MAXIMUM PRINCIPLE AND EXISTENCE RESULTS FOR ELLIPTIC SYSTEMS ON R N. 1. Introduction This work is mainly concerned with the elliptic system

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1 Electronic Journal of Differential Euations, Vol. 2010(2010), No. 60, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu MAXIMUM PRINCIPLE AND EXISTENCE RESULTS FOR ELLIPTIC SYSTEMS ON LIAMIDI LEADI, ABOUBACAR MARCOS 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. This rincile is then used to yield solvability for the cooerative ellitic systems by an aroximation method. 1. Introduction This work is mainly concerned with the ellitic system u = am(x) u 2 u + bm 1 (x) v β v + f in, v = cn 1 (x) u α u + dn(x) v 2 v + g in, u(x) 0, v(x) 0 as x +. (1.1) Here u := div( u 2 u), 1 < < +, is the so-called -Lalacian oerator; a, b, c, d, α and β are reals arameters; f, g, m, n, m 1 and n 1 are weights whose roerties will be secified later. We are concerned with the existence of ositive solutions and with the following form of maximum rincile: If f, g 0 in then u, v 0 in for any solution (u, v) of (1.1). It is well known that maximum rincile lays an imortant role in the theory on nonlinear euations. For instance, it is used to access existence results and ualitative roerties of solutions for linear and nonlinear differential euations, (see for instance [14] and [18] for a survey). Many works have been devoted to the study of linear and nonlinear ellitic systems either on a bounded domain or an unbounded domain of (in articular the whole R n ) (cf.[3, 5, 6, 7, 8, 9, 19]). In [12, 13] for the linear case (i.e = = 2), it was resented necessary and sufficient conditions for having maximum rincile and existence of ositive solutions. These results have been later extended in [9] to 2000 Mathematics Subject Classification. 35B50, 35J20, 35J55. Key words and hrases. Princial and nonrincial eigenvalues; ellitic systems; -Lalacian oerator; aroximation method. c 2010 Texas State University - San Marcos. Submitted June 5, Published May 5,

2 2 L. LEADI, A. MARCOS EJDE-2010/60 the nonlinear system n u i = a ij u j 2 u j + f i in Ω j=1 u i = 0 on Ω, i = 1, 2,... n where Ω is a bounded domain of. For secific interest for our uroses is the work in [19] where a study of roblems such as (1.1) was carried out in the case of in the resence of some weight functions. In our work we consider roblem (1.1) with coefficients b, c > 0, and the weight functions m(x), n(x), m 1 (x), n 1 (x) ositive. Here m belongs to L N/ ( ) L loc (RN ) and n belongs to L N/ ( ) L loc (RN ). Then we state necessary and sufficient conditions for a maximum rincile to hold. Moreover our techniue can be develoed to get a related result for the following class of cooerative systems u = am(x) u 2 u + bm 1 (x) u α v β v + f in, v = cn 1 (x) v β u α u + dn(x) v 2 v + g in, u(x) 0, v(x) 0 as x + (1.2) where the coefficients a, b, c, d, and the weights m(x), n(x), m 1 (x), n 1 (x) are as above. When a = b = c = d = 1, roblem (1.2) is relaxed to the articular case of system considered in [19] where the necessary condition for the maximum rincile to hold given by the authors is deend on x. The arguments develoed in this aer enable us to obtain a non deendance on x necessary condition. The remainder of the aer is organized as follows: In Section 3, the maximum rincile for (1.1) is given and is shown to be roven full enough to yield existence results of solutions for (1.1) in Section 4. In section 5, we briefly give a version of our result for the cooerative systems (1.2). In the reliminary Section 2, we collect some known results relative to the rincial ositive eigenvalue and to various Sobolev imbeddings. 2. Preliminaries Throughout this work, we will assume that 1 <, < N and (H1) m, n > 0; m L loc (RN ) L N/ ( ) and n L loc (RN ) L N/ ( ) (H2) 0 < m 1 (x) [m(x)] 1 [n(x)] and 0 < n 1 (x) [n(x)] 1 α+1 [m(x)] a.e. in (H3) f 0 and f L ( ) ( ); g 0 and g L ( ) ( ) (H4) b, c 0; α, β 0; α = 1 and + 1 = 1 Here = N N, = N N denote the critical Sobolev exonent of and resectively; is the Hölder conjugate of. It is clear that 1 = and 1 = α+1. We denote by D 1,s ( ) (with 1 < s < N) the comletion of C0 ( ) with resect to the norm ( ) 1/s. u D 1,s ( ) = u s It can be shown that (cf [16]) D 1,s ( ) = {u L s ( ) : u (L s ( )) N }

