Applied Mathematics and Computation

Existence of homoclinic solutions for a fourth order differential equation with a parameter

Tiexiang Li

^a

, Juntao Sun

^b

, Tsung-fang Wu

^c,⇑

aDepartment of Mathematics, Southeast University, Nanjing 211189, PR China

bSchool of Science, Shandong University of Technology, Zibo 255049, PR China

cDepartment of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

a r t i c l e i n f o

Keywords:

Fourth order differential equations Homoclinic solutions

Mountain pass theorem Variational methods

a b s t r a c t

In this paper, we study the existence of homoclinic solutions for a class of fourth order dif- ferential equations. By using variational methods, the existence and the non-existence of nontrivial homoclinic solutions are obtained, depending on a parameter.

Ó2014 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we consider a class of fourth-order differential equations with a parameter:

u^ð4Þþwu⁰⁰þkaðxÞufðx;uÞ ¼0; x2R; ð1:1Þ

wherewis a constant,k>0 is a parameter,f2CðRR;RÞand the functionasatisfies the following conditions:

(V1) a2CðR;RÞandaP0 onR;

(V2) there existsc>0 such that the setfa<cg:¼fx2RjaðxÞ<cgis nonempty and has finite measure;

(V3) X¼int a¹ð Þ0 is nonempty andX¼a¹ð Þ0 such thatXis a finite interval;

(V4) jfa<cgj<^c0

S²₁, wherej j is the Lebesgue measure,c0is defined by(2.1)in Section2andS1is the best constant for the embedding ofH²ðRÞinL¹ð Þ.R

As usual, we say that a solutionuðxÞof Eq.(1.1)is homoclinic (to 0) ifuðxÞ !0 asx! 1. In addition, ifuðxÞX0, thenuðxÞ is called a nontrivial homoclinic solution.

The above Eq. (1.1)has been put forward as mathematical model for the study of pattern formation in physics and mechanics. For example, the well-known Extended Fisher–Kolmogorov (EFK) equation proposed by Coullet et al. in 1987 [5]in study of phase transitions, and also by Dee and Van Saarlos in 1988[6], as well as the Swift–Hohenberg (SH) equation which is general model for pattern-forming process derived in[13]to describe random thermal fluctuations in the Boussin- esque equation and in the propagation of lasers in[8]. With appropriate changes of variables, stationary solutions of these equations lead to the following fourth order equation:

u^ð4Þþwu⁰⁰uþu³¼0;

http://dx.doi.org/10.1016/j.amc.2014.11.056 0096-3003/Ó2014 Elsevier Inc. All rights reserved.

⇑Corresponding author.

E-mail addresses:[email protected](T. Li),[email protected](J. Sun),[email protected](T.-f. Wu).

Contents lists available atScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

wherew>0 corresponds to EFK equation andw<0 to SH equation.

The study of homoclinic and heteroclinic solutions for the fourth order differential equations has attracted a lot of atten- tion by many researchers, see[1,3,4,9–12,14,15]. These works are mainly concerned on the autonomous case, such as the following equation:

u^ð4Þþwu⁰⁰þ

a

ubu²

c

u³¼0;

where

a

; b;

c

are nonnegative constants.

In 2001, Tersian and Chaparova[14]first considered a class of non-autonomous fourth order problems

u^ð4Þþwu⁰⁰þaðxÞubðxÞu²cðxÞu³¼0: ð1:2Þ

Applying the mountain pass theorem, they showed that Eq.(1.2)possesses one nontrivial homoclinic solutionu2H²ðRÞ whenaðxÞ; cðxÞanddðxÞare continuous periodic functions and satisfy some other assumptions. If there is no periodicity assumption ofaðxÞ; cðxÞanddðxÞ, then the case will be more difficult. In 2009, Li[9]studied the nonperiodic case of Eq.

