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December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

this integral converges to zero since a(x, T

), ∇T

(u)) converges to

a(x, T

(u), ∇T

(u)) strongly in (L

(Ω))

while ∇T

(u)(1 − h

)) con-

verges to zero strongly in (L

(Ω))

Finally, we conclude that

lim

n→∞

Ω

[a(x, T

), ∇T

))−a(x, T

), ∇T

(u))][∇T

)−∇T

(u)] = 0

then Lemma 5 of,

implies that,

) → T

(u) strongly in W

1,p

(Ω). (32)

And

∇u

→ ∇u a. e. in Ω.

Lemma 2.4. Let be u

a solution of (P

), then

(x, u

, ∇u

) → g(x, u, ∇u) strongly in L

(Ω) (33)

Proof. Let v = u

+ e

−

h(|s|)

h(|s|)χ

{s<−h}

ds as test function

in (P

), then

Ω

a(x, u

, ∇u

)∇



−e

−

h(|s|)

h(|s|)χ

{s<−h}



−

Ω

(x, u

, ∇u

−

h(|s|)

h(|s|)χ

{s<−h}

≤

Ω

|f|e

−

h(|s|)

h(|s|)χ

{s<−h}

which implies that,

Ω

a(x, u

, ∇u

)∇u

h(|u

−

h(|s|)

h(|s|)χ

{s<−h}

Ω

a(x, u

, ∇u

)∇u

−

h(|s|)

h(|u

|)χ

<−h}

≤

Ω

b(|u

|)|γ(x)|e

−

h(|s|)

h(|s|)χ

{s<−h}

Ω

h(|u

|)|∇u

−

h(|s|)

h(|s|)χ

{s<−h}

Ω

|f|e

−

h(|s|)

h(|s|)χ

{s<−h}

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

using (5) and since

h(|s|)χ

{s<−h}

ds ≤

−h

−∞

h(|s|) ds, we get

Ω

a(x, u

, ∇u

)∇u

−

h(|s|)

h(|u

|)χ

<−h}

≤ c

Ω

r−1

γ(x)

−h

−∞

h(|s|) ds + c

Ω

|f|

−h

−∞

h(|s|) ds dx

≤ c

−h

−∞

h(|s|) ds



|≥1

r−1

γ(x) dx

|≤1

r−1

γ(x) dx + c

Ω

|f| dx



≤ c

−h

−∞

h(|s|) ds (k∇u

1,p

+ c

+ kfk

1,p

)

≤ c

−h

−∞

h(|s|) ds

using again (5), we obtain

<−h}

h(|u

|)|∇u

≤ c

−h

−∞

h(|s|) ds

and since h ∈ L

(IR), we deduce that,

lim

h→+∞

sup

n∈IN

<−h}

h(|u

|)|∇u

dx = 0 (34)

consider v = u

− e

h(|s|)

h(|s|)χ

{s<−h}

ds as test function in

), then similarly to (34), we deduce that

lim

h→+∞

sup

n∈IN

>h}

h(|u

|)|∇u

dx = 0 (35)

assuming (34) and (35) we obtain

lim

h→+∞

sup

n∈IN

{|u

|>h}

h(|u

|)|∇u

dx = 0. (36)

On the other hand, we have

>h}

b(|u

|)|γ(x)| dx ≤

>h}

r−1

|γ(x)| dx

≤



Ω



>h}

|γ(x)|

p−r

≤ c

>h}

|γ(x)|

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

then, we conclude that

lim

h→+∞

sup

n∈IN

Ω

b(|u

|)|γ(x)| dx = 0 (37)

Combining (36), (37) and Vitali’s theorem, we conclude (33)

Proof of the theorem. Let ϕ ∈ W

1,p

(Ω) ∩ L

∞

(Ω) and take v = u

− ϕ) as test function in (P

), we get

Ω

a(x, u

, ∇u

)∇T

− ϕ) dx +

Ω

(x, u

, ∇u

− ϕ) dx

≤

Ω

− ϕ) dx.

(38)

Finally, from (18) and (33), we can pass to the limit in (38), this completes

the proof of the theorem.

References

1. A. Bensoussan, L. Boccardo and F. Murat On a non linear partial diﬀer-

ential equation having natural growth terms and unbounded solution, Ann.

