Ground state sign-changing solutions for a class of Schr

https://doi.org/10.4134/JKMS.j200497 pISSN: 0304-9914 / eISSN: 2234-3008

GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHR ¨ ODINGER-POISSON-KIRCHHOFF TYPE

PROBLEMS WITH A CRITICAL NONLINEARITY IN R ³

Aixia Qian and Mingming Zhang

Abstract. In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schr¨ odinger- Poisson-Kirchhoff system



 

 

− (1+b Z

R ³

|∇u| ² dx)4u+V (x)u+k(x)φu = λf (x)u+|u| ⁴ u, in R ³ ,

− 4φ = k(x)u ² , in R ³ ,

where b > 0, V (x), k(x) and f (x) are positive continuous smooth func- tions; 0 < λ < λ 1 and λ 1 is the first eigenvalue of the problem −4u + V (x)u = λf (x)u in H. With the help of the constraint variational method, we obtain that the Schr¨ odinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ 1 . Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.

1. Introduction

In this paper, we consider the existence of ground state sign-changing solu- tions for the following Schr¨ odinger-Poisson-Kirchhoff system

(SKP)







− (1+b Z

R ³

|∇u| ² dx)4u+V (x)u+k(x)φu = λf (x)u+|u| ⁴ u, in R ³ ,

− 4φ = k(x)u ² , in R ³ ,

where b > 0 and V (x), k(x), f (x) are nonnegative, 0 < λ < λ ₁ and λ ₁ is the first eigenvalue of the problem −4u + V (x)u = λf (x)u in H (see (1.2) for the definition of the space H). For more mathematical and physical interpretation of the problem (SKP), we refer to [1, 8, 10, 12–14, 16, 23] and the references therein.

Received September 2, 2020; Accepted January 22, 2021.

2010 Mathematics Subject Classification. Primary 35J05, 35J20, 35J60.

Key words and phrases. Schr¨ odinger-Poisson-Kirchhoff system, ground state sign- changing solution, critical nonlinearity, nonlocal term.

This work was financially supported by the Shandong Science Foundation ZR2020MA005.

c

2021 Korean Mathematical Society

1181

When b ≡ 0 and k(x) ≡ 0 in the system (SKP), it reduces to the classic semi- linear elliptic problem. Weth, Bartsch and Willem [2] obtained a ground state sign-changing solution. Remarkably, the system (SKP) is nonlocal because of the term k(x)φu and b( R

R 3 |∇u| ² dx)4u, which make the problem more compli- cated. This phenomenon provokes some mathematical difficulties, but it makes the study of the system (SKP) particulary interesting.

For the following Schr¨ odinger-Poisson-Kirchhoff system

(1.1)







− (1 + b Z

R ³

|∇u| ² dx)4u + u + k(x)φu = g(x, u), in R ³ ,

− 4φ = k(x)u ² , in R ³ ,

Wang and Zhou [20] obtained a sign-changing solution in the case when b = 0, k is a constant, and the nonlinearity g(u) = |u| ^p−1 u satisfies 4-superlinear, subcritical growth condition on u. Huang, Rocha and Chen [11] studied the existence of sign-changing solutions for (1.1) with b = 0, k(x) = 0 and critical growth condition. If b = 0, k(x) 6= 0, Zhong and Tang [26] studied the existence of ground state sign-changing solutions for the system (1.1) with critical growth.

If b 6= 0, k(x) = 0, Chen and Tang [10] studied the existence and the asymptotic behavior of ground state sign-changing solutions for the system (1.1). However, the above work obtained a sign-changing solution only in the case that the system (1.1) does not involve the nonlocal term or the nonlinearity does not satisfy the critical growth condition.

Comparing with the previous works, we investigate the existence of ground state sign-changing solutions for (SKP) with a critical nonlinearity. In this paper, we denote φ(u) = R

R ³ φ u u ² dx, therefore the system (SKP) possesses two nonlocal terms k(x)φ _u u and b( R

R ³ |∇|u|)4u. We adopt an idea from [26]

and [10]. However, [26] and [10] has only one non-local term and [10] is not the case of critical growth, so our construction methods are different from them. Moreover, regarding the existence of the ground state and sign-changing solutions for the Schr¨ odinger-Poisson-Kirchhoff systems with critical growth, to the best of our knowledge, few works concern on this up to now.

To avoid much details for checking the compactness, for V (x) not being a constant, we always assume that the potential function V (x) satisfies

(V ) V ∈ C(R ³ , R ⁺ ) such that H ⊂ H ¹ (R ³ ) and for 2 < s < 6 the embedding H ,→ L ^s (R ³ ) is compact, where

(1.2) H :=







H _r ¹ (R ³ ) = {u ∈H ¹ (R ³ ) : u(x) = u(|x|)}, if V (x) is a constant, {u ∈D ^1,2 (R ³ ) :

Z

R ³

(|∇u| ² +V (x)u ² )dx < ∞}, if V (x) is not a constant, with norm

kuk :=

Z

R ³

(|∇u| ² + V (x)u ² )dx

 ¹ ₂

;

H ¹ (R ³ ) := {u ∈ L ² (R ³ ) : ∇u ∈ L ² (R ³ )}

with norm

kuk H ¹ :=

Z

R ³

(|∇u| ² + u ² )dx

 ¹ ₂

; D ^1,2 (R ³ ) := {u ∈ L ^{2 ∗} (R ³ ) : ∇u ∈ L ² (R ³ )}

with norm

kuk _D 1,2 :=

Z

R ³

|∇u| ² dx

 ¹ ₂ . As usual, for 1 ≤ s < +∞,

kuk s :=

Z

R ³

|u| ^s dx

 ¹ _s

, u ∈ L ^s (R ³ ).

We first give assumptions about f (x) and k(x).

(K) k ∈ L ^p (R ³ ) ∩ L ^∞ (R ³ ) \ {0} for some p ∈ [2, +∞) and k(x) > 0 for ∀x ∈R ³ . (f 1 ) f ∈ L ^{3 2} (R ³ ) \ {0} and f (x) > 0 for ∀x ∈R ³ .

(f 2 ) there exist ρ > 0 and α > 0 such that f (x) ≥ C|x| ^−α for |x| < ρ.

Let S be the best Sobolev constant for the embedding of D ^1,2 (R ³ ) in L ^{2 ∗} (R ³ ).

Particularly,

(1.3) S := inf

D ^1,2 (R ³ )\{0}

R

R ³ |∇u| ² dx ( R

R ³ u ⁶ dx) ^{1 3} , |u| ² ₆ ≤ S ⁻¹ Z

R ³

|∇u| ² dx.

It is well known that S is achieved by the Talenti function [22]

(1.4) u _ = () ^{1 4}

( + |x| ² ) ^{1 2} ∈ D ^1,2 (R ³ ).

We can also get that R

R ³ |∇u  | ² dx = |u  | ⁶ ₆ = S ^{3 2} .

As is well known to us, for u ∈ H, Lax-Milgram theorem implies that there exists a unique φ _u ∈ D ^1,2 (R ³ ) such that

(1.5)

Z

R ³

∇φ u ∇v = Z

R ³

k(x)u ² vdx for ∀v ∈ D ^1,2 (R ³ ), that is, φ _u is the weak solution of −∆φ = k(x)u ² . Moreover we have

φ _u (x) = Z

R ³

k(x)u ² (y) 4π|x − y| dy, (1.6) T φ _u (u) =

Z

R ³

k(x)φ u u ² dx = 1 4π

Z

R ³

Z

R ³

k(x)k(y)u ² (x)u ² (y)

|x − y| dxdy, clearly φ u (x) ≥ 0 for any x ∈ R ³ .

