On Maximum Principle and Existence of Solutions for Nonlinear Cooperative Systems on R N

Transcription

1 ISS: On Maximum Princile and Existence of Solutions for onlinear Cooerative Systems on R M.Kasturi Associate Professor, Deartment of Mathematics, P.K.R. Arts College for Women, Gobi, Tamilnadu. ABSTRACT: In this work we give necessary and sufficient conditions for having a maximum rincile for cooerative ellitic systems involving - Lalacian oerator on the whole R. This rincile is then to yield solvability for the considered cooerative ellitic system by an aroximation method. KEYWORDS: Maximum Princile, - Lalacian Oerator, Ellitic System, Eigen Function, Aroximation method.. ITRODUCTIO This work is concerned with the general nonlinear cooerative ellitic system - u = a m (x) u 2 u + bm (x) h (u, v) + f in R - u = d n (x) v 2 v + cn (x) k (u, v) + g in R (.) u(x) 0, v (x) 0 as x + Here u : = div u 2 u, +, is the P- Lalacian oerator. The arameters a, b, c, d are nonnegative real arameter. The functions h, k: R 2 R are continuous and have some roerties like the weight functions m, m, n, n which will be secified later. The aim of this work is to construct a Maximum Princile with inverse ositivity assumtions which means that if f, g are nonnegative functions almost everywhere in R, then any solution (u, v) of (.) obey u 0; v 0 a.e in R. It is well known that the maximum rincile lays an imortant role in the theory on nonlinear euations. For instance it is used to access existence results of solutions for linear and nonlinear differential euations. [-5]. Many works have been devoted to the study of linear and nonlinear ellitic system on a bounded or an unbounded domain of R [-8]. Most of the work deals with Maximum Princile for a certain class of functions h and k. This work deals with a more general class of functions h, k. For secific interest for our uroses is the work in [7] where a study of roblems such as (.) was carried out in case of R in the resence of the weights m, m, n, n with the articular case h (s,t) = s t t and k (s, t)= s t s, and are some nonnegative real arameter in R. Clearly, our work extends the work [7] first by considering a roblem with weights and next by dealing with a more general class of function h, k in the of whole of R. For instance this result can aly for the case.. sin s α arctan t β t for t 0, sr h (s, t) = s t t for t 0, s R sin s α s arctan t β for s 0, t R k (s, t) = s s t for s 0, t R which is not taking into account in [7]. The remainder of the work is organized as follows: In Preliminary Section 2 we secify the reuired assumtions on the data of our considered roblem and we briefly give some known results relative to the rincial ositive eigenvalue Coyright to IJARSET

