TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 17 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 N R . In this paper we study the following equation: () () ()() NN ii0 i1 i1 ii u a x, u g x,u g x,u a x 0 x D, xx == ⎡⎤ ∂∂ ∇− + + = ∀∈ ⎢⎥ ∂∂ ⎣⎦ ∑∑ (1.1) The p − Laplace equation ( ) p ufx,u 0 − Δ+ = is a special case of ( ) 1.1 . If p2 = and () i i u ax,u x ∂ ∇= ∂ then () 1.1 has the form: () ()() N i0 i1 i u u g x,u g x,u a x 0 x = ⎡⎤ ∂ −Δ + + + = ⎢⎥ ∂ ⎣⎦ ∑ . (1.2) 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 p2≠ and () p2 i i u ax,u u x − ∂ ∇=∇ ∂ . The one we use here can solve the problem () 1.2 for all p1> . 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 ufx,u u0 ∂ −Δ =⎧ ⎪ ⎨ = ⎪ ⎩ in D . with the condition ( ) 10 that the function b is in ( ) p LD but not in () p loc LD. 2. TOPOLOGICAL DEGREE OF CLASS ( ) S + In this section, we recall the class ( ) S + introduced by Browder (see [ ] 1 , [ ] 2 ). Definition 2.1. Let D be 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: Science & Technology Development, Vol 9, No.9- 2006 Trang 18 ( ) i ( ) { } n n fx converges weakly to ( ) fx if { } n n x converges strongly to x in D , i.e. f is a demicontinuous mapping on D . ( ) ii { } n n x converges strongly to x if { } n n x converges weakly to x in D and ( ) nnn lim sup f x , x x 0 →∞ − ≤ . Definition 2.1. Let { } t g:0 t 1 ≤ ≤ be a one-parameter family of maps of D into * X . We say { } t g:0 t 1≤≤ is a homotopy of class ( ) S + , if the sequences { } n n x and ( ) { } n tn n gx converge strongly to x and ( ) t gx respectively for any sequence { } n n x in D converging weakly to some x in X and for any sequence { } n n t in [ ] 0,1 converging to t such that ( ) n ntnn 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 f on D at 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 0≠ then there exists xD ∈ such that ( ) fx y= . ( ) b If { } t g:0 t 1≤≤ is a homotopy of class ( ) S + and { } t y:0 t 1≤≤ is a continuous curve in * X such that ( ) tt ygD∉∂ for all [ ] t0,1∈ , then ( ) tt deg g ,D, y is constant in t on [ ] 0,1 . Proposition 2.2. Let * A:D X→ be a mapping of class ( ) S + . Suppose that 0D\D ∈ ∂ and Au 0 ≠ , Au, u 0≥ for uD ∈ ∂ . Then ( ) deg A, D,0 1= . Proposition 2.3. Let [ ] * t A:D X, t 0,1→∈ be the homotopy family of operators of class ( ) S + . Suppose that t Au 0 ≠ for [ ] uD, t0,1∈∂ ∈ . Then ( ) ( ) 01 deg A ,D,0 deg A ,D,0= . 