Define the energy functional I λ on the space H by I λ (u) = 1

2 kuk ² + b 4

Z

R ³

|∇u| ² dx

 2

+ 1 4

Z

R ³

k(x)φ u u ² dx (1.7)

− λ 2 Z

R ³

f (x)u ² dx − 1 6 Z

R ³

u ⁶ dx.

Then the I _λ is well defined on H and belongs to C ¹ (H, R). For each u, v ∈ H, we have

hI _λ ⁰ (u), vi = (1 + b|∇u| ² ₂ ) Z

R ³

∇u∇vdx + Z

R ³

V (x)uvdx + Z

R ³

k(x)φ _u uvdx

− λ Z

R ³

f (x)uvdx − Z

R ³

|u| ⁴ uvdx.

(1.8)

Critical points of the functional I λ correspond to the weak solutions for nonlocal problem (SKP). Furthermore, if u ∈ H is a critical point of I λ , (u, φ u ) is a solution of the system (SKP). Since φ u (x) ≥ 0, then (u, φ u ) is a sign-changing solution of the system (SKP) if and only if u is a critical point of I λ and u ^± 6= 0, where

u ⁺ (x) = max{u(x), 0}, u ⁻ (x) = min{u(x), 0}.

Next we give an essential decomposition, which is useful in finding the ground state sign-changing solutions of the system (SKP). Firstly, it follows from (1.8) and Fubini theorem that

T φ _u+ (u ⁻ ) = Z

R ³

k(x)φ _u + |u ⁻ | ² dx = Z

R ³

k(x)φ _u ⁻ |u ⁺ | ² dx = T φ _u− (u ⁺ ),

(1.9) T φ _u (u) = Z

R ³

k(x)φ u u ² dx = T φ _u+ (u ⁺ ) + T φ _u− (u ⁻ ) + 2T φ _u+ (u ⁻ ).

Then by a simple calculation, we can obtain (1.10) I λ (u) = I λ (u ⁺ ) + I λ (u ⁻ ) + b

2 k∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + 1

2 T φ _u+ (u ⁻ ), (1.11) hI _λ ⁰ (u), u ⁺ i = hI _λ ⁰ (u ⁺ ), u ⁺ i + T φ _u+ (u ⁻ ) + bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ , (1.12) hI _λ ⁰ (u), u ⁻ i = hI _λ ⁰ (u ⁻ ), u ⁻ i + T φ _u+ (u ⁻ ) + bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ . Clearly, when b = 0 and k(x) = 0, the system (SKP) doesn’t depend on the nonlocal term R

R ³ |∇u| ² dx4u and k(x)φ _u u any more, that is, it becomes (1.13)

( − 4u + V (x)u = λf (x)u + |u| ⁴ u, in R ³ , u ∈ H,

with energy functional (1.14) I λ (u) = 1

2 kuk ² − λ 2 Z

R ³

f (x)u ² dx − 1 6

Z

R ³

|u| ⁶ dx, ∀u ∈ H.

It is well defined on H and is of C ¹ with

I _λ (u) = I _λ (u ⁺ ) + I _λ (u ⁻ ), hI _λ ⁰ (u), u ⁺ i = hI _λ ⁰ (u ⁺ ), u ⁺ i, (1.15) hI _λ ⁰ (u), u ⁻ i = hI _λ ⁰ (u ⁻ ), u ⁻ i.

From (1.10), (1.11), (1.12) and (1.15), we can see that there are some essen-

tial differences in studying the sign-changing solutions of the problem (SKP)

between b = 0, k(x) = 0 and b 6= 0, k(x) 6= 0, because of the so called

nonlocal term R

R ³ |∇u| ² dx4u and k(x)φ _u u. Therefore, the methods of finding sign-changing solutions for the problem (1.13) seem to be not applicable to the system (SKP). In other words, our methods are different from those used in Zhang [24] and Huang etc [11]. Similarly, Zhong and Tang [26] studied the special case of (SKP), their methods of finding sign-changing solutions is also not applicable to the present problem.

In order to obtain the existence results of sign-changing solutions, we will consider the following minimization problem

M λ := {u ∈ H : u ^± 6= 0, hI _λ ⁰ (u), u ⁺ i = hI _λ ⁰ (u), u ⁻ i = 0}, m _λ := inf{I _λ (u) : u ∈ M _λ }.

We will show that the minimizer of M _λ is corresponding to a sign-changing solution for the system (SKP). If (u, φ _u ) is a sign-changing solution of the system (SKP), one can get u ∈ M _λ , and it is easier to study I _λ on M _λ .

Another aim of the article is to prove that the energy of any sign-changing solution in H of the system (SKP) is strictly lager than two times of the ground state solutions of (SKP). This property is called energy doubling by Weth in [21]. Motivated by [26] and [10], we can get the ground state solutions of the problem (SKP) as minimizers of the corresponding energy functional I λ on the following manifold

N λ := {u ∈ H, u 6= 0, hI _λ ⁰ (u), ui = 0}

with

c λ := inf{I λ (u) : u ∈ N λ },

which play an active role in finding the ground state solutions of Nehari type for (SKP).

Our main results can be stated by the following theorems.

Theorem 1.1. If the assumptions (V ), (K) and (f ₁ ), (f ₂ ) hold, b > 0, 0 <

λ < λ 1 and ³ ₂ < α < 2, then the system (SKP) has at least one ground state sign-changing solution which has precisely two nodal domains.

Theorem 1.2. Under the assumptions of Theorem 1.1, we can get that I λ (u λ )

> 2c λ , where u λ is the ground state sign-changing solution obtained in Theorem 1.1. Especially c λ is achieved either by a positive or a negative function.

In fact, Theorem 1.2 indicates that the energy of any sign-changing solution for (SKP) is strictly larger than twice of the least energy.

Remark 1.3. (i) Under the assumption (f 1 ), we can obtain that λ 1 > 0.

(ii) (f 2 ) can help us estimate the least energy level m λ of the functional I λ

on M λ and make I λ meet the (P S) m _λ -condition.

We organize this paper as follows. In Section 2, we present some notations

and prove some useful preliminary lemmas which provide the way for getting

one ground state sign-changing solution. In Section 3, we finish the proof of

Theorem 1.1 and Theorem 1.2.

2. Some preliminaries

In this section, we give some preliminary lemmas which are essential to demonstrate our results.

Lemma 2.1 ([3, 7, 18, 25]). For each u ∈ H, there exists a unique element φ u ∈ H such that −4φ u = k(x)u ² , especially φ u has the following properties.

(i) Suppose that k ∈ L ^∞ (R ³ ), then for each u ∈ H, there exists C > 0 such that kφ u k ≤ Ckuk ² and T φ _u (u) = R

R ³ |∇φ u | ² dx ≤ Ckuk ⁴ 12 5

≤ Ckuk ⁴ ; (ii) φ u ≥ 0 for a.e x ∈ R ³ , Moreover, φ u > 0 in R ³ when u 6= 0;

(iii) φ tu = t ² φ u , ∀t > 0;

(iv) φ maps bounded sets into bounded sets;

(v) If u n * u weakly in H ¹ (R ³ ), then φ u _n * φ u weakly in D ^1,2 (R ³ ); φ u _n → φ u strongly in L ⁶ _loc (R ³ ).

Lemma 2.2. If hypotheses (f 1 ) and (K) hold, then (i) the functional F : u ∈ H → R

R ³ f (x)u ² dx is weakly continuous; for any u ∈ H, G : u ∈ H → R

R ³ f (x)uvdx is weakly continuous;

(ii) the functional K : u ∈ H → R

R ³ k(x)φ u u ² dx is weakly continuous; Q : u ∈ H → R

R ³ k(x)φ u uvdx is weakly continuous.

Proof. Since part (i) is easier than (ii), we omit it. For part 2, in view of the Sobolev embedding theorems and (v) of Lemma 2.1, by u n * u weakly in H, we can deduce that

(2.1) (1) u n * u in L ⁶ (R ³ ), (2) u ² _n → u ² in L ³ _loc (R ³ ), (3) φ u _n * φ u in D ^1,2 (R ³ ), (4) φ u _n → φ u in L ⁶ _loc (R ³ ).