2 ISS: of the - Lalacian oerator. In section 3, the Maximum rincile for (.) is given and is shown to be roven full enough to yield existence results of solution for (.) in Section 4 by using a aroximation method. Throughout this work assume that,, n and II. PRELIMIARIES (B), 0; b, c 0 and α+ + + = ; (B2) f 0, f L ( ) (R ) ; g 0, gl ( ) (R ) with + = + =; (B3) m, m, n, n are smooth weights such that m, n0 ml loc (R ) L / (R ), nl loc (R ) L / (R ), and 0 m, n m (+)/ n (+) /. Here * =, P * = denote the critical Sobolev exonents of and resectively; is the Holder conjugate of. (B4) The functions h and k satisfy the sign conditions: t.h (s, t) 0, s.k (s, t) 0 for (s,t) R 2 and there exist 0 such that h (s, -t) - h (s, t) for t 0, s R h (s, t) = α+β+2 s α t t k (-s, t) -k (s, t) for t 0, s R for s 0, t R k (s, t) = ++2- s α s t for s 0, t R We denote by W, o (R ) (with s ) the comletion of C o (R ) with resect to the norm u,p W 0 R = u P R.,P wo (R ) is a reflexive Banach sace and it can be shown [8] that w O, (R ) = { ul P* (R ) : u (L P (R )) }. Here and henceforth the Lebesgue norm in L P (R ) will be denoted by. and the usual norm of w O, (R ) by.. The ositive and negative art of a function u are defined resective as u + = max {u, 0} andu - : = max {-u, 0}. Eualities (and ineualities) between two functions must be understood a.e. (R ) Consider the eigenvalue roblem with weight g. For a given gl /P (R - ) L (R - ),g (x) a.e in R - it was known that the eigenvalue roblem. - u = g (x) u 2 u in R - u (x) 0, as x + in R - (2.) admits an uniue ositive first eigen value (g, ) with a nonnegative eigenfunction. Moreover, this eigenvalue is isolated, simle and as a conseuence of its variational characterization one has (g, ) g R (x) u R u u W, 0 (R ). ow we denote by (resectively ) the ositive eigenfunction associated with (n, ) (resectively (n, )) normalized by R m (x) = (res n R (x) = ).The functions and belong to C,α (R - ) and by the weak maximum rincile, 0 and 0 on R - where v is the unit exterior normal. v v III. MAXIMUM PRICIPLE We assume that, and also that hyothesis (B 3 ) is satisfied. We begin by consider the roblem - u = µm (x) u 2 u+ h (x) in R - u (x) 0, as x + (3.) The following result was roved in [9, 0] Preosition 2. () Let h L ( ) (R - ). if (m, ) then (3.) admits a solution in D, (R ) Coyright to IJARSET

3 ISS: (2) let hl ( ) (R - ) with h 0 a.e in R - and h 0. (a) if [0, (m, ) [, then any solution u of (3.) is ositive in R -. (b) if = (m, ), then (3.) has no solution. (c) if (m, ), then (3.) has no ositive solution. Using [2,3] one also has the following regularity result. Preosition 2.2. For all r 0, any solution u of [2.] belongs to C,γ (B r ), where = (r) ]0,[ and B r is the ball of radius r centred at the origin. Let a (r) : = infk (x) B r and a 2 (r) : = suk 2(x) B r (3.2) where k (x) : = n (x) n(x) β + (x) α + β + + (x) ; k 2 (x) : = m (x) [ (x) α + β + ] m (x) (x). We denote by a = lim r + a (r) and a 2 = lim r + a 2 (r). Let set = a a 2. (3.3) The following ineualities can easily be roved = a (r) a 2 (r). for all r 0 and 0 (3.4) We now turn to our first main result A Maximum Princile holds for the system (.) if f 0 and g 0 imlies u 0 and v0 a.e in R. By a solution (u, v) of (.), we mean a weak solution i.e., (u, v) W 0, (R - ) x W 0, (R - ) such that u 2 R u. w = R [am (x) u 2 uw+ bm (x) h (u, v) w + fw ] (3.5) R 2 v. z = R [dn (x) v 2 vz+ cn (x) k (u, v) z + gz ] for all (w, z) W 0, (R - ) x W 0, (R - ). ote that