Multiplicity of Nontrivial Solutions for Kirchhoff Type Problems

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

Research Article

Multiplicity of Nontrivial Solutions for Kirchhoff Type Problems

Bitao Cheng,

Xian Wu,

and Jun Liu

1College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China

2Department 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

Δufx, u, inΩ, u0, on∂Ω,

1.1

whereΩis a smooth bounded domain inR^N N1,2, or 3,a, b >0, andf :Ω×R¹ →R¹ is a Carath´eodory function that satisfies the subcritical growth condition

fx, t≤C

1|t|^p−1

for some 2< p <2^∗

⎧⎨

⎩ 2N

N−2, N ≥3,

∞, N1,2,

1.2

whereCis a positive constant.

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

u_tt−

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:H₀¹Ωbe 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 usualL^r-norm. SinceΩis a bounded domain, it is well known thatX →L^rΩcontinuously forr ∈ 1,2^∗, compactly forr ∈ 1,2^∗. Hence, forr∈ 1,2^∗, there existsγ_rsuch 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

Ωfx, uv dx, ∀v∈X. 2.3

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−Ψu, 2.4

where

Ψu:

ΩFx, udx, ∀u∈X, Fx, t:

_t

0

fx, sds, ∀x, t∈Ω×R¹.

2.5

Moreover,

Φu, v

ab 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

Ωu³v dx, ∀v∈X. 2.10

Thisuis called an eigenvector corresponding to eigenvalueμ. Set Iu u ⁴, u∈S:

u∈X:

Ωu⁴1

. 2.11

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∈ C¹X, R¹. We say thatΦsatisfies theP S condition at the levelc∈R¹P 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 thatΦsatisfies theP Scondition.

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 >0 such that

Fu≥0, foru∈X1, u ≤r,

Fu≤0, foru∈X₂, u ≤r. 2.13

Assume also thatFis bounded below and

u∈XinfFu<0. 2.14

ThenFhas at least two nonzero critical points.

Theorem 2.3. LetX be an infinite dimensional real Banach space, and letF ∈ C¹X, R¹be even and satisfy theP Scondition andF0 0. LetX X1⊕X2, whereX2 is finite dimensional, andF satisfies that

ithere exist constantsρ, α >0 such thatF|_∂BρX1 ≥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.

f1fx, tis odd intfor allx∈Ω.

f2There existδ >0, >0 andλ∈λk, λ_k1,k∈N, such that

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

whereλkandλ_k1are two consecutive eigenvalues of the problem2.7.

f3There existδ >0 andλ∈ λk, λ_k1,k∈N such that

2Fx, t≤aλ|t|², ∀x∈Ω, |t| ≤δ, 3.2

whereλ_kandλ_k1are 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 ≥M_r 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 inL^rΩ, r∈ 1,2^∗. 3.6

We need to prove

Ψu≥lim sup

n→ ∞ Ψun inf

k∈Nsup

n≥kΨun. 3.7

If this is false, then

Ψu<lim sup

n→ ∞ Ψu_n inf

k∈Nsup

n≥kΨu_n, 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 2^p−1

|u|^p−1|un−u|^p−1 1

|un−u|dx

≤

ΩC2^p−1|u|^p−1|un−u|dx

ΩC2^p−1|un−u|^pdx

ΩC|un−u|dx

−→0, asn−→ ∞.

3.9

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|²C₁|t|^p, 3.12

Fx, t≥ a

2λk|t|²−C₂|t|^p. 3.13

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

Φu≥ a

2 u ²b

4 u ⁴−a 2λ

Ωu²dx−C1

Ω|u|^pdx

≥ a

2 u ²b

4 u ⁴−a 2

λ

λ_k1 u ²−C1γp u ^p a

2

1− λ λ_k1

u ²b

4 u ⁴−C₁γ_p u ^p.

3.14

Then, there existsr1>0 such that

Φu≥0, foru∈X1, u ≤r1, 3.15

due top >2 andλ < λ_k1. Moreover, foru∈X2, the combination of2.2and3.13implies that

Φu≤ a

2 u ²b

4 u ⁴−a

2λk

Ωu²dxC₂

Ω|u|^pdx

≤ a

2 u ²b

4 u ⁴−a 2

λk λk

u ²C3 u ^p −a

2

λ_k λk −1

u ² b

4 u ⁴C₃ 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

Ωu⁴dx−

Ωpx, udx, ∀u∈X. 3.19

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}andΩ²_n {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

Ωu⁴_ndx−

Ωpx, undx a

2 un 2 b 4

un 4−μ1

Ωu⁴_ndx

Ω¹n

−px, undx

Ω²n

−px, undx

≥ a 2 un 2−

Ω¹n

α|unx|^τdx−A

≥ a 2 un 2−

Ω¹n

α|unx|²dx−A

≥ a 2 un 2−

Ωα|unx|²dx−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

un u inX, un −→u inL^pΩ, unx−→ux for a.e. x∈Ω. 3.24

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

Ωfx, unu−undx≤

Ω

fx, un^p/p−1dx

_p−1/p

Ω|u−un|^pdx 1/p

≤2C

Ω

|un|^p1 dx

_p−1/p

· |u−un|_p

≤C₅|u−u_n|_p−→0, asn−→ ∞.

