Báo cáo nghiên cứu khoa học: " ON THE EXISTENCE OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS" ppt

ON THE EXISTENCE OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS Bui Boi Minh Anh 1 , Nguyen Minh Quan 1 , Tran Tuan Anh 2 , Vo Dang Khoa 3 1 State University o

ON THE EXISTENCE OF SOLUTIONS OF NONLINEAR ELLIPTIC

EQUATIONS WITH UNBOUNDED COEFFICIENTS

Bui Boi Minh Anh (1) , Nguyen Minh Quan (1) , Tran Tuan Anh (2) , Vo Dang Khoa (3)

(1) State University of New York at Buffalo, USA

(2) Georgia Institute of Technology, Atlanta, Georgia, USA

(3) University of Medicine and Pharmacy, Hochiminh City, Vietnam

( Manuscript Received on March 20 th , 2006, Manuscript Revised October 2 nd , 2006)

ABSTRACT : Using the topological degree of class ( )S + introduced by F E Browder in [ ]1 and [ ]2 , we extend some results of the papers [ ]3 and [ ]4 to the case of

Banach spaces with locally bounded conditions

1 INTRODUCTION

Let N be an integer 2≥ and D be a bounded open subset in R In this paper we N study the following equation:

u

The p− Laplace equation −Δ +pu f x, u( )= is a special case of 0 ( )1.1 If p 2= and

i

i

u

a x, u

x

∂

∇ =

∂ then ( )1.1 has the form:

N

u

x

=

∂

The problem ( )1.2 has been solved in [ ]4 (Theorem 3.1, p.514) by using the topological degree for operators of class ( )B + However, that method doesn’t work when

i

i

u

x

− ∂

∇ = ∇

∂ The one we use here can solve the problem ( )1.2 for all

p 1>

Moreover, our result is also stronger than Theorem 11 in [ ]3 (p.357) where the authors prove the existence result for the Dirichlet problem:

p D

u f x, u

u∂ 0

−Δ =

⎧⎪

with the condition ( )10 that the function b is in L Dp( ) but not in p ( )

loc

L D

In this section, we recall the class ( )S + introduced by Browder (see [ ]1 , [ ]2 )

Definition 2.1 Let Dbe a bounded open set of a reflexive Banach space X and f be

a mapping from D into the dual space X of X We say f is of class * ( )S + if f has the

following properties:

( )i {f x( )n }n converges weakly to f x( ) if { }xn n converges strongly to x in D, i.e f is a demicontinuous mapping on D

( )ii { }xn n converges strongly to x if { }xn n converges weakly to x in D and

( )

lim sup →∞ f x , x −x ≤ 0

Definition 2.1 Let {g : 0 t 1t ≤ ≤ be a one-parameter family of maps of } Dinto X *

We say {g : 0 t 1t ≤ ≤ is a homotopy of class } ( )S +, if the sequences { }xn n and {gt n( )xn }n

converge strongly to x and g x respectively for any sequence t( ) { }xn n in D converging weakly to some x in X and for any sequence { }tn n in [ ]0,1 converging to t such that

( )

n

lim sup →∞ g x , x −x ≤ 0

Let f be a mapping of class ( )S + on D and let p be in X \ f* ( )∂D By Theorems 4 and 5 in [ ]2 , the topological degree of fon Dat p is defined as a family of integers and is denoted by deg f , D, p In ( ) [ ]6 Skrypnik showed that this topological degree is single-valued (see also [ ]2 ) The following result was proved in [ ]2

Proposition 2.1 Let f be a mapping of class ( )S + from D into X , and let y be in *

( )

*

X \ f ∂D Then we can define the degree deg f , D, y( ) as an integer satisfying the following properties:

( )a If deg f , D, y( )≠ then there exists x D0 ∈ such that f x( )= y

( )b If {g : 0 t 1t ≤ ≤ is a homotopy of class } ( )S + and {y : 0 t 1t ≤ ≤ is a continuous }

curve in X such that * yt∉gt( )∂D for all t∈[ ]0,1 , then deg g , D, y( t t) is constant in t on