(1.2)and obtained the existence of nontrivial homoclinic solution by establishing a compactness lemma and using the mountain pass theorem. Furthermore, the author also studied a class of the nonhomogeneous fourth order equations with the general nonlinear termf:

u^ð4Þþwu⁰⁰þaðxÞu¼fðx;uÞ þhðxÞ;

and obtained the existence of homoclinic solution whenfsatisfied the well-known (AR) condition. Very recently, Sun and Wu[12]considered a class of fourth order differential equations with a perturbation:

u^ð4Þþwu⁰⁰þaðxÞu¼fðx;uÞ þkhðxÞjuj^p2u; ð1:3Þ

wherek>0 is a parameter, 16p<2 andh2L^2p2ðRÞ. By using variational methods, the existence result of two homoclinic solutions for Eq.(2.6)is obtained if the parameterkis small enough. In all these papers, in order to obtain an important inequality, the following condition

(A) there exists a constanta1>0 such that

0<a16aðxÞ ! þ1; asjxj ! þ1; ð1:4Þ

andw2 ffiffiffiffiffi a1

p ;

is required. However, if there exists the functionasatisfyinga¼0 in some finite intervalTofR, then the conditionðAÞdoes not hold.

Inspired by the above facts, the aim of this paper is to consider this case. We shall establish the existence result of non- trivial homoclinic solutions for Eq.(1.1)when the nonlinear termfsatisfies the asymptotically linear condition. Moreover, the non-existence of nontrivial homoclinic solutions will be discussed.

Before stating our result we need to introduce some notations.

Notation 1.1. Throughout this paper, we denote byj j_r theL^r-norm, 26r61 andh¼maxfh;0g. Also if we take a subsequence of a sequencefungwe shall denote it againfung.

Now we state our main result.

Theorem 1.1. Assume that the conditions (V1)–(V4) hold and w<2. In addition, we assume that the function f satisfy the following conditions:

(D1) f x;ð sÞis a continuous function onRRsuch that fðx;sÞ 0for all s<0and x2R. Moreover, there exists p2L¹ðR;R^þÞ with pj j₁<^H₂⁰such that

lim

s!0^þ

f x;ð sÞ

s ¼pðxÞuniformly inx2R and

f x;ð sÞ

s PpðxÞfor alls>0 andx2R;

whereH0:¼^c0c0S

2 1jfa<cgj

ð Þ

S²₁jfa<cgj ; (D2) there exist r>1and q2L¹X;R^þ

withjqj₁>0such that lims!1

f x;ð sÞ

s^r ¼0 uniformly inx2RnX and

lims!1

f x;ð sÞ

s ¼qðxÞuniformly inx2X;

(D3) there exist two constantsh;d0satisfyingh>2and06d0<^ðh2_2h^ÞH0such that Fðx;sÞ 1

hfðx;sÞs6d0s²for alls>0 andx2R;

(D4)

l

:¼inf R

Xu⁰⁰ðxÞ²wu⁰ðxÞ²

dxju2H²ðXÞ \H¹₀ðXÞ;R

XqðxÞu²dx¼1

n o

<1.

Then there existsK0>0such that for everyk>K0, Eq. (1.1) has at least one homoclinic solution.

Remark 1.1. By[14, Lemma 8]and Sobolev embedded theorem, it is not difficult to claim that

l

>0, which is achieved by some/₁2H²ðXÞ \H¹₀ðXÞwithR

Xq/²₁dx¼1; seeAppendix A Now, we consider the following the minimum problem:

l

₀¼inf Z

R

u⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²

dxju2H²ðRÞ;

Z

R

qðxÞu²dx¼1

: ð1:5Þ

Then we have the following result.

Theorem 1.2. Suppose that the conditions (V1)–(V4) and (D1)–(D2) hold. If

l

0> ¹

c0S²₁jfa<cgj and s#^{f x;sð}_s ^Þ is non-decreasing function for any fixed x2R, then for anykP¹_c, Eq.(1.1)does not admit any nontrivial homoclinic solution.

The remainder of this paper is organized as follows. In Section2, some preliminary results are presented. In Section 3, we give the proofs of our main results.

2. Variational setting and preliminaries

In this section, we give the variational setting for Eq.(1.1). We need the following result.

Lemma 2.1 [14, Lemma 8]. Assume that w<2. Then there exists a constant c0>0such that Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þuðxÞ²

h i

dxPc0kuk²_H² for allu2H²ðRÞ; ð2:1Þ

wherekuk_H2¼ R

Rhu⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²i

dx1=2

is the norm of Sobolev space H²ðRÞ.

