(bij(u)Diu)− 1

2
Dzbij(u)DiuDju

)
− V (εx)u+ λ|u|q−2u = 0,

u(x)→ 0 as |x| → ∞.

(1.2)

We are looking for weak solutions to (1.2), namely a function u ∈ H1(RN )∩L∞(RN )
satisfying∫

RN

N∑
i,j=1

(
bij(u)DiuDjϕ+

1

2
Dzbij(u)DiuDjuϕ

)
dx+

∫
RN

V (εx)uϕdx

= λ

∫
RN
|u|q−2uϕdx

for ϕ ∈ C∞0 (RN ). Formally problem (1.2) has a variational structure, given by the
functional

Iε(u) =
1

2

∫
RN

N∑
i,j=1

bij(u)DiuDju dx+
1

2

∫
RN

V (εx)u2 dx− λ

q

∫
RN
|u|q dx, u ∈ Y ,

where

Y =
{
u : u ∈ H1(RN ),

∫
RN

u2|∇u|2 dx < +∞
}
.



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 3

Now we define a truncation function and a perturbed functional. Let ϕ ∈
C∞0 (RN ) be such that ϕ(s) = 1 for |s| ≤ 1;ϕ(s) = 0 for |s| ≥ 2; |ϕ′(s)| ≤ 2, ϕ
is even and decreasing in the interval [1,2]. For µ ∈ (0, 1], x ∈ RN , z ∈ R define

bµ(x, z) = ϕ(µ exp{dist(µx,M)}z), mµ(x, z) =

∫ z

0

bµ(x, τ)dτ. (1.3)

Assume x = 0 ∈M . For x = 0 we simply use the notation

bµ(z) = bµ(0, z) = ϕ(µz), mµ(z) = mµ(0, z) =

∫ z

0

bµ(τ)dτ . (1.4)

Let

βij(z) = bij(z)− σ(1 + z2)δij , i, j = 1, . . . , N,

where σ > 0 is a fixed small positive constant so that βij , i, j = 1, . . . , N also satisfy
the assumptions (A1)-(A3) (with possibly different constants c0 and δ). Now we
define the perturbed functional Iµ,ε by

Iµ,ε(u) =
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−2

|∇u|2 dx+
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−4

u2|∇u|2 dx

+
1

2

∫
RN

N∑
i,j=1

βij(u)DiuDju dx+
1

2

∫
RN

V (εx)u2 dx− λ

q

∫
RN
|u|q dx

for µ ∈ (0, 1], u ∈ X = W 1,m(RN ) ∩ H1(RN ), where m > 4. Here we introduce
one additional coercive term for perturbation because the problem on unbounded
domain RN and the imbedding from W 1,m(RN ) to Lq(RN ) is not compact. More-
over, we use the penalization method due to[1, 2, 3] to localize the solutions. For
more results on standing waves, sign-changing solutions, ground state solutions and
asymptotic behavior of solutions to quasilinear Schrödinger equations, we refer the
reader to [1, 5, 10, 11].

Let ζ ∈ C∞0 (R) be such that ζ(t) = 0 for t ≤ 0, ζ(t) = 1 for t ≥ 1, and
0 ≤ ζ ′(t) ≤ 2. We define

χε(x) = ε−6ζ(dist(x,Mε)).

Let E(x) = V (x)− σ and define

Γµ,ε(u)

=
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−2

|∇u|2 dx+
1

2
σ

∫
RN

( u

mε(x, u)

)m−2

u2 dx

+
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−4

u2|∇u|2 dx+
1

2

∫
RN

N∑
i,j=1

βij(u)DiuDju dx

+
1

2

∫
RN

E(εx)u2 dx+
1

2β

(∫
RN

χε(x)u2 dx− 1
)β

+
− λ

q

∫
RN
|u|q dx

(1.5)

for u ∈ Xε = W 1,m
ε (RN ) ∩H1(RN ), 2 < β < q, and

W 1,m
ε (RN ) = W 1,m(RN ) ∩ Lmε (RN ),

where Lmε (RN ) is a weighted Lm-spaces

Lmε (RN ) =
{
u ∈ Lm(RN ),

∫
RN

exp{(m− 2) dist(εx,M)}|u|m dx < +∞
}



4 R. HE, X. LIU EJDE-2021/05

endowed with the norm

‖u‖Lmε (RN ) =
(∫

RN
exp{(m− 2) dist(εx,M)}|u|m dx

) 1
m

with a coercive weight. Then we know the space W 1,m
ε (RN ) is compactly imbedded

to Lp(RN ) for m ≤ p < m∗ = Nm
N−m , in particular the imbedding into Lq(RN ) is

compact. If

|u(x)| ≤ 1

ε
exp{− dist(εx,M)} for x ∈ RN and

∫
RN

χε(x)u2 dx < 1,

then Γµ,ε(u) = Iµ,ε(u). And if |∇u(x)| ≤ 1
µ for x ∈ RN , then Iµ,ε(u) = Iε(u). Here

no limit process µ → 0 is needed for the existence of critical point of the original
problem, and for small µ and ε, Γµ,ε shares critical points with Iε, resulting in
solutions of original equation for small µ and ε.

The article is organized as follows. In Section 2 we collect elementary properties
of the auxiliary functions involved in the perturbed functionals and prove some
technical results. In Section 3 we construct critical values of Γµ,ε by the method
of invariant sets with respect to the descending flow. In Section 4 we prove the
uniform bound for the gradient of the approximate sign-changing solutions obtained
in Section 3 and complete the proof of Theorem 1.1.

Also we fix some notations c, c0, c1, . . . denote possibly different positive con-
stants, and c(µ), if necessary, denotes constants depending on µ. In a given Banach
space, → and ⇀ denote the strong convergence and the weak convergence, respec-
tively.

2. Properties of auxiliary functions

In this section,we first recall some elementary properties and some estimates on
the auxiliary functions involved in the perturbations of the functionals, and the
following three lemmas whose proofs are quite the same as that of the results in [6]
and omit it here.

Lemma 2.1. For s > 0, z ∈ R, x ∈ RN , p = (pi) ∈ RN , ξ = (ξi) ∈ RN , the
following statements hold:

(1) 0 ≤ bµ(x, s) ≤ mµ(x,s)
s ≤ 1.

(2) mµ(x, s) = s, if s < µ−1 exp {−dist(µx,M)};

µ−1 exp{− dist(µx,M)} ≤ mµ(x, s) ≤ cµ−1 exp{−dist(µx,M)},

if µ−1 exp{− dist(µx,M)} ≤ s ≤ 2µ−1 exp{−dist(µx,M)};

mµ(x, s) = cµ−1 exp{− dist(µx, M)},

if s ≥ 2µ−1 exp{− dist(µx, M)}, where c =
∫∞

0
ϕ(τ) dτ .

