On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type

Boundary Value Problems

Volume 2011, Article ID 891430,10pages doi:10.1155/2011/891430

Research Article

On a Perturbed Dirichlet Problem for a Nonlocal

Differential Equation of Kirchhoff Type

Giovanni Anello

Department of Mathematics, University of Messina, S. Agata, 98166 Messina, Italy

Correspondence should be addressed to Giovanni Anello,[email protected]

Received 24 May 2010; Accepted 26 July 2010

Academic Editor: Feliz Manuel Minh ´os

Copyrightq2011 Giovanni Anello. 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.

We study the existence of positive solutions to the following nonlocal boundary value problem

−Ku2_Δuλus−1_{fx, uinΩ,u0 on∂Ω, wheres∈1,2,f:Ω×R}

→ Ris a Carath´eodory function,K:R → Ris a positive continuous function, andλis a real parameter. Direct variational

methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem−Ku2_{Δu λu}s−1 _{in Ω,} u|∂Ω0.

1. Introduction

This paper aims to establish the existence of positive solutions inW₀1,2Ωto the following problem involving a nonlocal equation of Kirchhofftype:

−Ku2Δuλus−1fx, u, inΩ,

u0 on∂Ω.

Pλ

HereΩis an open bounded set inRN_{with smooth boundary∂Ω,s∈1,2 ,f :Ω× 0,∞ →} 0,∞ is a Carath´eodory function,K :R → Ris a positive continuous function,λis a real parameter, andu _Ω|∇u|2_dx1/2_{is the standard norm inW}1,2

0 Ω. In what follows, for every real numbert, we putt |t|t/2.

By a positive solution ofPλ, we mean a positive functionu∈W01,2Ω∩C0Ωwhich is a solution ofPλin the weak sense, that is such that

Ku2

Ω∇ux∇vxdx−

Ω

for allv∈W₀1,2Ω. Thus, the weak solutions ofPλare exactly the positive critical points of the associated energy functional

Iu _u2

0

Kτdτ−

Ω

λuxs−1 _ux

0

fx, tdt

dx, u∈W₀1,2Ω. 1.2

WhenKt abta, b >0, the equation involved in problemPλis the stationary analogue of the well-known equation proposed by Kirchhoff in 1. This is one of the

motivations why problems like Pλ were studied by several authors beginning from the seminal paper of Lions 2. In particular, among the most recent papers, we cite 3–7and refer the reader to the references therein for a more complete overview on this topic.

The case λ 0 was considered in 3 and 4, where the existence of at least

one positive solution is established under various hypotheses on f. In particular, in 3

the nonlinearity f is supposed to satisfy the well-known Ambrosetti-Rabinowitz growth condition; in 4 f satisfies certain growth conditions at 0 and ∞, and fx, t/t is nondecreasing on 0,∞ for all x ∈ Ω. Critical point theory and minimax methods are used in 3and 4. For Kt abt andλ 0, the existence of a nontrivial solution as well as multiple solutions for problem Pλ is established in 5 and 7 by using critical point theory and invariant sets of descent flow. In these papers, the nonlinearityf is again satisfying suitable growth conditions at 0 and ∞. Finally, in 6, where the nonlinearity

ts−1

is replaced by a more generalhx, tand the nonlinearityf is multiplied by a positive parameterμ, the existence of at least three solutions for allλbelonging to a suitable interval depending onhand K and for allμsmall enoughwith upper bound depending on λis establishedsee 6, Theorem 1. However, we note that the nonlinearityts−1

does not meet the conditions required in 6. In particular, conditiona5of 6, Theorem 1is not satisfied byts−1

. Moreover, in 6 the nonlinearityf is required to satisfy a subcritical growth at ∞ and no other condition.

Our aim is to study the existence of positive solution to problemPλ, where, unlike previous existence resultsand, in particular, those of the aforementioned papers, no growth condition is required on f. Indeed, we only require that on a certain interval 0, C the function fx,· is bounded from above by a suitable constant a, uniformly in x ∈ Ω. Moreover, we also provide a localization of the solution by showing that for all r > 0 we can choose the constant ain such way that there exists a solution to Pλ whose distance inW₀1,2Ωfrom the unique solution of the unperturbed problemthat is problemPλwith f0is less thanr.