by assumtions (B) (B4), the integrals in (3.5) are well defined Regularity results from [2,3], weak solution (u,v) belong to c (R ) x c (R ). It is also know, that a weak of (.) decays to zero and infinity. [9, 4] ow ready to state the validity of the Maximum Princile for (.) Theorem 3. Assume (B) (B4). Then the Maximum Princile holds for (.) if (C ) (m, ) a, (C 2 ) (n, ) d, (C 3 ) (m, ) a) (α+)/ (n, ) d) (+)/ b (α+)/ c (+)/ Conversely if the Maximum Princile holds, then Conditions (C ) (C 4 ) are satisfied, where (C 4 ) ( (m, ) a) (α+)/ (n, ) d) (+)/ b (α+)/ c (+)/ Proof: The roof is arty adated from [, 6] Coyright to IJARSET 347

4 ISS: ecessity Part: Assume that the Maximum Princile holds for system (.) If (m, ) a then the functions f : (a- (m, ) m (x) - and g : = 0 are nonnegative, however (-, 0) satisfies (.), which contradicts the Maximum Princile. Similarly, if (n, ) d then f : = 0 and g : = (d- (n, )) n (x) are nonnegative functions and (0, -) satisfies (.), which is a contradiction with the Maximum Princile. ow, assume that (m, ) a, (n, ) d, and that (C4) does not hold; that is, (C4 ) (m, ) a) (+)/ (n, ) d) (+)/ b (+)/ C (+)/ Set A = m, a b α+ / B = n, a c + / Then by (C4 ) AB which imlies A 2, where infk (x), B = R (3.6) 2 = suk 2 (x) R Hence these exists R + such that A a 2 (/B) a. Let c, c 2 be two ositive real numbers such that = C 2 C + + Using (3.6), (B) and the above exression of we have [ (m, ) -a] m (x) [c (x)] ++2 bm (x) [c (x)] [c 2 (x)] + for all x R and [ (n, ) -d] n (x) [c 2 (x) ] ++2 cn (x) [c (x)] + [c 2 (x)] for all x R Furthermore, using the ineualities in (B4), we obtain -[ (m, ) a] m (x) [c (x)] bm (x) h(-c, -c 2 ) 0 -[ (n, ) d] n (x) [c 2 (x) ] - cn (x) k(-c, -c 2 ) 0 for all x R and for all x R Hence 0-[ (m, ) a] m (x) [c (x)] - -bm (x) h(-c, -c 2 ) = f, for all x R. 0- [ (n, ) d] n (x) [c 2 (x)] - - cn (x) k(-c, -c 2 )= g, for all x R. are nonnegative functions and (-c,-c 2 ) is a solution of (.).This is a contradiction with the Maximum Princile. Sufficiency Part : Assume that the conditions (C) (C3) are satisfied. So for f 0, g 0, suose that there exists a solution (u, v) of system (.). Multilying the first euation in (.) by u and the second on by v and integrating over R we have, Coyright to IJARSET

5 ISS: R u = a R m (x) u b R m (x) h (u, v) u - - R fu R v = d R n (x) v c R n (x) k (u, v) v - - R gv Then, using the sign conditions in (B4), we obtain R u a R m (x) u b R m (x) h (u, -v ) u R v d R n (x) v c R n (x) k (-u v ) v The following results are derived by using the sign conditions in (B4), h (u,-v - ) u = ++2 (u ) + (v ) +, and k (-u, v) v = ++2 (u ) -+ (v ) + hence R u a R m u + b ++2 R m (x)( u ) + (v ) + R v d R n v + c ++2 R n (x)( u ) + (v ) + Combining the variational characterization of (m, ) and (n, ) with the Holder ineuality and assumtion (B3), we have (m, ) -a ) m R (x) u b ++2 m R x u (+)/ n R x v (+)/ which imlies ( (n, ) -d ) n R (x) v c ++2 m R x u (+)/ n R x v (+)/ m R x u (+)/ [( (m, ) -a) m R x u (+)/ -b ++2 n R x v (+)/ ] 0 n R x v (+)/ [( (n, ) -d)] n R x v (+)/ -c ++2 m R x u (+)/ 0 (3.7) Let us show that u = v = 0 If m R (x) u =0 or n R (x) v = 0 then, using the fact that m o, n 0, and (3.7) we obtain u = v = 0, which imlies that the Maximum Princile holds. If R m(x) u - 0 and R n(x) v - 0, then we have (m, ) -a ) (n, ) -d) m R x u (+)/ b ++2 n R x v (+)/ n R x v (+)/ c ++2 m R x u (+)/ which imlies ( (m, ) a ) (+)/ m R x u + β + -b (+)/ ++2 (+)/ n R x v + β + Coyright to IJARSET

