Existence and multiplicity of systems of Kirchhoff ... - Wiley Online Library

Research Article Received 11 September 2015

Published online 28 June 2016 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.4007 MOS subject classification: 35B38; 35J20

Existence and multiplicity of systems of Kirchhoff-type equations with general potentials Guofeng Che and Haibo Chen *† Communicated by K. Gürlebeck This paper is concerned with the following systems of Kirchhoff-type equations: 8 Z > 2 > a C b jr uj dx u C V.x/u D Fu .x, u, v/, x 2 RN , > < RN R c C d RN jr vj2 dx v C V.x/v D Fv .x, u, v/, x 2 RN , > > > : u.x/ ! 0, v.x/ ! 0 as jxj ! 1. Under more relaxed assumptions on V.x/ and F.x, u, v/, we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: Kirchhoff-type equations; nontrivial solutions; Morse theory; local linking

1. Introduction Recently, many authors consider the following Kirchhoff-type problem: Z .a C b jruj2 dx/u C V.x/u D f .x, u/, x 2 RN ,

(1.1)

RN

where a >R 0, b 0 are constants. Problem (1.1) is an important nonlocal quasilinear problem because of the appearence of the term . RN jruj2 dx/u, which provokes some mathematical difficulties and also makes the study of such a class of problem particularly interesting. When V.x/ 0 and is a bounded domain of RN , problem (1.1) reduces to the following Dirichlet problem: Z 8 < .a C b jruj2 dx/u D f .x, u/, in , (1.2) : u D 0 on @, which is related to the stationary analog of the equation of Kirchhoff-type Z jruj2 dx/4u D f .x, u/, in , utt .a C b

(1.3)

where u denotes the displacement, f .x, u/ the external force, b the initial tension while a is related to the intrinsic properties of the string (such as Young’s modulus). Equations of this type were first proposed by Kirchhoff in [1] to describe the transversal oscillations

Copyright © 2016 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2017, 40 775–785

775

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China * Correspondence to: Haibo Chen, School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China. † E-mail: [email protected]

G. CHE AND H. CHEN of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations. For more details on the physical and mathematical background of problem (1.2), we refer the readers to the papers [1–4] and the references therein. More recently, Kirchhoff-type problems like (1.2) (in bounded domain) have been widely investigated by many authors [5–8]. In [5], Chen, Kuo, and Wu established the existence of multiple positive solutions for Kirchhoff-type equations that involve sign-changing weight functions by using Nehari manifold and fibering map. He and Zhou [6] obtained infinitely many solutions via the local minimum methods and the fountain theorems. Mao and Zhang [7] obtained multiple and sign-changing solutions via the invariant sets of decent flow; Perera and Zhang in [8] obtained a nontrivial solution via Young index and critical group. Equations like (1.1) in the whole space RN have also been studied extensively, for example, see [2, 9–19] and the references therein. In [9], Huang and Liu established the existence and nonexistence for Kirchhoff-type equations when f .x, u/ D jujp2 u and p 2 .2, 6/ via variational methods. Also, the ‘energy dobuling’ property of nodal solutions was discussed. Jin and Wu in [10] obtained three existence results of infinitely many radial solutions for a class of Kirchhoff-type problems by using the fountain theorem. In [11], Wu obtained four new existence results for nontrivial solutions and a sequence of high energy solutions for Schrödinger–Kirchhoff type equations by using a symmetric mountain pass theorem. Recently, Wu in [12] obtained five new critical point theorems on the product spaces and three existence theorems for the sequence of high energy solutions for the following systems of Kirchhoff-type equations: Z 8 ˆ ˆ .a C b jruj2 dx/u C V.x/u D Fu .x, u, v/, x 2 RN , ˆ < RN R .c C d RN jrvj2 dx/v C V.x/v D Fv .x, u, v/, x 2 RN , ˆ ˆ ˆ : u.x/ ! 0, v.x/ ! 0 as jxj ! 1,

(1.4)

where a, c > 0, b, d 0, they assumed that the potential V.x/ satisfied: .V0 / V.x/ 2 C.RN , R/ satisfies inf V.x/ b > 0, where b is a constant. x2RN