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 N R . We denote by ( ) 1,p 0 WD the completion of ( ) c CD, ∞  in the norm: () 1/p p c D D u u dx u C D, ∞ ⎛⎞ =∇ ∀∈ ⎜⎟ ⎝⎠ ∫  . Let k Ω be an increasing sequence of open subsets of Ω such that k Ω is contained in 1k+ Ω and 1 k k ∞ = Ω= Ω U . Put ( ) 1, 0 p XW = Ω , ( ) 1, 0 p kk XW = Ω . TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 19 We denote by p' and * p the conjugate exponent and the Sobolev conjugate exponent of p , i.e., 1 1 p' 1 p − ⎛⎞ =− ⎜⎟ ⎝⎠ and * Np if N p Np p if N p ⎧ ≥ ⎪ − = ⎨ ⎪ ∞ ≤ ⎩ . Let 01 N g , g , , g be real functions on Ω ×  satisfying the following conditions: () C1 The function ( ) i gx,t is measurable in x for fixed t in  and continuous in t for fixed x in Ω for any i 0, , N= . ( ) C2 ( ) 0 g x,0 0 x=∀∈Ω. () C3 () () () i s iii g x, t b x k t x, t , i 0, , N≤+ ∀∈Ω×= and ( ) C4 () () () () ()() ( ) N qr N ii0 i1 x z t x t c x g x, t z g x, t a x x, t,z = ⎡⎤ −α −β − ≤ + + ∀ ∈Ω× × ⎢⎥ ⎣⎦ ∑   where 0N0 N0N s , , s , k , , k , r , , r and r,q are non-negative real numbers and 0N b , , b and c, , αβ are measurable functions such that ( ) b Lα∈ Ω , () Np b , Np q 1 pq ⎛⎞ ∈∞ ⎜⎟ ⎜⎟ −− + ⎝⎠ , ( ) d L β ∈Ω, () Np d, Np r pr ⎛⎞ ∈ ∞ ⎜⎟ ⎜⎟ −+ ⎝⎠ , ( ) 1 cL∈Ω, ( ) r1,p∈ , ( ) q1,p1∈− , () 0 Np r, Np 1 p ⎛⎞ ∈∞ ⎜⎟ ⎜⎟ −+ ⎝⎠ , 1 00 Np sr, Np − ⎛⎞ − ∈ ∞ ⎜⎟ ⎝⎠ , ( ) 0 r aL ∈ Ω , () i Np r, Np 2 p ⎛⎞ ∈∞ ⎜⎟ ⎜⎟ −+ ⎝⎠ , 1 ii Np sr, Np − ⎛⎞ − ∈∞ ⎜⎟ ⎝⎠ and ( ) i r iloc bL ∈ Ω for any i 0, , N= . We assume that the functions ( ) i ax,s, i 1, , N = , ( ) N 1N s s , , s=∈ satisfy: () C5 () i ax,s is defined and differentiable w.r.t all of its arguments for x ∈ Ω , ( ) N 1N s s , , s=∈ . Moreover, ( ) i ax,0 0 = for all i 1, , N = , x ∈ Ω . ( ) C6 There exist positive constants 12 M,M such that the inequalities : ( ) () NN p2 i 2 ij 1 i i,j 1 i 1 j ax,s M1 s s − == ∂ ξ ξ≥ + ξ ∂ ∑∑ , ( ) () () p2 i j ax,s dx 1 s s − ∂ ≤+ ∂ and ( ) () p1 i 2 k ax,s M1s x − ∂ ≤+ ∂ are satisfied, where ( ) loc dL ∞ ∈Ω. Theorem 3.1. Under conditions ( ) ( ) C1 C6− , there exists u in X such that for any vY∈ , () () ()() NN ii0 i1 i1 ii vu ax,u dx gx,u gx,u axvdx 0 xx == ΩΩ ⎡⎤ ∂∂ ∇+ ++ = ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ . (3.1) To prove the theorem we need the following lemma. Science & Technology Development, Vol 9, No.9- 2006 Trang 20 Lemma 3.1. Let ( ) 1,p k0k XW=Ω . Under conditions ( ) ( ) C1 C6− there exists k u in k X such that for any k vX∈ , () () ()() NN k ik ik 0k i1 i1 ii uv ax,u dx gx,u gx,u ax vdx 0 xx == ΩΩ ⎡⎤ ∂∂ ∇+ ++ = ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ . Proof. Fix a u in k X . We will show that there exists a unique () k Tu in * k X satisfying () () () ()() kk NN k ki i0k i1 i1 ii uv T u ,v a x, u dx g x,u g x,u a x vdx 