3 EJDE-2010/60 MAXIMUM PRINCIPLE AND EXISTENCE RESULTS 3 and for any ositive weight g L loc (RN ) L N/s ( ) the following embeddings hold (cf.[10, 11, 15]) D 1,s ( ) L s ( ) and D 1,s ( ) L s (g, ) where L s (g, ) is the L s sace on with the weight g (cf. [11]). By solution (u, v) of (1.1) (or related euations), we mean a weak solution; i.e., (u, v) D 1, ( ) D 1, ( ) with u 2 u. w = [am(x) u 2 uw + bm 1 (x) v β vw + fw] R N (2.1) v 2 v. z = [cn 1 (x) u α uz + dn(x) v 2 vz + gz] for all (w, z) D 1, ( ) D 1, ( ). Note that by the above embeddings, every integral in (2.1) is well-defined. Regularity results from [20, 21] on general uasilinear euations imly that such a weak solution (u, v) belong to C 1 ( ) C 1 ( ). It is also known that a weak solution of (1.1) decays to zero at infinity (cf. [4, 10]). To conclude this introduction, let us briefly recall some roerties of the sectrum of with weight to be used later (cf. [1, 11]). We denote by λ 1 (m, ) := min { u : u D 1, ( ) and m u = 1 } (2.2) the uniue rincial eigenvalue of u = λm(x) u 2 u in (2.3) u(x) 0 as x + ; u > 0 in and by ϕ 1 (m) = ϕ 1 (m, ) D 1, ( ) C 1 ( ) the associated ositive eigenvalue such that m ϕ 1 (m) = 1. It is well known that λ 1 (m, ) is simle and isolated. Here and henceforth, we will denote by Φ = ϕ 1 (m, ) (resectively by Ψ = ϕ 1 (n, )) the ositive eigenfunction associated to λ 1 (m, ) (resectively λ 1 (n, )) and normalized by mφ(x) = nψ(x) = 1 (2.4) 3. Maximum rincile We assume that 1 <, < N and that hyothesis (H1), (H2), (H3) and (H4) are satisfied. We begin by consider the roblem u = µm(x) u 2 u + h(x) u(x) 0 as x + The following results were roved in [10, 11] in (3.1) Proosition 3.1. (1) Let h L ( ) ( ) and assume that (H1) is satisfied. If µ < λ 1 (m, ) then (3.1) admits a solution in D 1, ( ). (2) Let h L ( ) ( ) with h 0 a.e. in and h 0. (a) If µ [0, λ 1 (m, )[, then any solution u of (3.1) is ositive in. (b) If µ = λ 1 (m, ) then (3.1) has no solution (c) If µ > λ 1 (m, ) then (3.1) has no ositive solution. Using [20, 21], one also has a regularity result.

