Multiplicity of Nontrivial Solutions for Kirchhoff Type Problems

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Volume 2010, Article ID 268946,13pages doi:10.1155/2010/268946

Research Article

Multiplicity of Nontrivial Solutions for

Kirchhoff Type Problems

Bitao Cheng,

1

_{Xian Wu,}

2

_{and Jun Liu}

1

1_{College of Mathematics and Information Science, Qujing Normal University, Qujing,}

Yunnan 655011, China

2_{Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China}

Correspondence should be addressed to Xian Wu,[email protected]

Received 25 October 2010; Accepted 14 December 2010

Academic Editor: Zhitao Zhang

Copyrightq2010 Bitao Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using variational methods, we study the multiplicity of solutions for Kirchhoﬀtype problems −ab_Ω|∇u|2Δufx, u, inΩ;u0, on∂Ω. Existence results of two nontrivial solutions and infinite many solutions are obtained.

1. Introduction

Consider the following Kirchhoﬀtype problems

−

ab

Ω|∇u|

2_Δ_u_f_{x, u}_, _in_Ω_,

u0, on∂Ω,

1.1

whereΩis a smooth bounded domain inRN N1,2, or 3,a, b >0, andf :Ω×R1 →R1

is a Carath´eodory function that satisfies the subcritical growth condition

_f_{x, t}_≤_C₁_|_t_|p−1 _{for some 2}_{< p <}₂∗ ⎧ ⎨ ⎩

2N

N−2, N ≥3,

∞, N1,2,

1.2

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It is pointed out in 1that the problem1.1 model several physical and biological systems, whereudescribes a process which depends on the average of itselfe.g., population density. Moreover, this problem is related to the stationary analogue of the Kirchhoﬀ equation

utt−

ab

Ω|∇u|

2

Δugx, t, 1.3

proposed by Kirchhoff 2 as an extension of the classical D’ Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early studies of Kirchhoff equations were Bernstein 3 and Pohozaev 4. However, 1.3 received much attention only after Lions 5 proposed an abstract framework to the problem. Some interesting results can be found, for example, in 6–13. Specially, more recently, Alves et al. 14, Ma and Rivera 10, and He and Zou 9 studied the existence of positive solutions and infinitely many positive solutions of the problems by variational methods, respectively; Perera and Zhang 12 obtained one nontrivial solutions of 1.1 by Yang index theory; Zhang and Perera 13and Mao and Zhang 11got three nontrivial solutions a positive solution, a negative solution, and a sign-changing solution by invariant sets of descent flow.

In the present paper, we are interested in finding multiple nontrivial solutions of the problem1.1. We will use a three-critical-point theorem due to Brezis and Nirenberg 15and aZ2version of the Mountain Pass Theorem due to Rabinowitz 16to study the existence of multiple nontrivial solutions of problem1.1. Our results are diﬀerent from the above theses.

2. Preliminaries

LetX:H1

0Ωbe the Sobolev space equipped with the inner product and the norm

u, v

Ω∇u· ∇v dx, u u, u

1/2_. _2.1

Throughout the paper, we denote by| · |_rthe usualLr_{-norm. Since}_Ω_{is a bounded domain, it} is well known thatX →LrΩcontinuously forr ∈ 1,2∗, compactly forr ∈ 1,2∗. Hence, forr∈ 1,2∗, there existsγrsuch that

|u|_r ≤γr u , ∀u∈X. 2.2

Recall that a functionu∈Xis called a weak solution of1.1if

ab u 2

Ω∇u· ∇v dx

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Seeking a weak solution of problem 1.1 is equivalent to finding a critical point of C1 functional

Φu: a 2 u

2b 4 u

4₋_Ψ_u_, _2.4

where

Ψu:

ΩFx, udx, ∀u∈X,

Fx, t: _t

0

fx, sds, ∀x, t∈Ω×R1.

2.5

Moreover,

Φ_u_{, v}_a_b _u 2

Ω∇u∇v−

Ωfx, uv, ∀u, v∈X. 2.6

Our assumptions lead us to consider the eigenvalue problems

−Δuλu, inΩ,

u0, on∂Ω, 2.7

− u 2Δuμu3, inΩ,

u0, on∂Ω. 2.8

Denote by 0 < λ1 < λ2 < · · · < λk· · · the distinct eigenvalues of the problem2.7and by

V1, V2, . . . , Vk, . . .the eigenspaces corresponding to these eigenvalues. It is well known thatλ1 can be characterized as

λ1inf

u 2:u∈X,|u|₂1, 2.9

andλ1is achieved byϕ1>0.