Hence, given  > 0, by (2.1)(3), one has that for n large enough (2.2)

    Z

R ³

k(x)u ² (φ _{u n} − φ _u )dx    

≤ .

For each fixed v, by (2.1)(1), there holds (2.3)

    Z

R ³

k(x)φ u v(u n − u)dx    

< .

Especially, considering (2.1)(2) and (2.1)(4) respectively, we can claim that for each  > 0, ρ > 0 and for n large enough, there hold

(2.4) |u ² _n − u ² | _{3,B ρ (0)} < , (2.5) |φ u _n − φ u | _{6,B ρ (0)} < .

On the other hand, since {u n } is bounded in H and the Sobolev embedding D ^1,2 (R ³ ) in L ⁶ (R ³ ) is continuous, {φ u _n } is bounded in D ^1,2 (R ³ ) and in L ⁶ (R ³ ) by (iv) of Lemma 2.1. Moreover, by (K), ku ² _n and ku ² belong to L ^{6 5} (R ³ ), so for each  > 0, there exists ¯ ρ = ¯ ρ() > 0 such that

(2.6) |k| _2,R 3 \B ρ (0) < , ∀ρ ≥ ¯ ρ.

Therefore, by (2.2), (2.4), (2.6), we have that for n large enough 

   Z

R ³

k(x)φ _{u n} u ² _n dx − Z

R ³

k(x)φ _u u ² dx    

≤     Z

R ³

k(x)φ _{u n} (u ² _n − u ² )dx + Z

R ³

k(x)u ² (φ _{u n} − φ _u )dx    

≤ Z

R ³

|k(x)φ u _n (u ² _n − u ² )|dx +     Z

R ³

k(x)u ² (φ u _n − φ u )dx    

≤ |φ u _n | 6

Z

R ³

|k(x)(u ² _n − u ² )| ^{6 5} dx

 ⁵ ₆ + 

≤ C



 Z

R ³ \B ρ (0)

|k(x)(u ² _n − u ² )| ^{6 5} dx

! ⁵ ₆ +

Z

B _ρ (0)

|k(x)(u ² _n − u ² )| ^{6 5} dx

! ⁵ ₆ 

 + 

≤ C 

|k| _2,R ^{6 5} 3 \B ρ (0) · |u ² _n − u ² | ₃ ^{6 5} + |k| ₂ ^{6 5} · |u ² _n − u ² | _3,B ^{6 5}

ρ (0)

 ⁵ 6

+ 

≤ C ⁰ .

Similarly, by (2.3), (2.5), (2.6), one has that for n large enough 

   Z

R ³

k(x)φ _{u n} u _n vdx − Z

R ³

k(x)φ _u uvdx    

≤ Z

R ³

|k(x)u n v(φ u _n − φ u )|dx +     Z

R ³

k(x)φ u v(u n − u)dx    

≤ |u n | 6 · |v| 6

Z

R ³

|k(x)(φ u _n − φ u )| ^{3 2} dx

 ² ₃ + 

≤ C 

|k| _2,R ^{3 2} 3 \B ρ (0) · |φ u _n − φ u | ₆ ^{3 2} + |k|

3 2

2 · |φ u _n − φ u | _6,B ^{3 2}

ρ (0)

 ² ₃ + 

≤ ˜ C. 

Lemma 2.3. For each ¯ s, ¯ t > 0, if µ ∈ [ ¹ ₂ , 1], then the following system (Φ(¯s, ¯t) = ¯s − aSµ ^{− 1 3} (¯ s + ¯ t) ^{1 3} = 0,

Ψ(¯ s, ¯ t) = ¯ t − bS ² µ ^{− 2 3} (¯ s + ¯ t) ^{2 3} = 0

has a unique solution (¯ s 0 , ¯ t 0 ). Moreover, if Φ(¯ s, ¯ t) ≥ 0, Ψ(¯ s, ¯ t) ≥ 0, then ¯ s ≥ ¯ s 0 , t ≥ ¯ ¯ t 0 .

Proof. Similar to the proof of Lemma 3.6 [23], if Φ(¯ s 0 , ¯ t 0 ) = Ψ(¯ s 0 , ¯ t 0 ) = 0, then ¯ s 0 + ¯ t 0 = _a ^µ¯ 3 ^s S ^{3 0 3} , where S is the best Sobolev constant for the embedding D ^1,2 (R ³ ) in L ⁶ (R ³ ). So it follows from the second equality of system that

 µ¯ s ³ ₀ − a ³ S ³ ¯ s 0

a ³ S ³

 ³

= b ³ µ ⁻² S ⁶  µ¯ s ³ ₀ a ³ S ³

 ²

,

which implies that

¯

s 0 = abS ³ + a p

b ² S ⁶ + 4aµS ³ 2µ

and

t ¯ ₀ = b ² S ³ p

b ² S ⁶ + 4aµS ³

2µ ² + b ³ S ⁶

2µ ² + abS ³ µ .

If Φ(¯ s, ¯ t) ≥ 0, Ψ(¯ s, ¯ t) ≥ 0, then (¯ s + ¯ t) ≥ aSµ ^{− 1 3} (¯ s + ¯ t) ^{1 3} + bS ² µ ^{− 2 3} (¯ s + ¯ t) ^{2 3} . Define

e(l) = l − aSµ ^{− 1 3} l ^{1 3} − bS ² µ ^{− 2 3} l ^{2 3} , l > 0.

Then e(l) has a unique zero point l 0 > 0 and e(l) ≥ 0, hence, l ≥ l 0 . Namely

¯

s + ¯ t ≥ ¯ s 0 + ¯ t 0 . If ¯ s < ¯ s 0 , then

Φ(¯ s, ¯ t) = ¯ s − aSµ ^{− 1 3} (¯ s + ¯ t) ^{1 3} < ¯ s ₀ − aSµ ^{− 1 3} (¯ s ₀ + ¯ t ₀ ) ^{1 3} ,

which contradicts with Φ(¯ s, ¯ t) ≥ 0, so ¯ s ≥ ¯ s 0 . Similarly, ¯ t ≥ ¯ t 0 .  Lemma 2.4. If 0 < λ < λ ₁ , b > 0, u ∈ H with u ^± 6= 0, then there exists a unique pair of positive numbers (s _u , t _u ) such that s _u u ⁺ +t _u u ⁻ ∈ M λ . Moreover,

I λ (s u u ⁺ + t u u ⁻ ) = max

s,t≥0 I λ (su ⁺ + tu ⁻ ).

Proof. We prove the theorem by three steps.

Step 1. Let u ∈ H with u ^± 6= 0. We will first prove the existence of (s u , t u ).

Define

g u (s, t) = s ² ku ⁺ k ² + bs ⁴ k∇u ⁺ k ⁴ ₂ + bs ² t ² k∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + s ⁴ T φ _u+ (u ⁺ ) + s ² t ² T φ _u+ (u ⁻ )

− λs ² Z

R ³

f (x)|u ⁺ | ² dx − s ⁶ Z

R ³

|u ⁺ | ⁶ dx, (2.7)

e _u (s, t) = t ² ku ⁻ k ² + bt ⁴ k∇u ⁻ k ⁴ ₂ + bs ² t ² k∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + t ⁴ T _φ

u− (u ⁻ ) + s ² t ² T _φ

u+ (u ⁻ )

− λt ² Z

R ³

f (x)|u ⁻ | ² dx − t ⁶ Z

R ³

|u ⁻ | ⁶ dx.

(2.8)

For any fixed t ≥ 0, it is easy to see that g _u (0, t) = 0, g _u (s, s) > 0, e _u (s, s) > 0 for s > 0 small enough and g _u (t, t) < 0, e _u (t, t) < 0 for t > 0 large enough.