4 4 L. LEADI, A. MARCOS EJDE-2010/60 Proosition 3.2. For all r > 0, any solution (u, v) of (1.1)) belongs to C 1,γ (B r ) C 1,γ (B r ), where γ = γ(r) ]0, 1[ and B r is the ball of radius r centered at the origin. Let where a 1 (r) := inf B r k 1 (x), k 1 (x) := [ n 1 (x) n(x) k 2 (x) := [ m(x) m 1 (x) a 2 (r) := su B r k 2 (x), (3.2) ] [ Φ(x) ] α+1, ] α+1 Ψ(x) [ Φ(x) ] α+1 Ψ(x). We denote a 1 = lim r + a 1 (r) and a 2 = lim r + a 2 (r). Let One can easily rove that Θ a 1(r) a 2 (r) Θ = a 1 a 2. (3.3) for all r > 0 and 0 Θ 1 (3.4) We say that (1.1) satisfies the maximum rincile (in short (MP)) if for f, g 0 a.e in, any solution (u, v) of (1.1) is such that u > 0, v > 0 a.e. in. We now turn to our first main result, i.e., the validity of the (MP) which is stated as follows Theorem 3.3. Assume that hyothesis (H1) (H) are satisfied. Then the (MP) holds for (1.1) if (C1) λ 1 (m, ) > a (C2) λ 1 (n, ) > d (C3) [λ 1 (m, ) a] α+1 [λ1 (n, ) d] > b α+1 c Conversely, if the (MP) holds, then (C1), (C2) and (C4) are satisfied, where (C4) [λ 1 (m, ) a] α+1 [λ1 (n, ) d] > Θb α+1 c. Corollary 3.4. If = and m n a.e. in, then the (MP) holds for (1.1) if only if (C1), (C2) and (C4) are satisfied Proof of Theorem 3.3. The condition is necessary. The roof of (C1) or (C2) is standard (cf. for instance [2, 3, 19]). We give here the sketch of this roof. If λ 1 (m, ) a, then the functions f := [a λ 1 (m, )]mφ 1 and g := cn 1 Φ α+1 are nonnegative and ( Φ, 0) is a solution of (1.1), which contradicts the (MP). Similarly, if λ 1 (n, ) d, then the functions f := bm 1 Ψ and g := [d λ 1 (n, )]nψ 1 are nonnegative and (0, Ψ) is a solution of (1.1), a contradiction. The roof of (C4) can be adated from [19] as follow. We assume that λ 1 (m, ) > a and λ 1 (n, ) > d. If one of the coefficients Θ, b or c vanishes, then (C4) is satisfied. We will then assume that Θ 0, b 0, c 0 and that (C4) does not hold, i.e. Set A = ( λ 1(m,) a) α+1 b [λ 1 (m, ) a] α+1 [λ1 (n, ) d] Θb α+1 c (3.5) and B = ( λ 1(n,) d) c. Then, by (3.5), one has AB Θ, which clear imlies that Aa 2 1 B a 1. One deduces that there exists ξ R + such that Aa 2 ξ 1 B a 1.

5 EJDE-2010/60 MAXIMUM PRINCIPLE AND EXISTENCE RESULTS 5 Since the function a 1 (r) (resectively a 2 (r)) is decreasing (resectively increasing) on R +, one has Aa 2 (r) Aa 2 ξ 1 B a 1 1 B a 1(r), for all r > 0. But for any x, there exists r > 0 such that Conseuently we set for all x, i.e., Ak 2 (x) Aa 2 (r) and 1 B a 1(r) 1 B k 1(x). Ak 2 (x) Aa 2 (r) ξ 1 B a 1(r) 1 B k 1(x) Ak 2 (x) ξ x (3.6) B k 1 (x) 1 ξ x. (3.7) Next let we set ξ = ( c 1 ) α+1 c, where c1 and c 2 are ositive constants. 2 From (3.6) and (H4), one easily gets, [λ 1 (m, ) a]m(x)[c 2 Φ(x)] 1 + bm 1 (x)[c 1 Ψ(x)] 0 for all x. Similarly, using (3.7) and (H4), one has Hence [λ 1 (n, ) d]n(x)[c 1 Ψ(x)] 1 + cn 1 (x)[c 2 Φ(x)] α+1 0 for all x. f := [λ 1 (m, ) a]m(x)[c 2 Φ(x)] 1 + bm 1 (x)[c 1 Ψ(x)] 0 for all x and g := [λ 1 (n, ) d]n(x)[c 1 Ψ(x)] 1 + cn 1 (x)[c 2 Φ(x)] α+1 0 for all x are nonnegative functions and ( c 2 Φ, c 1 Ψ) is a solution of (1.1). This is a contradiction with the (MP). The condition is sufficient. A detailed roof of this art can be found in [3, 20]. We give a sketch here. Assume that the conditions (C1), (C2) and (C3) are satisfied. Let (u, v) be a solution of (1.1) for f, g 0. Moreover, suose that u 0 and v 0 and taking those functions as test function in (1.1), we find by Hölder ineuality that [(λ 1 (m, ) a) α+1 (λ1 (n, ) d) b α+1 c ] [( )( m u n v )] α+1 0, which contradicts assumtion (C4). By alying regularity results of [20, 21] and the maximum rincile of [22], one has in fact u > 0 and v > 0 a.e in.