0

.V0 / for any M > 0, measfx 2 RN : V.x/ Mg < 1, where meas denotes the Lebesgue measure in RN . 0 Later, under the conditions .V0 / and .V0 /, Zhou, Wu, and Wu in [13] presented a new proof technique to conclude the existence of high energy solutions for problem (1.4) under some assumptions that are weaker than those in [12], which unify and sharply improve the Theorems 3.1–3.3 in [12] as well as some results in other literature such as Theorems 1–4 in [11]. On the other hand, Morse theory and local linking theorem are powerful tools in modern nonlinear analysis [20–23], especially for the problems with resonance [24, 25]. However, to the best of our knowledge, there are no existed papers dealing with the existence of solutions for systems of Kirchhoff-type equations by using Morse theory. Inspired by the earlier works just described, the goal of this paper is to consider the existence and multiplicity of nontrivial solutions for problem (1.4). Under some natural assumptions, we prove the problem has at least two nontrivial solutions by using Morse theory and local linking arguments. On the other hand, by using the Clark theorem, the existence results of at least 2k distinct pairs of nontrivial solutions are obtained. It is worthy stressing that we will use a more general assumption on V.x/, which extends some recent results from the literature. Precisely, for the potential V.x/ and F.x, u, v/, we assume .V1 / V 2 C.RN , R/ satisfies inf V.x/ D a > 0. x2RN

2

.f1 / F 2 C.RN R2 , R/, and there exist 1 < ˛1 < ˛2 < < ˛m < 2, 1 < ˇ1 < ˇ2 < < ˇm < 2, ci .x/ 2 L 2˛i .RN , RC / and 2 di .x/ 2 L 2ˇi .RN , RC / such that

jFu .x, u, v/j

m X

˛i ci .x/j.u, v/j˛i 1 , jFv .x, u, v/j

iD1

m X

ˇi di .x/j.u, v/jˇi 1 , 8.x, u/ 2 RN R2 ,

iD1

1

where j.u, v/j D .u2 C v 2 / 2 . .f2 / There exist c1 > 0, 0 < c2
0 such that kukr Cr kukX , 8u 2 X and for any bounded sequence fun g X, there exists a subsequence of fun g such that un * u0 in X, un ! u0 in Lrloc .RN /, r 2 Œ2, 2 /. Therefore, for any .u, v/ 2 E, there exists C > 0 such that jj.u, v/jjrr C jjujjrr C jjvjjrr C jjujjrX C jjvjjrX Cjj.u, v/jjr , that is, jj.u, v/jjr Cjj.u, v/jj, that is, E ,! Lr .RN / Lr .RN / is continuous for 2 r 2 . On the other hand, suppose f.un , vn /g E are bounded, that is, fun g and fvn g are bounded in X, then there exist subsequences fun g and fvn g such that un ! u0 , vn ! v0 in Lrloc .RN /, r 2 Œ2, 2 /. Therefore,


777

0 jj.un , vn / .u0 , v0 /jjrr C.jjun u0 jjrr C jjvn v0 jjrr / ! 0, Math. Meth. Appl. Sci. 2017, 40 775–785

G. CHE AND H. CHEN as n ! 1, that is, .un , vn / ! .u0 , v0 /, in Lrloc .RN / Lrloc .RN /, r 2 Œ2, 2 /, that is, E ,! Lrloc .RN / Lrloc .RN / is compact for r 2 Œ2, 2 /. The proof is completed. Lemma 2.2 Assume that .V1 / and .f1 / hold. Then the functional I : E ! R defined by

I.u, v/ D

a 2

Z

b 4

jruj2 dx C

RN

c C 2

Z

Z

jruj2 dx

2 C

RN

Z

d jrvj dx C 4 RN 2

2

2

jrvj dx RN

Z

1 2

V.x/u2 dx

RN

Z

1 C 2

(2.1)

Z

2

V.x/v dx

F.x, u, v/dx

RN

RN

is well defined and of class C 1 .E, R/ and

hI0 .u, v/, .', /i D .a C b

Z

jruj2 dx/

Z

Z

RN

V.x/u'dx

RN

Z

jrvj2 dx/

C .c C d

Z

rur'dx C

RN

RN

Z

Fu .x, u, v/'dx RN

Z

(2.2)

Z

rvr dx C

V.x/v dx

RN

RN

Fv .x, u, v/ dx RN

Moreover, the critical points of I in E are solutions of problem (1.4). Proof R Set ˆ.u, v/ D RN F.x, u, v/dx, then by the definition of I, it suffices to show that ˆ.u, v/ 2 C 1 .E, R/ and

hˆ0 .u, v/, .', /i D

Z

Z Fu .x, u, v/'dx C RN

Fv .x, u, v/ dx.

