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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 PARAMETERDEPENDENT QUASILINEAR SCHRO¨ DINGER 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 Schr¨odinger equations, under a certain parametric conditions.

1. Introduction

In this article, we study the existence of localized nodal solutions for the parameter-dependent semiclassical quasilinear Schr¨odinger equation

N
ε2

1 Dj (bij (v)Div) − Dzbij (v)DivDj v

− V (x)v + λ|v|q−2v = 0,

2

i,j=1

(1.1)

v(x) → 0 as |x| → ∞,

where x ∈ RN , ε > 0 is a small parameter, Div = ∂∂xvi , Dzbij(z) = ddz 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

N

c−(1 + z2)|ξ|2 ≤

bij(z)ξiξj ≤ c+(1 + z2)|ξ|2

i,j=1

for z ∈ R, ξ = (ξi) ∈ RN ;

(A3) there exists δ > 0 such that

N

N

δ

bij (z)ξiξj ≤

i,j=1

i,j

1 bij(z) + 2 zDzbij(z) ξiξj ≤ q

1 −δ
2

N
bij (z)ξiξj
i,j=1

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−0 1, for x ∈ RN ;

2010 Mathematics Subject Classiﬁcation. 35B05, 35B45. Key words and phrases. Quasilinear Schro¨dinger equation; perturbation method; truncation technique; nodal solution. c 2021 Texas State University. Submitted October 2, 2020. Published January 25, 2021.
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(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 |x − y| < δ ,
y∈B
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, , 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 dist(x, Aδ) , for x ∈ RN , j = 1, . . . , k. ε
For small ε and 4 < q < 2 · 2∗, the authors in  established the existence of a sequence 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 perturbation developed in  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

1 Dj(bij(u)Diu) − 2 Dzbij(u)DiuDju

− V (εx)u + λ|u|q−2u = 0,

(1.2)

u(x) → 0 as |x| → ∞.

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

N RN i,j=1

1 bij(u)DiuDjϕ + 2 Dzbij(u)DiuDjuϕ

dx +

V (εx)uϕ dx
RN

= λ |u|q−2uϕ dx
RN

for ϕ ∈ C0∞(RN ). Formally problem (1.2) has a variational structure, given by the functional

1 Iε(u) = 2

N

1

bij(u)DiuDju dx +

RN i,j=1

2

V (εx)u2 dx − λ

RN

q

|u|q dx,
RN

u∈Y ,

where

Y = u : u ∈ H1(RN ), u2|∇u|2 dx < +∞ .
RN

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Now we deﬁne a truncation function and a perturbed functional. Let ϕ ∈ C0∞(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),

z
mµ(x, z) = bµ(x, τ )dτ.
0

(1.3)

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

bµ(z) = bµ(0, z) = ϕ(µz),

z
mµ(z) = mµ(0, z) = bµ(τ )dτ .
0

(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

1 Iµ,ε(u) = 2 σ RN

|∇u|

m−2
|∇u|2

dx

+

1 σ

mµ(|∇u|)

2 RN

|∇u|

m−4
u2|∇u|2 dx

mµ(|∇u|)

1

N

1

λ

+

βij(u)DiuDju dx +

V (εx)u2 dx −

|u|q dx

2 RN i,j=1

2 RN

q RN

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. Moreover, 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 Schr¨odinger equations, we refer the
reader to [1, 5, 10, 11]. Let ζ ∈ C0∞(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 =σ
2 RN

|∇u|

m−2
|∇u|2

dx

+

1 σ

mµ(|∇u|)

2 RN

u

m−2
u2 dx

mε(x, u)

1 +σ
2 RN

|∇u| mµ(|∇u|)

m−4
u2|∇u|2

dx

+

1

2

N
βij(u)DiuDju dx
RN i,j=1

1 +

E(εx)u2 dx + 1

2 RN

χε(x)u2 dx − 1

β

λ −

|u|q dx

RN

+ q RN

for u ∈ Xε = Wε1,m(RN ) ∩ H1(RN ), 2 < β < q, and

Wε1,m(RN ) = W 1,m(RN ) ∩ Lm ε (RN ),

where Lm ε (RN ) is a weighted Lm-spaces

(1.5)

Lm ε (RN ) = u ∈ Lm(RN ), exp{(m − 2) dist(εx, M )}|u|m dx < +∞
RN

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endowed with the norm u Lm(RN ) =
ε

1
exp{(m − 2) dist(εx, M )}|u|m dx m
RN

with a coercive weight. Then we know the space Wε1,m(RN ) is compactly imbedded to Lp(RN ) for m ≤ p < m∗ = NN−mm , in particular the imbedding into Lq(RN ) is compact. If

1 |u(x)| ≤ exp{− dist(εx, M )}
ε

for x ∈ RN and

χε(x)u2 dx < 1,

RN

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 constants, 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  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(sx,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) = 12 σ mµ|p(||p|) m−2|p|2. Then

(3.1) (3.2) (3.3)
(3.4)

c1(1 + µm−2|p|m−2)|p|2 ≤ fµ(p) ≤ c2(1 + µm−2|p|m−2)|p|2;

2fµ(p) ≤ ∇pfµ(p) · p ≤ |∇fµ(p)| · |p| ≤ mfµ(p);

N

∂2

i,j=1 ∂p ∂p

fµ(p)ξiξj ≥ σ

|p| m (|p|)

m−2|ξ|2 ≥ c(1 + µm−2|p|m−2)|ξ|2;

ij

µ

| ∂p∂i∂2pj fµ(p)| ≤ c mµ|p(||p|) m−2 ≤ c(1 + µm−2|p|m−2).