3.25

Moreover, since

ab un 2

Ω∇un∇u−un−

Ωfx, unu−undx

Φun,u−u_n

−→0, asn−→ ∞,

3.26

then

un −→ u , asn−→ ∞. 3.27

Hence,u_n → uinXdue to the uniform convexity ofX.

Now, the conclusion follows fromTheorem 2.2.

Corollary 3.2. If conditions (f2) and f₄

|t| → ∞lim

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

limt→0

fx, t

at λ, lim

|t| →∞

fx, t

bt³ μ, uniformly inx, 3.29

whereλ∈λk, λ_k1andμ∈μm, μ_m1is not an eigenvalue of2.8,k /m. We point out the condition

limt→0

fx, t

at λ, uniformly inx 3.30

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

|t| →lim∞

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, μ_m1is 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,λ_k< ξ < λ_k1,τ ∈1,2andr ≤μ₁. It is easy to verifyfx, tsatisfies conditions f2andf4, but

|t| →lim∞

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

|t| →lim∞

fx, t

bt³ μ < μ1, μ /0, C1

∃λ > λ2:Fx, t≥ aλ

2 t², |t|small. C2

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.

Theorem 3.6. Let conditionsf1, f3, and f5 hold, then the problem1.1 has infinite many solutions inX.

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|_∂BρX1 ≥ 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

Ωu²dx−C1

Ω|u|^pdx

≥ a

2 u ²b

4 u ⁴−a 2

λ

λ_k1 u ²−C₁γ_p u ^p a

2

1− λ λ_k1

u ²b

4 u ⁴−C₁γ_p u ^p.

3.35

Hence, we can choose smallρ >0 such that

Φu≥ a 4

1− λ λ_k1

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−C₅

Ω|u|^νdxC₆|Ω|. 3.38 SinceE1is finite dimensional, we can choosingRRE1>0 such that

Φu<0, ∀u∈E1\BR. 3.39

Moreover, by Lemma 2.2iiiin 13, we know thatΦsatisfiesP Scondition, andΦis even due tof1. Hence, the conclusion follows from Theorem 9.12 in 16.

Remark 3.7. Zhang and Perera 13obtained an existence theorem of three solutions for1.1 under the conditionf5and the condition

Fx, t≤ aλ1

2 t², |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.

References

1 M. Chipot and B. Lovat, “Some remarks on nonlocal elliptic and parabolic problems,” in Proceedings of the 2nd World Congress of Nonlinear Analysts, vol. 30, no. 7, pp. 4619–4627, 1997.

2 G. Kirchhoff, Mechanik, Teubner, leipzig, Germany, 1883.

3 S. Bernstein, “Sur une classe d’´equations fonctionnelles aux d´eriv´ees partielles,” Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, vol. 4, pp. 17–26, 1940.

4 S. I. Pohoˇzaev, “A certain class of quasilinear hyperbolic equations,” Matematicheskii Sbornik, vol.

96138, pp. 152–166, 1975.

5 J.-L. Lions, “On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, vol. 30 of North- Holland Mathematics Studies, pp. 284–346, North-Holland, Amsterdam, The Netherlands, 1978.

6 A. Arosio and S. Panizzi, “On the well-posedness of the Kirchhoffstring,” Transactions of the American Mathematical Society, vol. 348, no. 1, pp. 305–330, 1996.

7 M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation,” Advances in Differential Equations, vol. 6, no. 6, pp. 701–730, 2001.

8 P. D’Ancona and S. Spagnolo, “Global solvability for the degenerate Kirchhoffequation with real analytic data,” Inventiones Mathematicae, vol. 108, no. 2, pp. 247–262, 1992.

9 X. He and W. Zou, “Infinitely many positive solutions for Kirchhoff-type problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1407–1414, 2009.

10 T. F. Ma and J. E. Mu ˜noz Rivera, “Positive solutions for a nonlinear nonlocal elliptic transmission problem,” Applied Mathematics Letters, vol. 16, no. 2, pp. 243–248, 2003.

11 A. Mao and Z. Zhang, “Sign-changing and multiple solutions of Kirchhofftype problems without the P.S. condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1275–1287, 2009.

12 K. Perera and Z. Zhang, “Nontrivial solutions of Kirchhoff-type problems via the Yang index,” Journal of Differential Equations, vol. 221, no. 1, pp. 246–255, 2006.

13 Z. Zhang and K. Perera, “Sign changing solutions of Kirchhofftype problems via invariant sets of descent flow,” Journal of Mathematical Analysis and Applications, vol. 317, no. 2, pp. 456–463, 2006.

14 C. O. Alves, F. J. S. A. Corrˆea, and T. F. Ma, “Positive solutions for a quasilinear elliptic equation of Kirchhofftype,” Computers & Mathematics with Applications, vol. 49, no. 1, pp. 85–93, 2005.

15 H. Brezis and L. Nirenberg, “Remarks on finding critical points,” Communications on Pure and Applied Mathematics, vol. 64, no. 8-9, pp. 939–963, 1991.

16 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of Cbms Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.

17 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.