(2.3)

RN

First, we prove the existence of the Gateaux derivative of ˆ. From .f1 /, one has jF.x, u, v/j D jF.x, u, v/ F.x, 0, 0/j Z 1 Z 1 jFu .x, tu, tv/jjujdt C jFv .x, tu, tv/jjvjdt 0

(2.4)

0

m X

ci .x/j.u, v/j˛i C

m X

iD1

di .x/j.u, v/jˇi .

iD1

Then, for any .u, v/ 2 E, it follows from .V1 /, (2.4) and the Hölder inequality that "

Z

Z jF.x, u, v/jdx RN

RN

m X

a

m X

ci .x/j.u, v/j

˛i 2

# di .x/j.u, v/j

Z jci .x/j

a

ˇi 2

a

˛i 2

2 2˛i

dx

i Z 2˛ 2

jdi .x/j

2 2˛i

˛2i

V.x/j.u, v/j dx RN

Z

jjci jj

2

dx

2 2ˇi

i Z 2ˇ 2

dx

RN

iD1

ˇi

iD1

iD1

m X

RN

m X

m X

C

iD1

iD1

C

˛i

V.x/j.u, v/j2 dx

ˇ2i

(2.5)

RN

jj.u, v/jj˛i C

m X iD1

a

ˇi 2

jjdi jj

2 2ˇi

jj.u, v/jjˇi ,

778

which implies that I defined by (2.1) is well defined on E. Copyright © 2016 John Wiley & Sons, Ltd.


G. CHE AND H. CHEN For any function : RN ! .0, 1/, by .f1 / and the Hölder inequality, we have Z max jFu .x, u.x/ C t.x/'.x/, v.x/ C t.x/ .x//'.x/jdx Z max jFu .x, u.x/ C t.x/'.x/, v.x/ C t.x/ .x//jj'.x/jdx D

RN t2Œ0,1

RN t2Œ0,1

m X

Z

RN

iD1

C

.ci .x/j.u.x/ C t.x/'.x/, v.x/ C t.x/ .x//j˛i 1 /j'.x/jdx

˛i

" m Z X

C

.ci .x/.ju.x/j˛i 1 C j'.x/j˛i 1 /j'.x/jdx

N iD1 R Z m X

#

.ci .x/.jv.x/j

˛i 1

˛i 1

C j .x/j

/j'.x/jdx

RN

iD1

2 Z m X ˛i 4 2 C a

jci .x/j

2 2˛i

i Z 2˛ 2

dx

V.x/ju.x/j dx

RN

iD1

Z

RN

V.x/j'.x/j2 dx

˛i21

2

12

RN

C

m X

a

˛i 2

Z jci .x/j

i Z 2˛ 2

dx

m X

a

˛i 2

i Z 2˛ 2

2

jci .x/j 2˛i dx RN

V.x/jv.x/j2 dx

˛i21

RN

V.x/j'.x/j2 dx

(2.6)

˛2i

RN

Z

iD1

Z

2

V.x/j'.x/j dx

RN

iD1

C

2 2˛i

12

RN

C

m X

a

˛i 2

Z jci .x/j RN

iD1

Z

i Z 2˛ 2

2

˛i21

V.x/j .x/j dx

dx

12 #

2

2 2˛i

RN

V.x/j'.x/j dx RN

C

m X

jjci jj

iD1

2 2˛i

jjujj˛i 1 C jj'jj˛i 1 C jjvjj˛i 1 C jj jj˛i 1 jj'jj

< C1. Similarly, we have Z max jFv .x, u.x/ C t.x/'.x/, v.x/ C t.x/ .x// .x/jdx < C1.