6 ISS: ( (n, ) d ) (+)/ n R x v α + β + -c (+)/ ++2 (+)/ m R x u + β + Multilying the two ineualities above and using the fact that (++2-) α+ + (++2-) + = (++2) ( α+ + + ) - (+) - (+) = 0 (3.8) α + one has ( (m,) a ) ( (n, ) d ) x m R x u n R x v α + + α + + b c m R x u n R x v α + + α + + α + + and then ( (m, ) a ) ( (n, ) d ) b c x m R x u n R x v α + β + 0 Since (C ) (C 3 ) are satisfied and m, n 0 the ineuality above is not ossible. Conseuently u = v = 0 and the Maximum Princile holds. When = and m = n, the number is eual to and as a conseuence of theorem 3., we have the following result. Corollary 3.2 Consider the cooerative system (.) with = and m =n. Then the Maximum Princile holds if and only if (C ) (C 3 ) are satisfied. IV. EXISTECE OF SOLUTIOS In this section we rove that, under some conditions, system (.) admits atleast one solution. Theorem 4. Assume (B ), (B 2 ), (C ), (C 2 ), (C 3 ) are satisfied. Then for f L P (R ) and g L (R ), system (.) admits atleast one solution in W 0, (R ) W 0, (R ). The roof will be given in several stes and is artly adated from [,6,5]. To rove this theorem it reuires the lemmas state below. We chooser r 0 such that a + r 0 and d + r 0 Hence (.) reads as follows - u + rm (x) u 2 u =(a+r) m (x) u 2 u +bn (x) h (u, v) + f in R - v + rn (x) v 2 v = cn k (u, v) +(d+r)n (x) v 2 v +g in R (4.) u (x) 0, v (x) 0 For 0, now consider the system where Lemma 4.2 as x - u + rm (x) u 2 u = h ( x, u, v ) + f in R - v + rn(x) v 2 u = h ( x, u, v ) + g in R u v 0 as x (x) (4.2) h (x, s, t) = (a +r)m(x) s 2 s ( + / s ) - + bm (x) h (s, t) ( + h (s, t) ) -, k (x, s, t) = (d+r) n (x) t 2 t ( + / t ) - + cn (x)k (s, t) ( + k (s, t) ) - Coyright to IJARSET

7 ISS: Under the hyothesis of theorem (3.) System (4.2) has a solution in W 0, (R ) W 0, (R ). Proof : Let 0 be fixed.construction of sub solution and suer solution for system - u + rm(x) u 2 v = h ( x, u, v) + f in R - v + rn(x) v 2 v = k ( x, u, v) + g in R (4.3) u (x) 0, v (x) 0 as x From (B3), the functions h and are bounded ; that is, there exists a ositive constant M such that k h ( x, u, v) M, K ( x, u, v) M (u, v) W 0, (R ) W 0, (R ). Let u 0 W 0, (R ) (resectively v 0 W 0, (R ) be a solution of - u 0 + rm (x) u 0 2 u 0 = M + f (resectively - v 0 + rn (x) v 0 2 v 0 = M + g) It was known that u 0 u 0, v 0, v 0 are exists, moreover we have - u 0 + rm (x) u 0 2 u 0 - h ( x, u 0, v) - f 0 v [v 0, v 0 ] - u 0 + rm (x) u 0 2 u 0 - h ( x, u0, v) - f 0 v [v 0, v 0 ] - v 0 + rn (x) v 0 2 v 0 - k ( x, u, v 0 ) - g 0 u [u 0, u 0 ] - v 0 + rn (x) v 0 2 v 0 - k ( x, u, v 0 ) - g 0 u [u 0, u 0 ] So (u 0, u 0 ) and (v 0, v 0 ) are sub suer solutions of (4.3) Let K = [u 0, u 0 ] [v 0, v 0 ] and let T : (u, v) (w, z) the oerator