2. Results

Our first main result gives some conditions in order that the energy functional associated to the unperturbed problemPλhas a unique global minimum.

Theorem 2.1. Lets∈1,2 andλ >0. LetK: 0,∞ → Rbe a continuous function satisfying the following conditions:

a1inft≥0Kt>0;

a2the functiont → 1/2 _t

Then, the functional

Ψu 1_{2 u}2

0

Kτdτ− λ

s

Ωu s

dx, u∈W01,2Ω 2.1

admits a unique global minimum onW₀1,2Ω.

Proof. From conditiona3we find positive constantsC1, C2such that

1 2

_u2

0

Kτdτ ≥C1u2α−C2, for everyu∈W01,2Ω. 2.2

Therefore, by Sobolev embedding theorems, there exists a positive constantC3such that

Ψu≥C1u2α−C2−C3us, for everyu∈W₀1,2Ω. 2.3

Sinces∈0,2α , from the previous inequality we obtain

lim

u →∞Ψu ∞. 2.4

By standard results, the functional

u∈W₀1,2Ω−→1 s

Ωu s

dx 2.5

is of classC1_{and sequentially weakly continuous, and the functional}

u∈W₀1,2Ω−→ 1 2

_u2

0

Kτdτ 2.6

is of classC1_{and sequentially weakly lower semicontinuous. Then, in view of the coercivity} condition2.4, the functionalΨattains its global minimum onW₀1,2Ωat some pointu0 ∈ W₀1,2Ω.

Now, let us to show that

inf W01,2Ω

Ψ<0. _2.7

Indeed, fix a nonzero and nonnegative functionv∈C∞₀ Ω, and putvεεv. We have

Ψεv≤ε2 max

t∈ 0,ε2_v2Ktv

2₋λεs s

Ωv

Hence, taking into account thats < 2α < 2, for εsmall enough, one hasΨvε < 0. Thus, inequality2.7holds.

At this point, we show thatu0is unique. To this end, letv0∈W₀1,2Ωbe another global minimum forΨ. SinceΨis aC1_{functional with}

Ψ_{uv Ku}2

Ω∇u∇v dx−

Ωu s−1

v dx 2.9

for allu, v∈W₀1,2Ω, we have thatΨu0 Ψv0 0. Thus,u0andv0are weak solutions of the following nonlocal problem:

−Ku2Δuλus−1 in Ω,

u0 on∂Ω.

2.10

Moreover, in view of 2.7,u0 and v0 are nonzero. Therefore, from the Strong Maximum Principle,u0 andv0 are positive inΩas well. Now, it is well known that, for everyμ > 0, the problem

−Δuμus−1, inΩ,

u0, on∂Ω. 2.11

admits a unique positive solution inW₀1,2Ω see, e.g., 8, Lemma 3.3. Denote it byuμ. Then, it is easy to realize that for every couple of positive parametersμ1, μ2, the functionsuμ1, uμ2

are related by the following identity:

uμ1

μ1 μ2

1/s−1

uμ2. 2.12

From2.12and conditiona1, we infer thatu0andv0are related by

u0 ⎛ ⎜ ⎝K

v02

Ku02

⎞ ⎟ ⎠

1/s−1

v0. 2.13

Now, note that the identities

Ψ_u

0u0 Ψv0v0 0 2.14

lead to

Ku02

u02λ

Ωu s

0dx, K

v02

v02 λ

Ωv s

which, in turn, imply that

Ψu0 1

2 u02

0

Kτdτ−1

sK

u02

u02,

Ψv0 1 2

_v02

0

Kτdτ−1

sK

v02

v02.

2.16

Now, sinceu0andv0are both global minima forΨ, one hasΨu0 Ψv0. It follows that

1 2

_u02

0

Kτdτ−1

sK

u02

u02 1 2

_v02

0

Kτdτ−1

sK

v02

v02. 2.17

At this point, from conditiona2and2.17, we infer that

Ku02

Kv02

2.18

which, in view of2.13, clearly impliesu0v0. This concludes the proof.