Inst. Henri Poincar´e 5 N4 (1988) 149-169.

2. L. Boccardo, F. Murat and J.P. Puel, L

∞

estimate for some nonlinear

elliptic partial diﬀerential equations and application to an existence result,

SIAM J. Math. Anal. 2 (1992) 326-333.

3. L. Boccardo, F. Murat and J.P. Puel, Existence of bounded solutions for

nonlinear elliptic unilateral problems, Annali. Mat. Pura Appli. (1988) 183-

196.

4. J.L. Lions, Quelques m´ethodes de r´esolution des probl`emes aux limites non

lin´eaires, Dunod, Paris (1969).

5. A. Porretta, Nonlinear equations with natural growth terms and measure

data, Electronic J. Diﬀ. Equ., Conference 09 (2002) 181-202.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

On the principle eigencurve of a coupled system

A. El Khalil

Department of Mathematics, College of Sciences

Al-Imam Muhammad Ibn Saud Islamic

University P.O. Box 90950, Riyadh 11623, Saudi Arabia

E-mail: alakhalil@imamu.edu.sa

We prove that for any λ ∈ IR, there is a sequence of eigencurves (µ

(λ))

for

the nonlinear coupled elliptic system







−∆

u = λa(x)|u|

α−1

|v|

β+1

u + µ|u|

α−1

|v|

β+1

u in Ω

−∆

v = λa(x)|u|

α+1

|v|

β−1

v + µ|u|

α+1

|v|

β−1

v in Ω

(u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω),

by using min-max methods. We prove also via an homogeneity type condition

that the eigenvector corresponding to the principal µ

(λ) is bounded, positive

and smooth. We end this work by proving that µ

(λ) is simple.

Keywords: Quasilinear system; p-Laplacian operator; Principal eigencurve.

1. Introduction

Let Ω be a bounded domain in IR

not necessary regular; N > 1, p >

1, q > 1, and α , β > 0 satisfying the homogeneity assumption:

α + 1

β + 1

= 1, (H)

and a(x) ∈ L

∞

(Ω)\{0} be an indeﬁnite weight function which can change

the sign. Here, λ and µ are tow parameter eigenvalues. We consider the

following nonlinear elliptic system

p,q

(λ)







−∆

u = λa(x)|u|

α−1

|v|

β+1

u + µ|u|

α−1

|v|

β+1

u in Ω

−∆

v = λa(x)|u|

α+1

|v|

β−1

v + µ|u|

α+1

|v|

β−1

v in Ω

(u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω).

Problems where the operator −∆

is present arise both from pure and

applied mathematics. Like in the theory of quasiregular and quasiconformal

mapping, as well as from a variety of applications, e.g. Non-Newtonian

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

ﬂuids, reaction diﬀusion problems, ﬂow through porous media. Nonlinear

elasticity, glaciology, petroleum extraction, astronomy,. . . etc.

Many frameworks have been dealing with the corresponding equation

−∆

u = λ|u|

γ−1

one can cite for example Anane,

Binding,

elin et al.,

elin;

and in the particular case p = γ there are many works, we cite El

Khalil et al.,

Fleckinger et al.,

Hess et al.,

Kato,

Lindqvist,

Richardson,

. . . etc.

In the system case, we ﬁnd the work of De Th´elin (cf. Th

elin

) where

λ = 0 with Ω is regular bounded domain; also El Khalil, Ouanan and

Touzani studied the stability with respect to the rheological exponent p

and q of the ﬁrst eigenvalue see [11], Chabrowski Chabrowski

considers

particular case with λ = 0 introducing two perturbation functions in the

right hand side of the system; and Flekinger and coauthors studied this

problem in Fleckinger et al.

with µ is some function on x satisfying

some large hypothesis with Ω = IR

In our work, we investigate the situation improving the existence at least

a sequence of the eigencurves (eigenpair (λ, µ(λ)) for any bounded domain

by using the Ljusternik-Schnirelman theory on C

-manifolds cf. Szulkin.

So we give a new characterization of the principal eigencurve and also we

ﬁnd some result about the associated eigenvector when λ is in a suitable

range of IR depending of λ

= µ

(0) and the weight function a(x).