such that - w + rm (x) w 2 w= h ( x, u, v) + f in R - z + rn (x) z 2 z= k ( x, u, v) + g in R (4.4) w (x) 0, z (x) 0 as x + Let us rove that T (K) K. If (u, v) K, then (- w- 0 ) + rm (x) w 2 w =[ h ( x, u, v) - M] ) (4.5) Taking (w - 0 ) + as test function in (4.5), we have R ( w 2 w )(w - 0 ) + + r R m (x)( w 2 w ) (w - 0 ) + = R [h (x, u, v) M)] (w - 0 ) + 0. Since the weight m is ositive, by the monotonicity of the functions s s 2 and that of the Lalacian, we deduce that the last integral eual zero and the (w - 0 ) + = 0. That is w 0.Similarly we obtain 0 w by taking (w - 0 ) - as test function in (4, 5). So we have 0 w 0 and o z 0 and the ste is comlete. To show that T is comletely continuous we need the following lemma. Lemma 4.3 If (u n, v n ) (u, v) in L (R ) L (R ) as n then u n 2 u n u n 2 u () X n = m (x) + / u n Converges to X = m (x) + / u in L (R ) as n. where = h(u Y n = m (x) n v n ) h (u,v) Converges to Y = m (x) in L (R ) as n. +h (u n v n ) +h u,v (2) + Coyright to IJARSET

8 ISS: Proof : Since u n u in L (R ), there exist a subseuence still denoted (u n ) such that u n (x) u (x) a.e.on R u n (x) (x) a.e. on R with L (R ) (4.6) Let X n = m (x) u n 2 u + / u Then X n (x) X (x) = m (x) u (x) 2 u(x) + / u(x) a.e. on R, X n m u n m in L (R ) Thus, from Lebesue s dominated convergence theorem one has X n X = m (x) u 2 u + / u in L (R ) as n. So () is roved. Moreover, since v n v in L (R ) there exists a subseuences still denoted (v n ) such that v n (x) v (x) a.e on (R ), v n (x) (x) a.e on R with L (R ) (4.7) Using (B4), one has Y n m (u n, v n ) ++2 m + in L (R ), since α ++ Let Y n = m (x) h (u n,v n ) +h(u n,v n ) Then Y n (x) Y(x)= m (x) h (u x,v (x)) +h(u x,v (x)) So, we can aly the Lebesgue s dominated convergence theorem and then we obtain Y n (x) Y(x)= m (x) h (u x,v (x)) +h(u x,v (x)) Remark 4.4 We can similarly rove that, as n, in L (R ) as n n (x) v n 2 v n (+ / v n ) n (x) v 2 v (+ / v ) in L R, a.e. in R, as n n (x) k (u n, v n ) (+ k (u n, v n ) n (x) k (u, v) (+ k (u, v) in L R To comlete the continuity of T. Let us consider a seuence (u n, v n ) such that (u n, v n ) (u, v) in L (R ), L (R ), as. We will rove that (w n, z n ) = T (u n, v n ) (w, z) = T (u, v). ote that (w n, z n ) = T (u n, v n ) if and only if ( - w n + rm (x) w n 2 w n ) - ( - w + rm (x) w 2 w)= h ( x, u n, v n ) - (x, u,v) h = = (a+r) m x u n 2 u n + u n m x u 2 un + u + bm x h (u n, v n ) +h (u n, v n ) h (u,v) +h (u, v) (4.8) = (a+r) (X n - X) + b (Y n - Y) Multilying by (w n - w) and integrating over R one has ( R w n 2 w n - w 2 w) (w n - w) + r m R (x)(w n 2 w n - w n 2 w). (w n - w) (a +r) ( m R X n - X ) ( R w n w ) + b ( R Y n - Y ) Combining lemma 4.3 and the ineuality = (a+r) ( R X n - X) (w n - w)+ b ( R Y n - Y) (w n - w) ( R w n w ) x-y c ( x 2 x y 2 y (x y) s/2 x + y s/2 (4.9) where x, y R, c=c () 0 and s=2 if 2, s = if 2, We can conclude that w n w in W, 0 (R ) when n.similarly we show that z n z in W, 0 (R ) as n and then, the continuity of T is roved. Comactness of the oerator T. Suose (u n, v n ) a bounded seuence in W 0, (R ) W 0, (R ) and let (w n, z n ) = T(u n, v n ). Coyright to IJARSET