(3) We define fµ(p) = 1
2σ
( |p|
mµ(|p|)

)m−2|p|2. Then

(3.1) c1(1 + µm−2|p|m−2)|p|2 ≤ fµ(p) ≤ c2(1 + µm−2|p|m−2)|p|2;
(3.2) 2fµ(p) ≤ ∇pfµ(p) · p ≤ |∇fµ(p)| · |p| ≤ mfµ(p);

(3.3)
∑N
i,j=1

∂2

∂pi∂pj
fµ(p)ξiξj ≥ σ

( |p|
mµ(|p|)

)m−2|ξ|2 ≥ c(1 + µm−2|p|m−2)|ξ|2;

(3.4) | ∂2

∂pi∂pj
fµ(p)| ≤ c

( |p|
mµ(|p|)

)m−2 ≤ c(1 + µm−2|p|m−2).

(4) We define kε(x, z) = 1
2σ
(

z
mε(x,z)

)m−2
z2. Then



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 5

(4.1)

c1(1 + εm−2 exp{(m− 2) dist(εx,M)}|z|m−2)z2 ≤ kε(x, z)
≤ c2(1 + εm−2 exp{(m− 2) dist(εx,M)}|z|m−2)z2 ;

(4.2) 2kε(x, z) ≤ ∂
∂zkε(x, z)z ≤ mkε(x, z);

(4.3)

∂2

∂z2
kε(x, z) ≥ σ

(
z

mµ(x, z)

)m−2

≥ c(1 + εm−2 exp{(m− 2) dist(εx,M)}|z|m−2) ;

(4.4)

0 ≤ ∂2

∂z2
kε(x, z) ≤ c

(
z

mµ(x, z)

)m−2

≤ c(1 + εm−2 exp{(m− 2) dist(εx,M)}|z|m−2) .

(5) We define hµ(z, p) = 1
2σ
( |p|
mµ(|p|)

)m−4
z2|p|2. Then

(5.1) c1(1 + µm−4|p|m−4)z2|p|2 ≤ hµ(z, p) ≤ c2(1 + µm−4|p|m−4)z2|p|2;
(5.2)

4hµ(z, p) ≤ ∇phµ(z, p)p+
∂

∂z
hµ(z, p)z

≤ |∇phµ(z, p)| |p|+
∣∣ ∂
∂z
hµ(z, p)

∣∣ |z|
≤ mhµ(z, p) ;

(5.3)

N∑
i,j=1

∂2

∂pi∂pj
hµ(z, p)ξiξj ≥ σ

( |p|
mµ(|p|)

)m−4
z2|ξ|2

≥ c(1 + µm−4|p|m−4)z2|ξ|2,
∂2

∂z2
hµ(z, p) = σ

( |p|
mµ(|p|)

)m−4|p|2 ≥ c(1 + µm−4|p|m−4)|p|2 ;

(5.4)
∣∣ ∂2

∂pi∂pj
hµ(z, p)

∣∣ ≤ c( |p|
mµ(|p|)

)m−4
z2 ≤ c(1 + µm−4|p|m−4)z2,∣∣ ∂2

∂z2hµ(z, p)
∣∣ ≤ c( |p|

mµ(|p|)
)m−4|p|2 ≤ c(1 + µm−4|p|m−4)|p|2,∣∣∇p ∂

∂z
hµ(z, p)

∣∣ =
∣∣ ∂
∂z
∇phµ(z, p)

∣∣
≤ c
( |p|
mµ(|p|)

)m−4|z| |p|

≤ c(1 + µm−4|p|m−4)|z| |p| .

Lemma 2.2. For x ∈ RN , p, p ∈ RN , and z, z ∈ RN , the following three properties
hold:
(1)

〈∇pfµ(p)−∇pfµ(p), p− p〉 ≥ c(1 + µm−2(|p|m−2 − |p|m−2))|p− p|2

≥ c|p− p|2 + cµm−2|p− p|m,



6 R. HE, X. LIU EJDE-2021/05

|∇pfµ(p)−∇pfµ(p)| ≤ c(1 + µm−2(|p|m−2 − |p|m−2))|p− p|.

(2) ( ∂
∂z
kε(x, z)−

∂

∂z
kε(x, z)

)
(z − z)

≥ c
(
1 + εm−2 exp{(m− 2) dist(εx,M)}(|z|m−2 + |z|m−2)

)
|z − z|2

≥ c|z − z|2 + cεm−2 exp{(m− 2) dist(εx,M)}|z − z|m−2,∣∣ ∂
∂z
kε(x, z)−

∂

∂z
kε(x, z)

∣∣
≤ c
(
1 + εm−2 exp{(m− 2) dist(εx,M)}(|z|m−2 + |z|m−2)

)
|z − z|.

(3)

〈∇phµ(z, p)−∇phµ(z, p), p− p〉+
( ∂
∂z
hµ(z, p)− ∂

∂z
hµ(z, p)

)
(z − z)

≥ c
(
|p|2 + |p|2 + µm−4(|p|m−2 + |p|m−2)

)
|z − z|2

− ν
(
1 + µm−2(|p|m−2 + |p|m−2)

)
|p− p|2 − cνµ−2(|z|m−2 + |z|m−2)|z − z|2,∣∣∇phµ(z, p)−∇phµ(z, p)

∣∣
≤ c(1 + µm−4(|p|m−4 + |p|m−4))

(
(z2 + z2)|p− p|+ (|z|+ |z|)(|p|+ |p|)|z − z|

)
,∣∣ ∂

∂z
hµ(z, p)− ∂

∂z
hµ(z, p)

∣∣
≤ c(1 + µm−4(|p|m−4 + |p|m−4))

(
(|z|+ |z|)(|p|+ |p|)|p− p|+ (|p|2 + |p|2)|z − z|

)
,

where ν > 0 is any small constant, and cν depends on ν.

Lemma 2.3. Let Jµ,ε be the functional defined on Xε by

Jµ,ε(u) =
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−2

|∇u|2dx+
1

2
σ

∫
RN

( u

mε(x, u)

)m−2

u2dx

+
1

2
σ

∫
RN

( |∇u|
mµ(|∇u|)

)m−4

u2|∇u|2dx

+
1

2

∫
RN

N∑
i,j=1

βij(u)DiuDjudx+
1

2

∫
RN

E(εx)u2dx .