6 6 L. LEADI, A. MARCOS EJDE-2010/60 4. Existence of ositive solutions In this section, we rove the existence of ositive solutions for (1.1) under conditions (C1), (C2) and (C3), by an aroximation method used in [2, 3]. For ɛ ]0, 1[, we define the following exression u k 2 u k X k := 1 + ɛ 1/ u k, X := u 2 u ɛ 1/ u, 1 u k α u k Y k := 1 + ɛ 1/ u k, Y := u α u α ɛ 1/ u, α+1 X k := Y k := v k 2 v k 1 + ɛ 1/ v k, v 2 v 1 X := 1 + ɛ 1/ v, 1 v k β v k 1 + ɛ 1/ v k, Y := On has the following result which will be useful later. v β v 1 + ɛ 1/ v. Lemma 4.1. If (u k, v k ) converges to (u, v) in L ( ) L ( ) then (i) X k X in L 1 ( ), Y k Y in L (ii) X k X in L 1 ( ), Y k Y in L α+1 ( ) and in L (m, ). ( ) and in L (n, ). Proof. We give the roof for (i) and indicate that the same arguments hold for (ii). If u k u in L ( ), then there exists a subseuence denoted (u k ) such that u k u almost every where in and u k (x) l 1 (x) for some l 1 L ( ). Hence X k (x) X(x) a.e. in, X k (x) u k (x) 1 l 1 (x) 1 in L 1, which imlies, by dominated convergence Theorem, that X k X in L 1. Similarly, on deduces from the convergence of Y k to Y in L α+1 that Y k (x) Y (x) a.e in, Y k (x) u k (x) α+1 l 2 (x) α+1 in L α+1, and the conclusion follows. Moreover, using Hölder ineuality, we have Y k Y = m Y L (m, ) k Y m L N/ (R ) Y N k Y L α+1. We are now in osition to give the main result of this section. Theorem 4.2. Assume that (H1), (H2), (H3), (C1), (C2), (C3) are satisfied. Then for all f L ( ) ( ) and g L ( ) ( ), the system (1.1) has at least one solution (u, v) D 1, ( ) D 1, ( ).

7 EJDE-2010/60 MAXIMUM PRINCIPLE AND EXISTENCE RESULTS 7 The roof is artly adated from [2, 3]. We choose r > 0 such that a + r > 0 and d + r > 0. The system (1.1) is then euivalent to u + rm u 2 u = (a + r)m u 2 u + bm 1 v β v + f v + rn v 2 v = cn 1 u α u + (d + r)n v 2 v + g For ɛ ]0, 1[, let us introduce the system where u(x) 0, v(x) 0 as x + u ɛ + rm u ɛ 2 u ɛ = mh(u ɛ ) + m 1 h 1 (v ɛ ) + f v ɛ + rn v ɛ 2 v ɛ = n 1 k 1 (u ɛ ) + nk(v ɛ ) + g u ɛ (x) 0, v ɛ (x) 0 as x + in in in in u 2 u h(u) := (a + r) 1 + ɛ 1/ u, h v β v 1(v) := b ɛ 1/ v, u α u k 1 (u) := c 1 + ɛ 1/ u, k(v) := (d + r) v 2 v α ɛ 1/ v. 1 (4.1) (4.2) Lemma 4.3. Under hyothesis of Theorem 4.2, system (4.2) admits at least a coule of solution (u, v) in D 1, ( ) D 1, ( ). Proof. We give the roof in several stes. Ste 1. Construction of sub-suer solution for (4.2): Since the functions h, h 1, k and k 1 are bounded, there exists a constant M > 0 such that h(u) M, h 1 (v) M, k 1 (u) M, k(v) M for all (u, v) D 1, ( ) D 1, ( ). Let ξ 0 D 1, ( ) (resectively η 0 D 1, ( )) be a solution of u + rm u 2 u = (m + m 1 )M + f (resectively v + rm v 2 v = (n + n 1 )M + g), and let ξ 0 D 1, ( ) (resectively η 0 D 1, ( )) be solution of u + rm u 2 u = (m + m 1 )M + f (resectively v + rm v 2 v = (n + n 1 )M + g). Then (ξ 0, η 0 ) (resectively (ξ 0, η 0 )) is a suer solution (resectively sub solution) of system (4.2) since ξ 0 + rm ξ 0 2 ξ 0 mh(ξ 0 ) m 1 h 1 (η) f ξ 0 + rm ξ 0 2 ξ 0 (m + m 1 )M f = 0 η [η 0, η 0 ], η 0 + rn η 0 2 η 0 n 1 k 1 (ξ) nk(η 0 ) g η 0 + rn η 0 2 η 0 (n + n 1 )M g = 0 η [ξ 0, ξ 0 ], ξ 0 + rm ξ 0 2 ξ 0 mh(ξ 0 ) m 1 h 1 (η) f ξ 0 + rm ξ 0 2 ξ 0 (m + m 1 )M f = 0 η [η 0, η 0 ], η 0 + rn η 0 2 η 0 n 1 k 1 (ξ) nk(η 0 ) g η 0 + rn η 0 2 η 0 (n + n 1 )M g = 0 η [ξ 0, ξ 0 ].

