Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 117, pp. 1–17.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.117

CONSTRUCTION OF SINGLE-PEAK SOLUTIONS FOR GRUSHIN

EQUATIONS VIA REDUCTION METHOD

YAWEI WEI, XIAODONG ZHOU

Abstract. In this article, we construct single-peak solutions for a class of Grushin equations

by using the Lyapunov-Schmidt reduction method, whose concentration points locate on some

special critical points of the potential function.

1. Introduction

We consider the degenerate elliptic equation

−ε2∆γu+ a(z)u = uq−1, u > 0, in RN+l,

u ∈ H1,2
γ (RN+l),

(1.1)

where ε > 0 is a small parameter, 2 < q <
2Nγ

Nγ−2 , and a(z) ∈ C2(RN+l) satisfies 0 < a0 ≤ a(z) ≤ a1

in RN+l. Moreover, a0 and a1 satisfy 0 < a0 ≤ 1 ≤ a1, which are the lower and upper bounds of
a(z). The definition of H1,2

γ (RN+l) can be found in (1.7) below.
∆γ is the well-known Grushin operator given by

∆γu(z) = ∆xu(z) + |x|2γ∆yu(z). (1.2)

∆x and ∆y are the Laplace operators in the variable x and y respectively, with z = (x, y) ∈
RN × Rl = RN+l and N > 1, l > 1, N + l ≥ 3. Here, γ ≥ 0 is a real number and

Nγ = N + (1 + γ)l (1.3)

is the appropriate homogeneous dimension. It is worth noting that ∆γ is elliptic for x ̸= 0
and degenerates on {0} × Rl. When γ > 0 is an integer, ∆γ is the Hörmander operator and
is hypoelliptic. More about Hörmander condition and hypoellipticity could be found in [13].
Geometrically, ∆γ comes from a sub-Laplace operator on a nilpotent Lie group of step γ +1 by a
submersion. Specific explanation of geometric framework can be consulted in [3] by Bauer et al.

In [5], the authors studied the existence of solutions for the perturbed Schrödinger equation

−ε2∆u+ V (x)u = up, u > 0, in RN ,

u ∈ H1(RN ),
(1.4)

where ε > 0 is a parameter, N ≥ 3, p ∈ (2, 2N
N−2 ), and V (x) is a potential function in RN .

First, they determined the location of the concentration points for solutions of the above elliptic
problem by local Pohozaev identities. Second, they demonstrated the existence of peak solutions
at the concentration points. Third, they studied the local uniqueness of peak solutions for (1.4)
by applying local Pohozaev identities again. Similarly, in our research we consider the first two
issues for problem (1.1). In particular, the first issue has been well resolved in [25]. Here, we focus
primarily on the second issue for (1.1).

2020 Mathematics Subject Classification. 35J70, 35B40, 35B44.
Key words and phrases. Grushin equation; single-peak solution; Pohozaev identities;

Lyapunov-Schmidt reduction.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 6, 2025. Published December 18, 2025.

1



2 Y. WEI, X. ZHOU EJDE-2025/117

There are important results on the existence of solutions for problem (1.4). An earlier result is
[8], in which Floer and Weinstein obtained a solution concentrated at the global non-degenerate
minimum point when ε is small enough and N = 1, p = 3. Later, Oh [22] generalized Floer-
Weinstein’s results to higher dimension with 2 < p < 2N/(N − 2) and obtained the existence of
positive multi-peak solutions concentrated at any given set of non-degenerate critical points of
V (x) as ε → 0. In fact, the results in [8, 22] seem to rely essentially on the non-degeneracy of
the critical points. Also, Ambrosetti, Badiale, Cingolani [1] proved that there exists a single-peak
solution concentrated at the critical point of V (x) which may be degenerate as ε→ 0 by Lyapunov-
Schmidt reduction. On the other hand, by using variational methods, Rabinowitz [23] proved the
existence of a positive ground solution to (1.4) under some conditions on V (x). Rencently, Li
et al. [20] demonstrated an interesting dichotomy phenomenon for concentrating solutions of the
above Schrödinger equation. And one can refer to [6, 7, 10, 12] for further results.

Authors in [14] constructed the multi-peak solutions for the Chern-Simons-Schrödinger system.
The crucial step to apply implicit function theorem is to prove linear operator Lε restricted to
Eε,Y is invertible, where Lε is the linear operator of Chern-Simons-Schrödinger system

−ε2∆u+ V (x)u+ (A0 +A2
1 +A2

2)u = |u|p−2u, x ∈ R2,

∂1A0 = A2u
2, ∂2A0 = −A1u

2,

∂1A2 − ∂2A1 = −1

2
|u|2, ∂1A1 + ∂2A2 = 0,

(1.5)

and

Eε,Y := {φ ∈ Hε : ⟨
∂U i

ε,yi

∂yil
, φ⟩ε = 0, i = 1, . . . , k, l = 1, 2} (1.6)

is the complementary space of the approximate kernel of Lε. In (1.6),

Hε =
{
u ∈ H1(R2) : ∥u∥2ε =

∫
R2

ε2|∇u|2 + V (x)|u|2dx <∞
}
.

Thus it is vital to find the kernel or the approximate kernel of linear operator for the correspond-

ing equation. The approximate kernel of linear operator for (1.5) is Kε,Y = span{∂Ui
ε,yi

∂yi
l

, i =

1, . . . , k, l = 1, 2} exactly. This is the result for the Laplacian equation, which has been studied
wildly such as in [15, 19]. From [14], the core of finite dimension reduction is that one should first
derive the solution in Eε,Y by using the contraction mapping theorem, and then prove this solution
is the solution in whole space Hε, by using a topological degree theorem, where Hε = Eε,Y ⊕Kε,Y .
Existence of such Laplace equations has been widely studied in many literature. One can refer to
[9, 11, 16, 17, 19, 22, 24] and the references therein.

However there are few results for degenerate elliptic equations in this aspect. Liu and his
collaborators [18] proved that a critical Grushin-type problem has infinitely many positive multi-
bubbling solutions with arbitrarily large energy and cylindrical symmetry by mainly applying the
Lyapunov-Schmidt reduction argument. In this paper, we consider the existence of solutions for
Grushin equation (1.1), which is based on the kernel space of linear operator we have studied in
[26]. Now we introduce some notation and preliminaries that will be used later.

Definition 1.1. When γ ≥ 0, we define

H1,2
γ (RN+l) =

{
u ∈ L2(RN+l) | ∂u

∂xi
, |x|γ ∂u

∂yj
∈ L2(RN+l), i = 1, . . . , N, j = 1, . . . , l.

}
(1.7)

as a weighted Sobolev space.