(2.1)

Then for u, v, ϕ ∈ Xε, we have:
(1)

〈DJµ,ε(u)−DJµ,ε(v), u− v〉

≥ cµm−2

∫
RN
|∇u−∇v|mdx+ cεm−2

∫
RN

exp{(m− 2) dist(εx,M)}|u− v|m dx

+ c

∫
RN
|∇u−∇v|2 dx− cµ−2

∫
RN

(|u|m−2 + |v|m−2)(u− v)2 dx

− cµ−2

∫
RN

(u− v)2 dx

≥ cµ,ε‖u− v‖mW 1,m
ε (RN )

+ c‖u− v‖2H1(RN )

− cµ−2

∫
RN

(|u|m−2 + |v|m−2)(u− v)2 dx− cµ−2

∫
RN

(u− v)2dx ,



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 7

(2)

|〈DJµ,ε(u)−DJµ,ε(v), ϕ〉|
≤ c‖u− v‖H1(RN )‖ϕ‖H1(RN )

+ c
(
‖u‖m−2

W 1,m
ε (RN )

+ ‖v‖m−2

W 1,m
ε (RN )

)
‖u− v‖W 1,m

ε (RN )‖ϕ‖W 1,m
ε (RN ).

3. Construction of critical points of Γµ,ε

In this section, we will adopt the method of invariant sets of descending flow
developed in [4] to obtain multiple sign-changing critical points of the perturbed
functional Γµ,ε. For the reader’s convenience, we first give an abstract critical point
theorem, which has been proved in [9].

Let X be a Banach space, f be an even C1-functional on X. Let Pj , Qj , j =
1, . . . , k be a family of open convex sets of X,Qj = −Pj , j = 1, . . . , k. Set

W = ∪kj=1(Pj ∪Qj), Σ = ∩kj=1(∂Pj ∩ ∂Qj).
Assume

(A7) f satisfies the Palais-Smale condition,
(A8) c∗ = infx∈Σ f(x) > 0,

and assume there exists an odd continuous map A : X → X satisfying

(A9) For c0, b0 > 0, there exists b = b(c0, b0) > 0 such that if ‖Df(x)‖ ≥
b0, |f(x)| ≤ c0, then

〈Df(x), x−Ax〉 ≥ b‖x−Ax‖ > 0 .

(A10) A(∂Pj) ⊂ Pj , A(∂Qj) ⊂ Qj , j = 1, . . . , k.

We define

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

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

η(x) = x if f(x) < 0
}

where γ is the genus of symmetric sets,

γ(E) = inf
{
n : there exists an odd map η : E → Rn\{0}

}
.

We define the assumption

(A11) Γj is nonempty,

and the notation

cj = inf
A∈Γj

sup
x∈A\W

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

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

Theorem 3.1. Assume (A7)–(A11) hold. Then

(1) cj ≥ c∗, K∗cj 6= ∅ .
(2) cj →∞, as j →∞.
(3) If cj = cj+1 = · · · = cj+k−1 = c, then γ(K∗c ) ≥ k .

In the following we verify that the functional Γµ,ε satisfies all the assumptions
of Theorem 3.1. First we prove that the functional Γµ,ε satisfies the Palais-Smale
condition, i.e. assumption (A7).



8 R. HE, X. LIU EJDE-2021/05

Lemma 3.2. Γµ,ε is differentiable and satisfies the Palais-Smale condition.

Proof. For u, ϕ ∈ Xε, we have

〈DΓµ,ε(u), ϕ〉

=

∫
RN
∇pfµ(∇u)∇ϕdx+

∫
RN

∂

∂z
kε(x, u)ϕdx

+

∫
RN

(
∇phµ(u,∇u)∇ϕ+

∂

∂z
hµ(u,∇u)ϕ

)
dx

+

∫
RN

N∑
i,j=1

(
βij(u)DiuDjϕ+

1

2
Dzβij(u)DiuDjuϕ

)
dx+

∫
RN

E(εx)uϕdx

+
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)uϕdx− λ
∫
RN
|u|q−2uϕdx.

Since the imbedding from W 1,m
ε (RN ) to Lq(RN ) is compact, there exists c > 0 such

that

‖u‖Lq(RN ) ≤ c‖u‖W 1,m
ε (RN ).

Let {un} ⊂ Xε be a Palais-Smale sequence of Γµ,ε, namely, there exists L > 0 such
that |Γµ,ε(un)| ≤ L and DΓµ,ε(un)→ 0 as n→∞. By Lemma 2.1 and assumption
(A3), we deduce

Γµ,ε(un)

=

∫
RN

fµ(∇un) dx+

∫
RN

kε(x, un) dx+

∫
RN

hµ(un,∇un) dx

+
1

2

∫
RN

N∑
i,j=1

βij(un)DiunDjun dx+
1

2

∫
RN

E(εx)u2
n dx

+
1

2β

(∫
RN

χε(x)u2
ndx− 1

)β
+
− λ

q

∫
RN
|un|q dx

≥ c
{
µm−2

∫
RN
|∇un|mdx+ εm−2

∫
RN

exp{(m− 2) dist(εx,M)}|un|m dx

+ µm−4

∫
RN
|∇un|m−2u2

n dx
}

+ c
(∫

RN
(1 + u2

n)|∇un|2dx+

∫
RN

u2
n dx

)
+ c
(∫

RN
χε(x)u2

n dx− 1
)β

+
− c

∫
RN

uqn dx

≥ c
{
µm−2

∫
RN
|∇un|m dx+ εm−2

∫
RN

exp{(m− 2) dist(εx,M)}|un|m dx

+ µm−4

∫
RN
|∇un|m−2u2

ndx
}

+ c
(∫

RN
(1 + u2

n)|∇un|2dx+

∫
RN

u2
ndx

)
+ c
(∫

RN
χε(x)u2

n dx− 1
)β

+

− c
{∫

RN
|∇un|m dx+

∫
RN

exp{(m− 2) dist(εx,M)}|un|m dx
}q/m

(3.1)

which implies that {un} is bounded in Xε and
( ∫

RN χε(x)u2
ndx− 1

)β
+

is bounded.

Assume un ⇀ u in Xε and un → u in Ls(RN ), 2 ≤ s < 2 · 2∗. By Lemma 2.3, we



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 9

have

o(1) = 〈DΓµ,ε(un)−DΓµ,ε(um), un − um〉
= 〈DJµ,ε(un)−DJµ,ε(um), un − um〉

+
(∫

RN
χε(x)u2

n dx− 1
)β−1

+

∫
RN

χε(x)un(un − um) dx

−
(∫

RN
χε(x)u2

m dx− 1
)β−1

+

∫
RN

χε(x)un(un − um) dx

−
∫
RN

(|un|q−2un − |um|q−2um)(un − um) dx

≥ c
(
µm−2

∫
RN
|∇un −∇um|m dx

+ εm−2

∫
RN

exp{(m− 2) dist(εx,M)}|un − um|m dx
)

+ c

∫
RN
|∇un −∇um|2 dx− cµ−2

(∫
RN

(
|un|m−2 + |um|m−2

)
|un − um|2 dx

+

∫
RN
|un − um|2 dx

)
+ o(1)

≥ c‖un − um‖mW 1,m
ε (RN )

+ c‖un − um‖2H1(RN ) + o(1).