Let

X¼ fu2H²ðRÞj Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þaðxÞuðxÞ²

h i

dx<þ1g

be equipped with the inner product and norm u;

v

ð Þ ¼ Z

R

u⁰⁰ðxÞ

v

^{00ðxÞ wu0ðxÞ}

v

^{0ðxÞ þaðxÞuðxÞ}

v

^ðxÞ

½ dx

and corresponding normk ku ²¼ðu;uÞ. Fork>0, we also need the following inner product and norm ðu;

v

Þ_k¼

Z

R

u⁰⁰ðxÞ

v

^{00ðxÞ wu0ðxÞ}

v

^{0ðxÞ þkaðxÞuðxÞ}

v

^ðxÞ

½ dx

and corresponding normk ku ²_k¼ ðu;uÞ_k. It is clear thatk ku 6k ku _kforkP1. SetXk¼ðX;k ku _kÞ. From the conditions (V1)–(V4), (2.1)and the Sobolev inequality, we have

c0

Z

R

u⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²

h i

dx6 Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þuðxÞ²

h i

dx

¼ Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxþ Z

a<c

f g

uðxÞ²dxþ Z

aPc

f g

uðxÞ²dx 6

Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxþk ku ²₁jfa<cgj þ1 kc

Z

R

kaðxÞjuðxÞj²dx 6

Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxþ1 kc

Z

R

kaðxÞuðxÞ²dx þS²₁jfa<cgj

Z

R

u⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²

h i

dx:

Thus, Z

R

u⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²

h i

dx6 1

c0S²₁jfa<cgj Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxþ1 kc

Z

R

kaðxÞuðxÞ²dx

6 1

c0S²₁jfa<cgj Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þkaðxÞuðxÞ²

h i

dx

¼ 1

c0S²₁jfa<cgjk ku ²_k for allkP1

c; ð2:2Þ

which implies that the imbeddingXk,!H²ð ÞR is continuous for allkP¹_c, here the setfaPcg:¼fx2RjaðxÞPcg. Further- more, usingLemma 2.1again, one has

Z

R

uðxÞ²dx¼ Z

a<c

f g

uðxÞ²dxþ Z

faPcg

uðxÞ²dx6k ku ²₁jfa<cgj þ1 kc

Z

R

kaðxÞuðxÞ²dx 6S²₁jfa<cgj

Z

R

u⁰⁰ðxÞ²þu⁰ðxÞ²þuðxÞ²

h i

dxþ1 kc

Z

R

kaðxÞuðxÞ²dx

6S²₁jfa<cgj c0

Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þuðxÞ²

h i

dxþ1 kc

Z

R

kaðxÞuðxÞ²dx;

this implies that Z

R

uðxÞ²dx6 1 c0S²₁jfa<cgj

S²₁jfa<cgj c0

Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxþ1 kc

Z

R

kaðxÞuðxÞ²dx

" #

6

max ^S21jf_c^a<cgj

0 ;_kc¹

n o

c0S²₁jfa<cgj Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þkaðxÞuðxÞ²

h i

dx¼ S²₁jfa<cgj

c0c0S²₁jfa<cgjk ku ²_k

¼H¹₀ k ku ²_k; for allkP c0

cS²₁jfa<cgjP1

c: ð2:3Þ

Thus, by(2.2) and (2.3), for anyr2ð2;1ÞandkP ^c0

cS²₁jfa<cgj, one has Z

R

uðxÞ

j j^rdx6k ku ^r2₁ Z

R

uðxÞ²dx6S^r2₁ 1

c0S²₁jfa<cgjk ku ²_k

!^r2₂

S²₁jfa<cgj c0S²₁jfa<cgjk ku ²_k

¼ 1

a<c

f g

j j^r22

S²₁jfa<cgj c0S²₁jfa<cgj

!₂^r

k ku ^r_k¼ 1 a<c

f g

j j^r22H²₀^{rk ku}

r

k: ð2:4Þ

Now we begin describing the variational formulation of Eq.(1.1). Consider the functionalJ:Xk!Rdefined by JðuÞ ¼1

