ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD

ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD

FRANCISCO J ´ULIO S. A. CORR ˆEA AND GIOVANY M. FIGUEIREDO Received 18 November 2005; Revised 11 April 2006; Accepted 18 April 2006 Dedicated to our dear friend and collaborator Professor Claudianor O. Alves

We investigate the questions of existence of positive solution for the nonlocal problem

−M(u²)Δu=f(λ,u) inΩandu=0 on∂Ω, whereΩis a bounded smooth domain of R^N, andMand f are continuous functions.

Copyright © 2006 F. J. S. A. Corrˆea and G. M. Figueiredo. 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 prop- erly cited.

1. Introduction

In this paper, we study some questions related to the existence of positive solution for the nonlocal elliptic problem

−Mu²

Δu=f(λ,u) inΩ,

u=0 on∂Ω, (P)λ

whereΩis a bounded smooth domain,M:R⁺→Ris a function whose behavior will be stated later, f :R⁺×R→Ris a given nonlinear function, and · is the usual norm in H₀¹(Ω) given by

u²=

|∇u|² (1.1)

and finally, through this work,udenotes the integral_Ωu(x)dx.

The main goal of this paper is to establish conditions onMand f under which prob- lem(P)_λpossesses a positive solution.

Problem(P)λ is called nonlocal because of the presence of the termM(u²) which implies that the equation in(P)λis no longer a pointwise identity. This provokes some mathematical difficulties which make the study of such a problem particulary interesting.

Hindawi Publishing Corporation Boundary Value Problems

Volume 2006, Article ID 79679, Pages1–10 DOI10.1155/BVP/2006/79679

Besides, these kinds of problems have motivations in physics. Indeed, the operator M(u²)Δu appears in the Kirchhoffequation, by virtue of this (P)λ, is called of the Kirchhofftype, which arises in nonlinear vibrations, namely,

utt−Mu²

Δu=f(x,u) inΩ×(0,T), u=0 on∂Ω×(0,T),

u(x, 0)₌u0(x), u_t(x, 0)₌u1(x).

(1.2)

Hence, problem(P)λis the stationary counterpart of the above evolution equation.

Such a hyperbolic equation is a general version of the Kirchhoffequation ρ∂²u

∂t^{2 −} P0

h + E 2L

_L

0

∂u

∂x ²dx

∂²u

∂x^{2 =}0 (1.3)

presented by Kirchhoff[14]. This equation extends the classical d’Alembert’s wave equa- tion by considering the effects of the changes in the length of the strings during the vibra- tions. The parameters in (1.3) have the following meanings:Lis the length of the string,h is the area of cross-section,Eis the Young modulus of the material,ρis the mass density andP0is the initial tension.

Problem (1.2) began to call the attention of several researchers mainly after the work of Lions [15], where a functional analysis approach was proposed to attack it.

The reader may consult [1,2,8,16,18] and the references therein, for more informa- tion on(P)λ.

Actually, problem(P)_λis a particular example of a wide class of the so-called nonlocal equations whose study has deserved the attention of many researchers, mainly in recent years.

Let us cite some nonlocal problems in order to emphasize the importance of their studies.

First, we consider the problem

−a

|u|^qdx

Δu=H(x)f(u) inΩ,

u=0 on∂Ω, (1.4)

wherea:R⁺→R⁺is a given function, which does not have variational structure.

Such a problem appears in some physical situations related, for example, with biology in whichusometimes describes the population of bacteria, in caseq=1. In caseq=2, we get the well-known Carrier equation which is an appropriate model to study some ques- tions related to nonlinear deflections of beams. See [4–7,10] and the references therein, for more details related to problem (1.4).

Another relevant nonlocal problem is

−Δu=a(x,u)u^qp inΩ,

u=0 on∂Ω, (1.5)

wherea: ¯Ω×R→R⁺is a known function and · qis the usualL^q-norm, and its related system

−Δu^m= v^α_p inΩ,

−Δvⁿ= u^βq inΩ, u=v=0 on∂Ω

(1.6)

comes from a parabolic phenomenon. Such problems arise in the study of the flow of a fluid through a homogeneous isotropic rigid porous medium or in studies of popula- tion dynamics. It has been suggested that nonlocal growth terms present a more realistic model of population. See [9,11,12,20] and references therein.