So {un} is a Cauchy sequence in Xε, hence a convergent sequence. �

We define the operator A : Xε → Xε. Given u ∈ Xε, for a suitable constant
cµ > 0, we define v = Au ∈ Xε:

〈DJµ,ε(v), ϕ〉+
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)vϕ dx

+ cµ

∫
RN

(|v|m−2v + v)ϕdx

= λ

∫
RN
|u|q−2uϕdx+ cµ

∫
RN

(|u|m−2u+ u)ϕdx, for ϕ ∈ Xε

(3.2)

and

Jµ,ε(u) =

∫
RN

(
fµ(∇u) + kε(x,∇u) + hµ(u,∇u)

)
dx

+
1

2

∫
RN

N∑
i,j=1

βij(u)DiuDju dx+
1

2

∫
RN

E(εx)u2 dx, for u ∈ Xε.

(3.3)

In view of [6, Lemma 4.1], we know that for sufficiently large cµ > 0 the operator
A is well-defined and continuous. And similar to [6], we can prove the following
lemmas 3.3–3.6.

Lemma 3.3. There exist constants D > 0 and α ∈ ( 2
q , 1) such that∫

RN

N∑
i,j=1

(
βij(u) +

1

2
uDzβij(u)

)
DiuDju dx+

∫
RN

E(εx)u2 dx ≥ D
(∫

RN
|u|q dx

)α
.



10 R. HE, X. LIU EJDE-2021/05

Now we define

Q = Qδ =
{
u ∈ Xε :

1

2
D
(∫

RN
uq+ dx

)α
+
m− 1

m
cµ

∫
RN

um+ dx

+
1

2
cµ

∫
RN

u2
+ dx < δ

}
,

P = −Q =
{
u ∈ Xε :

1

2
D
(∫

RN
uq− dx

)α
+
m− 1

m
cµ

∫
RN

um− dx

+
1

2
cµ

∫
RN

u2
− dx < δ

}
.

Lemma 3.4. There exists δ0 = δ0(µ) such that for δ ≤ δ0
A(∂P ) ⊂ P, A(∂Q) ⊂ Q.

Lemma 3.5. There exist δ0 = δ0(µ), c∗ = c∗(δ, µ) such that

Γµ,ε(u) ≥ c∗ for u ∈ ∂P ∩ ∂Q.

Lemma 3.6. Let u ∈ Xε, v = Au, then it holds
(1)

〈DΓµ,ε(u), u− v〉 ≥ c
(
‖u− v‖m

W 1,m
ε (RN )

+ ‖u− v‖2H1(RN )

)
.

(2)

〈DΓµ,ε(u), ϕ〉 ≤ c
(
‖u‖m−2

W 1,m
ε (RN )

+ ‖v‖m−2

W 1,m
ε (RN )

)
‖u− v‖W 1,m

ε (RN )‖ϕ‖W 1,m
ε (RN )

+ c
(

1 +
(∫

RN
χε(x)u2dx− 1

)β−1

+

)
‖u− v‖H1(RN )‖ϕ‖H1(RN )

for all ϕ ∈ Xε.

Lemma 3.7. Let u ∈ Xε, v = Au. Assume |Γµ,ε(u)| ≤ c0, ‖DΓµ,ε(u)‖ ≥ b0. Then
there exists b = b(c0, b0) such that

〈DΓµ,ε(u), u− v〉 ≥ b‖u− v‖Xε > 0.

Proof. By Lemma 2.1, we have

Γµ,ε(u)− 1

2q
〈DJµ,ε(u)−DJµ,ε(v), u〉

= Γµ,ε(u)− 1

2q
〈DJµ,ε(u), u〉+

1

2q
〈DJµ,ε(v), u〉

=

∫
RN

(
fµ(∇u)− 1

2q
∇pfµ(∇u)∇u

)
dx+

∫
RN

(
kε(x, u)− 1

2q

∂

∂z
kε(x, u)u

)
dx

+

∫
RN

(
hµ(u,∇u)− 1

2q

(
∇phµ(u,∇u)∇u+

∂

∂z
hµ(u,∇u)u

))
dx

+

∫
RN

N∑
i,j=1

(1

2
βij(u)− 1

2q

(
βij(u) +

1

2
uDzβij(u)

))
DiuDju dx

+
(1

2
− 1

2q

) ∫
RN

E(εx)u2 dx

+
1

2β

(∫
RN

χε(x)u2 dx− 1
)β

+
− 1

2q

(∫
RN

χε(x)u2 dx− 1
)β−1

+

∫
RN

χε(x)uv dx



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 11

+
1

2q
cµ

∫
RN

(
|u|m−2u− |v|m−2v

)
u dx+

1

2q
cµ

∫
RN

(u− v)udx− λ

2q

∫
RN
|u|q dx

≥ c
(
‖u‖m

W 1,m
ε (RN )

+ ‖u‖2H1(RN )

)
− c+ c

(∫
RN

χε(x)u2 dx− 1
)β

+

− c
(∫

RN
χε(x)(u− v)2 dx

)β
+

+
1

2q
cµ

∫
RN

(
|u|m−2u− |v|m−2v

)
u dx

+
1

2q
cµ

∫
RN

(u− v)u dx.

The above estimate holds since

1

2β

(∫
RN

χε(x)u2 dx− 1
)β

+
− 1

2q

(∫
RN

χε(x)u2 dx− 1
)β−1

+

∫
RN

χε(x)uv dx

=
1

2β

(∫
RN

χε(x)u2 dx− 1
)β

+
+

1

2q

(∫
RN

χε(x)u2 dx− 1
)β−1

+

∫
RN

χε(x)u2 dx

− 1

2q

(∫
RN

χε(x)u2 dx− 1
)β−1

+

∫
RN

χε(x)u(u− v) dx

≥
( 1

2β
− η

2q

)( ∫
RN

χε(x)u2 dx− 1
)β

+

+ (
1

2q
− η

2q
)
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)u2 dx

− cη
(∫

RN
χε(x)(u− v)2 dx

)β
+

≥ c
(∫

RN
χε(x)u2 dx− 1

)β
+
− c− cη

(∫
RN

χε(x)(u− v)2 dx
)β

+
,

where 0 < η < 1, cη is a constant. By Lemma 2.3 (2), we obtain

‖u‖m
W 1,m
ε (RN )