(2.7)

RN t2Œ0,1

Then, by (2.1), (2.6), (2.7), and Lebesgue’s Dominated Convergence Theorem, we have ˆ.u C t', v C t / ˆ.u, v/ t Z Z Fu .x, u C t', v C t /'dx C D lim

hˆ0 .u, v/, .', /i D lim

t!0C

t!0C

Z

RN

RN

Z

Fu .x, u, v/'dx C

D RN

Fv .x, u C t', v C t / dx

(2.8)

Fv .x, u, v/ dx, RN

which implies (2.3) holds. Now, we show that ˆ.u, v/ 2 C 1 .E, R/. Let .un , vn / ! .u, v/ in E, then .un , vn / ! .u, v/ in L2 .RN / L2 .RN / and lim .un , vn / D .u, v/ a.e. x 2 RN RN .

n!1

(2.9)

Now, we claim that Z n!1 RN


jFu .x, un , vn / Fu .x, u, v/j2 dx D 0.

(2.10) Math. Meth. Appl. Sci. 2017, 40 775–785

779

lim

G. CHE AND H. CHEN Otherwise, there exist a constant "0 > 0 and a sequence f.uni , vni g such that Z jFu .x, uni , vni / Fu .x, u, v/j2 dx "0 , 8i 2 N.

(2.11)

RN 1 P In fact, because .un , vn / ! .u, v/ in L2 .RN / L2 .RN /, passing to a subsequence if necessary, it can be assumed that jj.uni , vni / iD1 1 12 P .u, v/jj22 < C1. Set !.x/ D jj.uni , vni / .u, v/jj22 , then !.x/ 2 L2 .RN /. Evidently iD1

jFu .x, uni , vni / Fu .x, u, v/j2 2jFu .x, uni , vni /j2 C 2jFu .x, u, v/j2 m h i X 2m1 ˛i2 jci .x/j2 j.uni , vni /j2.˛i 1/ C j.u, v/j2.˛i 1/ iD1

2m1 2m1

m X iD1 m X

h i .2˛i 1 C 1/˛i2 jci .x/j2 j.uni , vni / .u, v/j2.˛i 1/ C j.u, v/j2.˛i 1/

(2.12)

h i .2˛i 1 C 1/˛i2 jci .x/j2 j!.x/j2.˛i 1/ C j.u, v/j2.˛i 1/

iD1

:D h.x/, 8 i 2 N, x 2 RN , and

Z

h.x/dx D 2m1

RN

2m1

Z m X .2˛i 1 C 1/˛i2

h i jci .x/j2 j!.x/j2.˛i 1/ C j.u, v/j2.˛i 1/ dx

RN

iD1 m X

.2˛i 1 C 1/˛i2 jjci jj2

2 2˛i

iD1

i 1/ i 1/ C jj.u, v/jj2.˛ jj!jj2.˛ 2 2

(2.13)

< C1. It follows from (2.12), (2.13), and the Lebesgue’s Dominated Convergence Theorem that (2.10) holds. Analogously, we get Z lim jFv .x, un , vn / Fv .x, u, v/j2 dx D 0.

(2.14)

n!1 RN

Then, by (2.1), (2.10), (2.14), we have ˝ ˛ j ˆ0 .un , vn / ˆ0 .u, v/, .', / j

Z .jFu .x, un , vn / Fu .x, u, v/j/j'jdx RZN

C

.jFv .x, un , vn / Fv .x, u, v/j/j jdx "Z 12 jFu .x, un , vn / Fu .x, u, v/j2 dx jj'jj

RN

a

1 2

RN

Z C

jFv .x, un , vn / Fv .x, u, v/j2 dx

12

# jj jj

RN

! 0, as n ! 1, which implies that ˆ 2 C 1 .E, R/. Moreover, by a standard argument, it is easy to verify that the critical points of I in E are solutions of problem (1.1). ([27]). The proof is complete. We will use Morse theory in combination with local linking arguments to obtain the critical points of I, so we recall the following definitions and results. Definition 2.1 Let E be a real reflexive Banach space. We say that I satisfies the (PS)-condition, that is, every sequence fun g E satisfying I.un / is bounded and limn!1 I0 .un / D 0 contains a convergent subsequence. Let E be a real Banach space and I 2 C 1 .E, R/. K D fu 2 E : I0 .u/ D 0g, then the qth critical group of I at an isolated critical point u 2 K with I.u/ D c is defined by Cq .I, u/ :D Hq .Ic \ U, Ic \ U n fug/,

q 2 N :D f0, 1, 2, g,

780

where Ic D fu 2 E : I.u/ cg, U is a neighborhood of u, containing the unique critical point, H is the singular relative homology with coefficient in an Abelian group G. Copyright © 2016 John Wiley & Sons, Ltd.