9 ISS: Multilying the first euality in the definition of T by w n and integrating by arts on R, we notice the boundness of w n in W, 0 (R ) and then we use the comact imbedding of W, 0 (R ) in L R, to conclude. The same argument is valid with (z n ) in in L, (R ). Thus T is comletely continuous. Since the set K is convex, bounded and closed in L (R ) L (R ), the Schauder s fixed oint theorem, yields existence of a fixed oint for T and accordingly the existence of solution of system (4.2).Hence the lemma 4.2. Proof of Theorm 4.: The roof will be given in three stes. Ste : First to rove that (u, v ) is bounded in W, 0 (R ) W, 0 (R ). indeed assume that u or v as 0. Let t max {u ; v }, w = u t We have w and z with either w or z =. Dividing the first euation in (4.2) by (t ) / we obtain, z = v t - w + rm (x) w 2 w = (a+r) m (x) w 2 w ( + / u ) - + t / bm (x) h (t / w, t / z ) (+ h (u, v ) ) + t / f. Similarly dividing the second euation in (4.2) by (t ) / we obtain - z + rn (x) z 2 w = (d+r) n (x) w w (+ / u + ) / +t cn (x) k (t / w, t / z ) (+ k(u, v )) + t / g. Testing the first euation in the above system by w and using (B4), we obtain R w a m R (x) w + b ++2 m R (x) which, by the Holder ineuality, imlies + w + n (x) β+ / z + + t / R f w. R w a m R w + b ++2 ( m R w ) +/ ( n R w ) +/ / + t f ( ) z Using the variational characterization of (m, ) and the imbedding of W, 0 (R ) in L (R ). one has w a w + b ++2 w m, + Z + α + + (m,) (n,) where c (, ) is the imbedding constant. So, one gets ( m, )-a) ( w ) and accordingly ( m, )-a) +)/ m, + b α +β +2 ( z ) +)/ m, α + n, β + (lim su w + + ) m, In a similar way, it can be obtain )β + ( n, )-d α + (lim su z + + ) + c n, + α + b + c (, ) t ) +(t ) ( R f ) m, ( R + α +β +2 (α ) (lim su Z + ) α n, α + (α +β +2 ) + + (lim su w ) 2 n, + m, + + f ( ) z, Multilying term by term the exressions in (4.) and (4.2), and using (3.8) we obtain α + [( (m, ) a) ( (n, ) d) (+)/ - b α+ c (+)/ ] (lim su w + ) m, + w ) α (4.0) + + (lim su z ) + (4.) + (4.2) n, β + / 0. Since conditions (C ) (C 3 ) hold, one has Lim su w = Lim su z =0 This yields a contradiction since w = or w =, and conseuently (u, v ) is bounded in W 0, () W 0, (). Ste 2. ( u ; v ) converges strongly in W, 0 (R ) W, 0 (R ). when aroaches 0. It is obvious due to the boundness of (u, v ) in W, 0 (R ) W, 0 (R )). Ste 3. Let us rove that (u, v ) converges strongly in W 0, () W 0, () when aroaches 0. Since (u, v ) is bounded in W 0, (R ) W 0, (R ) we can extract a subseuence still denoted. (u, v ) which converges weakly to (u 0, v 0 ) in L (m, R ) L (n, R ) and strongly in L (R ) L (R ) when 0. Coyright to IJARSET