If u ∈ H1,2
γ (RN+l), we denote the gradient operator as

∇γu = (∇xu, |x|γ∇yu) = (ux1
, . . . , uxN

, |x|γuy1
, . . . , |x|γuyl

), (1.8)

and

|∇γu|2 = |∇xu|2 + |x|2γ |∇yu|2. (1.9)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 3

Then H1,2
γ (RN+l) is a Hilbert space, endowed with the inner product

⟨u, v⟩γ =

∫
RN+l

∇γu · ∇γv + uvdz, (1.10)

and the corresponding norm

∥u∥H1,2
γ (RN+l) =

(∫
RN+l

|∇γu|2 + |u|2dz
)1/2

. (1.11)

Remark 1.2. For studying problem (1.1), we define a new norm in H1,2
γ (RN+l) as follows:

∥u∥ε :=
(∫

RN+l

ε2|∇γu|2 + a(z)u2dz
)1/2

, (1.12)

which is induced by the inner product

⟨u, v⟩ε :=
∫
RN+l

ε2∇γu · ∇γv + a(z)uvdz, ∀u, v ∈ H1,2
γ (RN+l). (1.13)

We define

∥u∥1 :=
(∫

RN+l

ε2|∇γu|2 + u2dz
)1/2

. (1.14)

By our assumptions: 0 < a0 ≤ a(z) ≤ a1 and 0 < a0 ≤ 1 ≤ a1, we can find that ∥u∥ε and ∥u∥1
are equivalent norms in H1,2

γ (RN+l).

Definition 1.3. Let the following be a new distance on RN+l:

d(z, 0) =
( 1

(1 + γ)2
|x|2+2γ + |y|2

) 1
2+2γ

(1.15)

for z = (x, y) ∈ RN+l, and set

B̃r(0) := {z = (x, y) ∈ RN+l| d(z, 0) < r}. (1.16)

be a ball in the sense of this new distance.

Definition 1.4. We say that uε is a single-peak solution of Equation (1.1) concentrated at z0 =
(x0, y0) if uε satisfies

(i) uE has a local maximum point zε = (xε, yε) ∈ RN+l such that zε → z0 ∈ RN+l,as ε→ 0;
(ii) For any given τ > 0, there exists R≫ 1 such that

|uε(z)| ≤ τ for z ∈ RN+l\B̃Rε(zε);

(iii) There exists M > 0 such that

uε ≤M.

Remark 1.5. When performing blow-up analysis for the Grushin operator, the blow up points
must lie on the set {x = 0}, otherwise the term |x|γ would change. For example, in [2], for u ∈
D1,2

γ (RN+l) and ρ > 0, a rescaled sequence of functions of the form ue,ρ(z) := ρ
Nγ−2

2 u(ρx, ρ1+γy+

e) is also defined, where z = (x, y) ∈ RN+l and e ∈ Rl. In which, D1,2
γ (RN+l) is the closure

of C∞
0 (RN+l) with respect to the norm ∥v∥D1,2

γ (RN+l) :=
( ∫

RN+l |∇γv|2dz
)1/2

. Therefore, it is

reasonable to assume that the maximum value points of all solutions are located in {x = 0}. So
hereafter, we assume that the maximum point is in the form of zε = (0, yε).

Now we state our main result. For (1.1), we want to find a solution of the form

uε(z) = Uε,zε(z) + ωε(z), (1.17)

where Uε,zε(z) is the solution of

−ε2∆γu+ a(zε)u = uq−1, u > 0, in RN+l,

u(zε) = max
z∈RN+l

u(z), u ∈ H1,2
γ (RN+l),

(1.18)



4 Y. WEI, X. ZHOU EJDE-2025/117

and

zε → z0, as ε→ 0, where zε = (0, yε), z0 = (0, y0). (1.19)

We regard Uε,zε(z) as an approximate solution to Equation (1.1) and ωε(z) is a minor term in the
sense that

∥ωε∥ε = O(ε
Nγ
2 +1). (1.20)

Then our result is the following.

Theorem 1.6. Assume that z0 = (0, y0) are critical points of a(z) satisfying

deg
(
∇ya(z), B̃δ(z0), 0

)
̸= 0. (1.21)

Then there exists ε0 > 0, such that for any ε ∈ (0, ε0], (1.1) has solutions in the form of

uε(z) = Uε,zε(z) + ωε(z), (1.22)

for some zε = (0, yε) with zε → z0 and ∥ωε∥ε = O(ε
Nγ
2 +1), where Uε,zε(z) is the solution of (1.18).

The main contributions of this paper are summarized in the following three points. First, we
apply the Lyapunov-Schmidt reduction argument to degenerate elliptic equations. Second, we
construct solutions of (1.1) in the form of (1.22), where Uε,zε(z) is obtained by moving plane
method basically due to [4] we have studied. Third, we construct solutions of (1.1) in the form
of (1.22), where ωε(z) is obtained by reduction method and is based on the kernel space of linear
operator that we have studied in [26].

This article is organized as follows. In Section 2, we obtain the location of concentration points
for solutions of (1.1) by using Pohozaev identities generated from translations, which has been
studied by us in [25]. In Section 3, we get ready to construct single peak solutions of (1.1), which
uses the studies on the properties of main term in [4] and the kernel space in [26]. In Section 4,
we solve a finite dimension problem by applying the Lyapunov-Schmidt reduction argument. In
Section 5, we prove Theorem 1.6 to derive the existence of single-peak solutions for (1.1) by using
the topological degree method.

2. Locating the concentration points

The main tool to determine the location of the concentration points is the local Pohozaev
identities generated from translations for solutions of the following degenerate elliptic problem

−∆γu = f(z, u), in Ω, (2.1)

where Ω is a bounded open domain in RN+l and f : RN+l ×R → R is a continuous function. Let

F (z, u) =

∫ u

0

f(z, s)ds, (2.2)

then we have

Fxi
(z, u) =

∂F (z, u)

∂xi
+ f(z, u)

∂u

∂xi
, i = 1, . . . , N ; (2.3)

Fyj
(z, u) =

∂F (z, u)

∂yj
+ f(z, u)

∂u

∂yj
, j = 1, . . . , l. (2.4)

Let D ⊂⊂ Ω, we have established two Pohozaev identities generated from translations for
solutions of Equation (2.1) in [25] as follows.

Proposition 2.1 ([25, Theorem 1.1]). If u ∈ H1,2
γ (Ω) ∩ C2(Ω) is the solution of Equation (2.1),

then u satisfies

1

2

∫
∂D

|∇γu|2νixdS −
∫
∂D

( ∂u
∂νx

+ |x|2γ ∂u
∂νy

) ∂u
∂xi

dS − γ

∫
D

|∇yu|2|x|2(γ−1)xidz

=

∫
∂D

F (z, u)νixdS −
∫
D

∂F (z, u)

∂xi
dz, i = 1, . . . , N,

(2.5)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 5

and
1

2

∫
∂D

|∇γu|2νjydS −
∫
∂D

( ∂u
∂νx

+ |x|2γ ∂u
∂νy

) ∂u
∂yj

dS

=

∫
∂D

F (z, u)νjydS −
∫
D

∂F (z, u)

∂yj
dz, j=1,. . . ,l,

(2.6)

where ν = (νx, νy) is the unit outward normal of the point of ∂D.