2k ku ²_k Z

R

Fðx;uÞdx; ð2:5Þ

whereFis the primitive Fðx;uÞ ¼

Z u 0

fðx;sÞds:

Sincefis continuous, we deduce thatJis of classC¹and its derivative is given by hJ⁰ðuÞ;

u

i ¼

Z

R

u⁰⁰ðxÞ

v

^{00ðxÞ wu0ðxÞ}

v

^{0ðxÞ þkaðxÞuðxÞ}

v

^ðxÞ

½ dx

Z

R

fðx;uðxÞÞ

u

ðxÞdx;

for allu;

u

2Xk. Then, we can infer thatu2Xkis a critical point ofJif and only if it is a homoclinic solution of Eq.(1.1).

Furthermore, we have the following result.

Lemma 2.2. Suppose that the conditions (D1) and (D3) hold. Let u0 be a nontrivial homoclinic solution of Eq.(1.1), we have J uð 0Þ>0.

Proof. Sinceu0is a nontrivial homoclinic solution of Eq.(1.1), u0

k k²_k¼ Z

R

f x;ð u0Þu0dx: ð2:6Þ

By the conditions (D1) and (D3), the Hö lder inequality,(2.3) and (2.6), we have

J uð Þ ¼0 1 2k ku0 ²_k

Z

R

F x;ð u0ÞdxP1

2k ku0 ²_kd0

Z

R

u0ðxÞ²dx1 h Z

R

f x;ð u0Þu0dxP h2 2h d0

H0

u0

k k²_k>0;

whereH0:¼^1S_S2^21jfb<cgj

1jfb<cgj >0 as in the condition (D1). This completes the proof. h

Next, we give a useful theorem. It is the variant version of the mountain pass theorem, which allows us to find a so-called Cerami typeðPSÞsequence.

Theorem 2.1 ([7], Mountain Pass Theorem). Let E be a real Banach space with its dual space E, and suppose that I2C¹ðE;RÞ satisfies

maxfIð0Þ;IðeÞg6

l

<

g

⁶ inf

kuk¼qIðuÞ;

for some

l

<

g

;

q

>0and e2E withkek>

q

. Let^cP

g

be characterized by

^c¼inf

c2Cmax

06s61Ið

c

ð

s

ÞÞ;

where C¼ f

c

2Cð½0;1 ;EÞ:

c

ð0Þ ¼0;

c

ð1Þ ¼eg is the set of continuous paths joining 0 and e, then there exists a sequence fung E such that

IðunÞ !^cP

g

and ð1þ kunkÞkI⁰ðunÞkE!0; asn! 1:

3. Proofs of Theorems1.1 and 1.2

In what follows, we give the following two lemmas which ensure that the functionalJhas the mountain pass geometry, which will be used in the proof ofTheorem 1.1.

Lemma 3.1. Suppose that the conditions (V1)–(V4) and (D1)–(D2) hold. Then for everykP ^c0

cS²₁jfa<cgjthere exist two positive constants

q

;

g

such that JðuÞj_kuk

k¼qP

g

>0.

Proof. For any

>0, it follows from the conditions (D1) and (D2) that there existC>0 andr>2 such that Fðx;sÞ6j jp₁þ

2 s²þC

r jsj^r; for alls2R: ð3:1Þ

So that, fromLemma 2.1, (3.1)and the Sobolev inequality, we have for allu2Xk, Z

R

Fðx;uÞdx6j jp1þ

2 Z

R

u²dxþC r

Z

R

juj^rdx6j jp1þ

2H0

u k k²_kþC

r

c^r2₀² a<c

f g

j j^r22H^r2₀^{k ku}

r k;

which implies that JðuÞ ¼1

2k ku ²_k Z

R

Fðx;uÞdxP1

2k ku ²_kj jp1þ

2H0

u k k²_kC

r

c^r2₀² a<c

f g

j j^r22H²₀^{rk ku}

r k

¼k ku ²_k 1

2 1j jp1þ

H0

Cc^r2₀² r ajf <cgj^r22H^r2₀^{k ku}

r2 k

" #

: ð3:2Þ

Take

¼^H₂⁰j jp₁. It follows from(3.2)that there exist

q

;

g

>0 such thatJðuÞj_kuk

k¼qP

g

. h

Lemma 3.2. Suppose that the conditions (V1)–(V4), (D2) and (D4) hold. Let

q

>0be as inLemma3.1. Then there exists e2Xk

withkek_k>

q

such that JðeÞ<0for allkP ^c0

cS²₁jfa<cgj.