The rest of this paper is organized as follows. In section 2 we intro-

duce some deﬁnitions and prove some technical preliminary, in section 3

we prove that the principal eigencurve is well deﬁned and there exist at

least a sequence of the eigencurve, ﬁnally we prove some result about the

eigenvector associated of µ

(λ) when λ is in a suitable range.

2. Preliminaries

2.1. Functional framework

In the sequel, we shall use the standard notations.

1,p

(Ω) is the completion of C

∞

(Ω) in the space W

1,p

(Ω). In the other

words,

1,p

(Ω) =



u ∈ W

1,p

(Ω) | u

∂Ω

= 0



the value of u on Ω being understood in the trace sense.

It is well known that



1,p

(Ω), k∇.k



is separable, reﬂexive and uni-

formly convex, for 0 < p < +∞.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

The corresponding dual space will be denoted by W

−1,p

(Ω), where

= 1.

Remark that for each u ∈ W

1,p

(Ω), ∇u ∈ (L

(Ω))

and |∇u|

p−2

∇u ∈



(Ω)



. So, the operator p-Laplacian −∆

may be seen acting from

1,p

(Ω) into its dual by

h−∆

u, wi =

Ω

| ∇u |

p−2

∇u∇w, for u, w ∈ W

1,p

(Ω).

It’s well know (see e.g. El Khalil et al.

) that the inverse of the principal

eigenvalue of p-Laplacian is the best constant in the Poincar´e’s inequality

kuk

≤ ck∇uk

, ∀u ∈ W

1,p

(Ω).

we can see also that ∆

is the unique duality mapping corresponding

to the normalization numerical function f(t) = t

p−1

with respect to the

norm k∇.k

. Consequently the p-Laplacian is an homeomorphism between

1,p

(Ω) and W

−1,p

(Ω), with inverse is monotone; bounded and continu-

ous.

2.2. Deﬁnitions

In this article, all solutions are weak ones, i.e, (u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω)

is a solution of S

p,q

(λ) if for all (φ, ψ) ∈ C

∞

(Ω) × C

∞

(Ω),







Ω

|∇u|

p−2

∇u∇φ = λ

Ω

a(x)|u|

α−1

|v|

β+1

uφ + µ

Ω

|u|

α−1

|v|

β+1

uφ

Ω

|∇v|

q−2

∇v∇ψ = λ

Ω

a(x)|u|

α+1

|v|

β−1

vψ + µ

Ω

|u|

α+1

|v|

β−1

vψ.

In order to study the eigenvalue problem S

p,q

(λ), we introduce some func-

tional.

For (u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω), λ ∈ IR, we consider

A(u, v) =

α + 1

Ω

|∇u|

β + 1

Ω

|∇v|

− λ

Ω

a(x)|u|

α+1

|v|

β+1

;

B(u, v) =

Ω

|u|

α+1

|v|

β+1

We set

M =

(u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω)/B(u, v) = 1

and

(λ) = inf {A(u, v)/(u, v) ∈ M}, (2.1)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

and for k ∈ IN

∗

= {A ⊂ M : A is symmetric, compact, γ(A) = k},

where γ(A) is the genus of A, i.e, the smallest integer k such that there

exists an odd continuous map from A to IR

\{0}, see Rabinowitz

and

Szulkin

for details of genus.

Deﬁnition 2.1. T is said to be in the (S

) class, if for any sequence u

weakly convergent in X to u and lim sup

n→+∞

< T u

, u

− u >≤ 0, it follows

that u

→ u strongly inX.

Deﬁnition 2.2. The principal eigencurve of S

p,q

(λ) is the graph of the

numerical function µ

: λ −→ µ

(λ) from IR into IR.

2.3. Auxiliary lemmas

Now, we establish the following lemmas that we need in the proof of The-

orem 3.1.

Lemma 2.1. For all λ ∈ IR, we have

(i) M is a closed C

-manifold.

(ii) A and B are two even functionals and C

-diﬀerentiable in W

1,p

(Ω) ×

1,q

(Ω).

(iii) A is bounded from below in M.

Proof. Recall the following statements:

(i) B is submersion C

-diﬀerentiable and B

(u, v) 6= 0 ∀(u, v) ∈ M (be-

cause (B

(u, v), (u, v)) = α + β + 2 6= 0).