More generally, we consider the degenerate elliptic equation with Grushin type p-sub-Laplacian,

−∆p
γu = f(z, u), in Ω, (2.7)

where Ω is a bounded open domain in RN+l and f : RN+l ×R → R is a continuous function. For
any p > 1, Grushin type p-sub-Laplacian is denoted as

∆p
γu = divγ(|∇γu|p−2∇γu). (2.8)

Different from the method in [25], by applying domain variations we obtain the local Pohozaev
identities of translating type for solutions of Equation (2.7) in [27] as follows.

Proposition 2.2 ([27, Theorem 1.1]). If u ∈ W 1,p
λ,0 (Ω) ∩ C1(Ω) is the solution of equation (2.7),

then u satisfies

1

p

∫
∂D

|∇γu|pνixdS −
∫
∂D

|∇γu|p−2 ∂u

∂xi
⟨∇γu, νγ⟩dS −

∫
D

|∇γu|p−2γ|x|2(γ−1)xi|∇yu|2dz

=

∫
∂D

F (z, u)νixdS −
∫
D

∂F (z, u)

∂xi
dz, i = 1, . . . , N,

(2.9)

and
1

p

∫
∂D

|∇γu|pνjydS −
∫
∂D

|∇γu|p−2 ∂u

∂yj
⟨∇γu, νγ⟩dS

=

∫
∂D

F (z, u)νjydS −
∫
D

∂F (z, u)

∂yj
(z, u)dz, j = 1, . . . , l,

(2.10)

where ν = (νx, νy) is the unit outward normal of the point of ∂D and νγ = (νx, |x|γνy).

Especially, taking p = 2 in the above proposition, we return to the results in Proposition 2.1.
It follows from (2.5) and (2.6) that any solution u of Equation (1.1) in H1,2

γ (RN+l) ∩ C1(RN+l)
satisfies

1

2
ε2

∫
∂D

|∇γu|2νixdS − ε2
∫
∂D

( ∂u
∂νx

+ |x|2γ ∂u
∂νy

) ∂u
∂xi

dS − ε2γ

∫
D

|∇yu|2|x|2(γ−1)xidz

=
1

q

∫
∂D

uqνixdS − 1

2

∫
∂D

a(z)u2νixdS +
1

2

∫
D

∂a(z)

∂xi
u2dz, i = 1, . . . , N,

(2.11)

and
1

2
ε2

∫
∂D

|∇γu|2νjydS − ε2
∫
∂D

( ∂u
∂νx

+ |x|2γ ∂u
∂νy

) ∂u
∂yj

dS

=
1

q

∫
∂D

uqνjydS − 1

2

∫
∂D

a(z)u2νjydS +
1

2

∫
D

∂a(z)

∂yj
u2dz, j = 1, . . . , l,

(2.12)

where D is any domain of RN+l.
In this paper, our solution is assumed to satisfy Definition 1.4. Next, we determine the location

of the concentration point z0. For this purpose, we have estimated the behavior of the solution
uε and several surface integrals away from the maximum point zε in the following proposition by
using maximum principles for unbounded domain in [4] and L2 estimate for Grushin operator in
[21] respectively.

Proposition 2.3 ([25, Lemma 2.1]). Assume that uE is a solution of Equation (1.1) satisfying
Definition 1.4 in a subset of H1,2

γ (RN+l). Then for any α ∈ (0, infz∈RN+l a(z)), there exists a
constant C > 0, such that

uε(z) ≤ Ce−
√

αd(z,zε)
ε , ∀z ∈ RN+l, (2.13)



6 Y. WEI, X. ZHOU EJDE-2025/117

where C = (M + 1)e
√
αR. Moreover, if we denote

J1 =

∫
∂B̃δ(zε)

|∇γuε|2νixdS,

J2 =

∫
∂B̃δ(zε)

(∂uε
∂νx

+ |x|2γ ∂uε
∂νy

)∂uε
∂xi

dS,

J3 =

∫
∂B̃δ(zε)

|∇γuε|2νjydS,

J4 =

∫
∂B̃δ(zε)

(∂uε
∂νx

+ |x|2γ ∂uε
∂νy

)∂uε
∂yj

dS,

then we have

Ji ≤ C̃

∫
B̃δ(zε)

e−
√

αd(z,zε)
ε dz, i = 1, . . . , 4, (2.14)

where δ > 0 and C̃ > 0.

By using Pohozaev identities (2.11) and (2.12) together with Proposition 2.3, we have proved
that the location of the concentration point z0 is at some special critical points of the potential
function a(z) in the following proposition.

Proposition 2.4 ([25, Corollary 1.1]). Let zE be the local maximum point of the solution uE of
Equation (1.1) satisfying Definition 1.4 in a subset of H1,2

γ (RN+l). If a(z) has critical points in
the form of (0, y), then z0 = (x0, y0) satisfies ∇a(z0) = 0 and x0 = 0.

By now, we obtain a necessary condition for the concentration point z0 of the solution satisfying
Definition 1.4 for Equation (1.1), which basically due to our researches in [25]. In next section,
we will consider the converse of such problem, the existence of such solutions concentrating at the
points satisfying the necessary condition.

3. Preparations for constructing single peak solutions

In this section, we shall get ready to construct single peak solutions satisfying Definition 1.4
for Equation (1.1). More specifically, for (1.1), we want to find a solution of the form

uε(z) = Uε,zε(z) + ωε(z),

where Uε,zε(z) is the solution of (1.18). We regard Uε,zε(z) as an approximate solution to Equation
(1.1) and ωε(z) is a minor term in the sense that

∥ωε∥ε = O(ε
Nγ
2 +1).

Let w(z̃) be the solution of

−∆γw + w = wq−1, w > 0, in RN+l,

w(0) = max
z̃∈RN+l

w(z̃), w ∈ H1,2
γ (RN+l).

(3.1)

The properties of solutions for this equation is very clear. Actually in [4], we have showed the
partial radial symmetry and the decay rate at infinity of solutions to the equation (3.1) in the
following proposition.

Proposition 3.1 ([4, Theorem 1.1]). Let w(z̃) ∈ H1,2
γ (RN+l)∩C0(RN+l) be a solution to Equation

(3.1) with q > 2. Then w(z̃) = w(x̃, ỹ) is radially symmetric with respect to the variable ỹ and has
exponential decay at infinity.

By using scaling transformation, we can easily find that the relationship between Uε,zε(z) and
w(z̃) is

Uε,zε(z) =
(
a(zε)

) 1
q−2w

(√a(zε)

ε
x,

(√a(zε)

ε

)1+γ

(y − yε)
)
. (3.2)

Thus, the major term Uε,zε(z) is founded by Proposition 3.1 and relation (3.2). Next, we should
find the minor term ωε(z) in (1.17).