Remark 2.2. Note that conditiona2is satisfied if, for instance,Kis nondecreasing in 0,∞ and so, in particular, ifKt abtwitha, b >0.

From now on, whenever the functionK satisfies the assumption ofTheorem 2.1, we denote byusthe unique global minimum of the functionalΨdefined in2.1. Moreover, for everyu ∈ W₀1,2Ωandr > 0, we denote by Bruthe closed ball inW₀1,2Ωcentered atu with radiusr. The next result shows that the global minimumusis strictin the sense that the infimum ofΨon every sphere centered inusis strictly greater thanΨus.

Theorem 2.3. LetK,λ, andsbe asTheorem 2.1. Then, for every r >0one has

inf

vrΨusv>Ψus. 2.19

Proof. PutKt 1/2 ₀tKτdτfor everyt≥0, and letr >0. Assume, by contradiction, that

inf

vrΨusv Ψus. 2.20

Then,

inf W01,2Ω

Ψ Ψus inf

vr

Kr2_u

s22us, v

−λ

s

Ωusv s dx

. _2.21

Now, it is easy to check that the functional

Ju Kr2us22us, u

−λ

s

Ωusu s

is sequentially weakly continuous inW₀1,2Ω. Moreover, by the Eberlein-Smulian Theorem, every closed ball inW₀1,2Ωis sequentially weakly compact. Consequently,Jattains its global minimum inBr0, and

inf

u≤rJu infurJu. 2.23

Letv0 ∈Br0be such thatJv0 infurJu. From assumptiona1,K turns out to be a strictly increasing function. Therefore, in view of2.21, one has

Ψus Jv0≥K

v02us22us, u

− λ

s

Ωususdx Ψusv0. 2.24

This inequality entails thatusv0is a global minimum forΨ. Thus, thanks toTheorem 2.1,v0 must be identically 0. Using again the fact thatK is strictly increasing, from inequality2.24,

we would get

Ψus Jv0>Ψusv0 2.25

which is impossible.

Whenever the functionKis as inTheorem 2.1, we put

μr inf

vrΨusv−Ψus 2.26

for everyr >0.Theorem 2.3says that everyμris a positive number.

Before stating our existence result for problemPλ, we have to recall the following well-known Lemma which comes from 9, Theorems 8.16 and 8.30and the regularity results of 10.

Lemma 2.4. For everyh∈L∞Ω, denote byuhthe (unique) solution of the problem

−Δuhx in Ω,

u0 on∂Ω. 2.27

Then,uh∈C1Ω, and

sup h∈L∞_Ω\{0}

max_Ω|uh|

hL∞_Ω

def

C0<∞, 2.28

Theorem 2.5below guarantees, for everyr > 0, the existence of at least one positive solution ur for problem Pλ whose distance from us is less than r provided that the perturbation termfis sufficiently small inΩ× 0, Cwith

C >C0def

λC0 M

1/2−s

. 2.29

HereC0 is the constant defined inLemma 2.4andM inft≥0Kt>0. Note that no growth condition is required onf.

Theorem 2.5. Let K,λ, and sbe as in Theorem 2.3. Moreover, fix anyC > C0. Then, for every r >0, there exists a positive constantarsuch that for every Carath´eodory functionf:Ω× 0,∞ → 0,∞ satisfying

ess sup

x,t∈Ω×0,C fx, t< ar def

min

λC s−1 C₀2−s

C2−s−C2₀−s,μr γr

, 2.30

where μr is the constant defined in2.26and γ is the embedding constant ofW₀1,2Ω in L1Ω, problemPλadmits at least a positive solutionu∈W01,2Ω∩C1Ωsuch thatur−us< r.

