New York Journal of Mathematics. Positive Radial Solutions of Nonlinear Elliptic Systems

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1 ew York Journal of Mathematics ew York J. Math. 7 (21) Positive Radial Solutions of onlinear Elliptic Systems Abdelaziz Ahammou Abstract. In this article, we are concerned with the existence of positive radial solutions of the problem Δ pu = f(x, u, v) in Ω, (S + ) Δ qv = g(x, u, v) in Ω, u = v = on Ω, where Ω is a ball in R and f, g are positive functions satisfying f(x,, ) = g(x,, )=. Under some growth conditions, we show the existence of a positive radial solution of the problem S +. We use traditional techniques of the topological degree theory. When Ω = R, we give some sufficient conditions of nonexistence. Contents 1. Introduction and main result Preliminaries A priori bounds for positive solutions of (S + ) The blow up to isolate the trivial solution Proof of Theorem onexistence 276 References Introduction and main result In this work, we are concerned with the existence of positive radial solutions of the problem Δ p u = a(x)u u α 1 + b(x)v v β 1 in Ω, (S + ) Δ q v = c(x)u u γ 1 + d(x)v v δ 1 in Ω, u = v = on Ω, Received August 25, 2. Mathematics Subject Classification. 35J25, 35J6. Key words and phrases. Blow up argument, degree theory, Leray-Schauder theorem, excision property. 267 ISS /1

2 268 A. Ahammou where Ω := B R is the ball centered in zero and radius R>inR,a,b,cand d are given positive continuous functions. Our motivation for studying the system S + is based essentially from the fact that the problem has not necessarily a variational structure. We shall make recourse to topological degree methods by using the blowup technique introduced by Gidas and Spruck [1] in the scalar case. This method explores the different exponents (α, β, δ, γ). In the scalar case the interested reader may refer to [5], [6] and [16]. In the case of systems, many authors have extended this method to different situations (see [4], [3] and [15]). In recent years, for the scalar case the problems of existence and nonexistence have been studied by several authors by using different approaches (see[5], [6] and [16]). For the systems case, we mention the recent results of Boccardo, Fleckinger and de Thelin [2] where the authors prove the existence of the weak solutions of the following problem: Δ p u = a(x)u u α 1 + b(x)v v β 1 + h 1 (x) in Ω, (1.1) Δ q v = c(x)u u γ 1 + d(x)v v δ 1 + h 2 (x) in Ω, u = v = on Ω, under the following assumptions: (H1) max(p, q) <. (H2) (p 1)(q 1) >βγ. (H3) One of the following conditions holds: (i) p 1 >α, q 1 >δ. { p 1=α, q 1=δ, (ii) a < λ (1,p) and d < λ (1,q). { p 1=α, q 1 <δ, (iii) and a < λ (1,p). Here, Ω is smooth and bounded in R, λ (1,m) (m = p, q) is the first eigenvalue of the operator Δ m (m = p, q) onωandh 1 L p (Ω), h 2 L q (Ω). We observe that, with the same approach in [2], if h 1 and h 2 are identically zero, the solution (u, v) would be a trivial solution. Always in the system case, the interested reader may refer to [1], [4], [7], [8], [9], [11] and [12]. ow, we state our main result. Theorem 1.1. We assume that the hypotheses (H1), (H2) and (H3) hold. We also suppose that (H4) a, b, c, d C ([, + [) with inf (a(s),b(s),c(s),d(s)) >. s [,+ [ Then the problem (S + ) possesses a solution (u, v) in C 1 (B R ) C 2 (B R \{}), such that u>, v> in B R. The paper is organized as follows. At first, we consider the operator of solution S 1 associated to the problem (S + ) which allows us to seek solutions of the problem (S + ) as a fixed points of S 1. In Section 2 we introduce two families of operators, (S λ ) λ and (T μ ) μ, linked to the problem (S + ), acting in a suitable functional space and we give a fundamental lemma. In Section 3, we prove that for