To close this series of examples, we cite the problem Δu=

f(u)^α

f(u)^β inΩ, u=0 on∂Ω,

(1.7)

which arises in numerous physical models such as: systems of particles in thermodynam- ical equilibrium via gravitational (Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal runaway in ohmic heating , shear bounds in metal deformed under high strain rates, among others. References to these applications may be found in [21].

After these motivations, let us go back to our original problem(P)_λ. We impose the following conditions onMand f: M is a continuous function and satisfies

M(t)≥m0>0 ∀t≥0, (M1)

M(k)<μm0

2 for some 2< μ < p, for anyk >0, (M2) max M(k)^{(2−p+q)/(p−2)},M(k)^{2/ p−2}≤k

θ (M3)

for anyk >0, for someq≤p, 2< p <2^∗, andθ >0, where 2^∗=2N/(N−2) ifN≥3 and 2^∗= ∞ifN=2. We also suppose that f is a continuous function and satisfies

f(λ,t)− |t|^p−2t

λ :=g(t) withg(t)≥0. (f1) Note that by (f1), f(λ,t)≥0, for allλ >0 and assume that for allt≥0,

tlim→0⁺

g(t)

t ⁼0. (g1)

Moreover, we require that there exists 2< μ < psuch that 0< μG(t)=

_t

0g(s)ds≤g(t)t ∀t >0. (g2) Our main result is as follows.

Theorem 1.1. Let us suppose that the functionMsatisfies (M1), (M2), and (M3), f satisfies (f1), andg satisfies (g1) and (g2). Then there existsλ0>0 such that problem(P)λpossesses a positive solution for eachλ∈[0,λ0].

We point out that the functiong(t)= |t|^s−2twiths≥2^∗satisfies assumptions (g1) and (g2).

In the present paper, we continue the study from [2], because we consider supercriti- cal nonlinearities. In [2], the authors only consider nonlinearities with subcritical growth and so they are able to use a combination of the mountain pass theorem and an appro- priate truncation of the functionMto attack problem(P)λ.

In order to solve problem(P)_λ, we first consider a truncated problem which involves only a subcritical Sobolev exponent. We show that positive solution of truncated problem is a positive solution of(P)λ.

In Sections2and3, we study the truncated problem and inSection 4, we prove an existence result for problem(P)_λ.

2. The truncated problem

First of all, we have to note that because f has a supercritical growth, we cannot use directly the variational techniques, due to the lack of compactness of the Sobolev immer- sions.

So we construct a suitable truncation of f in order to use variational methods or, more precisely, the mountain pass theorem. This truncation was used in the paper [19]

(see [3,13]).

LetK >0 be a real number, whose precise value will be fixed later, and consider the functiong_K:R→Rgiven by

g_K(t)=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

0 ift <0, g(t) if 0≤t≤K,

g(K)

K^p−1t^p−1 ift≥K.

(2.1)

We also study the associated truncated problem

−Mu²

Δu=fK(u) inΩ,

u=0 on∂Ω, (T)λ

where f_K(t)=(t+)^p−1+λg_K(t). Such a function enjoys the following conditions:

f_K(t)=o(t) (ast−→0), (f_K,1) 0< μ

F_K(u)≤

f_K(u)u ∀u∈H₀¹(Ω),u >0, (f_K,2) whereμ >2 andF_K(t)=_t

0f_K(s)ds;

tlim→∞

fK(t)

t^{p−1 =}1 +λg(K)

K^p−1. (f_K,3)

3. Existence of solution for the truncated problem First, we note that

fK(t)≤C1|t|^q−1+C2|t|^p−1, (fK,4) whereC1≥0,C2>0, and for allq≥1. This is an immediate consequence of the definition of fK.

Hence, by using (fK,3), (fK,4), and (M1), we conclude from [2, Lemma 2] that there existsθ >0 such that

u_λ²≤max

Mu_λ^{(2−p+q)/(p−2)},Mu_λ^{22/ p−2}

θ (3.1)

for all classical solutionsu_λof (T)λ.

We now use (fK,1), (fK,2), (fK,3), (M1), (M2) (withμ >2 obtained from condition (fK,2)) and (M3) (withθ >0 obtained in (3.1)) to obtain, thanks to [2, Theorem 5], a positive solutionu_λofT0such thatI_λ(u_λ)=c_λ, wherec_λis the mountain pass level asso- ciated to the functional

I_λu_λ=1

2Mu_λ²−1 p

F_Ku_λ (3.2)

which is related to the problemT0, whereM(t) =_t

0M(s)ds.