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 7

It is easy to find that ωε(z) satisfies

L′
εωε = l′ε +R′

ε(ωε), z ∈ RN+l,

ωε ∈ H1,2
γ (RN+l),

(3.3)

where

L′
εωε = −ε2∆γωε(z) + a(z)ωε(z)− (q − 1)Uq−2

ε,zε (z)ωε(z), (3.4)

l′ε = (a(zε)− a(z))Uε,zε(z), (3.5)

R′
ε(ωε) = (Uε,zε(z) + ωε(z))

q−1 − Uq−1
ε,zε (z)− (q − 1)Uq−2

ε,zε (z)ωε(z). (3.6)

To employ the theory of weighted Sobolev spaces, we write the problem (3.3) in its weak form
as

Lεωε = lε +Rε(ωε), z ∈ RN+l,

ωε ∈ H1,2
γ (RN+l),

(3.7)

where Lε : H
1,2
γ (RN+l) → H1,2

γ (RN+l) is a bounded linear operator defined by

⟨Lεωε, ψ⟩ε =
∫
RN+l

ε2∇γωε(z) · ∇γψ(z) + a(z)ωε(z)ψ(z)− (q − 1)Uq−2
ε,zε (z)ωε(z)ψ(z)dz, (3.8)

lε ∈ H1,2
γ (RN+l) satisfies

⟨lε, ψ⟩ε =
∫
RN+l

(a(zε)− a(z))Uε,zε(z)ψ(z)dz, (3.9)

and Rε : H
1,2
γ (RN+l) → H1,2

γ (RN+l) is a nonlinear map defined by

⟨Rε(ωε), ψ⟩ε =
∫
RN+l

[
(Uε,zε(z) + ωε(z))

q−1 − Uq−1
ε,zε (z)− (q − 1)Uq−2

ε,zε (z)ωε(z)
]
ψdz, (3.10)

for any ψ ∈ H1,2
γ (RN+l). And Rε(ωε) satisfies

Rε(ωε) = o(ωε), as ωε → 0. (3.11)

Linear operator Lε in (3.8) is not always invertible, so we cannot directly use contraction
mapping theorem to derive ωε from (3.7). To construct a single peak solution for Equation (1.1),
we need to find out a kernel space or an approximate kernel of L′

ε.
Fortunately in [26], we have showed that the approximate kernel Kε of L′

ε in H1,2
γ (RN+l) is

given by

Kε := span
{∂Uε,zε(z)

∂y1
,
∂Uε,zε(z)

∂y2
, . . . ,

∂Uε,zε(z)

∂yl

}
, (3.12)

which is the kernel of linear operator

L̃εωε := −ε2∆γωε + a(zε)ωε − (q − 1)Uq−2
ε,zε ωε. (3.13)

We define

Eε = K⊥
ε := {ω ∈ H1,2

γ (RN+l) : ⟨ω, ∂Uε,zε(z)

∂yj
⟩ε = 0, j = 1, . . . , l}. (3.14)

In [26], we have proved that the linear operator LE is invertible when restricted to the space Eε

in the following proposition.

Proposition 3.2 ([4, Theorem 1.3]). Let Lε be the linear operator defined in (3.8), Kε be the
kernel space of linear operator (3.13) defined in (3.12), Eε be the the complement space of Kε

defined in (3.14), Qε be the projection from H1,2
γ (RN+l) to Eε as follows

Qεu = u−
l∑

j=1

bj
∂Uε,zε(z)

∂yj
, (3.15)

where Uε,zε(z) is defined in (3.2) and bj satisfies

⟨Qεu,
∂Uε,zε(z)

∂yk
⟩ε = 0, for k = 1, . . . , l. (3.16)



8 Y. WEI, X. ZHOU EJDE-2025/117

Then there exist ε0 > 0, θ0 > 0 and ρ > 0, such that for any ε ∈ (0, ε0] and zε ∈ B̃θ0(z0), QεLε is
a bijective mapping in Eε, moreover

∥QεLεωε∥ε ≥ ρ∥ωε∥ε, ∀ωε ∈ Eε. (3.17)

The above proposition plays an essential role in carrying out the reduction argument. In next
section, we will carry out the reduction for equation (3.7).

4. Reduction argument

In this section, we are now ready to carry out the reduction for (3.7). That is , we first consider
the equation (3.7) restricted to Eε to obtain the solution, and then prove that this solution holds
on the whole space H1,2

γ (RN+l). To use the the contraction mapping theorem to carry out the
reduction, we now estimate ∥lε∥ε and ∥Rε(ωε)∥ε.

Lemma 4.1. Under the assumption

a0 ≤ min{1, a(z)}, (4.1)

we have

∥lε∥ε = O
(
ε

Nγ
2

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

) )
. (4.2)

Proof. According to Remark 1.2, the assumption (4.1) indicates ∥η∥1 and ∥η∥ε are equivalent
norms in H1,2

γ (RN+l). For convenience, we denote them all as ∥η∥ε. Recall that

⟨lε, η⟩ε =
∫
RN+l

(
a(zε)− a(z)

)
Uε,zε(z)η(z)dz. (4.3)

Thus we have

|⟨lε, η⟩ε|

=

∣∣∣∣∫
RN+l

(
a(zε)− a(z)

)
Uε,zε(z)η(z)dz

∣∣∣∣
≤

(∫
RN+l

(
a(zε)− a(z)

)2
U2
ε,zε(z)dz

)1/2 (∫
RN+l

η2(z)dz

)1/2

≤
(∫

RN+l

(
a(zε)− a(z)

)2
U2
ε,zε(z)dz

)1/2 (∫
RN+l

ε2|∇γη(z)|2 + η2(z)dz

)1/2

(4.1)
=

(∫
RN+l

(
a(zε)− a(z)

)2
U2
ε,zε(z)dz

)1/2

∥η∥ε

(4.4)

On the other hand, according to (3.2), we let

x̃ =

√
a(zε)

ε
x, ỹ =

(√a(zε)

ε

)1+γ

(y − yε), (4.5)

then we derive

Uε,zε(z) =
(
a(zε)

) 1
q−2w(x̃, ỹ), (4.6)

a(z) = a(
ε√
a(zε)

x̃, (
ε√
a(zε)

)1+γ ỹ + yε). (4.7)

By Taylor expansion,

a(z)− a(zε) = ∇a(zε) · (z − zε) +O
(
(d(z, zε))

2
)

= ∇xa(zε) ·
ε√
a(zε)

x̃+∇ya(zε) · (
ε√
a(zε)

)1+γ ỹ +O
( ε2

a(zε)
(d(z̃, 0))2

)
.