(ii) It is obviously.

(iii) By H¨older’s inequality we have for all (u, v) ∈ M

A(u, v) =

α + 1

Ω

|∇u|

β + 1

Ω

|∇v|

− λ

Ω

a(x)|u|

α+1

|v|

β+1

≥

α + 1

Ω

|∇u|

β + 1

Ω

|∇v|

− |λ|kak

∞

B(u, v)

≥

α + 1

Ω

|∇u|

β + 1

Ω

|∇v|

− |λ|kak

∞

≥ λ

− |λ|kak

∞

, (2.2)

where λ

= µ

(0). Thus A is bounded from below.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Lemma 2.2. For all λ ∈ IR we have

i) B

is completely continuous in W

1,p

(Ω)×W

1,q

(Ω) and B is continuous

for the weakly convergence.

ii) A

maps the bounded sets in the bounded sets.

iii) If (u

, v

) * (u, v) weakly in W

1,p

(Ω) × W

1,q

(Ω) and A

, v

)

converges strongly in (W

1,p

(Ω) × W

1,q

(Ω))

, then (u

, v

) −→ (u, v)

strongly in W

1,p

(Ω) × W

1,q

(Ω).

iv) The restriction of A to M satisﬁes the Palais-Smale condition i.e,

for (u

, v

) ∈ M if A(u

, v

) is bounded and

)

, v

) −→ 0. (2.3)

Then (u

, v

) has strong convergent subsequence in W

1,p

(Ω)×W

1,q

(Ω).

Here (W

1,p

(Ω)×W

1,q

(Ω))

is the dual space of W

1,p

(Ω)×W

1,q

(Ω) with

respect to the standard product norm.

Proof. i) and ii) are obviously.

iii) Let (u

, v

) * (u, v) weakly in W

1,p

(Ω)×W

1,q

(Ω). and A

, v

) con-

verges strongly in (W

1,p

(Ω)×W

1,q

(Ω))

. Thus

∂A

∂u

, v

) (resp.

∂A

∂v

, v

))

converges strongly in W

−1,p

(Ω) (resp.W

−1,q

(Ω)).

On the other hand we have

< −∆

, u

− u >=

α+1

∂A

∂u

, v

), u

− u >

+λ < a|u

α−1

β+1

, u

− u >

and

< −∆

, v

− v >=

β+1

∂A

∂v

, v

), v

− v >

+λ < a|u

α+1

β−1

, v

− v > .

By passing to the limit we have

lim sup

n→+∞

< −∆

, u

− u > ≤ 0 and lim sup

n→+∞

< −∆

, v

− v >≤ 0.

Since p-Laplacian and q-Laplacian satisfy the condition (S

) we conclude

that

, v

) −→ (u, v)

strongly in W

1,p

(Ω) × W

1,q

(Ω) as n → +∞.

iv) From (2.3) we have

, v

) −

< A

, v

), (u

, v

) >

α + β + 2

, v

) −→ 0 as n → +∞, (2.4)

and thanks to (2.2) we have the boundness of {(u

, v

)}

in M.

Therefore, there is a subsequence of (u

, v

) still denoted (u

, v

), weakly

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

convergent in W

1,p

(Ω) × W

1,q

(Ω). Moreover, by i) we have B

, v

) →

(u, v) strongly and B(u, v) = 1 (because B(u

, v

) = 1) , then (u, v) ∈

M; and by ii) we have (A

, v

))

is bounded.

This and (2.4) implies that A

, v

) converges strongly in (W

1,p

(Ω) ×

1,q

(Ω))

. Finally, by iii), we conclude that (u

, v

) → (u, v) strongly in

1,p

(Ω) × W

1,q

(Ω).

3. Main results

Theorem 3.1. For any λ ∈ IR and any integer k ∈ IN

∗

(λ) = inf

A∈Γ

max

(u,v)∈A

A(u, v) (3.1)

is critical value of A(u, v) in M. More precisely, there exist (u

(λ), v

(λ)) ∈

M, µ

(λ) ∈ IR such that,

A(u

(λ), v

(λ)) = µ

(λ).

With (u

(λ), v

(λ)) is the eigenfunction of S

p,q

(λ) for the eigenpair eigen-

value (λ, µ

(λ)).