μis an eigenvalue of problem2.8means that there is a nonzerou∈Xsuch that

u 2

Ω∇u∇v dxμ

Ωu

3_{v dx,} _∀_v_∈_X. _2.10

Thisuis called an eigenvector corresponding to eigenvalueμ. Set

Iu u 4, u∈S:

u∈X:

Ωu

4₁

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Denote by 0< μ1< μ2<· · ·all distinct eigenvalues of the problem2.8. Then,

μ1:inf

u∈SIu, 2.12

μ1 > 0 is simple and isolated, andμ1 can be achieved at someψ1 ∈ Sandψ1 > 0 inΩ see

12,13.

We need the following concept, which can be found in 17.

Definition 2.1. LetX be a Banach space andΦ∈ C1_{X, R}1_{. We say that}_Φ_{satisfies the}_{P S} condition at the levelc∈R1_{P S}

ccondition for shortif any sequence{un} ⊂Xalong with

Φun → candΦun → 0 asn → ∞possesses a convergent subsequence. IfΦsatisfies

P S_ccondition for eachc∈R1_{, then we say that}_Φ_{satisfies the}_{P S}_condition.

In this paper, the following theorems are our main tools, which are Theorem 4 in 15

and Theorem 9.12 in 16, respectively.

Theorem 2.2. LetX be a real Banach space with a direct sum decompositionX X1⊕X2, where

kdimX2<∞. LetF∈C1X, R1and satisfyP Scondition. Assume that there isr >0such that

Fu≥0, foru∈X1, u ≤r,

Fu≤0, foru∈X2, u ≤r.

2.13

Assume also thatFis bounded below and

inf

u∈XFu<0. 2.14

ThenFhas at least two nonzero critical points.

Theorem 2.3. Let X be an infinite dimensional real Banach space, and letF ∈ C1_{X, R}1_{be even} and satisfy theP Scondition andF0 0. LetX X1⊕X2, whereX2 is finite dimensional, andF satisfies that

ithere exist constantsρ, α >0such thatF|_∂Bρ_X₁ ≥α, where

∂Bρ

u∈X : u ρ, 2.15

iifor each finite dimensional subspaceE1⊂X, the set{u∈E1 :Fu>0}is bounded.

Then,Fpossesses an unbounded sequence of critical values.

3. Main Results

We need the following assumptions.

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f2There existδ >0, >0 andλ∈λk, λk1,k∈N, such that

aλk|t|2≤2Fx, t≤aλ|t|2, ∀x∈Ω, |t| ≤δ, 3.1

whereλkandλk1are two consecutive eigenvalues of the problem2.7.

f3There existδ >0 andλ∈ λk, λk1,k∈Nsuch that

2Fx, t≤aλ|t|2, ∀x∈Ω, |t| ≤δ, 3.2

whereλkandλk1are two consecutive eigenvalues of the problem2.7.

f4

lim sup |t| → ∞

Fx, t−b/4μ1|t|4

|t|τ < α, uniformly inx∈Ω, 3.3

whereτ ∈ 0,2and 0<2α < aλ1.

f5∃ν >4 such thatνFx, t≤tfx, t,|t|large.

Now, we are ready to state our main results.

Theorem 3.1. If conditions (f2) and (f4) hold, then the problem 1.1 has at least two nontrivial solutions inX.

Proof. Set

X1

∞

ik1

Vi, X2

k

i1

Vi. 3.4

Then,Xhas a direct sum decompositionXX1⊕X2with dimX2<∞. LetMr be such that

|u|_r ≥Mr u , ∀u∈X2. 3.5

Step 1. Φis weakly lower semicontinuous.

Indeed, we only to showΨ:X → Ris weakly upper semicontinuous. Let{un} ⊂X,

u∈X,un uinX. Then, we may assume that

un−→u inLrΩ, r∈ 1,2∗. 3.6

We need to prove

Ψu≥lim sup

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If this is false, then

Ψu<lim sup

n→ ∞ Ψun kinf∈Nsup_n_≥_kΨun, 3.8

and hence there existε0 >0 and a subsequence of{un}, still denoted by{un}, such that

ε0<Ψun−Ψu

Ω Fx, un−Fx, udx

Ω

1

0

fx, usun−uun−uds dx

≤

Ω

₁

0

C|usun−u|p−11

|un−u|ds dx

≤

ΩC

2p−1|u|p−1|un−u|p−1

1|un−u|dx

≤

ΩC2

p−1_|_u_|p−1_|_u

n−u|dx

ΩC2

p−1_|_u

n−u|pdx

ΩC|un−u|dx

−→0, asn−→ ∞.