Then there exists 0 < y < Y , such that

(2.9) g u (y, y) > 0, e u (y, y) > 0, g u (Y, Y ) < 0, e u (Y, Y ) < 0.

Thus we can obtain from (2.7), (2.8), (2.9) that

(2, 10) g _u (y, t) > 0, g _u (Y, t) < 0, ∀t ∈ [y, Y ].

(2.11) e u (s, y) > 0, e u (s, Y ) < 0, ∀s ∈ [y, Y ].

By Miranda’s theorem [19], there exists a unique pair (s _u , t _u ) of positive num- bers with y < s u , t u < Y such that

g u (s u , t u ) = 0, e u (s u , t u ) = 0, which confirms that s _u u ⁺ + t _u u ⁻ ∈ M λ .

Step 2. We prove that the (s _u , t _u ) is unique. Suppose (¯ s, ¯ t) is another pair of positive numbers such that ¯ su ⁺ + ¯ tu ⁻ ∈ M _λ . Let m = su ⁺ + tu ⁻ ∈ M _λ . Then m ⁺ = su ⁺ , m ⁻ = tu ⁻ . Similarly, if v = s u u ⁺ + t u u ⁻ , then v ⁺ = s u u ⁺ , v ⁻ = t _u u ⁻ . Therefore w = ¯ su ⁺ + ¯ tu ⁻ = av ⁺ + dv ⁻ , where a = _s ^{s ¯}

u , d = _t ^{t ¯}

u . In view of the definition of the M λ , we have that

(2.12) (g _m (s, t) = hI _λ ⁰ (m), m ⁺ i = hI _λ ⁰ (m), (su ⁺ ) = 0i, e m (s, t) = hI _λ ⁰ (m), m ⁻ i = hI _λ ⁰ (m), (tu ⁻ ) = 0i.

By (2.12), we can deduce that

(2.13) (g _v (1, 1) = 0,

e v (1, 1) = 0.

Similarly there holds

(2.14) (g v (a, d) = g w (1, 1) = 0, e v (a, d) = e w (1, 1) = 0.

By (2.13), (2.14), we have that

(2.15) (g v (a, d) = g w (1, 1) = g v (1, 1) = 0, e _v (a, d) = e _w (1, 1) = e _v (1, 1) = 0.

So in order to show the uniqueness, we only need to prove that a = d = 1.

Suppose that 0 < a ≤ d, by (2.7), (2.8), we have that (g _v (a, a) ≤ g _v (a, d) = g _w (1, 1) = 0,

e _v (d, d) ≥ e _v (a, d) = e _v (1, 1) = 0, i.e.,

(2.16)



 

 

 

 

 

 

 

 

a ² kv ⁺ k ² + a ⁴ [bk∇v ⁺ k ⁴ ₂ + bk∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ + T φ _v (v ⁺ )]

≤ λa ² Z

R ³

f (x)|v ⁺ | ² dx + a ⁶ Z

R ³

|v ⁺ | ⁶ dx,

d ² kv ⁻ k ² + d ⁴ [bk∇v ⁻ k ⁴ ₂ + bk∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ + T φ _v (v ⁻ )]

≥ λd ² Z

R ³

f (x)|v ⁻ | ² dx + d ⁶ Z

R ³

|v ⁻ | ⁶ dx.

By a calculation, one has that (2.17)

a ² ≥ ^{b[k∇v + k 4 2 +k∇v} ₂ R ^{+ k 2 2 k∇v − k 2 2 ]+T φv (v + )}

R 3 |v ⁺ | ⁶ dx

+

q

[bk∇v ⁺ k ⁴ ₂ + bk∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ + T φ

(v ⁺ )] ² + 4 R

R

|v ⁺ | ⁶ dx(kv ⁺ k ² − λ R

R

f (x)|v ⁺ | ² dx) 2 R

R

|v ⁺ | ⁶ dx ,

(2.18)

d ² ≤ ^{b[k∇v − k 4 2 +k∇v} ₂ R ^{+ k 2 2 k∇v − k 2 2 ]+T φv (v − )}

R 3 |v ⁻ | ⁶ dx

+

q

[bk∇v ⁻ k ⁴ ₂ + bk∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ + T φ

(v ⁻ )] ² + 4 R

R

|v ⁻ | ⁶ dx(kv ⁻ k ² − λ R

R

f (x)|v ⁻ | ² dx) 2 R

R

|v ⁻ | ⁶ dx .

From the fact g v (1, 1) = e v (1, 1) = 0, we can derive that

(2.19)



 

 

 

 

 

 

 

  Z

R ³

|v ⁺ | ⁶ dx = (kv ⁺ k ² − λ Z

R ³

f (x)|v ⁺ | ² dx)

+ b[k∇v ⁺ k ⁴ ₂ + k∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ ] + T φ _v (v ⁺ ), Z

R ³

|v ⁻ | ⁶ dx = (kv ⁻ k ² − λ Z

R ³

f (x)|v ⁻ | ² dx)

+ b[k∇v ⁻ k ⁴ ₂ + k∇v ⁺ k ² ₂ k∇v ⁻ k ² ₂ ] + T φ _v (v ⁻ ).

It follows from (2.17), (2.18), and (2.19) that a ² ≥ 1, d ² ≤ 1. Together with the condition of 0 < a ≤ d, we can get that a = b = 1.

Step 3. We prove that I λ (s u u ⁺ + t u u ⁻ ) = max

s,t≥0 I λ (su ⁺ + tu ⁻ ). Since I λ (su ⁺ + tu ⁻ ) = s ²

2 ku ⁺ k ² + s ⁴

4 [bk∇u ⁺ k ⁴ ₂ + T φ _u+ (u ⁺ )]

− λ 2 s ²

Z

R ³

f (x)|u ⁺ | ² dx − 1 6 s ⁶

Z

R ³

|u ⁺ | ⁶ dx

+ t ²

2 ku ⁻ k ² + t ⁴

4 [bk∇u ⁻ k ⁴ ₂ + T _φ

u− (u ⁻ )]

− λ 2 t ²

Z

R ³

f (x)|u ⁻ | ² dx − 1 6 t ⁶

Z

R ³

|u ⁻ | ⁶ dx

+ s ² t ²

2 T φ _u+ (u ⁻ ) + s ² t ²

2 bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ ,

let I _λ (su ⁺ + tu ⁻ ) = h(s, t). It is easy to see that h(s, t) > 0 as |(s, t)| → 0; h(s, t) < 0 as |(s, t)| → ∞, which means the maximum point cannot be achieved on the boundary of R ² + . Without loss of generality, we only prove that I λ (s u u ⁺ ) < I λ (s u u ⁺ + t u u ⁻ ) or I λ (t u u ⁻ ) < I λ (s u u ⁺ + t u u ⁻ ). Here we prove that I λ (s u u ⁺ ) < I λ (s u u ⁺ + t u u ⁻ ). In fact

∂

∂t h(s _u , t) = tku ⁻ k ² + t ³ [bk∇u ⁻ k ⁴ ₂ + T _φ

u− (u ⁻ )] − λt Z

R ³

f (x)|u ⁻ | ² dx

− t ⁵ Z

R ³

|u ⁻ | ⁶ dx + s ² _u t[T φ _u+ (u ⁻ ) + bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ ] > 0 with t > 0 small enough, implies that h(s u , t) = I λ (s u u ⁺ +tu ⁻ ) is an increasing function with respect to t ∈ [0, ) for  small enough. As a result I λ (s u u ⁺ ) <

h(s u , t). So (s u , t u ) is a positive maximum point for h(s, t).