+ ‖u‖2H1(RN ) +
(∫

RN
χε(x)u2 dx− 1

)β
+

≤ c(1 + |Γµ,ε(u)|) + c|〈DJµ,ε(u)−DJµ,ε(v), u〉|+ c
∣∣ ∫

RN

(
|u|m−2 − |v|m−2v

)
u dx

∣∣
+ c
∣∣ ∫

RN
(u− v)u dx

∣∣+ c
(∫

RN
χε(x)(u− v)2 dx

)β
+

≤ c(1 + |Γµ,ε(u)|) + c
(
‖u‖m−2

W 1,m
ε (RN )

+ ‖u‖m−2

W 1,m
ε (RN )

)
‖u− v‖W 1,m

ε (RN )‖u‖W 1,m
ε (RN )

+ c‖u− v‖H1(RN )‖u‖H1(RN ) + c‖u− v‖2β
L2(RN )

≤ c
(

1 + |Γµ,ε(u)|+ ‖u− v‖m
W 1,m
ε (RN )

+ ‖u− v‖2H1(RN ) + ‖u− v‖2β
L2(RN )

)
+ τ
(
‖u‖m

W 1,m
ε (RN )

+ ‖u‖2H1(RN )

)
for τ ∈ (0, 1). Therefore

‖u‖m
W 1,m
ε (RN )

+ ‖u‖2H1(RN ) +
(∫

RN
χε(x)u2dx− 1

)β
+

≤ c
(

1 + |Γµ,ε(u)|+ ‖u− v‖m
W 1,m
ε (RN )

+ ‖u− v‖2H1(RN )

)
.

(3.4)

By Lemma 3.6 (2), we obtain

‖DΓµ,ε(u)‖



12 R. HE, X. LIU EJDE-2021/05

≤ c
(
‖u‖m−2

W 1,m
ε (RN )

+ ‖v‖m−2

W 1,m
ε (RN )

)
‖u− v‖W 1,m

ε (RN )

+ c
(

1 +
(∫

RN
χε(x)u2dx− 1

)β−1

+

)
‖u− v‖H1(RN )

≤ c
(
‖u‖m−2

W 1,m
ε (RN )

+ ‖v‖m−2

W 1,m
ε (RN )

+ 1 +
( ∫

RN
χε(x)u2dx− 1

)β−1

+

)
‖u− v‖Xε

≤ c
(
‖u‖m

W 1,m
ε (RN )

+ ‖v‖m
W 1,m
ε (RN )

+ 1 +
( ∫

RN
χε(x)u2dx− 1

)β
+

)
‖u− v‖Xε

≤ c
(

1 + |Γµ,ε(u)|+ ‖u− v‖m
W 1,m
ε (RN )

+ ‖u− v‖2H1(RN )

+ ‖u− v‖2β
L2(RN )

)
‖u− v‖Xε

≤ c
(
1 + |Γµ,ε(u)|+ ‖u− v‖αXε

)
‖u− v‖Xε

≤ c (1 + |Γµ,ε(u)|+ ‖u− v‖Xε)
α ‖u− v‖Xε

for α = max{m, 2, 2β}. Lemma 3.7 follows from the above inequality and Lemma
3.6 (1). �

Now we consider the assumption (A11).

Lemma 3.8. Assume B = {x ∈ RN : |x| ≤ r} ⊂ M . Let {ek}∞k=1 be a family
of linearly independent functions in C∞0 (B). There exist Rk, dk > 0 such that for
λ > dk,

J0(u) < 0 for u ∈ Ek, ‖u‖ = Rk

where Ek = span{e1, . . . , ek} and

J0(u) = σ

∫
RN

(
|∇u|m + e(m−2)|x||u|m + |u|m−2|∇u|2

)
dx

+
1

2

∫
RN

N∑
i,j=1

bij(u)DiuDju dx+
1

2

∫
RN

V∞u
2 dx− λ

q

∫
RN
|u|q dx

and V∞ = supx∈RN V (x).

Proof. Since Ek is finite-dimensional, all norms are equivalent, there exist constants
ck, dk > 0 such that

1

q

∫
RN
|u|q dx = ck,

J0(u) +
λ

q

∫
RN
|u|q dx < bk

for u ∈ Hk, ‖u‖ = Rk. It is easy to see that dk = bk/ck satisfies the condition. �

We define ϕk ∈ C(Bk, C
∞
0 (B)) as

ϕk(t) = Rk

k∑
i=1

tiei, t = (t1, . . . , tk) ∈ Bk = {t | t ∈ RN , |t| ≤ 1}.

Let

Γj = {E ⊂ Xε : E is compact − E = E, γ(E ∩ η−1(Σ)) ≥ j for η ∈ Λ}
Λ = {η ∈ C(Xε, Xε) : η is odd η(p) ⊂ P, η(Q) ⊂ Q, η(u) = u if Γµ,ε(u) ≤ 0}.

Moreover, we define Λk = max1≤i≤k+1{di}. Then we have the following result.



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 13

Lemma 3.9. For λ ≥ Λk, Ej = ϕj+1(Bj+1) ⊂ Γj , j = 1, . . . , k, so Γj is nonempty.

Proof. It is obviously that

ϕj+1(−t) = −ϕj+1(t), ϕj+1(0) = 0 ∈W.

For t ∈ Bj+1, u = ϕj+1(t),
( ∫

RN χε(x)u2 dx− 1
)β

+
= 0, we have

Γµ,ε(u) ≤ J0(u), u ∈ ϕj+1(Bj+1), µ, ε ∈ (0, 1].

By Lemmas 3.6 and 3.7, we obtain

ϕ(t) /∈W, if t ∈ ∂Bj+1,

sup
t∈∂Bj+1

Γµ,ε(ϕj+1(t)) ≤ 0 .

By [5, Lemma 4.2], we have γ(Ej ∪ η−1(Σ)) ≥ j for all η ∈ Λ, hence Ej ⊂ Γj . �

All the assumptions of Theorem 3.1 are satisfied, and we have the following
existence theorem.

Theorem 3.10. Assume (A1)–(A6). Then for any positive integer k, the functional
Γµ,ε(µ, ε ∈ (0, 1]) has k sign-changing critical points, the corresponding critical
values are defined as

cj(µ, ε) = inf
E∈Γj

sup
u∈E\M

Γµ,ε(u), j = 1, . . . , k. (3.5)

Moreover
(1) there exist mj , j = 1, . . . , k, independent of µ, ε such that

cj(µ, ε) ≤ mj , j = 1, . . . , k. (3.6)

(2) If cj(µ, ε) = · · · = cj+k−1(µ, ε) = c, then γ(K∗(c)) ≥ k.

Proof. By Theorem 3.1, we know that Γµ,ε(µ, ε ∈ (0, 1]) has k sign-changing critical
points, so we only need to prove estimate (3.6). Since Ej = ϕj+1(Bj+1) ∈ Γj and
Γµ,ε(u) ≤ J0(u) for u ∈ ϕj+1(Bj+1) and µ, ε ∈ (0, 1], , we have

cj(µ, ε) ≤ mj := sup
u∈Ej

J0(u).