10 ISS: As u u 0 in L (R ), v v 0 in L (R ) when 0 then there exists a function L (R ), L (R ) such that, u (x) v 0 (x) a.e as 0 and u in L (R ). v (x) v 0 (x) a.e as 0 and v in L ( (R ). Hence we have u 2 u (x) ( + / u ) u in L (R ) v 2 v (x) ( + / u ) v in L (R ) Since ( u ) 0, ( / v ) 0 a.e in R when 0, one can deduce that, u (x) 2 u (x) ( + / u (x) ) u 0 (x) 2 u 0 (x), v (x) 2 u (x) ( + / v (x) ) v 0 (x) 2 v 0 (x), a.e. in R as 0. Alying the dominated convergence theorem we obtain u 2 u ( + / u ) u 0 2 u o v 2 v ( + / u ) v 0 2 v o in LP (R ) as 0. h(u Similarly we have, v, ) +h(u, v, ) ++2 α + in L P (R ) since α ++ =, k(u, v, ) +k(u, v, ) ++2 α+ + in L (R ) since α+ + = and h(u x,v (x)) h (u 0 (x), v 0 (x)) a.e as 0, +h(u x,v, x,) k(u x,v (x)) +k(u x,v, x,) k (u 0 (x), v 0 (x)) a.e as 0, Again using the dominated convergence theorem we have h(u, v, ) +h(u, v, ) h (u 0, v 0 ) in L (R ) a.e as 0, k(u, v, ) +k(u, v, ) k (u 0, v 0 ) in L (R ) a.e as 0. ow, we conclude the strong convergence of (u, v ) in W 0, (R ) W 0, (R ) by alying (4.9). Finally, using a classical result in nonlinear analysis, we obtain - u o + r m (x) u 0 2 u 0 = (a+r) m(x) u 0 2 u 0 = bm m(x) h (u 0, v 0 ) + f in R - v o + r n (x) v 0 2 u 0 = (d+r) n(x) v 0 2 v 0 = cn (x) k(u 0, v 0 ) +g in R which can be written again as u 0 (x) 0, v 0 (x) 0 as x + - u o +am (x) u 0 2 u 0 + bm m(x) h (u 0 + v 0 ) + f in R - v o = dn (x) v 0 2 u 0 + cn k (x) h (u 0 + v 0 ) + g in R u 0 (x) 0, v 0 (x) 0, as x + This comletes the roof REFERECES [] Bouchekif M., Serag H., and De Th elin F., On maximum rincile and existence of solutions for some nonlinear ellitic systems, Rev. Mat. Al., 6-6. [2] De Figueiredo D.G.and Mitidieri E.,Maximum Princiles for Linear Ellitic Systems, Quaterno Matematico, 77, Di.Sc.Mat.,Univ Trieste,988. [3] De Figueiredo D.G. and Mitidieri E., Maximum rinciles for cooerative ellitic systems, Comtes Rendus Acad. Sc. Paris, 30 (990), [4] De Figueiredo D.G. and Mitidieri E., A maximum rincile for an ellitic system and alications to semilinear roblems, S. I. A. M. J. Math. Anal., 7 (986), [5] Fleckinger J., Hernandez J., and De Th elin F., Princie du maximum our un syst`eme ellitiue non lin eaire, Comtes Rendus Acad. Sc. Paris, 34 (992), [6] Leadi L. and Marcos A., Maximum rincile and existence results for ellitic systems on R, Elec. J. Diff. Eua., 200(60) (200), -3. [7] Leadi. L.and Marcos A., Maximum Princilefor on linear Co Oerative ellitic System on R, J. Part. Diff E., 24 (20), [8] Kozono H. and Sohr H., ew a riori estimates for the stokes euation in exterior domains, Indiana Univ. Math. J., 40 (99), -27. [9] Fleckinger J., Man asevich R. F., Stavrakakis. M., and De Th elin F., Princial eigenvalues for some uasilinear ellitic euations on R, Advances in Diff. Eua, 2(6) (997), [0] Fleckinger J., Gossez J. P., and De Th elin F., Antimaximum rincile in R: Local versus global, J. Diff. Eua., 96 (2004), [] Gossez J.-P. and Leadi L., Asymmetric ellitic roblems in R, Elec. J. Diff. Eua., 4 (2006), [2] Serrin J., Local behavior of solutions of uasilinear euations, Acta Math., (964), [3] Tolksdorf P., Regularity for a more general class of uasilinear ellitic euations, J. Diff. Euat., 5 (984), Coyright to IJARSET

11 ISS: [4] Drabek P., Kufner A., and icolosi F., Quasilinear Ellitic Euations with Degenerations and Singularities, De Gruyter, Walter, Inc., 997. [5] Boccardo L., Fleckinger J., and De Th elin F., Existence of solutions for some nonlinear cooerative Ssystems, Diff. and Int. Euations, 7 (994), Coyright to IJARSET