Proof. From Szulkin

and using Lemma 2.1 and Lemma 2.2 we need

only to prove that for all k in IN

∗

6= ∅.

Indeed, W

1,p

(Ω) × W

1,q

(Ω) is separable, thus for all k ∈ IN

∗

there exist

, e

, ...., e

k functions of W

1,p

(Ω) ×W

1,q

(Ω) linearly dense in W

1,p

(Ω) ×

1,q

(Ω), where e

= (e

, e

) with suppe

∩ suppe

= ∅ for all i 6= j and

B(e

) = 1.

Denote F

= span(e

, e

, ...., e

) is subspace of W

1,p

(Ω) × W

1,q

(Ω) and

dimF

= k; then if v ∈ F

there exist t

, t

, ...., t

real numbers such that

v = Σ

i=k

i=1

thus B(v) = Σ

i=k

i=1

α+β+2

Therefore the map v ∈ F

−→ B(v)

α+β+2

is a norm in F

Let k(., .)k

p,q

the norm in W

1,p

(Ω) × W

1,q

(Ω); hence, there is c > 0 such

that for all v ∈ F

, we have

ckvk

p,q

≤ B(v)

α+β+2

≤

kvk

p,q

This implies that the set

V = F

∩

(u, v) ∈ W

1,p

(Ω) × W

1,q

(Ω)/B(u, v) ≤ 1

6= ∅

is a bounded neighborhood symmetric of (0, 0) ∈ F

. Thus by (f) of Prop.

2.3 of Szulkin,

γ(F

∩ M) = k then Γ

6= ∅.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Corollary 3.1. For all λ ∈ IR we have

lim

k→+∞

(λ) = +∞

Proof. W

1,p

(Ω) ×W

1,q

(Ω) is separable, hence we can have bi-orthogonal

system (e

, e

∗

)

such that e

∈ W

1,p

(Ω) × W

1,q

(Ω), e

∗

∈ (W

1,p

(Ω) ×

1,q

(Ω))

with (e

)

is linearly dense in W

1,p

(Ω) × W

1,q

(Ω); and e

∗

are

total in (W

1,p

(Ω) × W

1,q

(Ω))

, see e.g Szulkin.

Set for k ∈ IN

∗

= span(e

, e

, . . . , e

), F

⊥

span(e

k+1

, e

k+2

, e

k+3

. . .);

Therefore, by (g) of proposition 2.3 in Szulkin,

for all A ∈ Γ

we have

A ∩ F

⊥

k−1

6= ∅ ∀k ≥ n hence

= inf

A∈Γ

sup

A∩F

⊥

k−1

A(u, v) −→ +∞ as n → +∞ (3.2)

indeed, if not, for k large enough, there is (u

, v

) ∈ F

⊥

k−1

such that

B(u

, v

) = 1 and

≤ A(u

, v

) ≤ m,

for some constant m independent of k. Thus

A(u

, v

) =

α+1

Ω

|∇u

β+1

Ω

|∇v

−λ

Ω

a(x)|u

α+1

β+1

≤ m.

This implies

α + 1

Ω

|∇u

β + 1

Ω

|∇v

≤ m + |λ|kak

∞

Hence (u

, v

) is bounded in W

1,p

(Ω) ×W

1,q

(Ω). Since (e

, e

∗

) = 0 ∀k ≥ n

and by the Sobolev’s imbedding we have (u

, v

) −→ (0, 0) as n → +∞,

strongly in L

(Ω) × L

(Ω). This is a contradiction, because B(u

, v

) = 1.

Since µ

(λ) ≥ m

so we conclude by (3.2) that

lim

k→+∞

(λ) = +∞.

Corollary 3.2. For all λ ﬁxed in IR µ

(λ) given by (2.1) is the smallest

eigenvalue of S

p,q

(λ) and there exists (u, v) ∈ M a solution of S

p,q

(λ)

associated for (λ, µ

(λ)).

Proof. Let (u, v) ∈ M and B = {(u, v), (−u, −v)} therefore γ(B) = 1

and B ∈ Γ

. Since A is even then µ

(λ) ≤ inf

(u,v)∈M

A(u, v), the reverse

inequality is obvious.