0 xx == ΩΩ ⎡⎤ ∂∂ =∇+ ++= ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ . (3.2) for all k vX∈ . Since ( ) i ax,0 0= for x ∈ Ω and condition ( ) C6 , () () kk 1 NNN i i i1 i1 j1 ijji 0 ax,tu vuv ax,u dx . dt dx xsxx === ΩΩ ⎡⎤ ∂∇ ∂∂∂ ∇= ⎢⎥ ∂∂∂∂ ⎢⎥ ⎣⎦ ∑∑∑ ∫∫∫ () k 1 N i j1 j 0 ax,tu dt u v dx s = Ω ⎡⎤ ∂∇ ≤∇∇ ⎢⎥ ∂ ⎢⎥ ⎣⎦ ∑ ∫∫ () k k k 1 p2 , 0 Nd 1 t u dt u vdx cv − ∞ ΩΩ Ω ⎡⎤ ≤+∇∇∇≤ ⎢⎥ ⎣⎦ ∫∫ , (3.3) where c is a positive number depending on k,N,u and d . Put ( ) ( ) ( ) ( ) k,i i k G u x g x,u x x , i 0, , N=∀∈Ω=. Then k,i G is a bounded, continuous mapping from ( ) ii rs k L Ω into ( ) i r 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 i0k i1 i u g x,u g x,u a x vdx x = Ω ⎡⎤ ∂ + +≤ ⎢⎥ ∂ ⎣⎦ ∑ ∫ () () 0k i0 N k,i k,0 k r r,k r ,k i=1 C G u u G u a v v X ΩΩ ⎡⎤ ≤++∀∈ ⎢⎥ ⎣⎦ ∑ . From this and ( ) 3.3 we get ( ) 3.2 . Next, we show that k T is of class ( ) S + . First, we check that k T is demicontinuous in k X. Let { } n n w be a sequence converging strongly to w in k X . Then for every v in in k X we have: () () ()() () k N kn k i n i i1 i v Tw Tw,v ax,w ax,w dx x = Ω ∂ −= ∇−∇+ ∂ ∑ ∫ () () ()() () () k N n in i 0n0 i1 ii ww g x,w g x,w g x,w g x,w a x vdx xx = Ω ⎡⎤ ⎛⎞ ∂∂ +−+−+ ⎢⎥ ⎜⎟ ∂∂ ⎝⎠ ⎣⎦ ∑ ∫ . (3.4) On the other hand: ()() () k N ini i1 i v ax,w ax,w dx x = Ω ∂ ∇− ∇ = ∂ ∑ ∫ TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 21 ( ) ( ) () k 1 NN in n n i1 j1 jji 0 ax,w t w w ww v .dtdx sxx == Ω ⎡⎤ ∂∇+∇− ∂− ∂ = ⎢⎥ ∂∂∂ ⎢⎥ ⎣⎦ ∑∑ ∫∫ () () () k k k 1 p2 nn n 3n , 0 Nd 1 w t w w dt w w vdx M w w − ∞Ω Ω Ω ⎡⎤ ≤ +∇+∇− ∇−∇≤ − ⎢⎥ ⎣⎦ ∫∫ . (3.5) where 3 M is a positive number depending on k, N,v and d . And: () () ()() () () k N n in i 0n0 i1 ii w w g x,w g x,w g x,w g x,w a x vdx xx = Ω ⎡⎤ ⎛⎞ ∂ ∂ − +−+= ⎢⎥ ⎜⎟ ∂∂ ⎝⎠ ⎣⎦ ∑ ∫ () () () () () k N n k,i n k,i k,0 n k,0 i1 ii ww Gw Gw Gw Gwvdx xx = Ω ⎡⎤ ⎛⎞ ∂∂ =−+− ⎢⎥ ⎜⎟ ∂∂ ⎝⎠ ⎣⎦ ∑ ∫ () () () ii N 4 k,i n k,i n k,i n r,k r,k i1 MGwGwwGwwwv ΩΩΩ = ⎡⎤ ≤−+−+ ⎢⎥ ⎣⎦ ∑ ( ) ( ) 0 4 k,0 n k,0 r,k MG w G w v Ω +− . (3.6) Since k,i G is a bounded, continuous mapping from ( ) ii rs k L Ω into ( ) i r k L Ω and { } n n w converges strongly to w in k X , from ( ) 3.4 and ( ) 3.6 , we have k T is demicontinuous in k X. Now let { } m m u be a sequence converging weakly to u in k X and ( ) km m m lim sup T u ,u u 0 →∞ − ≤ or () ( ) k n m im m i1 i uu limsup a x, u dx x →∞ = Ω ∂− ∇ + ∂ ∑ ∫ () ()()( ) k N m im 0m m i1 i u g x,u g x,u a x u u dx 0 x = Ω ⎡⎤ ∂ +++−≤ ⎢⎥ ∂ ⎣⎦ ∑ ∫ . (3.7) Since 11 ii Np rs pN −− − > for all i 0, , N = , the theorem of Rellich-Konkrachov gives us that the sequence ( ) { } k,i m m G u converges to ( ) k,i Gu in () i r k L Ω . Thus, ( ) { } k,0 m m G u converges to ( ) k,0 Gu in ( ) i r k L Ω . This implies : ( ) ( ) ( ) k 0m m gx,u ax u udx 0. Ω +−→ ⎡⎤ ⎣⎦ ∫ On the other hand, since { } m m u converges to u in p L, m u ∂ converges weakly to u ∂ and ( ) { } k,i m m G u converges to ( ) k,i Guin ( ) i r k L Ω ,we get ()() k N m mm i1 i u gx,u u udx 0. x = Ω ∂ −→ ∂ ∑ ∫ Hence () ()()( ) k N m im 0m m m i1 i u lim g x,u g x,u a x u u dx 0. x →∞ = Ω ⎡⎤ ∂ + +−= ⎢⎥ ∂ ⎣⎦ ∑ ∫ (3.8) Science & Technology Development, Vol 9, No.9- 2006 Trang 22 So, it follows from ( ) 3.7 and ( ) 3.8 that () ( ) k n m im m i1 i uu limsup a x, u dx 0 x →∞ = Ω ∂− ∇ ≤ ∂ ∑ ∫ or ()() ( ) k n m imi m i1 i uu limsup ax,u ax,u dx 0 x →∞ = Ω ∂− ∇ −∇ ≤ ⎡⎤ ⎣⎦ ∂ ∑ ∫ . (3.9) By condition ( ) C6 ()() () ( ) k N ii i1 i vu ax,v ax,u dx x = Ω ∂− ∇− ∇ ∂ ∑ ∫ ( ) ( ) ()() k 1 NN i i1 j1 jji 0 ax,ut vu vu vu .dt dx sxx == Ω ⎡⎤ ∂∇+∇− ∂− ∂− = ⎢⎥ ∂∂∂ ⎢⎥ ⎣⎦ ∑∑ ∫∫ () () () () k 1 p2 2p 1 5 0 M 1 ut vu dt vu M vu . − Ω ⎡⎤ ≥ +∇+∇− ∇− ≥ ∇− ⎢⎥ ⎣⎦ ∫∫ (3.10) Combining ( ) 3.9 and ( ) 3.10 , we have the conclusion that the sequence { } m m u converges to u in k X . Thus, k T is of class ( ) S + in k X . Next we calculate the topological degree of the operator k T. By condition (C4), the Holder enequality and ( ) 3.10 , we have: ( ) k Tu,u Ω = () ( ) k n i i1 i u ax,u x = Ω ∂ ∇ + ∂ ∑ ∫ () ( ) () () k N i0 i1 i u g x,u g x,u a x u x dx x = Ω ⎡⎤ ∂ ++ ⎢⎥ ∂ ⎣⎦ ∑ ∫ 11 pqr 5 b pqb bdr 1 Mu u u u c. Ω ≥−α∇−β− where 11 b ,d are positive numbers such that 111 1 b pb1 −−− + +=, 11 1 dd1 −− +=. From conditions of b ,d we have: * 1 1qb p < < , * 1 1rd p < < . By Poincare inequalities, the Sobolev embedding theorem there exists C0> such that: () 1 pq1r k5 b qb d 1 Tu,u Mu C u C u c. + ΩΩ Ω ≥−α−β− Since () r,q 1 1,p+∈ , we can choose s0> such that : q1p rp p 5 bd1 M Cs Cs cs 2 +− − − α−β−<. Let { } GwX:w s Ω =∈ <and kk GGX = I . Then k G is an open bounded set in k Xand () p 5 kkk M Tu,u s,u G 2 Ω ≥∀∈∂ . Since k T satisfies condition ( ) S + on k X , by Proposition 2.2 we conclude that ( ) k X kk deg T ,G ,0 1 = . Then there exists k kXk uG∈∂ such that ( ) kk Tu 0 = , i.e. () () ()() NN k ik ik 0k k i1 i1 ii uv ax,u dx gx,u gx,u ax vdx 0,vX xx == ΩΩ ⎡⎤ ∂∂ ∇+ ++ =∀∈ ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ which completes the proof of the lemma . TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 23 Proof of Theorem 3.1. By Lemma 3.1, there exists a sequence { } k kXk k uG⊂∂ such that: ( ) kk Tu 0 = . (3.11) Since { } kX k uG⊂∂ , it is bounded in X . Let u be the weak limit of { } k k u in ( ) 1,p 