8 8 L. LEADI, A. MARCOS EJDE-2010/60 Ste 2. Definition of oerator T. Denote by K = [ξ 0, ξ 0 ] [η 0, η 0 ] and define the oerator T : (u, v) (w, z) such that w + rm w 2 w = mh(u) + m 1 h 1 (v) + f in z + rm z 2 z = n 1 k 1 (u) + nk(v) + g in w(x) 0, z(x) 0 as x + (4.3) Ste 3. Let us rove that T (K) K. If (u, v) K then we have ( w ξ 0 ) + rm( w 2 w ξ 0 2 ξ 0 ) = m[h(u) M] + m 1 [h 1 (v) M]) (4.4) Taking (w ξ 0 ) + as test function in (4.4), we have ( w 2 w ξ 0 2 ξ 0 ) (w ξ 0 ) + + r m( w 2 w ξ 0 2 ξ 0 )(w ξ 0 ) + R N = [m(h(u) M) + m 1 (h 1 (v) M)](w ξ 0 ) + 0. Hence by the monotonicity of the function x x 2 x and by the monotonicity of the -Lalacian, we deduce that (w ξ 0 ) + = 0 and then w ξ 0. Similarly we get ξ 0 w by taking (w ξ 0 ) as test function in (4.4). So we have ξ 0 w ξ 0 and η 0 z η 0 and the ste is comlete. Ste 4. T is comletely continuous: We will first rove that T is continuous. Indeed let (u k, v k ) (u, v) D 1, ( ) D 1, ( ), we will rove that (w k, z k ) = T (u k, v k ) converges to (w, z) = T (u, v). ( w k + rm w k 2 w k ) ( w + rm w 2 w) = m[h(u k ) h(u)] + m 1 [h 1 (v k ) h 1 (v)] = (a + r)m(x k X) + bm 1 (Y k Y ), (4.5) where X k, X, Y k and Y are reviously define in Lemma 4.1. Then taking (w k w) as test function in (4.5), we get ( w k 2 w k w 2 w) (w k w) ( w k 2 w k w 2 w) (w k w) + r m( w k 2 w k w 2 w)(w k w) = (a + r) m(x k X)(w k w) + b m 1 (Y k Y )(w k w). Using Hölder ineuality, we obtain m(x k X)(w k w) m L N/ ( ) X k X L /( 1) ( ). w k w L ( )

9 EJDE-2010/60 MAXIMUM PRINCIPLE AND EXISTENCE RESULTS 9 and m 1 (Y k Y )(w k w) [m 1/ (w k w)][n ()/ (Y k Y )] w k w L (m, ). Y k Y L (n, ), since = 1. Conseuently 0 ( w k 2 w k w 2 w) (w k w) (a + r) m L N/ (R ) X N k X L /( 1) (R ) w N k w L ( ) + b Y k Y L (n, ). w k w L (m, ) Using then the ineuality x y c[( x 2 x y 2 y)(x y)] s/2 [ x + y ] 1 s/2, (4.6) where x, y, c = c() > 0 and s = 2 if 2, s = if 1 < < 2 (cf. e.g. [17]), one easily obtains that w k w in D 1, ( ). Similarly, we have z k z in D 1, ( ). We now rove that oerator T is comact. Let (u k, v k ) be a bounded seuence in D 1, ( ) D 1, ( ) and set (w k, z k ) = T (u k, v k ). We have w k + rm w k 2 w k = mh(u k ) + m 1 h 1 (v k ) + f (4.7) Taking w k as test function in (4.7), we get w k + r m w k R N = mh(u k )w k + m 1 h 1 (v k )w k + fw k R N (a + r) m u k 1 w k + b m 1 v k w k + f w k [ (a + r) u k 1 L (m, ) + b v k L (n,r )]. wk N L (m, ) + f L ( ) ( ) w k L (R ). N Hence w k is bounded in D 1, ( ) and conseuently, u to a subseuence w k converges to w weakly in D 1, ( ) and strongly in L (m, ). Now taking (w k w ) as test function in (4.7), we have w k 2 w k (w k w ) + r m w k 2 w k (w k w ) R N = [mh(u k ) + m 1 h 1 (v k )](w k w ) + f(w k w )