[ ]0,1

Proposition 2.2 Let A : D→X* be a mapping of class ( )S + Suppose that 0 D \ D∈ ∂

and

Au 0≠ , Au, u ≥ for u0 ∈∂ D

Then deg A, D, 0( )= 1

t

A : D→X , t∈ 0,1 be the homotopy family of operators of class ( )S + Suppose that A u 0t ≠ for u∈∂D, t∈[ ]0,1 Then deg A , D, 0( 0 )=deg A , D,0( 1 )

3 NONLINEAR ELLIPTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS

Let p be a real number 2≥ , N be an integer 2≥ , Ω and D be bounded open subsets

in R We denote by N 1,p( )

0

W D the completion of C D,c ∞( ) in the norm:

1/ p p

c D

D

u ⎛ u dx⎞ u C D,∞

Let Ω be an increasing sequence of open subsets of k Ω such that Ω is contained in k 1

k+

Ω and

1

k k

∞

=

Ω =UΩ Put 1, ( )

0

p

0

p

X =W Ω

We denote by p ' and p the conjugate exponent and the Sobolev conjugate exponent *

of p , i.e.,

1 1

p ' 1

p

−

= −⎜ ⎟

*

Np

if N p

N p p

if N p

⎪ −

= ⎨

⎩

Let g , g , ., g be real functions on 0 1 N Ω× satisfying the following conditions:

( )C1 The function g x, ti( ) is measurable in x for fixed t in and continuous in t for fixed x in Ω for any i 0, ., N=

( )C2 g x, 00( )=0 x∀ ∈Ω

g x, t ≤b x +k t ∀ x, t ∈Ω× , i 0, , N= and

( )C4

i 1

=

where s , ., s , k , ., k , r , ., r and 0 N 0 N 0 N r,q are non-negative real numbers and

b , ., b and c, , α β are measurable functions such that α ∈Lb( )Ω ,

− − +

− +

⎝ ⎠, c L∈ 1( )Ω , r∈( )1, p ,

q∈ 1, p 1− ,

0

Np

− +

1

N p

Np

− ∈⎛ − ∞⎞

⎝ ⎠, a L∈ r 0( )Ω ,

i

Np

− +

1

N p

Np

− ∈⎛ − ∞⎞

i loc

b ∈L Ω for any i 0, ., N=

We assume that the functions a x,si( ), i 1, ., N= , ( ) N

s= s , ., s ∈ satisfy:

( )C5 a x,si( ) is defined and differentiable w.r.t all of its arguments for x∈Ω,

s= s , ., s ∈ Moreover, a x,0i( )= for all 0 i 1, ., N= , x∈Ω

( )C6 There exist positive constants M , M such that the inequalities : 1 2

a x,s

M 1 s s

−

∂

∂

i j

a x,s

d x 1 s s

−

∂

i

2 k

a x,s

M 1 s x

−

∂

∂ are satisfied, where d L∈ ∞ loc( )Ω

Theorem 3.1 Under conditions ( ) ( )C1 − C6 , there exists u in X such that for any

v Y∈ ,

To prove the theorem we need the following lemma

Lemma 3.1 Let 1,p( )

X =W Ω Under conditions ( ) ( )C1 − C6 there exists u in k X k

such that for any v X∈ k,

k

u v

∂

Proof Fix a u in X We will show that there exists a unique k T u in k( ) *

k

X satisfying

k

u v

∂

for all v X∈ k

Since a x,0i( )= for 0 x∈Ω and condition ( )C6 ,

1

i i

a x, t u

( )

k

1 N i

0

a x, t u

s

= Ω

∂

∫ ∫

k

1

p 2 ,

0

Ω

where c is a positive number depending on k, N, u and d

Put Gk,i( )( )u x =g x, u x xi( ( ) ) ∀ ∈Ωk, i 0, ., N= Then G is a bounded, k,i continuous mapping from r s i i( )