0 W Ω . By ( ) 3.11 we have ( ) kk k T u ,v 0, v X=∀∈. (3.12) Fix l + ∈ . We consider the function ( ) lc C ∞ ρ ∈Ω which satisfies l 01≤ρ ≤ and () l1 l l 1 if x x 0 if x − ∈ Ω ⎧ ρ= ⎨ ∉ Ω ⎩ . For all k l≥ we have lk l k uuXρ−ρ∈. Then, ( ) 3.12 implies ( ) kk lk l Tu,u u 0 ρ −ρ = . (3.13) This yields ( ) kk lk l k lim T u , u u 0 →∞ ρ −ρ = , that is () ( ) l N lk l ik k i1 i uu lim a x, u dx x →∞ = Ω ∂ρ −ρ ∇ + ∂ ∑ ∫ () ()()( ) l N k ik 0k lkl i1 i u gx,u gx,u ax u udx 0 x = Ω ⎡⎤ ∂ +++ρ−ρ= ⎢⎥ ∂ ⎣⎦ ∑ ∫ . (3.14) Since { } lk k uρ converges weakly to l u ρ in X , arguing as in the Lemma 3.1 (the proof of k T satisfying condition ( ) S + , we have () ()()( ) l N k ik 0k lkl k i1 i u lim g x,u g x,u a x u u dx 0 x →∞ = Ω ⎡⎤ ∂ + +ρ−ρ= ⎢⎥ ∂ ⎣⎦ ∑ ∫ . (3.15) Therefore, ( ) 3.14 and ( ) 3.15 imply () ( ) l N lk l ik k i1 i uu lim a x, u dx 0 x →∞ = Ω ∂ρ −ρ ∇ = ∂ ∑ ∫ , or ()() ( ) l N k l ikk l k i1 ii uu lim a x, u u u dx 0 xx →∞ = Ω ∂− ⎡⎤ ∂ρ ∇ −+ρ = ⎢⎥ ∂∂ ⎣⎦ ∑ ∫ . (3.16) Since { } k k u converges to u in ( ) p L Ω , it is easily seen that ()() l N l ikk k i1 i lim a x, u u u dx 0 x →∞ = Ω ∂ρ ∇ −= ∂ ∑ ∫ . Combining this and () 3.16 we obtain () ( ) l N k li k k i1 i uu lim a x, u dx 0 x →∞ = Ω ∂− ρ ∇= ∂ ∑ ∫ , or Science & Technology Development, Vol 9, No.9- 2006 Trang 24 ()() ( ) l N k li ki k i1 i uu lim a x, u a x, u dx 0 x →∞ = Ω ∂− ρ∇−∇ = ⎡⎤ ⎣⎦ ∂ ∑ ∫ . (3.17) On the other hand, by ( ) 3.10 : ()() ( ) () N p k iki 8k i1 i uu ax,u ax,u M u u x = ∂− ∇− ∇ ≥ ∇−⎡⎤ ⎣⎦ ∂ ∑ . Hence () l p lk k lim u u dx 0 →∞ Ω ρ∇ − = ∫ . This means that { } k k u strongly converges to u on l Ω for all l + ∈ . Now fix vY ∈ . Our goal is to show that () () ()() NN k ii0 i1 i1 ii uv ax,u dx gx,u gx,u axvdx 0 xx == ΩΩ ⎡⎤ ∂∂ ∇+ ++ = ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ . (3.18) Indeed, since vY∈ , there exists a positive integer m such that ( ) m sup p v ⊂Ω . Then k vX∈ for all k m≥ . By Lemma 3.1: () () ()() mm NN k ik ik 0k i1 i1 ii u v ax,u dx gx,u gx,u ax vdx 0 xx == ΩΩ ⎡⎤ ∂ ∂ ∇+ ++ = ⎢⎥ ∂∂ ⎣⎦ ∑∑ ∫∫ . Since { } k k u strongly converges to u on m Ω , it follows from the above equality that ( ) 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. TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 25 [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). . Trang 17 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). 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,. have the conclusion that the sequence { } m m u converges to u in k X . Thus, k T is of class ( ) S + in k X . Next we calculate the topological degree of the operator k T. By condition