The second part of the theorem is the direct result of the Theorem 3.1. �

4. Proof of Theorem 1.1

In this section, we prove that the perturbed functionals shares critical points with
the original problem for small parameters. Therefore, we first prove the uniform
bound for the gradient of the sign-changing solutions obtained in Section 3. In
order to prove the following Theorem 4.9, we need some propositions and lemmas.

Proposition 4.1. Assume Γµ,ε(u) ≤ L,DΓµ,ε(u) = 0. Then

(1) There exists K = K(L) such that |u(x)| ≤ K for x ∈ RN .
(2) For any δ > 0 there exists c = c(δ, L) such that |u(x)| ≤ cε3 for x ∈

RN\(Mε)
δ.



14 R. HE, X. LIU EJDE-2021/05

Proof. We apply Moser’s iteration to obtain the L∞-bound.
(1) For T > 0, let uT (x) = u(x) if |u(x)| ≤ T ; uT (x) = ±T if ±u(x) ≥ T . Take

ϕ = |uT |2k−2u as test function in 〈DΓµ,ε(u), ϕ〉 = 0 where k ≥ 1, we have∫
RN
∇pfµ(∇u)∇ϕdx+

∫
RN

∂

∂z
kε(x, u)ϕdx

+

∫
RN

(
∇phµ(u,∇u)∇ϕ+

∂

∂z
hε(u,∇u)ϕ

)
dx+

∫
RN

E(εx)uϕdx

+
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)uϕdx ≥ 0.

Since 〈DΓµ,ε(u), ϕ〉 = 0, we have∫
RN

N∑
i,j=1

(
βij(u)DiuDjϕ+

1

2
Dzβij(u)DiuDjuϕ

)
dx+

∫
RN

E(εx)uϕdx

≤ λ
∫
RN
|u|q−2uϕdx.

(4.1)

For the right-hand side of (4.1), for any ν > 0 there exists cν > 0 such that∫
RN
|u|q−2uϕdx ≤ ν

∫
RN

u2|uT |2k−2 dx+ cν

∫
RN

(u2|uT |k−1)2 dx. (4.2)

For the left-hand side of (4.1), by the conditions (A5) and (A3), we have∫
RN

N∑
i,j=1

(
βij(u)DiuDjϕ+

1

2
Dzβij(u)DiuDjuϕ

)
dx+

∫
RN

E(εx)uϕdx

≥
∫
RN

N∑
i,j=1

(
βij(u) +

1

2
uDzβij(u)

)
DiuDju|uT |2k−2 dx

+ c1

∫
RN

u2|uT |2k−2 dx

≥ c
∫
RN
|∇u|2u2|uT |2k−2dx+ c1

∫
RN

u2|uT |2k−2 dx

≥ c

k2

∫
RN
|∇(u2|uT |k−1)|2 dx+ c1

∫
RN

u2|uT |2k−2 dx

≥ c

k2

(∫
RN

(u2|uT |k−1)2∗ dx
)2/2∗

+ c1

∫
RN

u2|uT |2k−2 dx.

(4.3)

By (4.1)–(4.3), with λν < c1, we have(∫
RN

(u2|uT |k−1)2∗ dx
)2/2∗

≤ ck2

∫
RN

(u2|uT |k−1)2 dx. (4.4)

Assume
∫
RN (u2|uT |k−1)2 dx < +∞. Let T →∞ in (4.4) we obtain(∫

RN
|u|(k+1)·2∗ dx

)2/2∗

≤ ck2

∫
RN
|u|2(k+1) dx. (4.5)

Denote d = 2∗/2, then(∫
RN
|u|(k+1)·2∗ dx

) 1
2∗(k+1) ≤ (ck2)

1
2(k+1)

(∫
RN
|u|2

∗(k+1) 1
d dx

) d
2∗(k+1)

.



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 15

Let 2∗(1 + k1) 1
d = 2∗, i.e., k1 = 2∗−2

2 > 0. Starting from k = k1, by iteration we
have

‖u‖L∞(RN ) ≤ c‖u‖L2∗ (RN ) ≤ c,

since (∫
RN
|u|2

∗
dx
)1/2∗

≤ c
(∫

RN
|∇u|2 dx

)1/2

and by (3.1) we know that
∫
RN |∇u|

2 dx is bounded.

(2) For x0 ∈ RN , 0 < ρ < R ≤ 1. Let η ∈ C∞0 (RN , [0, 1]) such that η(x) = 1
for x ∈ Bρ = Bρ(x0), η(x) = 0 for x /∈ BR = BR(x0) and |∇η| ≤ c

R−ρ . Take

ϕ = u|u|2k−2ηm, k ≥ 1 as test function in 〈DΓµ,ε(u), ϕ〉 = 0, we have∫
RN
∇pfµ(∇u)∇ϕdx+

∫
RN

∂

∂z
kε(x, u)ϕdx

+

∫
RN

(
∇phµ(u,∇u)∇ϕ+

∂

∂z
hµ(u,∇u)ϕ

)
dx

+

∫
RN

N∑
i,j=1

(
βij(u)DiuDjϕ+

1

2
Dzβij(u)DiuDjuϕ

)
dx+

∫
RN

E(εx)uϕdx

+
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)uϕdx

= λ

∫
RN
|u|q−2uϕdx.

(4.6)

By the definition of ϕ, we have∫
RN

∂

∂z
kε(x, u)ϕdx+

∫
RN

∂

∂z
hµ(u,∇u)ϕdx+

∫
RN

E(εx)uϕdx

+
(∫

RN
χε(x)u2 dx− 1

)β−1

+

∫
RN

χε(x)uϕdx ≥ 0.

(4.7)

And by Lemma 2.1, we obtain∫
RN
∇pfµ(∇u)∇ϕdx

= (2k − 1)

∫
RN

(∇fµ(∇u),∇u)|u|2k−2ηm dx

+m

∫
RN
∇pfµ(∇u)|u|2k−2uηm−1∇ηdx

≥ c
∫
RN

(1 + µm−2|∇u|m−2)|∇u|2|u|2k−2ηm dx

− c
∫
RN

(1 + µm−2|∇u|m−2)|∇u||u|2k−1ηm−1|∇η| dx

≥ −c
∫
RN
|u|2kηm−2|∇η|2 dx− cµm−2

∫
RN
|u|2k−2|u|m|∇η|m dx

≥ − c

(R− ρ)m

∫
BR

|u|2k dx.

