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Introduction This paper aims to establish the existence of positive solutions in W01,2Ω to the following problem involving a nonlocal equation of Kirchhoff type: −Ku2 Δu λu s−1 fx, u, i

Volume 2011, Article ID 891430, 10 pages

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,ganello@unime.it

Received 24 May 2010; Accepted 26 July 2010

Academic Editor: Feliz Manuel Minh ´os

Copyrightq 2011 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

−Ku2Δu  λu s−1 fx, u in Ω, u  0 on ∂Ω, where s ∈1, 2 , f : Ω×R → R is a Carath´eodory

function, K :R → R is a positive continuous function, and λ is 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−Ku2Δu  λu s−1 in Ω,

u |∂Ω 0

1 Introduction

This paper aims to establish the existence of positive solutions in W01,2Ω to the following problem involving a nonlocal equation of Kirchhoff type:

−Ku2

Δu  λu s−1  fx, u, in Ω,

u  0 on ∂Ω.

P λ

HereΩ is an open bounded set in RN with smooth boundary ∂Ω, s ∈1, 2 , f : Ω × 0, ∞ →

0, ∞ is a Carath´eodory function, K : R → R is a positive continuous function, λ is a real

parameter, andu  Ω|∇u|2dx 1/2 is the standard norm in W01,2Ω In what follows, for

every real number t, we put t  |t|  t/2.

By a positive solution ofP λ , we mean a positive function u ∈ W 1,2

0 Ω ∩ C0Ω which

is a solution ofP λ  in the weak sense, that is such that

K

u2 

Ω∇ux∇vxdx −



Ω



λux s−1  fx, uxv xdx  0 1.1

for all v ∈ W01,2Ω Thus, the weak solutions of P λ are exactly the positive critical points of the associated energy functional

I u 

u2 0

K τdτ −



Ω



λux s−1

ux

0

f x, tdt



dx, u ∈ W01,2 Ω. 1.2

When Kt  a  bt a, b > 0, the equation involved in problem  P λ is the stationary analogue of the well-known equation proposed by Kirchhoff in 1 This is one of the motivations why problems like P λ were studied by several authors beginning from the seminal paper of Lions 2 In particular, among the most recent papers, we cite 3 7 and 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 3 and 4 For Kt  a  bt and λ  0, the existence of a nontrivial solution as

well as multiple solutions for problem P λ is established in 5 and 7 by using critical

point theory and invariant sets of descent flow In these papers, the nonlinearity f is again

satisfying suitable growth conditions at 0 and ∞ Finally, in 6, where the nonlinearity

t s−1

 is replaced by a more general hx, t and the nonlinearity f is multiplied by a positive parameter μ, the existence of at least three solutions for all λ belonging to a suitable interval depending on h and K and for all μ small enough with upper bound depending on λ is

establishedsee 6, Theorem 1 However, we note that the nonlinearity ts−1

 does not meet the conditions required in 6 In particular, condition a5 of 6, Theorem 1 is not satisfied

by t s−1

 Moreover, in 6 the nonlinearity f is required to satisfy a subcritical growth at ∞

and no other condition

Our aim is to study the existence of positive solution to problemP λ, 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 a in such way that there exists a solution to  P λ whose distance

in W01,2Ω from the unique solution of the unperturbed problem that is problem P λ with

f  0 is less than r.

2 Results

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

Theorem 2.1 Let s ∈1, 2 and λ > 0 Let K : 0, ∞ → R be a continuous function satisfying the

following conditions:

a1 inft≥0 Kt > 0;

a2 the function t → 1/2t

0Kτdτ − 1/sKtt is strictly monotone in 0, ∞ ;

a3 lim inft → ∞ t −2αt

0Kτdτ > 0 for some α ∈s/2, 1

Then, the functional

Ψu  12

u2 0

K τdτ − λ

s



Ωu sdx, u ∈ W01,2Ω 2.1

admits a unique global minimum on W01,2 Ω.

Proof From condition a3 we find positive constants C1, C2such that

1 2

u2 0

K τdτ ≥ C1u 2α − C2, for every u ∈ W01,2 Ω. 2.2

Therefore, by Sobolev embedding theorems, there exists a positive constant C3such that

Ψu ≥ C1u 2α − C2− C3u s

, for every u ∈ W01,2 Ω. 2.3

Since s ∈0, 2α , from the previous inequality we obtain

lim

By standard results, the functional

u ∈ W01,2Ω −→ 1

s



is of class C1and sequentially weakly continuous, and the functional

u ∈ W01,2Ω −→ 1

2

u2 0

is of class C1and sequentially weakly lower semicontinuous Then, in view of the coercivity condition2.4, the functional Ψ attains its global minimum on W 1,2