(4.8)

Since ω(z̃) has exponential decay at infinity by Proposition 3.1, for any α > 0, it holds that∫
RN+l

(d(z̃, 0))
α
w2(z̃)dz̃ ≤ C < +∞. (4.9)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 9

Continuing with the last row of (4.4), we have

|⟨lε, η⟩ε|

≤
(∫

RN+l

(
a(zε)− a(z)

)2
U2
ε,zε(z)dz

)1/2

∥η∥ε

≤
(∫

RN+l

(
|∇xa(zε)|

ε√
a(zε)

|x̃|+ |∇ya(zε)|(
ε√
a(zε)

)1+γ |ỹ|+ ε2

a(zε)
(d(z̃, 0))2

)2

×
(
a(zε)

) 2
q−2w2(z̃)

( ε√
a(zε)

)Nγ

dz̃
)1/2

∥η∥ε

≤ Cε
Nγ
2

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

)
∥η∥ε.

(4.10)

Therefore, (4.2) holds. □

Lemma 4.2. We have the estimate

∥Rε(ωε)∥ε = O
(
εNγ(1−min{2,q−1}+1

2 )∥ωε∥min{2,q−1}
ε

)
. (4.11)

Proof. Recall that

⟨Rε(ωε), η⟩ε =
∫
RN+l

[(
Uε,zε(z) + ωε(z)

)q−1 − Uq−1
ε,zε (z)− (q − 1)Uq−2

ε,zε (z)ωε(z)
]
ηdz. (4.12)

We denote

R̂ε(ωε) =
(
Uε,zε(z) + ωε(z)

)q−1 − Uq−1
ε,zε (z)− (q − 1)Uq−2

ε,zε (z)ωε(z). (4.13)

First, we consider the case of 2 < q ≤ 3. Note that for any a > 0 and b > 0, if p < 1, one has

(a+ b)p < ap + bp. (4.14)

By the mean value theorem and the above inequality, we have

R̂ε(ωε) = (q − 1)
(
Uε,zε(z) + θωε(z)

)q−2
ωε(z)− (q − 1)Uq−2

ε,zε (z)ωε(z)

≤ (q − 1)
(
Uq−2
ε,zε (z) + θq−2ωq−2

ε (z)
)
ωε(z)− (q − 1)Uq−2

ε,zε (z)ωε(z)

= (q − 1)θq−2ωq−1
ε (z)

= O(ωq−1
ε (z)),

(4.15)

where θ ∈ (0, 1). Thus we obtain

|⟨Rε(ωε), η⟩ε| =
∣∣∣ ∫

RN+l

R̂ε(ωε)η(z)dz
∣∣∣

≤ C

∫
RN+l

|ωε(z)|q−1|η(z)|dz

≤ C
(∫

RN+l

|ωε(z)|qdz
) q−1

q
(∫

RN+l

|η(z)|qdz
)1/q

(4.16)

Next according to the fact that the embedding H1,2
γ (RN+l) ↪→ Lp(RN+l) is continue for every

p ∈ [2, 2∗γ ] and by blow up, we can obtain the relationship between norm ∥η∥Lp(RN+l) and norm
∥η∥1 (we still denote ∥η∥1 by ∥η∥ε for convenience). Let

x̃ =
x

ε
, ỹ =

y − yε
ε1+γ

, (4.17)

and

b(z̃) = η(εx̃, ε1+γ ỹ + yε).



10 Y. WEI, X. ZHOU EJDE-2025/117

Then we derive ∫
RN+l

|η(z)|qdz = εNγ

∫
RN+l

|b(z̃)|qdz̃

≤ CεNγ

(∫
RN+l

|∇γb(z̃)|2 + |b(z̃)|2dz̃
)q/2

= CεNγ

(∫
RN+l

(ε2|∇γη(z)|2 + |η(z)|2) 1

εNγ
dz

)q/2

= CεNγ(1− q
2 )
(∫

RN+l

ε2|∇γη(z)|2 + |η(z)|2dz
)q/2

.

(4.18)

Continuing with the last row of (4.16), we have

|⟨Rε(ωε), η⟩ε| ≤ C
(∫

RN+l

|ωε(z)|qdz
) q−1

q
(∫

RN+l

|η(z)|qdz
)1/q

(4.18)

≤ CεNγ(1− q
2 )

q−1
q

(∫
RN+l

ε2|∇γωε(z)|2 + |ωε(z)|2dz
) q

2 ·
q−1
q

× εNγ(1− q
2 )

1
q

(∫
RN+l

ε2|∇γη(z)|2 + |η(z)|2dz
) q

2 ·
1
q

= CεNγ(1− q
2 )∥ωε∥q−1

ε ∥η∥ε.

(4.19)

Now, we consider the case of q > 3. Note that for any a > 0 and b > 0, if p > 2, one has

(a+ b)p = ap + pap−1b+
p(p− 1)

2
ap−2b2 +O(bp). (4.20)

According to (4.20), we have(
Uε,zε(z) + ωε(z)

)q−1
= Uq−1

ε,zε (z) + (q − 1)Uq−2
ε,zε (z)ωε(z)

+
(q − 1)(q − 2)

2
Uq−3
ε,zε (z)ω

2
ε(z) +O

(
ωq−1
ε (z)

)
.

(4.21)

Then

R̂ε(ωε) ≤ CUq−3
ε,zε (z)ω

2
ε(z) + Cωq−1

ε (z). (4.22)

By using the Hölder’s inequality and (4.18), we obtain∣∣∣ ∫
RN+l

Uq−3
ε,zε (z)ω

2
ε(z)η(z)dz

∣∣∣
≤

(∫
RN+l

Uq
ε,zε(z)dz

) q−3
q
(∫

RN+l

|ωε(z)|qdz
)2/q(∫

RN+l

|η(z)|qdz
)1/q

≤ CεNγ(1− q
2 )

q−3
q ∥Uε,zε∥q−3

ε · εNγ(1− q
2 )

2
q ∥ωε∥2ε · ε

Nγ(1− q
2 )

1
q ∥η∥ε

= Cε−
Nγ
2 ∥ωε∥2ε∥η∥ε,

(4.23)

which uses

∥Uε,zε∥ε = O
(
ε

Nγ
2

)
. (4.24)

As same as (4.19), we can obtain∣∣∣ ∫
RN+l

ωq−1
ε (z)η(z)dz

∣∣∣ ≤ CεNγ(1− q
2 )∥ωε∥q−1

ε ∥η∥ε, (4.25)

and it can be absorbed in (4.23).
By combining (4.19) and (4.23), we complete the proof. □

Now we use the contraction mapping theorem to carry out the reduction. Namely, we solve the
problem

QεLεωε = Qεlε +QεRε(ωε), ωε ∈ Eε. (4.26)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 11

Lemma 4.3. There exists ε0 > 0, such that for any ε ∈ (0, ε0], and any zε with zε → z0, there is
a unique solution ωε ∈ Eε satisfying (4.26). In addition, we have the estimate