3 Positive radial solutions 269 any positive solution (u, v) of the problem, it is bounded. By using the theory of degree, we show that there exists a positive number ρ 1 > sufficiently large such that deg(s 1,B(,ρ 1 ))=1. On the other hand, in Section 4 by means of the argument blow-up, we show that there exists a number ρ 2 > sufficiently small such that deg(s 1,B(,ρ 2 ))=. In Section 5 by the excision property we deduce the existence of the nontrivial positive solutions of (S + ) stated in Theorem 1.1. Finally, in Section 6 we give sufficient conditions for the nonexistence of positive radial solutions of the problem (S + )onω=r. 2. Preliminaries We now consider χ the space χ = {(u, v) C (Ω) C (Ω) u = v =on Ω} equipped with the norm (u, v) = u + v, which makes it a Banach space. Let S λ and T τ : χ χ be the operators defined by S λ (u, v) =(S 1 (u, v); S 2 (u, v)) and T τ (u, v) =(T 1 (u, v); T 2 (u, v)) such that R t ] S 1 (u, v)(r) = λ 1 [t 1 1 s 1 (a(s) u(s) α + b(s) v(s) β )ds dt, r R t ] S 2 (u, v)(r) = λ 1 [t 1 1 s 1 (c(s) u(s) γ + d(s) v(s) δ )ds dt, and T 1 (u, v)(r) = T 2 (u, v)(r) = R r R r r [t 1 t [t 1 t ] 1 s 1 (a(s) u(s) α + b(s) (v(s)+τ) β )ds dt, ] 1 s 1 (c(s) u(s) γ + d(s) v(s) δ )ds dt. It is well know that, for all λ [, 1] and for all τ [, [, S λ and T τ are completely continuous operators on χ. From the Maximum principle this implies that S λ (χ) χ and that the problem (S + ) is equivalent to find some non trivial fixed point (u, v) χ of the operator S 1 (by taking λ = 1) such that u () = v ()=. We make use in a fundamental way of the following lemma (cf. [3, Lemma 2.1, p. 276]): Lemma 2.1. Let u C 1 ([.R]) C 2 (],R]), u, satisfying (2.1) (r 1 u (r) p 2 u (r)) on [,R]. Then, for any r ], R 2 [ we have : (2.2) u(r) C,p r u (r) where (2.3) C,p = p 1 p ) (1 2 p. Proof. Integrating (2.1) from r to s [r, R 2 [wehave: (2.4) s 1 u (s) r 1 u (r)

4 27 A. Ahammou and therefore: (2.5) u (s) r 1 u (r) s 1. Integrating again from r to 2r with respect to s, we obtain: (2.6) u(r) u(r) u(2r) r 1 u (r) 2r r s 1 ds. Since 2r s 1 r ds = C,p r p, we obtain the Lemma. In the following sections, we do not distinguish notationally between a sequence and one of its subsequences, to keep the notation simple. 3. A priori bounds for positive solutions of (S + ) Proposition 3.1. Under the hypotheses (H1), (H2), (H3) and (H4) there exists some C > such that λ [, 1] if (u, v) χ is a fixed point of the operator S λ then (u, v) C. This implies that ρ 1 >C, λ ], 1[ we have (3.1) deg(i S λ,b(,ρ 1 ), ) = const = 1, where B(,ρ 1 )={(u, v) χ (u, v) ρ 1 }. Proof. We suppose by contradiction that there exist λ [, 1] and (u, v) χ such that (3.2) (u, v) =S λ (u, v) with (u, v) = c>. otice that by definition of S λ we get u, v in [,R]. Hence (u, v) = u() + v(). Thus, since (3.3) u() = λ 1 v() = λ 1 R R t ] [t 1 1 s 1 (a(s) u(s) α + b(s) v(s) β )ds dt, t ] [t 1 1 s 1 (c(s) u(s) γ + d(s) v(s) δ )ds dt, we have [ u() Cλ 1 (u()) α +(v()) β] 1 (3.4) [ v() Cλ 1 (u()) γ +(v()) δ] 1 (3.5). Moreover, from (H3), there exist two numbers l> and k> such that (3.6) β p 1 < l k < q 1 γ. Denote σ =(u()) 1 1 (3.7) l +(v()) k, Hence, from (3.4) and (3.5), we get (u()) 1 1 [ l Cλ l() σ lα + σ kβ] 1 l() (3.8)