Proof. FixC > C0. For every fixedr > 0 which, without loss of generality, we can suppose such thatr≤ us, letar be the number defined in2.30. Letf :Ω× 0,∞ → 0,∞ be a Carath´eodory function satisfying condition2.30, and put

fCx, t ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

fx,0, ifx, t∈Ω×−∞,0 ,

fx, t, ifx, t∈Ω× 0, C,

fx, C, ifx, t∈Ω×C,∞ ,

2.31

as well as

a ess sup

Moreover, for every u ∈ W₀1,2Ω, put Φu _Ω0uxfCx, tdtdx. By standard results, the functionalΦis of classC1_inW1,2

0 Ωand sequentially weakly continuous. Now, observe that thanks to2.30, one has

sup

v≤r

Φusv−Φus sup

v≤r

Ω

usxvx

usx

fCx, tdt

dx

≤ sup

v≤r

Ω

usx|vx|

usx

fCx, tdt

dx

≤asup

v≤r

Ω|vx|dx < arγrμr.

2.33

Then, we can fix a number

σ∈Ψus,Ψus μr

2.34

in such way that

sup_v≤rΦusv−Φus σ−Ψus <1.

2.35

Applying 11, Theorem 2.1to the restriction of the functionalsΨand−Φto the ballBrus, it follows that the functionalΨ−Φadmits a global minimum on the setBrus∩Ψ−1− ∞, σ . Let us denote this latter byur. Note that the particular choice ofσforcesurto be in the interior ofBrus. This means thatur is actually a local minimum forΨ−Φ, and soΨ−Φur 0. In other words,ur is a weak solution of problemPλwithfC in place off. Moreover, since r ≤ us and us−ur < r, it follows that ur is nonzero. Then, by the Strong Maximum Principle,ur is positive inΩ, and, by 10,ur ∈ C1Ωas well. To finish the proof is now suffice to show that

max

Ω u≤C. 2.36

Arguing by contradiction, assume that

max

Ω u > C. 2.37

FromLemma 2.4and condition2.30we have

max Ω u≤

C0 Ku2

λmax

Ω u s−1_a

Therefore, using2.30 and recalling the notationMinft≥0Kt>0, one has

max Ω u

2−s_≤ C0 M

λ ar max_Ω us−1

≤ C0 M

λ ar

Cs−1 ≤C

2−s_{, 2.39}

that is absurd. The proof is now complete.

Remarks 2.6. To satisfy assumption 2.30ofTheorem 2.5, it is clearly useful to know some lower estimation ofar. First of all, we observe that by standard comparison results, it is easily seen that

C0 max

x∈Ω u0x, 2.40

whereu0is the unique positive solution of the problem

−Δu1, inΩ,

u0, on∂Ω. 2.41

WhenΩis a ball of radiusR >0 centered atx0 ∈RN, thenu0x 1/2NR2− |x−x0|2, and soC0R2/2N. More difficult is obtaining an estimate from below ofμr: ifr >us, one has

inf

vrΨusv≥ 1

2inft≥0Ktr− us 2₋λ

sγ s

srs, 2.42

whereγsis the embedding constant ofLsΩinW₀1,2Ω. Therefore,μr grows asr2at∞. If r ≤ us, it seems somewhat hard to find a lower bound forμr. However, with regard to this question, it could be interesting to study the behavior ofμron varying of the parameterλfor every fixedr >0. For instance, how doesμrbehave asλ → ∞? Another question that could be interesting to investigate is finding the exact value ofμr at least for some particular value ofrfor instancer useven in the case ofK≡1.

References

1 G. Kirchhoff,Mechanik, Teubner, Leipzig, Germany, 1883.

2 J.-L. Lions, “On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, G. M. de la Penha and L. A. J. Medeiros, Eds., vol. 30 ofNorth-Holland Mathematics Studies, pp. 284–346, North-Holland, Amsterdam, The Netherlands, 1978.

3 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.

4 A. Bensedik and M. Bouchekif, “On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity,”Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 1089–1096, 2009.

5 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. 6 B. Ricceri, “On an elliptic Kirchhoff-type problem depending on two parameters,”Journal of Global

7 Y. Yang and J. Zhang, “Positive and negative solutions of a class of nonlocal problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 122, no. 1, pp. 25–30, 2010.

8 A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,”Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994.

9 D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, vol. 22, Springer, Berlin, Germany, 1977.

10 E. DiBenedetto, “C1α_{local regularity of weak solutions of degenerate elliptic equations,”Nonlinear}

Analysis: Theory, Methods & Applications, vol. 7, no. 8, pp. 827–850, 1983.