∥ωε∥ε ≤ C∥lε∥ε ≤ Cε
Nγ
2

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

)
. (4.27)

Proof. According to Proposition 3.2, QεLε is invertible on Eε, we can rewrite (4.26) as

ωε = (QεLε)
−1Qεlε + (QεLε)

−1QεRε(ωε) := Aωε. (4.28)

It follows from Proposition 3.2 and Lemma 4.1 that

∥(QεLε)
−1Qεlε∥ε ≤ C∥lε∥ε ≤ Cε

Nγ
2 +1. (4.29)

Let

B := {ωε ∈ Eε : ∥ωε∥ε ≤ ε
Nγ
2 +1−µ}, (4.30)

where µ > 0 is a fixed small constant.
We will apply the contraction mapping theorem in ball B.
(i) A maps from B to B. According to Lemma 4.1 and Lemma 4.2, for any ωε ∈ B, it holds

∥Aωε∥ε ≤ C∥lε∥ε + C∥Rε(ωε)∥ε

≤ Cε
Nγ
2 +1 + CεNγ(1−min{2,q−1}+1

2 )∥ωε∥min{2,q−1}
ε

≤ Cε
Nγ
2 +1 + CεNγ(1−min{2,q−1}+1

2 )ε(
Nγ
2 +1−µ)min{2,q−1}

= Cε
Nγ
2 +1 + Cε

Nγ
2 +(1−µ)min{2,q−1}

≤ 1

2
ε

Nγ
2 +1−µ +

1

2
ε

Nγ
2 +1−µ

= ε
Nγ
2 +1−µ.

(4.31)

(ii) A is a contraction map. For any ωε,1, ωε,2 ∈ B,

∥Aωε,1 −Aωε,2∥ε ≤ C∥Rε(ωε,1)−Rε(ωε,2)∥ε. (4.32)

By (4.12), we have

⟨Rε(ωε,1)−Rε(ωε,2), η⟩ε =
∫
RN+l

(
R̂ε(ωε,1)− R̂ε(ωε,2)

)
η(z)dz, (4.33)

where

R̂ε(ωε,1)− R̂ε(ωε,2) =
(
Uε,zε(z) + ωε,1(z)

)q−1 −
(
Uε,zε(z) + ωε,2(z)

)q−1

− (q − 1)Uq−2
ε,zε (z)

(
ωε,1(z)− ωε,2(z)

)
.

(4.34)

First, we consider the case of 2 < q ≤ 3. By the mean value theorem and the inequality (4.14),
we have

R̂ε(ωε,1)− R̂ε(ωε,2)

= (q − 1)
[(
Uε,zε(z) + ωε,2(z) + θ

(
ωε,1(z)− ωε,2(z)

))q−2

− Uq−2
ε,zε (z)

](
ωε,1 − ωε,2

)
≤ (q − 1)

[
(θωε,1(z))

q−2
+ ((1− θ)ωε,2(z))

q−2
](
ωε,1(z)− ωε,2(z)

)
≤ (q − 1)

(
ωq−2
ε,1 (z) + ωq−2

ε,2 (z)
) (
ωε,1(z)− ωε,2(z)

)
,

(4.35)



12 Y. WEI, X. ZHOU EJDE-2025/117

where θ ∈ (0, 1). Thus we have

|⟨Rε(ωε,1)−Rε(ωε,2), η⟩ε|

≤ (q − 1)

∫
RN+l

(
|ωε,1(z)|q−2 + |ωε,2(z)|q−2

)
|ωε,1(z)− ωε,2(z)||η(z)|dz

Hólder
≤ (q − 1)

((∫
RN+l

|ωε,1(z)|qdz
) q−2

q

+
(∫

RN+l

|ωε,2(z)|qdz
) q−2

q
)

×
(∫

RN+l

|ωε,1(z)− ωε,2(z)|qdz
)1/q(∫

RN+l

|η(z)|qdz
)1/q

(4.18)

≤ CεNγ(1− q
2 )
(
∥ωε,1(z)∥q−2

ε + ∥ωε,2(z)∥q−2
ε

)
∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε

(4.30)

≤ CεNγ(1− q
2 )ε(

Nγ
2 +1−µ)(q−2)∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε

= Cε(1−µ)(q−2)∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε

≤ 1

2
∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε.

(4.36)

Next, we consider the case of q > 3. Applying (4.21) and according to (4.34), we can obtain

R̂ε(ωε,1)− R̂ε(ωε,2) ≤
(q − 1)(q − 2)

2
Uq−3
ε.zε (z)

(
ω2
ε,1(z)−ω2

ε,2(z)
)
+C

(
ωq−1
ε,1 (z)− ωq−1

ε,2 (z)
)
. (4.37)

Moreover, it holds that

|R̂ε(ωε,1)− R̂ε(ωε,2)| ≤ CUq−3
ε.zε (z)

(
|ωε,1(z)|+ |ωε,2(z)|

)
|ωε,1(z)− ωε,2(z)|

+ C
(
|ωε,1(z)|q−2 + |ωε,2(z)|q−2

)
|ωε,1(z)− ωε,2(z)|.

(4.38)

Because the second term of (4.38) is the same as in (4.36), we estimate the first term of (4.38).∫
RN+l

Uq−3
ε.zε (z)

(
|ωε,1(z)|+ |ωε,2(z)|

)
|ωε,1(z)− ωε,2(z)||η(z)|dz

Hölder
≤

(∫
RN+l

Uq
ε.zε(z)dz

) q−3
q
((∫

RN+l

|ωε,1(z)|q
)1/q

+
(∫

RN+l

|ωε,2(z)|q
)1/q)

×
(∫

RN+l

|ωε,1(z)− ωε,2(z)|qdz
)1/q(∫

RN+l

|η(z)|qdz
)1/q

(4.18)

≤ Cε−
Nγ
2 (∥ωε,1(z)∥ε + ∥ωε,2(z)∥ε) ∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε.

(4.30)

≤ Cε1−µ∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε

≤ 1

2
∥ωε,1(z)− ωε,2(z)∥ε∥η(z)∥ε,

(4.39)

which uses (4.24).
Combining the expressions above, we have proved

∥Aωε,1 −Aωε,2∥ε ≤
1

2
∥ωε,1(z)− ωε,2(z)∥ε, ∀ωε,1, ωε,2 ∈ B. (4.40)

By contraction mapping theorem, we conclude that there exists ε0 > 0, such that for any ε ∈ (0, ε0],
and any zε with zε → z0, there is a unique ωε ∈ Eε, which depend ε and zε, satisfying

QεLεωε = Qεlε +QεRε(ωε). (4.41)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 13

At last, similar to (4.31), we obtain that

∥ωε∥ε = ∥Aωε∥ε
≤ C∥lε∥ε + C∥Rε(ωε)∥ε

≤ C∥lε∥ε + CεNγ(1−min{2,q−1}+1
2 )∥ωε∥min{2,q−1}

ε

(4.30)

≤ C∥lε∥ε + CεNγ(1−min{2,q−1}+1
2 )

(
ε

Nγ
2 +1−µ

)min{2,q−1}−1

∥ωε∥ε

= C∥lε∥ε + Cε(1−µ)(min{2,q−1}−1)∥ωε∥ε.