In view of above fact, we can know that _∂t ^∂

u h(s u , t u ) = _∂s ^∂

u h(s u , t u ) is equiv- alent to g _u (s _u , t _u ) = e _u (s _u , t _u ) = 0. By the definition of M _λ , one has that s _u u ⁺ + t _u u ⁻ ∈ M λ . Together with the uniqueness of the (s _u , t _u ), we can obtain that

I _λ (s _u u ⁺ + t _u u ⁻ ) = max

s,t≥0 I _λ (su ⁺ + tu ⁻ ).

 Lemma 2.5. If 0 < λ < λ 1 , b > 0, then for ∀u ∈ H with u ^± 6= 0, we have

(i) if g u (1, 1) ≤ 0, e u (1, 1) ≤ 0, then there exists unique pair (s u , t u ) ∈ (0, 1] × (0, 1], such that s _u u ⁺ + t _u u ⁻ ∈ M λ .

(ii) if g _u (1, 1) ≥ 0, e _u (1, 1) ≥ 0, then there exists unique pair (s _u , t _u ) ∈ [1, +∞] × [1, +∞), such that s _u u ⁺ + t _u u ⁻ ∈ M _λ , where g _u (s, t), e _u (s, t) are given as (2.7) and (2.8).

Proof. (i) If u ∈ H, u ^± 6= 0, we can get that there exists a unique pair (s _u , t _u ) of positive numbers such that s _u u ⁺ + t _u u ⁻ ∈ M _λ . Assume that 0 < s _u ≤ t _u , then one has that

t ² _u ku ⁻ k ² + t ⁴ _u [b(k∇u ⁻ k ⁴ ₂ + k∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ ) + T _{φ u} (u ⁻ )]

≥ t ² _u ku ⁻ k ² + t ⁴ _u [bk∇u ⁻ k ⁴ ₂ + T _φ

u− (u ⁻ )] + s ² _u t ² _u [bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + T _φ

u+ (u ⁻ )]

= λt ² _u Z

R ³

f (x)|u ⁻ | ² dx + t ⁶ _u Z

R ³

|u ⁻ | ⁶ dx.

Therefore

t ² _u ≤ ^{b[k∇u − k 4 2 +k∇u} ₂ R ^{+ k 2 2 k∇u − k 2 2 ]+T φu (u − )}

R 3 |u ⁻ | ⁶ dx

+

q

[bk∇u ⁻ k ⁴ ₂ + bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + T φ

(u ⁻ )] ² + 4 R

R

|u ⁻ | ⁶ dx(ku ⁻ k ² − λ R

R

f (x)|u ⁻ | ² dx) 2 R

R

|u ⁻ | ⁶ dx .

Since g u (1, 1) ≤ 0, e u (1, 1) ≤ 0, we have that ku ⁻ k ² − λ

Z

R ³

f (x)|u ⁻ | ² dx

≤ Z

R ³

|u ⁻ | ⁶ dx − b[k∇u ⁻ k ⁴ ₂ + k∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ ] − T φ _u (u ⁻ ), thus t ² _u ≤ 1. Together with 0 < s _u ≤ t _u , we have that 0 < s _u ≤ t _u ≤ 1.

Similarly, we can prove (ii). 

Lemma 2.6. Define functions s, t: H → (0, +∞) by s(u) = s u , t(u) = t u . Under the assumption of Lemma 2.5, we have

(i) s, t are continuous in H.

(ii) if u ⁺ _n → 0 strongly in H as n → ∞, one has s _{u n} → +∞; if u ⁻ _n → 0 strongly in H as n → ∞, one has t u _n → +∞.

(iii) if {u n } ⊂ M λ , lim

n→∞ I λ (u n ) = m λ , then m λ > 0 and C ⁰ ≤ ku n k ≤ C ⁰⁰ for some C ⁰ , C ⁰⁰ ≥ 0.

Proof. (i) By Lemma 2.4, there exist (¯ s _{u n} , ¯ t _{u n} ) and (¯ s _u , ¯ t _u ), such that ¯ s _{u n} u ⁺ _n + t ¯ _{u n} u ⁻ _n ∈ M _λ , ¯ s _u u ⁺ + ¯ t _u u ⁻ ∈ M _λ . Then we have that

ku ⁺ _n k ² + ¯ s ² _{u n} [bk∇u ⁺ _n k ⁴ ₂ + T φ

u + n

(u ⁺ _n )]

+ ¯ t ² _u

n [bk∇u ⁺ _n k ² ₂ k∇u ⁻ _n k ² ₂ + T _φ

u + n

(u ⁻ _n )]

= λ Z

R ³

f (x)|u ⁺ _n | ² dx + ¯ s ⁴ _u

n

Z

R ³

|u ⁺ _n | ⁶ dx, (2.20)

ku ⁻ _n k ² + ¯ t ² _{u n} [bk∇u ⁻ _n k ⁴ ₂ + T φ

u − n

(u ⁻ _n )]

+ ¯ s ² _{u n} [bk∇u ⁺ _n k ² ₂ k∇u ⁻ _n k ² ₂ + T φ

u + n

(u ⁻ _n )]

= λ Z

R ³

f (x)|u ⁻ _n | ² dx + ¯ t ⁴ _{u n} Z

R ³

|u ⁻ _n | ⁶ dx.

(2.21)

Take a sequence {u _n } in H, such that u n → u strongly in H, then u ^± _n → u ^± strongly in H. We claim that {¯ s _{u n} }, {¯ t _{u n} } are bounded in R ⁺ . Otherwise, if

¯

s _{u n} → +∞ as n → +∞, let

y(u n ) = ku ⁺ _n k ² + ¯ s ² _{u n} [bk∇u ⁺ _n k ⁴ ₂ + T φ

u + n

(u ⁺ _n )]

+ ¯ t ² _{u n} [bk∇u ⁺ _n k ² ₂ k∇u ⁻ _n k ² ₂ + T φ

u + n

(u ⁻ _n )]

− λ Z

R ³

f (x)|u ⁺ _n | ² dx = ¯ s ⁴ _u

n

Z

R ³

|u ⁺ _n | ⁶ dx := h(u _n ), (2.22)

together with u ^± _n → u ^± 6= 0 in H, we can get that y(u n )

¯ s ⁴ _u

n

→ 0, h(u n )

¯ s ⁴ _u

n

→ C,

as ¯ s u _n → +∞, which leads to a contradiction. So going if necessary to a subsequence, still denoted by {¯ s u _n } and {¯ t u _n }, suppose that there is a pair of nonnegative number (¯ s, ¯ t) such that

lim

n→∞ ¯ s _{u n} = ¯ s, lim

n→∞

¯ t _{u n} = ¯ t.

Passing to the limit in (2.20) and (2.21), we get that ku ⁺ k ² + ¯ s ² [bk∇u ⁺ k ⁴ ₂ + T φ _u+ (u ⁺ )]

+ ¯ t ² [bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + T φ _u+ (u ⁻ )]

= λ Z

R ³

f (x)|u ⁺ | ² dx + ¯ s ⁴ Z

R ³

|u ⁺ | ⁶ dx, (2.23)

ku ⁻ k ² + ¯ t ² [bk∇u ⁻ k ⁴ ₂ + T φ _u− (u ⁻ )]

+ ¯ s ² [bk∇u ⁺ k ² ₂ k∇u ⁻ k ² ₂ + T _φ

u+ (u ⁻ )]

= λ Z

R ³

f (x)|u ⁻ | ² dx + ¯ t ⁴ Z

R ³

|u ⁻ | ⁶ dx.

(2.24)

It follows from b > 0, 0 < λ < λ 1 and u ^± 6= 0, that ¯ s > 0, ¯ t > 0. Therefore,

¯

su ⁺ + ¯ tu ⁻ ∈ M λ . By the uniqueness of (s u , t u ), we have that s u = ¯ s, t u = ¯ t.

Namely the functions s, t are continuous in H.