(4) We deﬁne kε(x, z) = 12 σ mε(zx,z) m−2z2. Then

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(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) ≤

∂ ∂z

k

ε

(

x,

z

)z

≤ mkε(x, z);

(4.3)

∂2 ∂z2 kε(x, z) ≥ σ

z

m−2

mµ(x, z)

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

(4.4)

∂2

z

m−2

0 ≤ ∂z2 kε(x, z) ≤ c mµ(x, z)

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

(5) We deﬁne hµ(z, p) = 21 σ mµ|p(||p|) m−4z2|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 ∂2 hµ(z, p)ξiξj ≥ σ |p| m−4z2|ξ|2

i,j=1 ∂pi∂pj

mµ(|p|)

≥ c(1 + µm−4|p|m−4)z2|ξ|2,

(5.4)

∂2

|p| m−4 2

m−4 m−4 2

∂z2 hµ(z, p) = σ mµ(|p|) |p| ≥ c(1 + µ |p| )|p| ;

∂p∂i∂2pj hµ(z, p) ≤ c mµ|p(||p|) m−4z2 ≤ c(1 + µm−4|p|m−4)z2, ∂∂z22 hµ(z, p) ≤ c mµ|p(||p|) m−4|p|2 ≤ c(1 + µm−4|p|m−4)|p|2,

∂ ∇p ∂z hµ(z, p)

∂ = ∂z ∇phµ(z, p)

|p| m−4

≤c

|z| |p|

mµ(|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,

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|∇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 − cν µ−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

1 Jµ,ε(u) = 2 σ RN

|∇u|

m−2
|∇u|2dx

+

1 σ

mµ(|∇u|)

2 RN

u mε(x, u)

1 +σ
2 RN

|∇u|

m−4
u2|∇u|2dx

mµ(|∇u|)

1

N

1

+

βij(u)DiuDjudx +

E(εx)u2dx .

2 RN i,j=1

2 RN

m−2
u2dx

Then for u, v, ϕ ∈ Xε, we have: (1)

(2.1)

DJµ,ε(u) − DJµ,ε(v), u − v

≥ cµm−2 |∇u − ∇v|mdx + cεm−2 exp{(m − 2) dist(εx, M )}|u − v|m dx

RN

RN

+ c |∇u − ∇v|2 dx − cµ−2 (|u|m−2 + |v|m−2)(u − v)2 dx

RN

RN

− cµ−2 (u − v)2 dx
RN
≥ cµ,ε u − v m Wε1,m(RN ) + c

u−v

2 H1(RN )

− cµ−2 (|u|m−2 + |v|m−2)(u − v)2 dx − cµ−2 (u − v)2dx ,

RN

RN

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(2)

| DJµ,ε(u) − DJµ,ε(v), ϕ |

≤ c u − v H1(RN ) ϕ H1(RN )

+c

u + v m−2

W

1 ε

,m

(R

N

)

m−2

W

1,m ε

(R

N

)

u−v

W

1,m ε

(R

N

)

ϕ

Wε1,m(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  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 .
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 sup f (x), j = 1, 2, . . . ,
A∈Γj x∈A\W
Kc = {x : Df (x) = 0, f (x) = c}, Kc∗ = Kc \ W .
Theorem 3.1. Assume (A7)–(A11) hold. Then (1) cj ≥ c∗, Kc∗j = ∅ . (2) cj → ∞, as j → ∞. (3) If cj = cj+1 = · · · = cj+k−1 = c, then γ(Kc∗) ≥ 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).

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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)ϕ dx

∂ + RN ∇phµ(u, ∇u)∇ϕ + ∂z hµ(u, ∇u)ϕ dx

N

1

+

βij (u)DiuDj ϕ + Dzβij (u)DiuDj uϕ

RN i,j=1

2

dx +

E(εx)uϕ dx
RN

β−1

+

χε(x)u2 dx − 1

χε(x)uϕ dx − λ |u|q−2uϕ dx.

RN

+

RN

RN

Since the imbedding from Wε1,m(RN ) to Lq(RN ) is compact, there exists c > 0 such that

u

Lq(RN ) ≤ c

u

W

1,m ε

(R

N

)

.