k

L Ω into r i( )

k

L Ω by conditions ( ) ( )C2 , C3 and by a result

in [ ]5 , p.30 Moreover, by Sobolev embedding theorem there exists a positive C such that:

k

N

k

u

x

= Ω

∫

N

i=1

From this and ( )3.3 we get ( )3.2 Next, we show that T is of class k ( )S + First, we check that T is demicontinuous in k X Let k { }wn n be a sequence converging strongly to

w in X Then for every v in in k X we have: k

k

N

v

x

= Ω

∂

∂

∑

∫

k

N

n

=

Ω

On the other hand:

k

N

v

x

=

Ω

∂

∂

∑

∫

( ( ) ) ( )

k

1

Ω

∑ ∑

k

,

0

Ω

where M is a positive number depending on k, N, v and d 3

And:

k

N

n

=

Ω

∫

k

N

n

=

Ω

∫

N

4 k,i n k,i r ,k n k,i r ,k n

i 1

=

0

4 k,0 n k,0 r ,k

Since G is a bounded, continuous mapping from k,i r s i i( )

k

L Ω into r i( )

k

L Ω and { }wn n converges strongly to w in X , from k ( )3.4 and ( )3.6 , we have T is demicontinuous in k k

X

Now let { }um m be a sequence converging weakly to u in X and k

( )

m

lim sup T u , u u 0

k

n

m

x

→∞ Ω =

∂

∑

k

N

m

u

x

= Ω

∂

Since 1 1

i i

N p

r s

pN

− − > −

for all i 0, ., N= , the theorem of Rellich-Konkrachov gives us that the sequence {Gk,i( )um }m converges to Gk,i( )u in r i( )

k

( )

{Gk,0 um }mconverges to Gk,0( )u in r i( )

k

k

Ω

∫

On the other hand, since { }um mconverges to u in L ,p ∂ converges weakly to uum ∂ and

( )

{Gk,i um }m converges to Gk,i( )u in r i( )

k

k

N

m

u

x

= Ω

∂

∑

Hence

k

N

m

m i 1

i

u

x

→∞ = Ω

So, it follows from ( )3.7 and ( )3.8 that ( ) ( )

k

n

m

x

→∞ Ω =

∂

∑

k

n

m

x

→∞ Ω =

∑

By condition ( )C6

k

N

v u

x

= Ω

∂

∑

∫

k

1

i

Ω

∑ ∑

k

0

Ω

Combining ( )3.9 and (3.10), we have the conclusion that the sequence { }um m

converges to u in X Thus, k T is of class k ( )S + in X Next we calculate the topological k

degree of the operator T k

By condition (C4), the Holder enequality and (3.10 , we have: )

( )

k

T u , u

k

n i

u

a x, u

x

= Ω

∂

∂

∑

k

N

u

x

= Ω

∫

where b ,d are positive numbers such that 1 1 1 1 1

1

b− +p− +b− = , 1 1 1

1

d− +d− = From 1 conditions of b,d we have: *

1

1 qb< < , p *

1

1 rd< < By Poincare inequalities, the Sobolev p embedding theorem there exists C 0> such that:

( )

1

T u , u Ω≥M u Ω−C α u + − βC u Ω− c Since r,q 1+ ∈( )1, p , we can choose s 0> such that :

M

2

Let G={w X : w∈ Ω< and s} Gk = IG Xk Then G is an open bounded set in k

k

M

2

Since T satisfies condition k ( )S +on X , by Proposition 2.2 we conclude that k

k k deg T ,G ,0 = 1 Then there exists uk∈∂XkGk such that T uk( )k = , i.e 0

k

u v

∂

which completes the proof of the lemma

Proof of Theorem 3.1 By Lemma 3.1, there exists a sequence { }uk k ⊂ ∂XkGk such

that:

( )

k k

Since { }uk k ⊂ ∂XG, it is bounded in X Let u be the weak limit of { }uk kin

( )

1,p

0

By (3.11 we have )

( )

Fix l∈ + We consider the function ρ ∈l Cc ∞( )Ω which satisfies 0≤ ρ ≤ and l 1

l

l

1 if x x

0 if x

−

∈Ω

⎧

For all k l≥ we have ρl ku − ρ ∈lu Xk Then, (3.12) implies

( )

k k l k l

This yields klim T u , uk( )k l k lu 0

l

N

l k l

k

x

→∞

= Ω

∂

∑

∫

l

N

k

u

x

= Ω

∂

Since {ρl k ku } converges weakly to ρ in X , arguing as in the Lemma 3.1 (the proof lu

of T satisfying condition k ( )S +, we have

l

N

k

k

u

x

→∞

= Ω

Therefore, (3.14 and ) (3.15 imply ) ( ) ( )

l

N

l k l

k

x

→∞

= Ω

∂

∑

l

N

k l

k i 1

→∞ = Ω

∑

Since { }uk k converges to u in Lp( )Ω , it is easily seen that

l

N

l

k i 1

i

x

→∞ = Ω

∂ρ

∂

∑

l

N

k

k

x

→∞

= Ω

∂

∑

( ) ( ) ( )

l

N

k

i

x

→∞ = Ω

∂

∑

k

x

=

l

p

k

→∞

Ω

This means that { }uk k strongly converges to u on Ω for all ll ∈ + Now fix v Y∈ Our goal is to show that

k

u v

∂

Indeed, since v Y∈ , there exists a positive integer m such that sup p v( )⊂ Ω Then m k

v X∈ for all k m≥ By Lemma 3.1:

k

u v

∂

Since { }uk k strongly converges to u on Ω , it follows from the above equality that m

(3.18) holds We now N complete the proof of the theorem

VỀ SỰ TỒN TẠI NGHIỆM CỦA PHƯƠNG TRÌNH ELLIPTIC PHI TUYẾN

VỚI CÁC HỆ SỐ KHÔNG BỊ CHẶN

Bùi Bội Minh Anh (1) , Nguyễn Minh Quân (1) , Trần Tuấn Anh (2) , Võ Đăng Khoa (3)

(1) Trường Đại học NewYork tại Buffalo, Hoa Kỳ (2) Viện Công nghệ Georgia, Hoa Kỳ (3) Trường Đại học Dược Tp.HCM, Việt Nam

TÓM TẮT : Sử dụng bậc tôpô của lớp ( )S + được giới thiệu bởi F E Browder trong các bài báo [ ]1 và [ ]2 , chúng tôi mở rộng một số kết quả của các bài báo [ ]3 và [ ]4 sang

trường hợp không gian Banach với các điều kiện bị chặn địa phương

TÀI LIỆU THAM KHẢO

[1] F E Browder, Nonlinear elliptic boundary value problems and the generalized

topological degree , Bull Amer Math Soc., 76pp 999-1005, (1970)

[2] F E Browder, Fixed point theory and nonlinear problems, Proc Symp Pure Math,

39, 49-86, (1983)

[3] G Dinca, P Jebelean and J Mawhin, Variational and topological methods for

Dirichlet problems with p-Laplacian , Portugaliae mathematica 58 Fasc 3-2001

[4] D M Duc, N H Loc , P V Tuoc, Topological degree for a class of operators and

applications , Nonlinear Analysis 57, 505-518, (2004)

[5] M.A Krasnosel´kii, Topological methods in the theory of nonlinear integral

equations, Pergamon Press, Oxford, (1964)

[6] I.V Skrypnik, Nonlinear Higher Order Elliptic Equations (in Russian), Noukova

Dumka Kiev, (1973)

[7] I.V Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems,

Am Math Soc Transl., Ser II 139 (1994)