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This is a contradiction. Hence,Ψis weakly upper semicontinuous, and henceΦis weakly lower semicontinuous.

Step 2. There existsr >0, such that

Φu≥0, foru∈X1, u ≤r,

Φu≤0, foru∈X2, u ≤r.

3.10

Particularly,

Φu<0, foru∈X2, 0< u ≤r. 3.11

Indeed, by1.2andf2, there exist two positive constantsC1,C2such that

Fx, t≤ a

2λ|t| 2_C

1|t|p, 3.12

Fx, t≥ a

2λk|t| 2₋_C

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Thus, foru∈X1, the combination of2.2and3.12implies that

Φu≥ a

2 u 2b

4 u 4₋a

2λ

Ωu

2_dx₋_C 1

Ω|u|

p_dx

≥ a 2 u

2b 4 u

4₋a 2

λ λk1

u 2−C1γp u p

a

2

1− λ

λk1

u 2b

4 u 4₋_C

1γp u p.

3.14

Then, there existsr1>0 such that

Φu≥0, foru∈X1, u ≤r1, 3.15

due top >2 andλ < λk1. Moreover, foru∈X2, the combination of2.2and3.13implies

that

Φu≤ a

2 u 2b

4 u 4₋a

2λk

Ωu

2_dx_C 2

Ω|u|

p_dx

≤ a 2 u

2b 4 u

4₋a 2

λk

u 2C3 u p

−a 2

λk

λk −

1

u 2 b

4 u 4_C

3 u p,

3.16

whereC3C2γp. Hence, there existsr2>0 such that

Φu≤0, foru∈X2, u ≤r2,

Φu<0, for u∈X2, 0< u ≤r2.

3.17

Lastly, the conclusion follows from choosingrmin{r1, r2}.

Step 3. Φis coercive onX, that is,Φu → ∞asn → ∞, andΦis bounded from below. In fact, set

px, t:Fx, t−b

4μ1|t|

4_. _3.18

Then,

Φu a

2 u 2b

4 u 4₋b

4μ1

Ωu

4_dx₋

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Conditionf4implies that

lim sup |t| → ∞

px, t

|t|τ < α, uniformly inx∈Ω, 3.20

whereτ ∈ 0,2and 0<2α < aλ1. By contradiction, ifΦis not coercive onX, then there exist a sequence{un} ⊂Xand some constantC4 ∈R1such that

un −→ ∞, asn−→ ∞, butΦun≤C4. 3.21

By virtue of3.20, there exist some constantM >1 such that

−px, t>−α|t|τ, ∀x∈Ω, |t|> M. 3.22

SetΩ1

n {x∈Ω:|unx|> M}andΩ2n {x∈Ω :|unx| ≤M}. Then, the combination of

3.19–3.22and1.2implies that there existsAAM>0 such that

C4 ≥Φun

a

2 un 2b

4 un 4₋b

4μ1

Ωu

4 ndx−

Ωpx, undx

a

2 un 2 b

4

un 4−μ1 Ωu 4 ndx Ω1 n

−px, undx

Ω2 n

−px, undx

≥ a 2 un

2₋

Ω1 n

α|unx|τdx−A

≥ a 2 un

2₋

Ω1 n

α|unx|2dx−A

≥ a 2 un

2₋

Ωα|unx|

2_dx₋_A

≥ a 2 − α λ1

un 2−A−→∞, asn−→ ∞.

3.23

This is a contradiction. Therefore,Φis coercive onX and soΦis bounded from blew due to

Φis weakly lower semicontinuous.

Step 4. ΦsatisfiesP Scondition; that is, anyP Ssequence has a convergent subsequence. Indeed, let{un} ⊂Xbe aP Ssequence ofΦ. By the coerciveness ofΦwe know that {un}is bounded inX. By the reflexivity ofX, we can assume that there existsu∈Xsuch that

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Hence, by1.2, we know that there isC5>0 such that

Ωfx, unu−undx≤

Ω

_f_{x, u}_np/p−1_dxp−1/p

Ω|u−un|

p_dx 1/p

≤2C

Ω

|un|p1

dx

p−1/p

· |u−un|p

≤C5|u−un|p−→0, asn−→ ∞.

3.25

Moreover, since

ab un 2

Ω∇un∇u−un−

Ωfx, unu−undx

Φ_u

n,u−un

−→0, asn−→ ∞,

3.26

then

un −→ u , asn−→ ∞. 3.27

Hence,un → uinXdue to the uniform convexity ofX.

Now, the conclusion follows fromTheorem 2.2.

Corollary 3.2. If conditions (f2) and

f₄

lim |t| → ∞

Fx, t−b

4μ1|t| 4

−∞, uniformly inx∈Ω 3.28

hold, then the problem1.1has at least two nontrivial solutions inX.