0 Ω at some point u0 ∈

W01,2Ω

Now, let us to show that

inf

W01,2ΩΨ < 0. 2.7 Indeed, fix a nonzero and nonnegative function v ∈ C∞0 Ω, and put v ε  εv We have

Ψεv ≤ ε2 max

t∈ 0,ε2v2K tv

2−λε s

s



Ωv s dx. 2.8

Hence, taking into account that s < 2α < 2, for ε small enough, one has Ψv ε  < 0 Thus,

inequality2.7 holds

At this point, we show that u0is unique To this end, let v0∈ W 1,2

0 Ω be another global minimum forΨ Since Ψ is a C1functional with

Ψuv  Ku2 

Ω∇u∇v dx −



Ωu s−1

for all u, v ∈ W01,2Ω, we have that Ψu0  Ψv0  0 Thus, u0and v0are weak solutions

of the following nonlocal problem:

−Ku2

Δu  λu s−1

u  0 on ∂Ω.

2.10

Moreover, in view of 2.7, u0 and v0 are nonzero Therefore, from the Strong Maximum

Principle, u0 and v0 are positive inΩ as well Now, it is well known that, for every μ > 0,

the problem

−Δu  μu s−1

admits a unique positive solution in W01,2Ω see, e.g., 8, Lemma 3.3 Denote it by u μ Then,

it is easy to realize that for every couple of positive parameters μ1, μ2, the functions u μ1, u μ2

are related by the following identity:

u μ1



μ1

μ2

1/s−1

From2.12 and condition a1, we infer that u0and v0are related by

u0

⎛

⎜K



v02

K

u02

⎞

⎟

1/s−1

Now, note that the identities

Ψu0u0  Ψv0v0  0 2.14 lead to

K

u02

u02 λ



v02

v02  λ



Ωv s0dx 2.15

which, in turn, imply that

Ψu0  1

2

u02 0

K τdτ −1

s K



u02

u02

,

Ψv0  1

2

v02 0

K τdτ −1

s K



v02

v02.

2.16

Now, since u0and v0are both global minima forΨ, one has Ψu0  Ψv0 It follows that

1

2

u02

0

K τdτ − 1

s K



u02

u02 1

2

v02 0

K τdτ −1

s K



v02

v02

. 2.17

At this point, from conditiona2 and 2.17, we infer that

K

u02

 Kv02

2.18 which, in view of2.13, clearly implies u0 v0 This concludes the proof

Remark 2.2 Note that condition a2 is satisfied if, for instance, K is nondecreasing in 0, ∞ and so, in particular, if Kt  a  bt with a, b > 0.

From now on, whenever the function K satisfies the assumption ofTheorem 2.1, we

denote by u sthe unique global minimum of the functionalΨ defined in 2.1 Moreover, for

every u ∈ W01,2 Ω and r > 0, we denote by B r u the closed ball in W 1,2

0 Ω centered at u with radius r The next result shows that the global minimum u s is strict in the sense that the

infimum ofΨ on every sphere centered in u sis strictly greater thanΨu s

Theorem 2.3 Let K, λ, and s be as Theorem 2.1 Then, for every r > 0 one has

inf

Proof Put  Kt  1/2t

0Kτdτ for every t ≥ 0, and let r > 0 Assume, by contradiction, that

inf

Then,

inf

W01,2ΩΨ  Ψu s  inf

vr





K

r2 u s2 2u s , v−λ

s



Ωu s  v s

dx



. 2.21

Now, it is easy to check that the functional

J u   K

r2 u s2 2u s , u−λ

s



Ωu s  u s

dx, u ∈ W01,2Ω 2.22

is sequentially weakly continuous in W01,2Ω Moreover, by the Eberlein-Smulian Theorem,

every closed ball in W01,2 Ω is sequentially weakly compact Consequently, J attains its global minimum in B r0, and

inf

u≤r J u  inf

Let v0 ∈ B r 0 be such that Jv0  infur Ju From assumption a1, K turns out to be a

strictly increasing function Therefore, in view of2.21, one has

Ψu s   Jv0 ≥ K

v02 u s2 2u s , u− λ

s



Ωu s  u s

dx  Ψ u s  v0. 2.24

This inequality entails that u s v0is a global minimum forΨ Thus, thanks toTheorem 2.1, v0

must be identically 0 Using again the fact that K is strictly increasing, from inequality 2.24,

we would get

Ψu s   Jv0 > Ψu s  v0 2.25 which is impossible

Whenever the function K is as inTheorem 2.1, we put

μ r  inf

for every r > 0.Theorem 2.3says that every μ ris a positive number

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

of 10

Lemma 2.4 For every h ∈ L∞Ω, denote by u h the (unique) solution of the problem

−Δu  hx in Ω,

Then, u h ∈ C1Ω, and

sup

h∈L∞ Ω\{0}

maxΩ|u h|

h L∞ Ω

def

where the constant C0depends only on N, |Ω|.