(4.42)

Taking ε > 0 small enough, such that Cε(1−µ)(min{2,q−1}−1) < 1
2 , gives that

∥ωε∥ε ≤ C∥lε∥ε ≤ Cε
Nγ
2

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

)
. □

5. Existence of single peak solutions

In this section, we are committed to proving Theorem 1.6 to obtain a solution in the form of
(1.17) for equation (1.1). Lemma 4.3 tells us that

Qε (Lεωε − lε −Rε(ωε)) = 0, (5.1)

i.e., for some constants aj ,

Lεωε − lε −Rε(ωε) =

l∑
j=1

aj
∂Uε,zε(z)

∂yj
. (5.2)

In next step, we need to choose the appropriate zε = (0, yε), such that aj = 0, which is
dependent in zε. Therefore, if we take zε such that

⟨Lεωε − lε −Rε(ωε),
∂Uε,zε(z)

∂yj
⟩ε = 0, for j = 1, . . . , l, (5.3)

the right-hand side of (5.2) must be equal to 0; then we achieve our goal. Therefore, if zε satisfies∫
RN+l

ε2∇γuε(z) · ∇γ
∂Uε,zε(z)

∂yj
+ a(z)uε(z)

∂Uε,zε(z)

∂yj
− uq−1

ε (z)
∂Uε,zε(z)

∂yj
dz = 0, (5.4)

for j = 1, . . . , l, then we have aj = 0, j = 1, . . . , l.

Proof of Theorem 1.6. We only need to solve zε from (5.4) by using degree theorem. The main
idea is that we simply the LHS of (5.4) and find major term with Uε,zε(z), then we prove that the
influence of ωε(z) is negligible and will not destroy the major term without ωε(z). Now we insert
Uε,zε(z) into the LHS of (5.4), then we have∫

RN+l

ε2∇γUε,zε(z) · ∇γ
∂Uε,zε(z)

∂yj
+ a(z)Uε,zε(z)

∂Uε,zε(z)

∂yj
− Uq−1

ε,zε (z)
∂Uε,zε(z)

∂yj
dz

:= I1 + I2 − I3.

(5.5)

For first term, since Uε,zε(z) is the solution of (1.18), we have

−ε2∆γUε,zε(z) + a(zε)Uε,zε(z) = Uq−1
ε,zε (z). (5.6)

Taking the derivative with respect to yj on both sides of (5.6) gives

−ε2∆γ
∂Uε,zε(z)

∂yj
+ a(zε)

∂Uε,zε(z)

∂yj
= (q − 1)Uq−2

ε,zε (z)
∂Uε,zε(z)

∂yj
. (5.7)

From (5.7), the symmetry of Uε,zε(z) with respect to the second variable and let

ỹ = (

√
a(0)

ε
)1+γy, r = |ỹ|, (5.8)



14 Y. WEI, X. ZHOU EJDE-2025/117

we obtain

I1 =

∫
RN+l

ε2∇γUε,zε(z) · ∇γ
∂Uε,zε(z)

∂yj
dz

=

∫
RN+l

(q − 1)Uq−1
ε,zε (z)

∂Uε,zε(z)

∂yj
− a(zε)Uε,zε(z)

∂Uε,zε(z)

∂yj
dz

=

∫
RN+l

[
(q − 1)

((
a(0)

) 1
q−2w(

√
a(0)

ε
x, r)

)q−1

−
(
a(0)

) q−1
q−2w(

√
a(0)

ε
x, r)

]
× C

∂w(

√
a(0)

ε x, r)

∂r

yj
r
ε(1+γ)(l−1) dx dy = 0,

(5.9)

where C = C(a(0)) and we use the fact that
∫
RN+l f(x, |y|)yjdz = 0, f(x, |y|) is radially symmetric

with respect to the variable y.
Similarly, for the third term, we obtain

I3 =

∫
RN+l

Uq−1
ε,zε (z)

∂Uε,zε(z)

∂yj
dz

=

∫
RN+l

((
a(0)

) 1
q−2w(

√
a(0)

ε
x, r)

)q−1

×
(
a(0)

) 1
q−2

∂w(

√
a(0)

ε x, r)

∂r
(

√
a(0)

ε
)1+γ yj

r

( ε√
a(0)

)(1+γ)l

dx dy = 0.

(5.10)

For the second term, let (4.5), and denote b(z̃) := a( ε√
a(zε)

x̃, ( ε√
a(zε)

)1+γ ỹ + yε), we derive

I2 =

∫
RN+l

a(z)Uε,zε(z)
∂Uε,zε(z)

∂yj
dz

=
1

2

∫
RN+l

a(z)
∂U2

ε,zε(z)

∂yj
dz

= −1

2

∫
RN+l

∂a(z)

∂yj
U2
ε,zε(z)dz +O(e−

τ
ε )

(4.5)
= −1

2

∫
RN+l

∂b(z̃)

∂ỹj

(√a(zε)

ε

)1+γ(
a(zε)

) 2
q−2w2(z̃)

( ε√
a(zε)

)Nγ

dz̃ +O(e−
τ
ε )

= −1

2

(
a(zε)

)α

εNγ−1−γ

∫
RN+l

∂b(z̃)

∂ỹj
w2(z̃)dz̃ +O(e−

τ
ε )

Taylor
= −1

2
(a(zε))

α
εNγ−1−γ ∂a(z)

∂yj

∣∣∣
z=zε

∫
RN+l

w2(z̃)dz̃ +O
(
εNγ−γ

)
,

(5.11)

where ∫
RN+l

w2(z̃)dz̃ ≤ C < +∞,

α =
2

q − 2
+

1 + γ −Nγ

2
.

By combining (5.9), (5.10), and (5.11), we obtain∫
RN+l

ε2∇γUε,zε(z) · ∇γ
∂Uε,zε(z)

∂yj
+ a(z)Uε,zε(z)

∂Uε,zε(z)

∂yj
− Uq−1

ε,zε (z)
∂Uε,zε(z)

∂yj
dz

= −1

2
(a(zε))

α
εNγ−1−γ ∂a(z)

∂yj

∣∣∣
z=zε

∫
RN+l

w2(z̃)dz̃ +O
(
εNγ−γ

)
.