Proof. By the conditionðD4ÞandRemark 1.1, we can choose a nonnegative function/₁2H²ð Þ \X H¹₀ð ÞX with Z

X

qðxÞ/²₁ðxÞdx¼1 such that Z

X

/⁰⁰₁ðxÞ²w/⁰₁ðxÞ²

h i

dx¼

l

<1:

Therefore, by the condition (D2) and Fatou’s lemma, we have

t!þ1lim Jðt/₁Þ

t² ¼1

2k/1k²_klim

t!þ1

Z

R

Fðx;t/₁Þ

t²/²₁ /²₁dx¼1 2 Z

X

/⁰⁰₁ðxÞ²w/⁰₁ðxÞ²

h i

dx lim

t!þ1

Z

X

Fðx;t/₁Þ t²/²₁ /²₁dx 61

2

l

1 2 Z

R

qðxÞ/²₁ðxÞdx¼1

2ð

l

1Þ<0:

So, ifJðt/₁Þ ! 1ast! þ1, then there existse2Xkwithkek_k>

q

such thatJðeÞ<0. h Next, we define

a

¼inf

c2C_k max

06t61Jð

c

ðtÞÞ and

a

0ð Þ ¼X inf

c2C_kð ÞT max

06t61Jj_H²_{ð Þ\HX} ¹

0ð ÞXð

c

ðtÞÞ;

whereJj_H2ð Þ\HX ¹₀ð ÞX is a restriction ofJonH²ð Þ \X H¹₀ð Þ,X C¼ f

c

2Cð½0;1 ;XkÞ:

c

ð0Þ ¼0;

c

ð1Þ ¼eg and

Cð Þ ¼ fT

c

2Cð½0;1 ;H²ð Þ \X H¹₀ð ÞÞX :

c

ð0Þ ¼0;

c

ð1Þ ¼eg:

Note that Jj_H²_{ð Þ\HX} ¹

0ð ÞXð Þ ¼u 1 2 Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dx Z

R

Fðx;uÞdx;

foru2H²ð Þ \X H¹₀ð ÞX and

a

0ð ÞX independent ofk. Moreover, if the conditions (D1)–(D4) hold, then by the proofs ofLemmas 2.1 and 3.1, we can conclude that Jj_H2ð Þ\HX ¹₀ð ÞX satisfies the mountain pass hypothesis as in Theorem 2.1. Since

H²ð Þ \X H¹₀ð ÞX

Xkfor allk>0, we have 0<

g

⁶

a

⁶

a

0ð ÞX for allkP ¹

cS²1jfa<cgj. TakeD0>

a

0ð Þ. Thus,T 0<

g

⁶

a

k6

a

0_{ð Þ}X <D0for allkP 1

cS²₁jfa<cgj: ð3:3Þ

FromLemmas 2.1 and 3.1andTheorem 2.1, we obtain that for eachkP ¹

cS²1jfa<cgj, there existsfung Xksuch that JðunÞ !

a

>0 and ð1þ kunk_kÞkJ⁰ðunÞk_X¹

k !0; asn! 1; ð3:4Þ

where 0<

g

⁶

a

⁶

a

0ð ÞX <D0. Furthermore, we have the following results.

Lemma 3.3.Suppose that the conditions (V1)–(V4) and (D1)–(D3) hold. Then fungdefined by(3.4)is bounded in Xk for all kP ¹

cS²₁jfb<cgj.