(ii) We only need to show that t u _n → +∞, if u ⁻ _n → 0 in H as n → ∞.

Similarly, we can prove that s u _n → +∞, if u ⁺ _n → 0 in H. On the contrary, if there exists M > 0, such that t u _n ≤ M , then it follows from the Sobolev inequality that

o(ku ⁻ k ² ) = t u _n 4

Z

R ³

|u ⁻ | ⁶ dx ≤ Cku ⁻ _n k ⁶ . By (2.21), b > 0 and 0 < λ < λ ₁ , we deduce that

ku ⁻ _n k ² + t ² _{u n} [bk∇u ⁻ _n k ⁴ ₂ + T φ

u − n

(u ⁻ _n )] + s ² _{u n} [bk∇u ⁺ _n k ² ₂ k∇u ⁻ _n k ² ₂ + T φ

u + n

(u ⁻ _n )]

− λ Z

R ³

f (x)|u ⁻ _n | ² dx − t ⁴ _{u n} Z

R ³

|u ⁻ _n | ⁶ dx

≥ (1 − λ λ 1

)ku ⁻ _n k ² − o(ku ⁻ _n k ² ) > 0,

which contradicts with the fact that s u _n u ⁺ _n + t u _n u ⁻ _n ∈ M λ , so t u _n → +∞.

(iii) Since {u n } ⊂ M λ , we have

ku ^± _n k ² + [bk∇u ^± _n k ⁴ ₂ + k∇u ⁺ _n k ² ₂ k∇u ⁻ _n k ² ₂ ] + T _{φ un} (u ^± _n )

= λ Z

R ³

f (x)|u ^± _n | ² dx + Z

R ³

|u ^± _n | ⁶ dx.

By 0 < λ < λ 1 , b > 0, Sobolev’s inequality and H¨ older’s inequality, one has that

ku ^± _n k ² ≤ λ Z

R ³

f (x)|u ^± _n | ² dx + Z

R ³

|u ^± _n | ⁶ dx ≤ λ

λ ₁ ku ^± _n k ² + S ⁻³ ku ^± _n k ⁶ , then

(2.25) ku ^± _n k ≥ [S ³ (1 − λ

λ ₁ )] ^{1 4} > 0.

Furthermore, it follows from {u n } ⊂ M λ ⊂ N λ that m λ + o(1)

(2.26)

= I λ (u n ) = I λ (u n ) − 1

6 hI _λ ⁰ (u n ), (u n )i

= 1

3 (ku _n k ² − λ Z

R ³

f (x)u ² _n dx) + 1

12 [bk∇u _n k ² ₂ Z

R ³

|∇u _n | ² dx + T _{φ un} (u ^± _n )]

≥ 1 3 (1 − λ

λ 1

))ku _n k ² ,

which means that m _λ > 0 and {u _n } is bounded in H. By (2.25) and (2.26), we have that C ⁰ ≤ ku n k ≤ C ⁰⁰ for some C ⁰ , C ⁰⁰ > 0.  Inspired by [5] and [22], with the help of Nehari manifold and Minimax methods in critical point theory, the following results hold.

Lemma 2.7. Suppose (f ₁ ), (f ₂ ), (K), (V ) hold and let b > 0, we have that (i) for each u ∈ H \ {0}, there exists a unique ˜ t _u > 0, such that ˜ t _u u ∈ N _λ and I _λ ( ˜ t _u u) = max

t≥0 I _λ (tu);

(ii) if 0 < λ < λ 1 , ³ ₂ < α < 2, the system (SKP) has a positive ground state solution u ₀ ∈ N _λ , and I _λ (u ₀ ) = c _λ , then c _λ < c ^∗ = ₄ ^b S ³ + ^{[b 2 S 4 +4S]}

3 2

24 + ^{b 3} ₂₄ ^{S 6} . Proof. Since the proof of (i) is standard, we omit it here (you can see [11]).

Next we give the proof for (ii).

Recall that S is attained by the Tulenti function u _ = ^

1 4

(+|x| ² ) ¹ 2

. Define w  = ϕ ◦ u  , where ϕ ∈ C ₀ ^∞ (R ³ ) : R ³ → [0, 1] satisfies

(2.27) ϕ(x) =

(1, x ∈ B 2R (0), 0, x ∈ R ³ \B 2R (0).

Similar to the calculation of [4], we have the following estimate as  → 0, (2.28)

Z

R ³

|∇w  | ² dx = K 1 + O( ^{1 2} ), Z

R ³

|w  | ⁶ dx = K 2 + O( ^{1 2} ), S = K ₁ K

1 3

2

,

(2.29)

Z

R ³

|w  | ^s dx =



 



 



O( ^{s 4} ), s ∈ [2, 3), O( ^{3 4} | ln |), s = 3, O( ^{6−s 4} ), s ∈ (3, 6),

where K 1 ,K 2 are positive constants. According to (2.28) and (2.29) we have (2.30)

R

R ³ |∇w  | ² dx ( R

R ³ |w _ | ⁶ dx) ^{1 3} = S + O( ^{1 2} ).

In fact, for any  > 0, there exists t  > 0 such that t  w  ∈ N λ . Furthermore, {t  } >0 has a positive lower bound. Otherwise, there exists a subsequence

 n , such that t  _n → 0 as n → ∞. By the definition of c λ , we have that 0 < c λ ≤ lim

n→∞ I λ (t  _n w  _n ) = 0, which leads to a contradiction.

Therefore, in view of the definition of c λ and (i), we have c λ ≤ max

t≥0 I λ (tw  ).

Define a function e(t) = 1

2 t ² kw  k ² ₂ + b 4 t ⁴

Z

R ³

|∇w  | ² dx

 2

− 1 6 t ⁶

Z

R ³

|w  | ⁶ dx.

By e ⁰ (t) = 0, it is easy to see that e(t) attains its maximum at

t ₀ =



 b( R

R ³ |∇w  | ² dx) ² + q b ² ( R

R ³ |∇w  | ² dx) ⁴ + 4kw  k ² R

R ³ |w  | ⁶ dx 2 R

R ³ |w  | ⁶ dx





1 2

and (2.31)

e(t 0 ) = bkw  k ² k∇w  k ⁴ ₂ 4 R

R ³ |w  | ⁶ dx + [b ² k∇w  k ⁸ ₂ + 4kw  k ² R

R ³ |w  | ⁶ dx] ^{3 2} + b ³ k∇w  k ¹² ₂ 24( R

R ³ |w  | ⁶ dx) ² . (2.29) and (2.30) imply that

e(t 0 ) = b

4 S ³ + [b ² S ⁴ + 4S] ^{3 2} + b ³ S ⁶

24 + O( ^{1 2} ) = c ^∗ + O( ^{1 2} ).

By (i), we have that I _λ (tw _ ) ≤ max

t≥0 I _λ (tw _ )

= max

t≥0

h 1

2 t ² kw  k ² + b

4 t ⁴ k∇w  k ⁴ ₂ − 1 6 t ⁶

Z

R ³

|w  | ⁶ dx + 1

4 t ⁴ T ϕ _w (w  )

− 1 2 t ² λ

Z

R ³

f (x)|w  | ² dx i .

By (2.30) and (2.31), it is not difficult to get that as  → 0,

(2.32)

max

t≥0

h 1

2 t ² kw _ k ² + b

4 t ⁴ k∇w _ k ⁴ ₂ − 1 6 Z

R ³

|w _ | ⁶ dx i

= e(t ₀ ) = b

4 S ³ + [b ² S ⁴ + 4S] ^{3 2} + b ³ S ⁶

24 + O( ^{1 2} ).

By H¨ older’s inequality, the boundedness of s and t, (2.29) and Lemma 1.2(i), we can deduce that as  → 0,

(2.33) 1

4 t ⁴ T ϕ _w (w  ) ≤ 1

4 t ⁴ |ϕ w _ | 6 |w  | ² 12 5

≤ C|w  | ⁴ 12 5

≤ Ckw  k ⁴ ≤ C ⁰ .