G. CHE AND H. CHEN We say that u 2 E is a homological nontrivial critical point of I if at least one of its critical groups is nontrivial. Now, we present the following Theorems that will be used later. Theorem 2.1 ([28], Proposition 2.1) Assume that I has a critical point u D 0 with I.0/ D 0. Suppose that I has a local linking at 0 with respect to E D V ˚ W, k D dim V < 1, that is, there exists > 0 small such that ( I.u/ 0, u 2 V, kuk ; I.u/ > 0,

u 2 W,

0 < kuk .

Then Ck .I, 0/ © 0, hence 0 is a homological nontrivial critical point of I. Theorem 2.2 ([28], Proposition 2.1) ( Let E be a real Banach space and let I 2 C 1 .E, R/ satisfy the (PS)-condition and is bounded from below. If I has a critical point that is homological nontrivial and is not a minimizer of I, then I has at least three critical points. Theorem 2.3 ([29], Theorem 9.1) Let E be a real Banach space, I 2 C 1 .E, R/ with I even, bounded from below, and satisfying (PS)-condition. Suppose I.0/ D 0, there is a set K E such that K is homeomorphic to Sj1 by an odd map, and supK I < 0. Then I possesses at least j distinct pairs of critical points.

3. Proofs of main results In this section, we will prove Theorem 1.1 and Theorem 1.2. To complete the proof, we need the following lemmas. Lemma 3.1 Assume that .V1 / and .f1 / hold, then I is bounded from below and satisfies the .PS/ condition. Proof By Lemma 2.1, .f1 /, the Sobolev embedding theorem and the Hölder inequality, we have I.u, v/ D

a 2 C

Z

jruj2 dx C

RN

c 2

Z

jrvj2 dx C RN

Z

jruj2 dx

d 4

Z

jrvj2 dx

ZR

iD1

Z

C

V.x/u2 dx

RN

1 2

Z

V.x/v 2 dx

Z F.x, u, v/dx

RN

ci .x/j.u, v/j˛i dx

RN

1 minfa, c, 1gjj.u, v/jj2 a 2 iD1 ˇi 2

2

Z

RN

F.x, u, v/dx RN m Z X

m X

a

1 2

C

N

1 minfa, c, 1gjj.u, v/jj2 2 iD1

m X

2

RN

1 minfa, c, 1gjj.u, v/jj2 2

b 4

˛i 2

iD1

Z

2

di .x/j.u, v/jˇi dx

RN

V.x/j.u, v/j2 dx

jci .x/j 2˛i dx

˛2i

RN

V.x/j.u, v/j2 dx

jdi .x/j 2ˇi dx RN

(3.1)

i Z 2˛ 2

RN i Z 2ˇ 2

2

m Z X

ˇ2i

RN m X

m

X ˇi ˛i 1 minfa, c, 1gjj.u, v/jj2 a 2 jjci jj 2 jj.u, v/jj˛i a 2 jjdi jj 2 jj.u, v/jjˇi , 2˛ 2ˇi i 2 iD1 iD1

which implies that I.u, v/ ! C1, as n ! 1, because a, c > 0, ˛i , ˇi 2 .1, 2/. Consequently, I is bounded from below. Next, we prove that I satisfies the .PS/ condition. Assume that f.un , vn /g is a (PS) sequence of I such that I.un , vn / is bounded and jjI0 .un , vn /jj ! 0, as k.un , vn /k ! 1. Then, it follows from (3.1) that there exists a constant C > 0 such that jj.un , vn /jj2 a

1 2

jj.un , vn /jj C, n 2 N.

(3.2)

Then by Lemma 2.1, there exists .u, v/ 2 E such that .un , vn / * .u, v/ in E, .un , vn / ! .u, v/ in Lsloc .RN / Lsloc .RN / s 2 Œ2, 2 /,

(3.3)

.un , vn / ! .u, v/ a.e. RN RN .

781

On the other hand, for any given " > 0, by .f1 / , we can choose R" > 0 such that Copyright © 2016 John Wiley & Sons, Ltd.


G. CHE AND H. CHEN Z

i 2˛ 2

2

jci .x/j 2˛i dx

< ", i D 1, 2, : : : , m.