Proof. Fornlarge enough, byðD3Þ, the Hölder inequality and Lemmas2.1 and 3.1, one has

a

þ1PJ uð Þ n 1

hhJ⁰ðunÞ;uni ¼ 1 21

h

kunk²_k Z

R

Fðx;unÞ 1 hfðx;unÞ

dxPh2

2h kunk²_kd0

Z

R

u²_ndx

P h2 2h d0

H0

kunk²_k;

which implies thatfungis bounded inXk. h

Proposition 3.4. Suppose that the conditions (V1)–(V4) and (D1)–(D4) hold. Let D0>0 be as in (3.3). Then there exists K¼KðD0ÞP ¹

cS²₁jfb<cgjsuch that J satisfies theðCÞ_a–condition in Xkfor all

a

<D0andk>K.

Proof. Letfu_ngbe aðCÞ_a–sequence with

a

<D₀. ByLemma 3.3, there exist a subsequencefu_ngandu₀inXksuch that un*u0weakly inXk;

un!u0strongly inL^rlocð Þ;R for 26r<1: ð3:5Þ

Now we prove thatun!u0strongly inXk. Let

v

n¼unu0. Then by the conditions (D1)–(D4) and Brezis–Lieb Lemma[2], we have

J^ð

v

^{nÞ ¼J uð Þ n J uð Þ þ0 oð Þ1 andJ0ð}

v

^{nÞ ¼oð Þ:1}

It follows from (V2) and(3.5)that Z

R

v

²nðxÞdx¼ Z

aPc

f g

v

²nðxÞdxþ Z

a<c

f g

v

²nðxÞdx6 1 kc

Z

aPc

f g

kaðxÞ

v

²nðxÞdxþoð Þ1 6 1

kck

v

ⁿk²_kþoð Þ:1 ð3:6Þ

Then, by the Sobolev inequality and(2.2), we have Z

R

v

nðxÞ

j j^rdx6j

v

nj^r2₁ Z

R

v

nðxÞ²dx6S^r2₁

kc k

v

nk^r2_H²k

v

nk²_kþoð Þ1 6S^r2₁ kc

1 c0S²₁jfa<cgj

!^r2₂

v

n

k k^r_kþoð Þ1: ð3:7Þ

By the conditions (D1)–(D3) and Brezis–Lieb Lemma[2], we have Jð

v

nÞ ¼J uð Þ n J uð Þ þ0 oð Þ1 andJ⁰ð

v

nÞ ¼oð Þ1:

Consequently, this together with the condition (D3),Lemma 2.2 and (2.3), we obtain D0>

a

J uð Þ0 PJð

v

nÞ 1

h J⁰ð

v

nÞ;

v

n

þoð Þ1 P h2 2h d0

H0

v

n

k k²_kþoð Þ;1

which implies that

v

ⁿ

k k²_k6 2hH0D0

H0ðh2Þ 2hd0

þoð Þ:1 Moreover, by(2.4), one has

Z

R

v

nðxÞ

j j^rdx6 c^r2₀² a<c

f g

j j^r22H^r2₀k

v

nk^r_k6 1 a<c

f g

j j^r22

2hH0D0

H0ðh2Þ 2hd0

₂^r

þoð Þ1: ð3:8Þ

Since J⁰ð

v

nÞ;

v

n

¼oð Þ1 and Z

R

f x;ð

v

nÞ

v

ndx6jp^þj₁þ

^Z

R

v

nðxÞ²dxþC Z

R

v

nðxÞ

j j^rdx; ð3:9Þ

it follows from(3.6) and (3.8) and (3.9)that oð Þ ¼1 k

v

nk²_k

Z

R

fðx;

v

nÞ

v

ndxPk

v

nk²_kjp^þj₁þ

^Z

R

v

²nðxÞdxC Z

R

v

nðxÞ j j^rdx

Pk

v

nk²_kjp^þj₁þ

kc k

v

nk²_kC Z

R^N

v

n

j j^rdx

ðr2Þ=r Z

R^N

v

n

j j^rdx

2=r

Pk

v

^nk2k 1jp^þj₁þ

kc 2hH0D0

H0ðh2Þ 2hd0

ð Þjfa<cgj^r2r

!^r22

S^r2₁ kc

1 c0S²₁jfa<cgj

!^r22

2 4

3 5 8 2=r

><

>:

9>

=

>;: Thus, there existsK¼KðD0ÞPmax ¹_c;_cS2 ¹

1jfa<cgj

n o

such that

v

n!0 strongly inXkfork>K. This completes the proof.

h

Now we give the proof ofTheorem 1.1: ByProposition 3.4and 0<

g

⁶

a

⁶

a

0ð ÞX for allkP ¹

cS²₁jfa<cgj, for eachD₀>

a

0ð ÞX there exists

KPmax 1

c; 1

cS²₁jfa<cgj

( )

>0

such that for everyk>KandðCÞ_a–sequencefu_ngforJonXkthere exist a subsequencefu_nganduk2Xksuch thatu_n!uk

strongly inXk. Moreover,J uð Þ ¼k

a

andukis a nontrivial homoclinic solution of Eq.(1.1).

Now we give the proof ofTheorem 1.2:Letu02H²ð ÞR be a nontrivial homoclinic solution of Eq.(1.1). Then by(2.2), for kP¹_cwe have

Z

R

u⁰⁰₀ðxÞ²wu⁰₀ðxÞ²þkaðxÞuðxÞ²₀

h i

dx¼ Z

R

fðx;u0Þu0dx6 Z

R

qu²₀dx6 1

l

₀^{k ku0}

2 H²

6 1

l

₀c0S²₁jfa<cgj Z

R

u⁰⁰₀ðxÞ²wu⁰₀ðxÞ²þkaðxÞuðxÞ²₀

h i

dx

<

Z

R

u⁰⁰₀ðxÞ²wu⁰₀ðxÞ²þkaðxÞuðxÞ²₀

h i

dx;

which is a contradiction.

Acknowledgment

T. Li was supported by the NSFC (Grand No. 11471074, No. 91330109) and the Fundamental Research Funds for the Central Universities. J. Sun was supported by the NSFC (Grant No. 11201270, No. 11271372), Shandong Natural Science

Foundation (Grant No. ZR2012AQ010), and Young Teacher Support Program of Shandong University of Technology. T. F. Wu was supported by the National Science Council, Taiwan.

Appendix A

Consider the minimum problem

l

¼inf Z

X

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dxju2H²ðX_{Þ \}H¹₀ðX_Þ; Z

X

qu²dx¼1

: ð4:1Þ

Then we have the following result.

Lemma 4.1.There exist a constant c1>0and/₁2H²ðXÞ \H¹₀ðXÞwithR

Xq/²₁dx¼1such that

l

¼ Z

X

/⁰⁰₁ðxÞ²w/⁰₁ðxÞ²

h i

dxPc1;

i.e., the minimum problem(4.1)is achieved.

Proof. For anyu2H²ðXÞ \H¹₀ðXÞwithR

Xqu²dx¼1, byLemma 2.1and Sobolev embedded theorem, we have Z

X

u⁰⁰ðxÞ²wu⁰ðxÞ²

h i

dx¼ Z

R

u⁰⁰ðxÞ²wu⁰ðxÞ²þaðxÞuðxÞ²

h i

dxPc0kuk²_H²Pc0S²₂ Z

X

u²dxPc0S²₂ jqj₁>0:

Therefore,

l

P^c_jqj^0S22

1>0. Letfu_ng H²ðXÞ \H¹₀ðXÞbe a minimizing sequence of(4.1). Clearly,R

Xqu²_ndx¼1 andfu_ngis bounded. Then by the compact imbedding theorem, there exist a subsequence fung and /₁2H²ðXÞ \H¹₀ðXÞsuch that un*/₁weakly inH²ðXÞ \H¹₀ðXÞandun!/₁strongly inL²ðRÞ. So it is easy to verify thatR

Xqu²_ndx!R

Xq/²₁dxasn! 1 andR

Xq/²₁dx¼1. Therefore,

l

^6Z

X

/⁰⁰₁ðxÞ²w/⁰₁ðxÞ²

h i

dx6lim inf

n!1

Z

X

u⁰⁰_nðxÞ²wu⁰_nðxÞ²

h i

dx¼

l

; which implies thatR

Xh/⁰⁰₁ðxÞ²w/⁰₁ðxÞ²i

dx¼

l

. This completes the proof. h

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