In view of (f 2 ), if 0 < λ < λ 1 ,  ∈ (0, τ ² ] and ³ ₂ < α < 2, we can have 1

2 λ Z

R ³

f (x)|w _ | ² dx ≥ C ^{1 2} Z

|x|≤τ

|x| ^−α

 + |x| ² dx + 1 2 λ

Z

|x|≥τ

f (x)|w _ | ² dx

≥ C ^{1 2} Z τ

0

r ² r ^α ( + r ² ) dr

= C ^{1− α 2} Z τ  ^{− 1} 2

0

ρ ² ρ ^α (1 + ρ ² ) dρ (2.34)

≥ C ⁰  ^{1− α 2} Z 1

0

ρ ² 2ρ ^α dρ

= C ⁰  ^{1− α 2} .

It follows from (2.32), (2.33), (2.34) and ³ ₂ < α < 2 that, as  → 0,

(2.35)

I λ (tw  ) ≤ max

t≥0 I λ (tw  ) ≤ e(t 0 ) + C ⁰ ( −  ^{1− α 2} )

= b

4 S ³ + [b ² S ⁴ + 4S] ^{3 2}

24 + +b ³ S ⁶

24 + O( ^{1 2} ) + C ⁰ ( −  ^{1− α 2} ).

That is

max

t≥0 I _λ (tw _ ) < b

4 S ³ + [b ² S ⁴ + 4S] ^{3 2}

24 + +b ³ S ⁶ 24 = c ^∗ .

Combining (2.35), we deduce that c _λ < c ^∗ .  3. Proof of the main results

In this section, we will claim the existence of sign-changing solutions for the system (SKP). Since the problem (SKP) involves bi-nonlocal terms and critical nonlinearity, we need to construct a sign-changing (P S) m _λ -sequence. Inspired by the method of [6], we give some definitions.

Let P denote the cone of nonnegative functions in H, Q = [0, 1] × [0, 1] and Σ be the set of continuous map δ with s, t ∈ [0, 1], namely

Σ = (δ ∈ C(Q, H); (a) : δ(s, 0) = 0, δ(0, t) ∈ P, δ(1, t) ∈ −P ; (b) : I λ (δ(s, 1)) ≤ 0, f (δ(s, 1)) ≥ 2,

where f (δ(s, 1)) =

R

R

|δ(s, 1)| ⁶ dx kδ(s, 1)k ² − λ R

R

f (x)|δ(s, 1)| ² dx + T φ

δ(s, 1) + bk∇δ(s, 1)k ² 2

R

R

|∇δ(s, 1)| ² dx . Choosing u ∈ H with u ^± 6= 0. Let δ(s, t) = mt(1 − s)u ⁺ + mtsu ⁻ , where m > 0, s, t ∈ [0, 1]. It is easy to see that δ ∈ Σ for m > 0 large enough, which means Σ 6= ∅. Define

Γ λ (u, v) =







R

R 3 |u| ⁶ dx kuk ² −λ R

R 3 f (x)|u| ² dx+T _φu (u)+T _φv (u)+b[k∇uk ⁴ ₂ +k∇uk ² ₂ k∇vk ² ₂ ] , u 6= 0,

0, u = 0.

Clearly, Γ λ (u, v) > 0 if 0 < λ < λ 1 , b > 0 and u 6= 0; u ∈ M λ if and only if Γ λ (u ⁺ , u ⁻ ) = Γ λ (u ⁻ , u ⁺ ) = 1. Set

Θ λ = {u ∈ H : |Γ λ (u ⁺ , u ⁻ ) − 1| < 1

2 , |Γ λ (u ⁻ , u ⁺ ) − 1| < 1 2 }.

3.1. Main lemmas and theirs proof Lemma 3.1. inf

δ∈Σ sup

u∈δ(Q)

I λ (u) = inf

u∈M λ

I λ (u) = m λ .

Proof. On the one hand, for any u ∈ M _λ , there exists δ(s, t) = mt(1 − s)u ⁺ + mtsu ⁻ ∈ Σ for m > 0 large enough. By Lemma 2.4, we can deduce that

I _λ (u) = max

s,t≥0 I _λ (su ⁺ + tu ⁻ ) ≥ sup

u∈δ(Q)

I _λ (u) ≥ inf

δ∈Σ sup

u∈δ(Q)

I _λ (u).

Consequently,

inf

u∈M _λ I _λ (u) ≥ inf

δ∈Σ sup

u∈δ(Q)

I _λ (u).

On the other hand, from the definition of Σ, we claim that for each δ ∈ Σ, there exists u δ ∈ δ(Q) T M λ , which implies that

sup

u∈δ(Q)

I λ (u) ≥ I λ (u δ ) ≥ inf

u∈M λ

I λ (u).

Therefore

δ∈Σ inf sup

u∈δ(Q)

I λ (u) ≥ inf

u∈M _λ I λ (u).

In fact, for each δ ∈ Σ, t ∈ [0, 1], it is easy to check that δ(0, t) ∈ P , δ(1, t) ∈ −P, so we have that

(3.1) Γ _λ (δ ⁺ (0, t), δ ⁻ (0, t))−Γ _λ (δ ⁻ (0, t), δ ⁺ (0, t)) = Γ _λ (δ ⁺ (0, t), δ ⁻ (0, t)) ≥ 0, (3.2) Γ λ (δ ⁺ (1, t), δ ⁻ (1, t))−Γ λ (δ ⁻ (1, t), δ ⁺ (1, t)) = −Γ λ (δ ⁻ (1, t), δ ⁺ (1, t)) ≤ 0.

Meanwhile, we derive from the definition of Σ that for all δ ∈ Σ and s ∈ [0, 1], Γ λ (δ ⁺ (s, 1), δ ⁻ (s, 1)) + Γ λ (δ ⁻ (s, 1), δ ⁺ (s, 1)) ≥ f (δ ⁺ (s, 1)) ≥ 2, which follows from ^d _c + ^f _e ≥ ^d+f _c+e for all c, d, e, f > 0. Consequently (3.3) Γ λ (δ ⁺ (s, 1), δ ⁻ (s, 1)) + Γ λ (δ ⁻ (s, 1), δ ⁺ (s, 1)) − 2 ≥ 0, (3.4) Γ λ (δ ⁺ (s, 0), δ ⁻ (s, 0)) + Γ λ (δ ⁻ (s, 0), δ ⁺ (s, 0)) − 2 ≤ 0.

So by (3.1), (3.2), (3.3), (3.4) and Miranda’s theorem [15], one has that there exists (ˆ s, ˆ t) ∈ Q such that

Γ λ (δ ⁺ (ˆ s, ˆ t), δ ⁻ (ˆ s, ˆ t)) − Γ λ (δ ⁻ (ˆ s, ˆ t), δ ⁺ (ˆ s, ˆ t)) = 0, Γ λ (δ ⁺ (ˆ s, ˆ t), δ ⁻ (ˆ s, ˆ t)) + Γ λ (δ ⁻ (ˆ s, ˆ t), δ ⁺ (ˆ s, ˆ t)) = 2.

Then

Γ _λ (δ ⁺ (ˆ s, ˆ t), δ ⁻ (ˆ s, ˆ t)) = Γ _λ (δ ⁻ (ˆ s, ˆ t), δ ⁺ (ˆ s, ˆ t)) = 1,

which means that for all δ ∈ Σ, there exists u δ = δ(ˆ s, ˆ t) ∈ δ(Q) T M λ . There- fore,

inf

δ∈Σ sup

u∈δ(Q)

I λ (u) = inf

u∈M λ

I λ (u) = m λ .