5 (3.9) Positive radial solutions 271 (v()) 1 1 [ k Cλ k() σ lγ + σ kδ] 1 k(). Summing (3.8) and (3.9), we deduce that σ satisfies (3.1) 1 Cλ 1 l() [σ l(α p+1) + σ kβ l()] 1 l() + Cλ 1 k() [σ lγ k() + σ k(δ q+1)] 1 k(). First Case: (H3)(i) is satisfied. Here, (3.1) leads us to a contradiction for σ sufficiently large. Second Case: (H3)(ii) or (H3)(iii) is satisfied. In this case we suppose that there exist some sequences {λ n } and {(u n,v n )} satisfy (3.2), this implies that Δ p u n = λ n a(x)u n u n α 1 + λ n b(x)v n v n β 1 in B(,R), (3.11) Δ q v n = λ n c(x)u n u n γ 1 + λ n d(x)v n v n δ 1 in B(,R), u n = v n = on B(,R), and we suppose that c n = (u n,v n ) + as n +. Then, from (3.1), we deduce easily that λ n λ>asn +. We introduce new functions ũ n and ṽ n in the following way: ũ n (r) = u n(r) σ l, ṽ n(r) = v n(r) n σ k n where, σ n =(u n ()) 1 l +(vn ()) 1 k. Taking (ũ n, ṽ n ) in (3.11) we get, in B(,R) (3.12) Δ p ũ n (x) =σ l(α+1 p) n λ n a(x) ũ n (x) α + σ l()+kβ n λ n b(x) ṽ n (x) β (3.13) Δ q ṽ n (x) =σ n k()+lγ λ n c(x) ũ n (x) γ + σ n k(δ+1 q) λ n d(x) ṽ n (x) δ, ũ n =ṽ n = on B(,R), Multiplying (3.12) by ũ n, (3.13) by ṽ n and by integrating, we infer ũ n (x) p = σ l(δ+1 p) n λ n a(x) ũ n (x) α+1 dx B B +σ l()+kβ n λ n b(x) ṽ n (x) δ ũ n (x)dx B ṽ n (x) q = σ k()+lγ n λ n c(x) ũ n (x) γ ṽ n (x)dx B B +σ k(δ+1 q) n λ n d(x) ṽ n (x) δ+1 dx. Observe that (ũ n ()) 1 l +(ũn ()) 1 k =1. Consequently, from (H3)(ii) or (H3)(iii), (H4) and (3.6) we deduce that (ũ n, ṽ n )is bounded in W 1,p (B(,R)) W 1,q (B(,R)). B

6 272 A. Ahammou Thus (ũ n, ṽ n ) converges weakly to some (ũ, ṽ) W 1,p (B(,R)) W 1,q (B(,R)). On the other hand, it easy to see that Δ p ũ n C, n, Δ q ṽ n C, n with some positive constant C>depending on (, p, q, a, b, c, d). Therefore, for all n we have (ũ n, ṽ n ) C 1 (B(,R)) C 1 (B(,R)) and ũ n K and ṽ n K. ow since (ũ n, ṽ n ) = 1 for all n, the Arzelà-Ascoli theorem together with the weak convergence of (ũ n, ṽ n )to(ũ, ṽ) ensure that (ũ n, ṽ n ) converges uniformly to (ũ, ṽ) and that (ũ, ṽ) is not identically zero. Consequently, by passing to the limit it follows that: 1. If (H3)(ii) is satisfied Δ p ũ(x) =λa(x) ũ(x) p 2 ũ(x) in B(,R), Δ q ṽ(x) =λd(x) ṽ(x) q 2 ṽ(x) in B(,R). But from a <λ (1,p) and d <λ (1,q) we get the contradiction. 2. If (H3)(iii) is satisfied, we obtain Δ p ũ(x) =λa(x) ũ(x) p 2 ũ(x) in B(,R), Δ q ṽ(x) = in B(,R), ũ =ṽ = on B(,R). Then from a <λ (1,p), we deduce the contradiction. So, in the different cases there exists C > sufficiently large such that ρ 1 >C we have deg(i S λ,b(,ρ 1 ), ) = const λ [, 1]. Hence (3.14) deg(i S 1,B(,ρ 1 ), ) = deg(i S,B(,ρ 1 ), )=1 ρ 1 >C. The proof of Proposition 3.1 is complete. 4. The blow up to isolate the trivial solution We shall prove, under (H1), (H2), and (H4), that there exists some ρ 2 > such that deg(i T τ,b(,ρ 2 ), )= τ [, [. Proposition 4.1. Under the assumptions (H1), (H2) and (H4) there exists some ρ> such that for all τ [, [ and for all fixed points (u, v) χ\{(, )} of T τ we have (u, v) >ρ.this implies that, for ρ 2 sufficiently small, deg(i T τ,b(,ρ), ) = const = τ [, [. Proof. Firstly, from the maximum principle, it follows that the problem (4.1) (u, v) =T τ ((u, v))