10 10 L. LEADI, A. MARCOS EJDE-2010/60 and conseuently ( w k 2 w k w 2 w ) (w k w ) ( w k 2 w k w 2 w ) (w k w ) + r m( w k 2 w k w 2 w )(w k w ) R N = m[h(u k ) h(u )](w k w ) + m 1 [h 1 (v k ) h 1 (v )](w k w ) [ u k 1 L (m, ) + u 1 L (m, ) + v k L (n, ) + v L (n, )] wk w L (m, ). We then deduce that ( w k 2 w k w 2 w ) (w k w ) 0. From (4.6), we conclude that w k converges to w in D 1, ( ). Similarly, we rove that z k converges to z in D 1, ( ). Since the set K is convex, bounded and closed in D 1, ( ) D 1, ( ), alying Schauder s fixed oint theorem, then there exists a fixed oint for T which gives the existence of solution of system (4.2). Proof of Theorem 4.2. The roof will be given by three stes. Ste 1. We show that (u ɛ, v ɛ ) is bounded in D 1, ( ) D 1, ( ). Indeed denoting by t ɛ = max( u ɛ D 1, ( ), v ɛ D 1, ( ) ), z ɛ = t 1/ ɛ u ɛ and w ɛ = t 1/ ɛ v ɛ. Since (u ɛ, v ɛ ) is solution of (4.2), we have z ɛ + rm z ɛ 2 z ɛ = (a + r)m z ɛ 2 z ɛ 1 + t 1/ w ɛ + rm w ɛ 2 w ɛ = ɛ ɛ 1/ z ɛ + bm 1 w ɛ β w ɛ t 1/ ɛ ɛ 1/ w ɛ cn 1 z ɛ α z ɛ ɛ 1/ z ɛ + (d + r)n wɛ 2 w ɛ α t 1/ ɛ 1/ w ɛ 1 + t 1/ ɛ ɛ + t 1/ ɛ f, + t 1/ 1 Hence taking z ɛ as test function in the first euation, we get z ɛ + r m z ɛ R N (a + r) m z ɛ + b m 1 w ɛ z ɛ + t 1/ ɛ f z ɛ, which imlies, by Hölder ineuality and (H2), ( ) 1/ ( z ɛ a m z ɛ + b m z ɛ n w ɛ R ( ) N + t 1/ ɛ f L ( ) (R n ) z ɛ 1/ ) ()/ ɛ g.