 Lemma 3.2. Suppose that 0 < λ < λ ₁ , b > 0. Then there exists a (P S) _{m λ} - sequence {u _n } ⊂ Θ λ for I _λ .

Proof. Firstly, we find a (P S) _{m λ} -sequence {u _n } ⊂ H for I λ . Consider a mini- mizing sequence {v _n } ⊂ M λ , δ _n ∈ Σ, where δ n (s, t) = mt(1−s)v _n ⁺ +mtsv ⁻ _n ∈ Σ, then

n→+∞ lim max

v∈δ n (Q)

I _λ (v) = lim

n→∞ I _λ (v _n ).

By classical deformation lemma [17], we can derive that there exists {u _n } ⊂ H such that

(3.5) I λ (u n ) → m λ , I _λ ⁰ (u n ) → 0, dist(u n , δ n (Q)) → 0 as n → ∞.

Suppose the thesis is false. Then it is possible to find an r > 0, such that δ n (Q) T V r = ∅ for n large enough, where

V r = {u ∈ H : ∃w ∈ H s.t. kw − uk ≤ r, kI _λ ⁰ (w)k ≤ r, |I λ (w) − m λ | ≤ r}.

By deformation lemma in [17,22], there exists a continuous map η : [0, 1]×H → H satisfying, for some  ∈ (0, ^m ₂ ^λ ) and each t ∈ [0, 1],

(a) η(0, u) = u; η(t, −u) = −η(t, u);

(b) η(t, u) = u; ∀u ∈ I _λ ^{m λ −} S(H\I _λ ^{m λ +} );

(c) η(1, I _λ ^{m λ +  2} \V r ) ⊂ I _λ ^{m λ −  2} ;

(d) η(1, (I _λ ^{m λ + 2 } T P )\V r ) ⊂ I _λ ^{m λ −  2} T P , where I _λ ^d = {u ∈ H : I _λ (u) ≤ d}.

Since lim

n→+∞ max

v∈δ n (Q)

I _λ (v) = m _λ , we can select n such that (3.6) δ n (Q) ⊂ I ^{m λ +}

 2

λ , δ n (Q) \ V r = ∅.

Define ¯ δ n : Q → H by ¯ δ n (s, t) = η(1, δ n (s, t)) for ∀(s, t) ∈ Q. Then similar proof as that of [26], ¯ δ n ∈ Σ. By (3.6) and property (c) of η, we can derive that ˜ δ n (Q) ⊂ I ^{m λ −}

 2

λ , which leads to a contradiction, m λ = inf

δ∈Σ sup

v∈δ(Q)

I λ (v) ≤ max

v∈δ _n (Q) I λ (v) ≤ m λ −  2 . So we can find a (P S) m _λ -sequence {u n } for I λ .

Next, we prove that {u _n } ⊂ Θ λ for n large enough. We only need to show that u ^± _n 6= 0, namely that Γ λ (u ⁺ _n , u ⁻ _n ) → 1, Γ _λ (u ⁻ _n , u ⁺ _n ) → 1. By (3.5), there exists a sequence {w _n }, such that

(3.7) w n = γ n v _n ⁺ + β n v ⁻ _n ∈ δ n (Q), I λ (w n ) → m λ , kw n − u n k → 0.

So in order to get u ^± _n 6= 0, we can prove that γ n v ⁺ _n 6= 0 and β n v ⁻ _n 6= 0 for n large enough. By {v _n } ⊂ M λ and Lemma 2.6(iii), we only need to show that

n→∞ lim γ _n 6= 0, lim

n→∞ β _n 6= 0. Otherwise, if lim

n→∞ γ _n 6= 0, β _n → 0, it follows from the continuity of I _λ and (3.7) that

m λ = lim

n→∞ I λ (w n ) = lim

n→∞ I λ (γ n v ⁺ _n + β n v _n ⁻ ) = lim

n→∞ I λ (γ n v ⁺ _n ).

Let t ⁴ _n = (1 − _λ ^λ

1 ) ^kv

− n k ² R

R 3 |v n ⁻ | ⁶ dx , by Lemma 2.4, b > 0 and 0 < λ < λ 1 , we have that

m λ = lim

n→∞ I λ (v n ) = lim inf

n→∞ max

s,t>0 I λ (sv _n ⁺ + tv ⁻ _n )

≥ lim inf

n→∞ I _λ (γ _n v ⁺ _n + t _n v _n ⁻ )

= lim inf

n→∞

h

I λ (γ n v ⁺ _n ) + t ² _n

2 kv _n ⁻ k ² + t ⁴ _n

4 (bk∇v ⁻ _n k ⁴ ₂ + T φ

v − n

(v ⁻ _n ))

− λ 2 t ² _n

Z

R ³

f (x)|v ⁻ _n | ² dx − 1 6 t ⁶ _n

Z

R ³

|v _n ⁻ | ⁶ dx + γ ² _n t ² _n

2 (bk∇v _n ⁺ k ² ₂ k∇v _n ⁻ k ² ₂ + T _φ

v + n

(v ⁻ _n )) i

≥ lim inf

n→∞

h t ² _n 2 (1 − λ

λ 1

)kv ⁻ _n k ² − 1 6 t ⁶ _n

Z

R ³

|v _n ⁻ | ⁶ dx + I λ (γ n v _n ⁺ ) i

= lim inf

n→∞

h 1 3 (1 − λ

λ 1

) ^{3 2} kv ⁻ _n k ³ ( R

R ³ |v ⁻ n | ⁶ dx) ^{1 2} + I λ (γ n v _n ⁺ ) i

≥ lim inf

n→∞ I λ (γ n v ⁺ _n ) + 1

3 S ^{3 2} (1 − λ λ 1

) ^{3 2}

= lim

n→∞ I λ (γ n v _n ⁺ ) + 1

3 S ^{3 2} (1 − λ λ 1

) ^{3 2}

= m λ + 1

3 S ^{3 2} (1 − λ λ 1

) ^{3 2} ,

which leads to a contradiction. Thus {u n } ⊂ Θ λ for n large enough.  Lemma 3.3. Under conditions (f 1 ), (K) and b > 0, if 0 < λ < λ 1 , then we have that any bounded sequence {u n } ⊂ Θ λ ⊂ H such that

I λ (u n ) → d ∈ (0, c λ + c ^∗ ), I _λ ⁰ (u n ) → 0,

contains a convergent subsequence. In other words, there exists u ∈ H with u 6= 0, such that I _λ ⁰ (u) = 0 and I λ (u) = d, where c ^∗ = ₄ ^b S ³ + ^{[b 2 S 4 +4S]}

3 2

24 + ^+b ₂₄ ^{3 S 6} . Proof. Since {u n } is bounded in H, up to a subsequence, still denoted by {u n }.

We can suppose that there exists u ∈ H such that

(3.8)

u n * u weakly in H,

u n → u strongly in L ^s _loc (R ³ ), s ∈ [1, 6), u n (x) → u(x) a.e on R ³ .

It follows from Lemma 2.2 that I _λ ⁰ (u) = 0. Setting w n = u n − u, then w n * 0 weakly in H. By the well-known Br´ ezis-Lieb Lemma [22], one has that

(3.9)

ku n k ² = kw n k ² + kuk ² + o(1), k∇u n k ² ₂ = k∇w n k ² ₂ + k∇uk ² ₂ + o(1),

k∇u n k ⁴ ₂ = k∇w n k ⁴ ₂ + k∇uk ⁴ ₂ + 2k∇w n k ² ₂ k∇uk ² ₂ + o(1), ku n k ⁶ ₆ = kw n k ⁶ ₆ + kuk ⁶ ₆ + o(1).

It follows from Lemma 2.2 that (3.10) F (w _n ) =

Z

R ³

f (x)|w _n | ² dx → 0, K(w _n ) = Z

R ³

k(x)φ _{w n} w _n ² dx → 0.