7 is equivalent to find solutions u, v of (4.2) (4.3) (4.4) Positive radial solutions 273 ( r 1 u (r) p 2 u (r) ) = r 1 [ a(r) u(r) α + b(r) v(r)+τ β], ( r 1 v (r) q 2 v (r) ) = r 1 [ c(r) u(r) γ + d(r) v(r) δ], u () = v () = u(r) =v(r) =. By integrating on [,r]we get u (r) Cr 1 β (4.5) (v(r)+τ) v (r) Cr 1 δ (4.6) (u(r))., Hence, u < and v < and it follows that u(r), v(r). Thus, from (4.5), we have (4.7) u (r) Cr 1 τ β. By integrating (4.7) from to R, we obtain that (4.8) u() CR p τ β. ow, we introduce new functions ũ and ṽ in the following way: (4.9) and make the change of variables ũ(r) = u(r) σ l ṽ(r) = v(r) σ k, (4.1) y = r σ, on [,R] where (4.11) σ =(u()) 1 l +(v()) 1 k and l, k are positive numbers to be chosen below. In this way we obtain the following equations for ũ(y) and ṽ(y) defined on interval [, R σ ]: ( d ) y 1 dũ dy dy (y) p 2 dũ (4.12) dy (y) = y 1 F (ũ(y), ṽ(y)), ( d ) y 1 dṽ dy dy (y) q 2 dṽ (4.13) dy (y) = y 1 G(ũ(y), ṽ(y)), dũ dṽ (4.14) () = dy dy () = ũ(r σ)=ṽ(r σ )=, where [ F (ũ(y), ṽ(y)) = a(σy)a ũ(y) α + b(σy)b ṽ(y)+ τ ] β (4.15) σ k, [ (4.16) G(ũ(y), ṽ(y)) = c(σy))c ũ(y) γ + d(σy))d ṽ(y)) δ],

8 274 A. Ahammou and (4.17) A = σ p+l(α p+1) C = σ q+k()+lγ B = σ p l()+kβ, D = σ q+k(δ q+1), By choosing (4.18) we obtain (4.19) l = p(q 1) + βq (p 1)(q 1) βγ R σ = R σ. and k = q(p 1) + pγ (p 1)(q 1) βγ, A = σ lα kβ, B =1, C =1, D = σ kδ lγ. ote that (ũ, ṽ) satisfies (4.2) dũ dy (y), ũ(y) 1 y [,R σ], (4.21) and dṽ dy (y), ṽ(y) 1 y [,R σ] (4.22) (ũ()) 1 1 l +(ṽ()) k =1. Thus, we have (4.23) (y 1 ũ (y) p 2 ũ (y)) y 1 b(σy) ṽ(y) β, on [,R σ ] (y 1 ũ (y) q 2 ũ (y)) y 1 c(σy) ũ(y) γ, on [,R σ ] ũ () = ṽ ()=. Integrating (4.23) on (,y) and taking into account that (H4) holds, we have y [,R σ ] ( y ) 1 ũ (4.24) (y) b 1 (ṽ(y)) β, (4.25) ṽ (y) From Lemma 2.1, we have for y ] ], Rσ 2 (4.26) ( y ) 1 c 1 (ũ(y)) γ. ũ(y) C,p y ũ (y) C,p ( 1 ) 1 y p b1 ṽ(y) β, (4.27) ṽ(y)) C,q y ṽ (y) C,q ( 1 ) 1 y q c1 ũ(y) γ. Thus, from (4.26) and (4.27), we obtain (4.28) (4.29) (ṽ(y)) ()() βγ q()+pγ C y, y (ũ(y)) ()() βγ p()+qβ C y, y ], R ] σ, 2 ], R ] σ, 2