(3.4)

jxj>R"

It follows from (3.3) that there exists n0 > 0 such that Z

j.un , vn / .u, v/j2 dx < "2 , for n n0 .

(3.5)

jxjR"

Therefore, by .f1 /, (3.2), (3.5) and the Hölder inequality, for any n n0 , one has Z jFu .x, un , vn / Fu .x, u, v/jj.un , vn / .u, v/jdx jxjR"

Z

12 Z

2

2

jFu .x, un , vn / Fu .x, u, v/j dx jxjR"

12

j.un , vn / .u, v/j dx jxjR"

Z

2.jFu .x, un , vn /j2 C jFu .x, u, v/j2 /dx

"

12

jxjR"

" 2"

m X

˛i2

2"

m X

2"

(3.6)

/dx

˛i2 jjci jj2

#1

2 2.˛i 1/ 2.˛i 1/ C jj.u, v/jj2 jj.un , vn /jj2

˛i2 jjci jj2

#1

2 2.˛i 1/ 2.˛i 1/ C jj.u, v/jj2 . C

2 2˛i

m X

C j.u, v/j

2.˛i 1/

jxjR"

iD1

"

2.˛i 1/

2

jci .x/j .j.un , vn /j

iD1

"

# 12

Z

2 2˛i

iD1

For another, for n 2 N, it follows from .f1 /, (3.2), (3.4), and Hölder inequality that Z jFu .x, un , vn / Fu .x, u, v/jj.un , vn / .u, v/jdx jxj>R"

m X

Z jxj>R"

iD1

2

jci .x/j j.un , vn /j˛i 1 C j.u, v/j˛i 1 .j.un , vn /j C j.u, v/j/ dx

˛i

m X

Z jxj>R"

iD1

2

m X

Z ˛i

jci .x/j

2

Z

i 2˛ 2

dx

2

˛i

m X

jci .x/j 2˛i dx jxj>R"

iD1

2"

2 2˛i

jxj>R"

iD1 m X

jci .x/j j.un , vn /j˛i C j.u, v/j˛i dx

˛i

i 2˛ 2

˛i i jj.un , vn /jj˛ 2 C jj.u, v/jj2

i C ˛i C jj.u, v/jj˛ 2

(3.7)

i ˛i C ˛i C jj.u, v/jj˛ 2 .

iD1

Because " is arbitrary, combining (3.6) and (3.7), we have Z lim

n!1 RN

.Fu .x, un , vn / Fu .x, u, v//..un , vn / .u, v//dx D 0.

(3.8)

.Fv .x, un , vn / Fv .x, u, v//..un , vn / .u, v//dx D 0.

(3.9)

Arguing by the same way, we have Z

782

lim

n!1 RN



G. CHE AND H. CHEN Then by (2.2), (3.8), (3.9), and the weak convergence of f.un , vn /g, one has on .1/ D hI0 .un , vn / I0 .u, v/, .un u, vn v/i Z Z Z D aCb jrun j2 dx jr.un u/j2 dx C V.x/jun uj2 dx RN RN RN Z ŒFu .x, un , vn / Fu .x, u, v .un u/dx N R Z Z Z C cCd jrvn j2 dx jr.vn v/j2 dx C V.x/jvn vj2 dx RN RN RN Z ŒFv .x, un , vn / Fv .x, u, v/ .vn v/dx RN Z Z Z b jruj2 dx jrun j2 dx rur.un u/dx RN

Z

Z

2

RN 2

jrvj dx

d RN

Z

jrvn j dx RN

(3.10)

RN

ZR

N

rvr.vn v/dx Z Z jruj2 dx jrun j2 dx

rur.un u/dx minfa, c, 1gjj.un u, vn v/jj2 b N RN RN ZR Z Z jrvj2 dx jrvn j2 dx rvr.vn v/dx d RN RN RN Z Z ŒFu .x, un , vn / Fu .x, u, v .un u/dx ŒFv .x, un , vn / Fv .x, u, v/ .vn v/dx, RN

RN

For another, the boundedness of fun g and fvn g imply Z Z Z jruj2 dx jrun j2 dx b RN

Z d

RN

jrvj2 dx

RN

Z

rur.un u/dx ! 0, as n ! 1.

(3.11)

rvr.vn v/dx ! 0, as n ! 1.