Furthermore, Iλ

uλ

−1 μI_λuλ

uλ≥ m0

2 ⁻

Muλ² μ

uλ²+ 1

μ fK

uλ

uλ−FK uλ

≥m0

2 uλ²+ 1

μ fK

uλ

uλ−FK

uλ

.

(3.3)

4. Proof ofTheorem 1.1

In the proof ofTheorem 1.1, we need the following estimate.

Lemma 4.1. Ifu_λis a solution (positive) of problemT0, thenu_λ ≤Cfor allλ≥0, where C >0 is a constant that does not depend onλ.

Proof. SinceFk(t)≥t+^p/ p, one hascλ≤c0, wherec0is the mountain pass level related to the functional

I0(u)=1

2Mu²

−1 p

|u|^p (4.1)

which is associated to the problem

−Mu²

Δu= |u|^p−2u inΩ,

u=0 on∂Ω. (T0)

Furthermore,

c0≥cλ=Iλ

uλ

=Iλ

uλ

−1 μI_λuλ

uλ (4.2)

and from (3.3),

c0≥m0

2 uλ²+ 1 μfK

uλ

uλ−FK uλ

. (4.3)

From (f_K,2), we get

uλ≤ 2c0

m0 :=C (4.4)

for allλ≥0.

Next, we are going to use the Moser iteration method [17](see [3,13]).

Proof ofTheorem 1.1. Letu_λbe a solution of problemT0. We will show that there isK0

such that for allK > K0, there exists a correspondingλ0for which u_λL∞(Ω)≤K ∀λ∈

0,λ0

. (4.5)

If this is the case, one has f_K(u_λ)=u_λ^p−1+λg(u_λ) and sou_λis a solution of problem(P)_λ for allλ∈[0,λ0].

For the sake of simplicity, we will use the following notation:

u_λ:=u. (4.6)

ForL >0, let us define the following functions:

uL=

⎧⎨

⎩

u ifu≤L, L ifu > L, z_L=u^2(β_L ⁻¹⁾u, w_L=uu^β_L⁻¹,

(4.7)

whereβ >1 will be fixed later. Let us usez_Las a test function, that is, Mu²

∇u∇zL=

fK(u)zL (4.8)

which implies Mu²

u^2(β_L ⁻¹⁾|∇u|²= −2(β−1)

u^2β_L⁻³u∇u∇uL+

fK(u)uu^2(β_L ⁻¹⁾. (4.9) Because of the definition ofuL, we have

2(β−1)

u^2β_L⁻³u∇u∇uL=2(β−1)

{u≤L}u^2(β−1)|∇u|²≥0 (4.10)

and using the fact

f_K(u)≤

1 +λg(u) K^p−1

|u|^p−1 (4.11)

together with (M1)

u^2(β_L ⁻¹⁾|∇u|²≤

1 +λg(K) K^p−1

1 m0

u^pu^2(β_L ⁻¹⁾, (4.12) we obtain

u^2(β_L ⁻¹⁾|∇u|²≤Cλ,K

u^pu^2(β_L ⁻¹⁾, (4.13) whereCλ,K=(1 +λ(g(u)/K^p−1))(1/m0).

On the other hand, from the continuous Sobolev immersion, one gets w_L²₂∗≤C1 ∇w_L²=C1 ∇

uu^β_L⁻¹². (4.14) Consequently,

w_L²₂∗≤C1

u^2(β_L ⁻¹⁾|∇u|²+C1(β−1)²

u^2(β_L ⁻²⁾u²∇u_L² (4.15) which gives

w_L²₂∗≤C2β²

u^2(β_L ⁻¹⁾|∇u|². (4.16) From (4.13) and (4.16), we get

w_L²₂∗ ≤C2β²C_λ,K

u^pu^2(β_L ⁻¹⁾ (4.17)

and hence,

w_L²₂∗≤C2β²C_λ,K

u^p−2uu^β_L⁻¹²=C2β²C_λ,K

u^p−2w_L². (4.18) We now use H¨older inequality, with exponents 2^∗/[p−2] and 2^∗/[2^∗−(p−2)], to ob- tain