Theorem 2.5below guarantees, for every r > 0, the existence of at least one positive solution u r for problem P λ  whose distance from u s is less than r provided that the perturbation term f is sufficiently small in Ω × 0, C with

C >  C0def λC0

M

1/2−s

Here C0 is the constant defined inLemma 2.4and M  inf t≥0 Kt > 0 Note that no growth

condition is required on f.

Theorem 2.5 Let K, λ, and s be as in Theorem 2.3 Moreover, fix any C >  C0 Then, for every

r > 0, there exists a positive constant a r such that for every Carath´eodory function f : Ω × 0, ∞ →

0, ∞ satisfying

ess sup

x,t∈Ω× 0,C f x, t < a r def min



λ C

s−1



C02−s



C2−s− C2−s0 

, μ r γr



where μ r is the constant defined in2.26 and γ is the embedding constant of W 1,2

0 Ω in L1Ω,

problemP λ  admits at least a positive solution u ∈ W 1,2

0 Ω ∩ C1Ω such that u r − u s  < r.

Proof Fix C >  C0 For every fixed r > 0 which, without loss of generality, we can suppose such that r ≤ us, let a r be the number defined in2.30 Let f : Ω × 0, ∞ → 0, ∞ be a

Carath´eodory function satisfying condition2.30, and put

f C x, t 

⎧

⎪

⎨

⎪

⎩

f x, 0, ifx, t ∈ Ω × −∞, 0 ,

f x, t, ifx, t ∈ Ω × 0, C,

f x, C, ifx, t ∈ Ω × C, ∞ ,

2.31

as well as

a  ess sup

Moreover, for every u ∈ W01,2 Ω, put Φu  Ωux

0 f C x, tdtdx By standard

results, the functionalΦ is of class C1in W01,2Ω and sequentially weakly continuous Now, observe that thanks to2.30, one has

sup

v≤r Φu s  v − Φu s  sup

v≤r



Ω

u s xvx

u s x f C x, tdt



dx

≤ sup

v≤r



Ω

u s x|vx|

u s x f C x, tdt



dx

≤ asup

v≤r



Ω|vx|dx < a r γr  μ r

2.33

Then, we can fix a number

σ ∈

Ψu s , Ψu s   μ r



2.34

in such way that

supv≤r Φu s  v − Φu s

Applying 11, Theorem 2.1 to the restriction of the functionals Ψ and −Φ to the ball B r u s, it follows that the functionalΨ − Φ admits a global minimum on the set B r u s ∩ Ψ−1 − ∞, σ  Let us denote this latter by u r Note that the particular choice of σ forces u rto be in the interior

of B r u s  This means that u r is actually a local minimum forΨ − Φ, and so Ψ − Φu r  0

In other words, u r is a weak solution of problemP λ  with f C in place of f Moreover, since

r ≤ u s  and u s − u r  < r, it follows that u r is nonzero Then, by the Strong Maximum

Principle, u r is positive inΩ, and, by 10, u r ∈ C1Ω as well To finish the proof is now suffice to show that

max

Arguing by contradiction, assume that

max

FromLemma 2.4and condition2.30 we have

max

K

u2



λmax

Ω u s−1  a r

Therefore, using2.30 and recalling the notation M  inf t≥0 Kt > 0, one has

max

Ω u2−s ≤ C0

M



λ  a r

maxΩ u s−1



≤ C0

M



λ  a r

C s−1

≤ C2−s, 2.39

that is absurd The proof is now complete

Remarks 2.6 To satisfy assumption 2.30 ofTheorem 2.5, it is clearly useful to know some

lower estimation of a r First of all, we observe that by standard comparison results, it is easily seen that

C0 max

where u0is the unique positive solution of the problem

−Δu  1, in Ω,

WhenΩ is a ball of radius R > 0 centered at x0 ∈ RN , then u0x  1/2NR2− |x − x0|2, and

so C0 R2/2N More difficult is obtaining an estimate from below of μ r : if r > u s, one has

inf

vr Ψu s  v ≥ 12inf

t≥0 K tr − u s2−λ

s γ

s

s r s , 2.42

where γ s is the embedding constant of L s Ω in W 1,2

0 Ω Therefore, μ r grows as r2at∞ If

r ≤ u s , it seems somewhat hard to find a lower bound for μ r However, with regard to this

question, it could be interesting to study the behavior of μ r on varying of the parameter λ for every fixed r > 0 For instance, how does μ r behave as λ → ∞? Another question that could

be interesting to investigate is finding the exact value of μ r at least for some particular value

of r for instance r  u s  even in the case of K ≡ 1.

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