(3.12)

RN

jrvn j2 dx

Z

RN

RN

Then, by (3.8)–(3.12), we have .un , vn / ! .u, v/ in E. Therefore, I satisfies the .PS/ condition. The proof is complete. We choose an orthogonal basis fej g of X and define Xj :D spanfej g, j D 1, 2, : : :, Yk :D ˚kjD1 Xj , Zk D ˚1 jDkC1 Xj , then X D Yk ˚ Zk and E D .Yk Yk / ˚ .Zk Zk /. Lemma 3.2 Suppose that the conditions of Theorem 1.1 are satisfied, then there exists k0 2 N such that Ck0 .I, .0, 0// © 0. Proof It follows from .f1 / that the zero function is a critical point of I. So we only need to prove that I has a local linking at 0 with respect to E D .Yk Yk / ˚ .Zk Zk /. Step 1 Take .u, v/ 2 Yk Yk , because Yk Yk is finite dimensional, we have that for given r0 , there exists 0 < < 1 small such that .u, v/ 2 Yk Yk ,

k.u, v/k ) j.u, v/j < r0 ,

x 2 RN .

For 0 r0 g, then RN D 3iD1 i . For the sake of simplicity, let G.x, u, v/ D F.x, u, v/ c1 j.u, v/j . Therefore, from .f2 /, it follows that 2 Z Z Z 1 c d V.x/u2 dx C jrvj2 dx C jrvj2 dx N N 2 RN 2 4 RN RN Z Z Z R Z Z R 1 C V.x/v 2 dx c1 j.u, v/j dx C C G.x, u, v/dx 2 RN RN 1 2 3 Z Z .b C d/ 1 jj.u, v/jj4 c1 j.u, v/j dx G.x, u, v/dx. maxfa, c, 1gjj.u, v/jj2 C 2 4 RN 1 R Note that the norms on Yk Yk are equivalent to each other, k.u, v/k is equivalent to k.u, v/k and 1 G.x, u, v/dx ! 0 as r ! 0. Because 0 < < 2, then I.u, v/ 0, for all .u, v/ 2 Yk Yk with k.u, v/k . I.u, v/ D

a 2

Z

jruj2 dx C

b 4

Z

jruj2 dx

2

C



783

Step 2 Take .u, v/ 2 Zk Zk , because the embedding E ,! Lp .RN / Lp .RN / is continuous, we have that for given r0 , there exists 0 < < 1 small such that

G. CHE AND H. CHEN .u, v/ 2 Zk Zk ,

k.u, v/k ) j.u, v/j < r0 ,

x 2 RN .

Therefore, it follows from .f2 / that I.u, v/ D

a 2 C

Z

jruj2 dx C

RN

c 2

Z

RN

b 4

jrvj2 dx C

Z

jruj2 dx

2

RN

d 4

Z

jrvj2 dx

ZR

C 2

1 2

C

N

Z

V.x/u2 dx

RN

1 2

Z

V.x/v 2 dx

RN

Z F.x, u, v/dx RN

1 minfa, c, 1gjj.u, v/jj2 c2 j.u, v/j2 dx 2 RN 1 1 > minfa, c, 1gjj.u, v/jj2 minfa, c, 1gjj.u, v/jj2 D 0 2 2

Therefore, we complete the proof due to Theorem 2.1. Proof of Theorem 1.1. By Lemma 3.1, I satisfies the (PS)-condition and is bounded from below. By Lemma 3.2 and Theorem 2.1, the trivial solution .u, v/ D .0, 0/ is homological nontrivial and is not a minimizer. Then Theorem 1.1 follows immediately from Theorem 2.2. Proof of Theorem 1.2. By .f3 /, we can easily check that the functional I is even. Lemma 3.1 shows that I satisfies the (PS)-condition and is bounded from below. For > 0, let K D S D f.u, v/ 2 Yk Yk : k.u, v/k D g. Thus, just as shown in the proof of Lemma 3.2, if > 0 is small enough, we have that sup I.u, v/ < 0. K

By the definition of Yk Yk , we have dim.Yk Yk / D 2k, then by Theorem 2.3, we have that I has at least 2k distinct pairs of critical points. Therefore, problem (1.4) has at least 2k distinct pairs of solutions.

Acknowledgements This work is partially supported by Natural Science Foundation of China 11271372 and Mathematics and Interdisciplinary Sciences Project of CSU.

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