w_L²₂∗≤C2β²C_λ,K

u^2∗

(p−2)/2^∗

w^2.2_L ^{∗/[2∗−(p−2)]}

[2^∗₋(p−2)]/2^∗

, (4.19)

where 2<2.2^∗/(2^∗−(p−2))<2^∗. Considering the continuous Sobolev immersion H₀¹(Ω)L^q(Ω), 1≤q≤2^∗, we obtain

w_L²₂∗≤C₂β²C_λ,Ku^p−2w_L²_α∗, (4.20)

whereα^∗=2.2^∗/(2^∗−(p−2)). UsingLemma 4.1, we get wL²

2^∗≤C3β²Cλ,KC^p−2wL²

α^∗. (4.21)

SincewL=uu^β_L⁻¹≤u^βand supposing thatu^β∈L^α∗(Ω), we have from (4.21) that uu^β_L^−12∗

2/2^∗

≤C4β²Cλ,K

u^βα∗

2/α^∗

<+∞. (4.22) We now apply Fatou’s lemma with respect to the variableLto obtain

|u|^2β_β·2^∗≤C4Cλ,Kβ²|u|^2β_βα∗ (4.23) so

|u|β.2^∗≤ C4Cλ,K

_1/β2

β^1/β|u|βα^∗. (4.24) Furthermore, by consideringχ=2^∗/α^∗, we have 2^∗=χα^∗andβχα^∗=2^∗·βfor allβ >1 verifyingu^β∈L^α∗(Ω).

Let us consider two cases.

Case 1. First, we considerβ=2^∗/α^∗and note that

u^β∈L^α∗(Ω). (4.25)

Hence, from the Sobolev immersions,Lemma 4.1, and inequality (4.24), we get

|u|(2^∗)²/α^∗≤ C4Cλ,K

1/2β

β^1/βCC5, (4.26)

so

|u_|χ²_α^∗_≤C6

C_λ,K^1/χ2χ^1/χ. (4.27)

Case 2. We now considerβ=(2^∗/α^∗)²and note again that

u^β∈L^α∗(Ω). (4.28)

From inequality (4.24), we obtain

|u|(2^∗)³/(α^∗)²≤C6

C_λ,K^1/β2β^1/β|u|(2^∗)^2/α∗, (4.29) which implies

|u|χ³α^∗≤C6

C_λ,K^1/χ2χ^21/χ2|u|χ²α^∗ (4.30) or

|u|χ³α^∗≤C7

C_λ,K^1/χ2+1/χ2χ^22/χ2+1/χ. (4.31)

An iterative process leads to

|u|χ^(m+1)α^∗≤C8

Cλ,K

^m_i=1χ²⁽⁻ⁱ⁾

χ^2mi=1iχ−i. (4.32) Taking limit asm→ ∞, we obtain

|u|L^∞(Ω)≤C8

C_λ,K^σ1χ^σ2, (4.33)

whereσ1=_∞

i=1χ²⁽⁻ⁱ⁾andσ2=2^∞_i=1iχ⁻ⁱ.

In order to chooseλ0, we consider the inequality C8

C_λ,K^σ1 χ^σ2=C8

1 +λg(K) K^p−1

1 m0

σ1

χ^σ2≤K, (4.34)

from which

1 +λg(K) K^p−1

σ1

≤Km^σ₀¹

χ^σ2C8. (4.35)

Choosingλ0, verifying the inequality λ0≤

K^1/σ1m0

C9 −1 K^p−1

g(K), (4.36)

and fixingKsuch that

K^1/σ1m0

C9 −1

>0, (4.37)

we obtain

u_λL∞(Ω)≤K _∀λ_∈0,λ0

, (4.38)

which concludes the proof.

Acknowledgments

We would like to thank the two anonymous referees whose suggestions improved this work. The first author was partially supported by Instituto do Milˆenio-AGIMB, Brazil.

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Francisco J ´ulio S. A. Corrˆea: Departamento de Matem´atica-CCEN, Universidade Federal do Par´a, 66.075-110 Bel´em-Par´a, Brazil

E-mail address:[email protected]

Giovany M. Figueiredo: Departamento de Matem´atica-CCEN, Universidade Federal do Par´a, 66.075-110 Bel´em-Par´a, Brazil

E-mail address:[email protected]