(4.8)



16 R. HE, X. LIU EJDE-2021/05

Similarly, we estimate the integral
∫
RN ∇phµ(u,∇u)∇ϕdx and obtain∫

RN
∇phµ(u,∇u)∇ϕdx ≥ − c

(R− ρ)m

∫
BR

|u|2k dx . (4.9)

Also, we have ∫
RN

N∑
i,j=1

(
βij(u)DiuDjϕ+

1

2
Dzβij(u)DiuDjuϕ

)
dx

≥ c
∫
RN

(1 + u2)|∇u|2|u|2k−2ηmdx

− c
∫
RN

(1 + u2)|∇u||∇η||u|2k−1ηm−1 dx

≥ c

k2

∫
RN
|∇(|u|kηm2 )|2dx− c

∫
RN
|u|2k|∇η|2 dx

≥ c

k2

(∫
Bρ

|u|k·2
∗
dx
)2/2∗

− c

(R− ρ)m

∫
BR

|u|2k dx.

(4.10)

For the right-hand side of (4.6) we have

λ

∫
RN
|u|q−2uϕdx = λ

∫
RN
|u|q−2|u|2kηm dx ≤ c

∫
BR

|u|2k dx. (4.11)

By (4.6)-(4.11), we have(∫
Bρ

|u|2
∗·kdx

)2/2∗

≤ ck2

(R− ρ)m

∫
BR

|u|2kdx for k ≥ 1.

Applying iteration we obtain

‖u‖L∞
R/2

(x) ≤ c‖u‖L2
R(x).

Because ∫
RN\(Mε)δ

u2 dx ≤ cδε6,

we have |u(x)| ≤ cδε3 for x ∈ RN\(Mε)
δ. �

Lemma 4.2 (Profile decomposition). Fix µ and let εn → 0. Assume un ∈ Xεn ,
DΓµ,εn(un) = 0, Γµ,εn(un) ≤ L. Then there exist Uk, rn ∈ X = W 1,m(RN ) ∪
H1(RN ), yn,k ∈ RN such that

un =
∑
k

Uk(· − yn,k) + rn . (4.12)

(1) un(·+ yn,k) ⇀ Uk in X as n→∞.
(2) |yn,k − yn,l| → ∞ as n→∞ for k 6= l.
(3) ‖un‖2H1(RN ) =

∑
k ‖u‖2H1(RN ) + ‖rn‖H1(RN ) + o(1),

‖un‖mW 1,m(RN ) ≥
∑
k ‖Uk‖mW 1,m(RN ) + ‖rn‖mW 1,m(RN ) + o(1).

(4) ‖rn‖Ls(RN ) → 0 as n→∞, 2 < s < 2 · 2∗,
‖un‖sLs(RN ) =

∑
k ‖Uk‖sLs(RN ) + o(1) as n→∞.

Proof. By Lemma 3.2 we know that {un} is bounded in X, so the result of the
lemma follows from [8]. �



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 17

By Proposition 4.1 (2) limn→∞ dist(yn,k,Mεn) < +∞. We denote

y∗k = lim
n→∞

εnyn,k.

Since dist(yn,k,Mε) = ε−1
n dist(εnyn,k,M), we have

dist(y∗n,k,M) = 0, y∗k ∈M. (4.13)

Similar to the proof of [6, Lemma 3.2 and Corollary 3.1 ], we can obtain that the
summation in the profile decomposition (4.12) has only finitely many terms and
there exists m > 0 such that

∫
RN |Un|

q dx ≥ m. Assume that the sequence {un}
has the profile decomposition (4.12). We denote

Ω
(n)
R = RN\

{
∪k BR(yn,k) ∪BR(0)

}
.

Proposition 4.3. There exist c, α, independent of n, such that∫
Ω

(n)
R

Fµ,εn(x, un,∇un) dx ≤ c exp{−αR}, λ
∫

Ω
(n)
R

|un|q dx ≤ c exp{−αR}

where

Fµ,εn(x, z, p) = fµ(p) + kεn(x, z) + hµ(z, p) +
1

2

N∑
i,j=1

βij(z)pipj

+
1

2
E(εnx)z2 +

1

2
ξnχεn(x)z2,

ξn =
(∫

RN
χεn(x)u2

ndx− 1
)β−1

+
.

Proposition 4.4. Assume the profile decomposition (4.12) holds, and denote y∗k =

limn→∞ εnyn,k. Then y∗k ∈ A, i.e., y∗k is a critical point of V in M .

The proofs of Propostions 4.3 and 4.4 are similar to the corresponding results in
[6], and we omit them.

Lemma 4.5. Assume Γµ,ε(u) ≤ L, DΓµ,ε(u) = 0. Then there exist constants
α = α(µ,L) and c = c(µ,L) such that

|u(x)| ≤ c exp{−α dist(x,Mε)}, for x ∈ RN .

Proof. Assume un ∈ Xεn , DΓµ,εn(un) = 0, Γµ,εn(un) ≤ L. By Lemma 3.2, {un} is
bounded in X = W 1,m(RN ) ∩H1(RN ). Suppose the profile decomposition (4.12)
holds,

un =

k0∑
k=1

Uk(· − yn,k) + rn .

By Proposition 4.3, there exist α, c such that∫
RN\{∪k0k=1BR(yn,k)∪BR(0)}

u2
n dx ≤ c exp{−αR} .

By Moser’s iteration,

|un(x)| ≤ c exp{−αR} for x ∈ RN \ {∪k0k=1BR(yn,k) ∪BR(0)} .
Let Rn(x) = min{|x|, |x− yn,k|, k = 1, . . . , k0}. Then

|un(x)| ≤ c exp{−αRn(x)} .



18 R. HE, X. LIU EJDE-2021/05

Since εnyn,k → y∗k ∈ A, k = 1, . . . , k0, for any δ > 0 there exists ε such that for
εn < ε, dist(εnyn,k,A) < δ; hence Rn(x) ≥ dist(x, (Aδ)εn) and

|un(x)| ≤ c exp{−α dist(x, (Aδ)εn)} ≤ c exp{−α dist(x,Mεn)} . (4.14)

�

To prove Lemma 4.8, to apply the regularity theory for elliptic equations (see
[7]), and write down the divergence form of the equation, which is satisfied by the
critical points of the functional Iµ,ε,

Qu = divA(u,∇u) +B(x, u,∇u) = 0, (4.15)

where

Ai(z, p) = Aiµ(z, p) =
∂

∂pi
fµ(p) +

∂

∂pi
hµ(z, p) +

N∑
j=1

βij(z)pj

=
1

2
σ
( |p|
mµ(|p|)

)m−2(
m− (m− 2)

|p|bµ(|p|)
mµ(|p|)

)
pi

+
1

2
σ
( |p|
mµ(|p|)

)m−4(
(m− 2)− (m− 4)

|p|bµ(|p|)
mµ(|p|)

)
z2pi

+

N∑
j=1

βij(z)pj

(4.16)

and

B(x, z, p)

= Bµ(x, z, p) = − ∂

∂z
hµ(z, p)− 1

2

N∑
i,j=1

Dzβij(z)pipj − V (εx)z + λ|z|q−2z

= −σ
( |p|
mµ(|p|)

)m−4

z|p|2 − 1

2

N∑
i,j=1

Dzβij(z)pipj − V (εx)z + λ|z|q−2z .