(5.12)



EJDE-2025/117 SINGLE-PEAK SOLUTIONS FOR GRUSHIN EQUATIONS 15

Next we prove that the influence of ωε(z) is negligible and will not destroy the major term
without ωε(z). We insert Uε,zε(z) + ωε(z) into the LHS of (5.4), then we have∫

RN+l

ε2∇γ

(
Uε,zε(z) + ωε(z)

)
· ∇γ

∂Uε,zε(z)

∂yj
+ a(z)

(
Uε,zε(z) + ωε(z)

)∂Uε,zε(z)

∂yj

−
(
Uε,zε(z) + ωε(z)

)q−1 ∂Uε,zε(z)

∂yj
dz

:= J1 + J2 − J3.

(5.13)

For the first two terms, we have

J1 + J2

=

∫
RN+l

ε2∇γ

(
Uε,zε(z) + ωε(z)

)
· ∇γ

∂Uε,zε(z)

∂yj
+ a(z)

(
Uε,zε(z) + ωε(z)

)∂Uε,zε(z)

∂yj

=

∫
RN+l

ε2∇γUε,zε(z) · ∇γ
∂Uε,zε(z)

∂yj
+ a(z)Uε,zε(z)

∂Uε,zε(z)

∂yj

+

∫
RN+l

ε2∇γωε(z) · ∇γ
∂Uε,zε(z)

∂yj
+ a(z)ωε(z)

∂Uε,zε(z)

∂yj

(5.9)
= 0 +

∫
RN+l

a(z)Uε,zε(z)
∂Uε,zε(z)

∂yj
dz + ⟨ωε(z),

∂Uε,zε(z)

∂yj
⟩ε

=

∫
RN+l

a(z)Uε,zε(z)
∂Uε,zε(z)

∂yj
dz

(5.11)
= −1

2
(a(zε))

α
εNγ−1−γ ∂a(z)

∂yj

∣∣∣
z=zε

∫
RN+l

w2(z̃)dz̃ +O
(
εNγ−γ

)
,

(5.14)

which we have used the fact ⟨ωε(z),
∂Uε,zε (z)

∂yj
⟩ε = 0 since ωε ∈ Eε and

∂Uε,zε (z)
∂yj

∈ Kε.

Finally, we estimate the last term

J3 =

∫
RN+l

(
Uε,zε(z) + ωε(z)

)q−1 ∂Uε,zε(z)

∂yj
dz

=

∫
RN+l

Uq−1
ε,zε (z)

∂Uε,zε(z)

∂yj
+ (q − 1)Uq−2

ε,zε (z)ωε(z)
∂Uε,zε(z)

∂yj
dz

+

O
( ∫

RN+l ω
q−1
ε (z)

∂Uε,zε (z)
∂yj

dz
)
, 2 < q ≤ 3,∫

RN+l U
q−3
ε,zε (z)ω

2
ε(z)

∂Uε,zε (z)
∂yj

dz +O
( ∫

RN+l ω
q−1
ε (z)

∂Uε,zε (z)
∂yj

dz
)
, q > 3.

(5.15)

In which as for (5.10), ∫
RN+l

Uq−1
ε,zε (z)

∂Uε,zε(z)

∂yj
dz = 0. (5.16)

Similar to (4.10) and by using (1.20), we have

(q − 1)

∫
RN+l

Uq−2
ε,zε (z)

∂Uε,zε(z)

∂yj
ωε(z)dz

(5.7)
=

∫
RN+l

ε2∇γ
∂Uε,zε(z)

∂yj
· ∇γωε(z) + a(zε)

∂Uε,zε(z)

∂yj
ωε(z)dz

= ⟨ωε(z),
∂Uε,zε(z)

∂yj
⟩ε +

∫
RN+l

(a(zε)− a(z))
∂Uε,zε(z)

∂yj
ωε(z)dz

=

∫
RN+l

(a(zε)− a(z))
∂Uε,zε(z)

∂yj
ωε(z)dz

= O
(
ε

Nγ
2 −1−γ

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

)
∥ωε∥ε

)
(1.20)
= O

(
εNγ−γ

(
ε|∇xa(zε)|+ ε1+γ |∇ya(zε)|+ ε2

))
.

(5.17)



16 Y. WEI, X. ZHOU EJDE-2025/117

For the other terms,∫
RN+l

∂Uε,zε(z)

∂yj
ωq−1
ε (z)dz

Hölder
≤

(∫
RN+l

(∂Uε,zε(z)

∂yj

)q

dz
)1/q(∫

RN+l

ωq
ε(z)dz

) q−1
q

(4.18)

≤ CεNγ(1− q
2 )∥∂Uε,zε

∂yj
∥ε∥ωε∥q−1

ε

≤ CεNγ−γ+q−2,

(5.18)

which uses (1.20) and

∥∂Uε,zε

∂yj
∥ε = O

(
ε

Nγ
2 −1−γ

)
. (5.19)

This equation is proved in [26, Proposition 4.1]. Similarly, we conclude that∫
RN+l

Uq−3
ε,zε (z)ω

2
ε(z)

∂Uε,zε(z)

∂yj
dz

Hölder
≤

(∫
RN+l

Uq
ε,zε(z)dz

) q−3
q
(∫

RN+l

ωq
ε(z)dz

)2/q(∫
RN+l

(∂Uε,zε(z)

∂yj

)q

dz
)1/q

(4.18)

≤ CεNγ(1− q
2 )∥Uε,zε∥q−3

ε ∥ωε∥2ε∥
∂Uε,zε

∂yj
∥ε

≤ CεNγ−γ+1,

(5.20)

which uses (1.20), (4.24) and (5.19).
From (5.14)-(5.20), we have proved that∫

RN+l

ε2∇γ

(
Uε,zε(z) + ωε(z)

)
∇γ

∂Uε,zε(z)

∂yj
+ a(z)

(
Uε,zε(z) + ωε(z)

)∂Uε,zε(z)

∂yj

−
(
Uε,zε(z) + ωε(z)

)q−1 ∂Uε,zε(z)

∂yj
dz.

=

∫
RN+l

ε2∇γUε,zε(z)∇γ
∂Uε,zε(z)

∂yj
+ a(z)Uε,zε(z)

∂Uε,zε(z)

∂yj
− Uq−1

ε,zε (z)
∂Uε,zε(z)

∂yj
dz

+ εNγ−γ−1O(εmin{1,q−1}).

(5.21)

By (5.4), (5.12), and (5.21), we know that

∇ya(zε) = O(εmin{1,q−1}). (5.22)

According to our assumption deg
(
∇ya(z), B̃δ(z0), 0

)
̸= 0, there exists such a zε which satisfies

(5.22). Moreover, d(zε, z0) = O(εmin{1,q−1}). This completes the proof. □

Acknowledgments. This work is supported by the NSFC under the grands 12271269 and the
Fundamental Research Funds for the Central Universities.

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Yawei Wei

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China.
orcid: 0000-0002-9743-917X

Email address: weiyawei@nankai.edu.cn

Xiaodong Zhou
School of Mathematical Sciences, Nankai University, Tianjin 300071, China.

orcid: 0009-0000-7962-6521
Email address: 1120210030@mail.nankai.edu.cn