11 EJDE-2010/60 MAXIMUM PRINCIPLE AND EXISTENCE RESULTS 11 By the imbedding of D 1, ( ) in L ( ), we deduce that z ɛ a D 1, λ 1 (m, ) z ɛ z ɛ D + b D 1, w ɛ D 1, 1, [λ 1 (m, )] 1/ [λ 1 (n, )] ()/ + c 1 t 1/ ɛ f L ( ) z ɛ D 1,, where c 1 = c 1 (, N) is the constant of the imbedding of D 1, ( ) into L ( ), and conseuently ( zɛ ) D 1, 1 ( wɛ ) b D1, [λ 1 (m, )] 1/ [λ 1 (n, )] 1/ Similarly, we obtain ( wɛ D 1, [λ 1 (n, )] 1/ ) 1 c ( zɛ D1, [λ 1 (m, )] 1/ ) α+1 + c 1 tɛ 1/ [λ 1 (m, )] 1/ f L ( ). (4.8) + c 2 t 1/ ɛ [λ 1 (n, )] 1/ g L ( ) (4.9) Now assume that u ɛ (or v ɛ )is unbounded in D 1, ( ) ( in D 1, ( )). Then t ɛ + and it follows from (4.8) and (4.9), that ( [λ 1 (m, ) a] α+1 [λ1 (n, ) d] zɛ ) (α+1)() D 1, [λ 1 (m, )] 1/ ( wɛ ) (α+1)() D 1, [λ 1 (n, )] 1/ ( b α+1 zɛ ) (α+1)() ( c D 1, wɛ ) (α+1)() D 1,, [λ 1 (m, )] 1/ [λ 1 (n, )] 1/ which imlies [(λ 1 (m, ) a) α+1 (λ1 (n, ) d) b α+1 c ] ( zɛ D 1, w ɛ ) (α+1)() D1, 0. λ 1 (m, ) 1/ λ 1 (n, ) 1/ But this is a contradiction since conditions (C1), (C2) and (C3) hold. Ste 2. Using the same arguments as in [19], we easily rove that (ɛ 1 uɛ, ɛ 1 vɛ ) (0, 0) in D 1, ( ) D 1, ( ). Ste 3. Now we rove that (u ɛ, v ɛ ) converges strongly in D 1, ( ) D 1, ( ) as ɛ 0. Indeed from Ste 1 and Ste 2, we have (u ɛ, v ɛ ) is bounded in D 1, ( ) D 1, ( ) and ɛ 1 uɛ 0 a.e in R n. So u to a subseuence (u ɛ, v ɛ ) (u 0, v 0 ) in L (m, ) L (n, ) and conseuently u ɛ 2 u ɛ u 1 + ɛ 1 ɛ 1 l 1 uɛ 1 1 in L (m, ), u ɛ (x) 2 u ɛ (x) 1 + ɛ 1 uɛ (x) 1 u 0(x) 2 u 0 (x) a.e. in. From the dominated convergence theorem, we have h(u ɛ ) h(u 0 ) in L (m, ) as ɛ 0. Similarly, we get h 1 (v ɛ ) h 1 (v 0 ) in L (n, ), k 1 (u ɛ ) k 1 (u 0 ) in L (m, ) and k(v ɛ ) k(v 0 ) in L (n, ). We finally use (4.6) to deduce that

12 12 L. LEADI, A. MARCOS EJDE-2010/60 (u ɛ, v ɛ ) (u 0, v 0 ) in D 1, ( ) D 1, ( ) as ɛ 0. Therefore, assing to the limit in (4.2), we obtain u 0 = am u 0 2 u 0 + bm 1 v 0 β v 0 + f in, v 0 = cn 1 u 0 α u 0 + dn v 0 2 v 0 + g in which imlies that (u 0, v 0 ) is solution of (1.1). We remark that when α = β = 0 and = = 2, we obtain the results resented in [12, 13]. 5. Related results u = am(x) u 2 u + bm 1 (x) u α v β v + f v = cn 1 (x) u α u v β + dn(x) v 2 v + g u(x) 0, v(x) 0 as x + in R n in R n where we assume that the conditions (H1),(H2 ),(H3) and (H4 ) hold, with (H2 ) 0 < m 1 (x) m(x) α+1 n(x) and 0 < n 1 (x) m(x) α+1 n(x) (H4 ) b, c 0; α, β 0; α+1 + = 1. Under these assumtions, one has the following results. (5.1) a.e. in Theorem 5.1. Assume that hyothesis (H1), (H2 ), (H3), (H4 ) are satisfied. Then the (MP) holds for (5.1) if (C1 ) λ 1 (m, ) > a; (C2 ) λ 1 (n, ) > d; (C3 ) [λ 1 (m, ) a] α+1 [λ1 (n, ) d] > b α+1 c ; Conversely, if the (MP) holds, then (C1 ), (C2 ) and (C4 ) are satisfied, where (C4 ) [λ 1 (m, ) a] α+1 [λ1 (n, ) d] > Θb α+1 c. The tools used to establish the above results can be easily adated for the roblem Theorem 5.2. Assume that (H1), (H2 ), (H3), (C1 ), (C2 ), (C3 ) hold. Furthermore assume that m L ( ) ( ) and m L ( ) ( ). Then for all f L ( ) ( ) and g L ( ) ( ), the system (5.1) has at least one solution (u, v) D 1, ( ) D 1, ( ). The roofs of theorems 5.1 and 5.2 can be adated from those of theorems 3.3 and 4.2 resectively. Acknowledgements. The authors would like to exress their gratitude to Professor F. de Thélin for his constructive remarks and suggestions. L. Leadi is grateful to the International Centre for Theoretical Physics (ICTP) for financial suort during his visit

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