Proof. Note that the condition f₄ implies f4. Hence, the conclusion follows from

Theorem 3.1.

Remark 3.3. Perera and Zhang 12only obtained one nontrivial solution of Kirchhoﬀtype problem1.1by Yang index under the conditions

lim t→0

fx, t

at λ, |t| →lim∞

fx, t

bt3 μ, uniformly inx, 3.29

whereλ∈λk, λk1andμ∈μm, μm1is not an eigenvalue of2.8,k /m. We point out the condition

lim t→0

fx, t

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implies the conditionf2, and asm0, that is,μ < μ1, the condition

lim |t| →∞

fx, t

bt3 μ, uniformly inx 3.31

implies the conditionf4. Moreover, we allowμ≡μ1is an eigenvalue of2.8. Whenm≥1, The following example shows that there are functions which satisfyf2andf4and do not satisfy the condition

f6μ∈μm, μm1is not an eigenvalue of2.8.

Example 3.4. Set

fx, t

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−sτ|t|τ−1−br|t|3sτbr−aξ, t <−1,

aξt, |t| ≤1,

sτ|t|τ−1br|t|3−sτ−braξ, t >1,

3.32

wheres < α,λk< ξ < λk1,τ ∈1,2andr ≤μ1. It is easy to verifyfx, tsatisfies conditions

f2andf4, but

lim |t| →∞

fx, t

bt3 r≤μ1, uniformly inx. 3.33

Certainly, ourTheorem 3.1cannot contain Theorem 1.1 in 12completely.

Remark 3.5. Zhang and Perera 13obtained a existence theoremTheorem 1.1iiof three solutions a positive solution, a negative solution, and a sign-changing solution for 1.1

under the conditions

lim |t| →∞

fx, t

bt3 μ < μ1, μ /0, C1

∃λ > λ2:Fx, t≥ aλ 2 t

2_, _|_t_|_small_. _C

2

But, our conditionf4is weaker than the conditionC1and the left hand of our condition

f2 is weaker than the condition C2. Moreover, we allow μ ≡ μ1 is an eigenvalue of

2.8. The aboveExample 3.4withk 1 i.e,λ1 < ξ < λ2shows that there are functions which satisfy all conditions ofTheorem 3.1and do not satisfy Theorem 1.1iiin 13. Hence, Theorem 1.1iiin 13cannot contain ourTheorem 3.1.

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Proof. Set

X1

∞

ik1

Vi, X2

k

i1

Vi. 3.34

Then,Xhas a direct sum decompositionXX1⊕X2with dimX2<∞.

Step 1. There exist constantsρ > 0 andα >0 such thatΦ|_∂Bρ_X₁ ≥ α, whereBρ {u ∈X :

u ρ}.

Indeed, foru∈X1, by1.2andf3, we know3.12holds. Hence, by2.2, we have

Φu≥ a

2 u 2b

4 u 4₋a

2λ

Ωu

2_dx₋_C 1

Ω|u|

p_dx

≥ a 2 u

2b 4 u

4₋a 2

λ λk1

u 2−C1γp u p

a

2

1− λ

λk1

u 2b

4 u 4₋_C

1γp u p.

3.35

Hence, we can choose smallρ >0 such that

Φu≥ a

4

1− λ

λk1

ρ2:α >0, 3.36

wheneveru∈X1with u ρ.

Step 2. For each finite dimensional subspaceE1⊂X, the set{x∈E1 :Φx≥0}is bounded. Indeed, by1.2andf5, we know that there exist constantsC5, C6>0 such that

Fx, t≥C5|t|ν−C6. 3.37

Hence, for everyu∈E1\ {0}, one has

Φu≤ a

2 u 2b

4 u 4₋_C

5

Ω|u|

ν_dx_C

6|Ω|. 3.38

SinceE1is finite dimensional, we can choosingRRE1>0 such that

Φu<0, ∀u∈E1\BR. 3.39

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Remark 3.7. Zhang and Perera 13obtained an existence theorem of three solutions for1.1

under the conditionf5and the condition

Fx, t≤ aλ1

2 t

2_, _|_t_|_small_, _3.40

which implies our condition f3. OurTheorem 3.6 obtains the existence of infinite many solutions of1.1in the case adding the conditionf1.

Acknowledgments

The authors would like to thank the referee for the useful suggestions. This work is supported in partly by the National Natural Science Foundation of China 10961028, Yunnan NSF Grant no. 2010CD080, and the Foundation of young teachers of Qujing Normal University

2009QN018.

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