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)

= fµ(∇un) dx + kε(x, un) dx + hµ(un, ∇un) dx

RN

RN

RN

1

N

1

+

βij (un)DiunDj un dx +

E(εx)u2n dx

2 RN i,j=1

2 RN

1 +

χε(x)u2ndx − 1 β+ − λq

|un|q dx
N

RN

R

≥ c µm−2 |∇un|mdx + εm−2 exp{(m − 2) dist(εx, M )}|un|m dx

RN

RN

+ µm−4 |∇un|m−2u2n dx + c
RN

(1 + u2n)|∇un|2dx +
RN

β

+c

χε(x)u2n dx − 1 − c uqn dx

RN

+

RN

u2n dx
RN

(3.1)

≥ c µm−2 |∇un|m dx + εm−2 exp{(m − 2) dist(εx, M )}|un|m dx

RN

RN

+ µm−4 |∇un|m−2u2ndx + c
RN

(1 + u2n)|∇un|2dx + u2ndx

RN

RN

β

+c

χε(x)u2n dx − 1

RN

+

q/m

−c

|∇un|m dx + exp{(m − 2) dist(εx, M )}|un|m dx

RN

RN

which implies that {un} is bounded in Xε and RN χε(x)u2ndx − 1 β+ is bounded. Assume un u in Xε and un → u in Ls(RN ), 2 ≤ s < 2 · 2∗. By Lemma 2.3, we

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have

o(1) = DΓµ,ε(un) − DΓµ,ε(um), un − um

= DJµ,ε(un) − DJµ,ε(um), un − um

β−1

+

χε(x)u2n dx − 1

χε(x)un(un − um) dx

RN

+

RN

β−1

χε(x)u2m dx − 1

χε(x)un(un − um) dx

RN

+

RN

− (|un|q−2un − |um|q−2um)(un − um) dx
RN

≥ c µm−2 |∇un − ∇um|m dx
RN

+ εm−2 exp{(m − 2) dist(εx, M )}|un − um|m dx
RN

+ c |∇un − ∇um|2 dx − cµ−2
RN

|un|m−2 + |um|m−2 |un − um|2 dx
RN

+ |un − um|2 dx + o(1)
RN
≥ c un − um m Wε1,m(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 +

β−1

χε(x)u2 dx − 1

χε(x)vϕ dx

RN

+

RN

+ cµ (|v|m−2v + v)ϕ dx
RN

= λ |u|q−2uϕ dx + cµ (|u|m−2u + u)ϕ dx, for ϕ ∈ Xε

RN

RN

(3.2)

and

Jµ,ε(u) =
RN
1 +
2

fµ(∇u) + kε(x, ∇u) + hµ(u, ∇u) dx

N

1

βij(u)DiuDju dx +

RN i,j=1

2

E(εx)u2 dx,
RN

(3.3) for u ∈ Xε.

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 , we can prove the following lemmas 3.3–3.6.

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

N RN i,j=1

1 βij(u) + 2 uDzβij(u)

DiuDju dx +

E(εx)u2 dx ≥ D
RN

α
|u|q dx .
RN

10

R. HE, X. LIU

EJDE-2021/05

Now we deﬁne

1 Q = Qδ = u ∈ Xε : 2 D

uq+ dx
RN

1 + 2 cµ

u2+ dx < δ ,
N

R

1 P = −Q = u ∈ Xε : 2 D

uq− dx
RN

1 + 2 cµ

u2− dx < δ .
N

R

α m−1 + m cµ
α m−1 + m cµ

um + dx
RN
um − dx
RN

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 Wε1,m(RN ) + u − v 2H1(RN ) . (2)

DΓµ,ε(u), ϕ for all ϕ ∈ Xε.

≤c

u + v m−2
Wε1,m(RN )

m−2

W

1,m ε

(R

N

)

u−v

W

1,m ε

(R

N

)

ϕ

Wε1,m(RN )

+c 1+

β−1

χε(x)u2dx − 1

RN

+

u − v H1(RN ) ϕ H1(RN )

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

1 Γµ,ε(u) − 2q DJµ,ε(u) − DJµ,ε(v), u

1

1

= Γµ,ε(u) − 2q DJµ,ε(u), u + 2q DJµ,ε(v), u

1

1∂

= RN fµ(∇u) − 2q ∇pfµ(∇u)∇u dx + RN kε(x, u) − 2q ∂z kε(x, u)u dx

1

+ RN hµ(u, ∇u) − 2q ∇phµ(u, ∇u)∇u + ∂z hµ(u, ∇u)u dx

N
+
RN i,j=1

1

1

1

2 βij(u) − 2q βij(u) + 2 uDzβij(u)

DiuDju dx

11 +−
2 2q

E(εx)u2 dx
RN

1 +

χε(x)u2 dx − 1 β − 1

RN

+ 2q

β−1

χε(x)u2 dx − 1

χε(x)uv dx

RN

+

RN
Nodal SolutionsExistenceSolutionsSetsAssumption