(4.17)

For µ ∈ (0, 1], we define

gµ(t) = µm−2tm−1 + t, t > 0 . (4.18)

Then

1 ≤
g′µ(t) · t
gµ(t)

≤ m− 1 . (4.19)

We apply the regularity theory for elliptic equations ([7]) to prove Lemma 4.8 and
the theory needs the following two propositions, which are similar to the proof in
[6].

Proposition 4.6. It holds that

(1) p ·A(z, p) ≥ φ(K)gµ(|p|)|p|,
(2) |A(z, p)| ≤ Φ(K)gµ(|p|),
(3) |B(x, z, p)| ≤ Φ(K)(1 + gµ(|p|)|p|)

for x ∈ RN , z ∈ R, |z| ≤ K, p ∈ RN , where φ, Φ are two functions from R+ to R+

such that φ is decreasing and Φ is increasing.



EJDE-2021/05 QUASILINEAR SCHRÖDINGER EQUATIONS 19

Proposition 4.7. Let

aij = aijµ =
∂

∂pj
Aiµ(z, p) .

Then

(1)
∑N
i,j=1 a

ijξiξj ≥ φ(K)
gµ(|p|)
|p| |ξ|

2;

(2) |aij(z, p)| ≤ Φ(K)
gµ(|p|)
|p| ;

(3) |A(z, p)−A(w, p)| ≤ Φ(K)|z − w| gµ(|p|);
(4) |B(x, z, p)| ≤ Φ(K)(1 + gµ(|p|)|p|).

Lemma 4.8. Assume Iµ,ε(u) ≤ L, DIµ,ε(u) = 0. Then there exists a constant
H = H(L) such that

|∇u(x)| ≤ H for x ∈ RN .

Proof. By Proposition 4.1, there exists K = K(L) such that if DIµ,ε(u) = 0 and
Iµ,ε(u) ≤ L, then u is bounded, |u(x)| ≤ K for x ∈ RN . By [7, Corollary 1.5 and
Theorem 1.7], Propositions 4.6 and 4.7, we have

‖u‖C1,β(RN ) ≤ H (4.20)

where β = β(K) ∈ (0, 1), H = H(K), β, H are independent of µ, ε. �

Theorem 4.9. Assume Γµ,ε(u) ≤ L,DΓµ,ε(u) = 0. Then there exists ε = ε(µ,L),
µ = µ(L) such that Γµ,ε(u) = Iε(u) and DIε(u) = 0 if 0 < ε ≤ ε and 0 < µ ≤ µ.

Proof. Assume DΓµ,ε(u) = 0, Γµ,ε(u) ≤ L. By Lemma 4.5 there exist c = c(δ, L),
α = α(δ, L) such that

|u(x)| ≤ c exp{−α dist(x, (Aδ)ε)} ≤ c exp{−α dist(x,Mε)}. (4.21)

Then for ε ≤ ε(µ), we have

|u(x)| ≤ 1

ε
exp{−εdist(x,Mε)} =

1

ε
exp{−dist(εx,M)} for x ∈ RN .

So
mε(x, u(x)) ≡ u(x) for x ∈ RN . (4.22)

Also, if we denote d = dist(Aδ, ∂M), then for x 6∈Mε, we obtain

dist(x, (Aδ)ε) ≥ dist(x,Mε) + dε−1 . (4.23)

Therefore, ∫
RN

χε(x)u2 dx ≤ ε−6

∫
RN\Mε

u2 dx

≤ cε−6

∫
RN\Mε

exp{−2α dist(x, (Aδ)ε)} dx

≤ cε−6

∫
‖x‖≥dε−1

exp{−α|x|} dx

≤ cε−N−5 exp{−αdε−1} → 0 as ε→ 0 .

Moreover, for ε ≤ ε(µ), we have(∫
RN

χε(x)u2 dx− 1
)

+
= 0 . (4.24)

By (4.22) and (4.24), Iµ,ε(u) = Γµ,ε(u) and DIµ,ε(u) = DΓµ,ε(u) = 0. By Propo-
sition 4.1 there exists K = K(L) such that |u(x)| ≤ K for x ∈ RN . By Lemma 4.8



20 R. HE, X. LIU EJDE-2021/05

there exist β = β(K) ∈ (0, 1), H = H(K) such that ‖u‖C1,β(RN ) ≤ H. For

µ ≤ µ(K) := 1
H , we have |∇u(x)| ≤ H ≤ 1

µ for x ∈ RN . Hence mµ(|∇u|) = |∇u|,
Iε(u) = Iµ,ε(u) = Γµ,ε(u) and DIε(u) = DIµ,ε(u) = DΓµ,ε(u) = 0. �

Proof of Theorem 1.1. Given an integer k, for λ > Λk, by Theorem 3.10, the func-
tional Γµ,ε(µ, ε ∈ (0, 1]) has k pairs of sign-changing critical points ±uj(µ, ε),
j = 1, . . . , k, the corresponding critical values satisfy

0 < c1(µ, ε) ≤ · · · ≤ ck(µ, ε) ≤ mk. (4.25)

By Theorem 4.9 there exist µk = µk(mk) and εk = εk(µ,mk) > 0 such that if
0 < µ < µk, 0 < ε < εk, Γµ,ε(u) ≤ mk, and DΓµ,ε(u) = 0, then Γµ,ε(u) = Iε(u),
and DIε(u) = 0.

For 0 < ε < εk, uj,ε = uj(µ, ε), j = 1, . . . , k are critical points of the functional
Iε. Also, by (4.14), for any δ > 0 there exists εk(δ) such that for 0 < ε < εk(δ) it
holds

|uj,ε(x)| ≤ c exp{−α dist(x, (Aδ)ε)}, x ∈ RN ,
so

|vj,ε(x)| ≤ c exp
{
− α

ε
dist(x,Aδ)

}
, x ∈ RN ,

where c = ck(µ,mk). �

References

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equation in RN , Nonlinear Anal., 182 (2019), 113–142.
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Rui He

Department of Mathematics, Yunnan Normal University, Kunming 650500, China
Email address: 493202750@qq.com

Xiangqing Liu (corresponding author)

Department of Mathematics, Yunnan Normal University, Kunming 650500, China
Email address: lxq8u8@163.com


	1. Introduction
	2. Properties of auxiliary functions
	3. Construction of critical points of , 
	4. Proof of Theorem ??
	References