9 Positive radial solutions 275 where here and henceforth C>denotes a positive constant depending only of (a, b, c, d,, p, q). Taking into account (4.2), (4.21) and since (ũ, ṽ) are non increasing functions on [,R σ ], we obtain [ y C, y, R ] σ (4.3), 2 where C := C(a, b, c, d,, p, q). Then, as R σ when σ, (4.3) it is not true for σ sufficiently small. Consequently, since σ ρ 1 l + ρ 1 k where (u, v) = ρ, it follows, according the above argument, that for ρ sufficiently small the equation (u, v) =T τ ((u, v)) has no solution on B(,ρ) for τ [, + [. Then, deg(i T τ,b(,ρ), ) is well-defined and by properties of topological degree, we get that (4.31) deg(i T τ,b(,ρ), ) = const, τ. Moreover, from (4.8), T τ1 than ρ, then we get has no solution in B(,ρ) when τ 1 it is sufficiently large deg(i T τ1,b(,ρ), )=. Consequently, from of the Leray-Schauder degree properties, we deduce that deg(i T τ,b(,ρ), ) = deg(i T τ1,b(,ρ), )=. 5. Proof of Theorem 1.1 The proof is an immediate consequence of Proposition 3.1 and Proposition 4.1. By taking ρ 2 sufficiently small, we may assume, from Proposition 4.1 and Leray- Schauder degree properties, that (5.1) deg(i T τ,b(,ρ), ) = deg(i T,B(,ρ), )=. Thus, from Proposition 3.1, for ρ 1 > sufficiently large we have (5.2) Then, since deg(i S 1,B(,ρ 1 ), )=1. S 1 = T, by excision property we obtain (5.3) deg(i S 1,B(,ρ 1 )\B(,ρ 2 ), )=+1. Consequently S 1 admits at least one fixed point (u, v) (, ). Hence, we obtain the results of Theorem 1.1.

10 276 A. Ahammou 6. onexistence In this section we study some nonexistence result for positive radial solutions for quasilinear system of the form { Δ (S p,q ) p u a(x)u u α 1 + b(x)v v β 1 in R, Δ q v c(x)u u γ 1 + d(x)v v δ 1 in R, First consider the semilinear case, i.e., p = q = 2. When, b = c =, the system (S p,q ) reduced simply to the case of two single equations Δu u α, Δv v δ on R. This prototype model has been studied quite extensively. For example, we survey some results on a single equation, namely Δu = u α on R. In this case we give the results of Gidas and Spruck [1] where the authors prove that if <α< +2 2 then u =. A very elementary proof valid for <α< 2 was given by Souto [15]. In fact his proof is valid for the case of u being a nonnegative supersolution, i.e., Δu u α on R. Always in the semilinear case, if a = d = the system (S p,q ) becomes Δu v β, Δv u γ, which is natural extension of the well known Lane-Emden equation and thus is referred to as the Lane-Emden system. This case is studied by Serrin and Zou [13]; the authors give a nonexistence of positive solutions for system (S 2,2 ) when the exponents β and γ are subcritical in the sense 1 β γ +1 > 2. Moreover, in [14] the same authors prove the existence of positive (radial) solution (u, v) onr for the system under the following assumption 1 β γ Let us now mention the key of our result concerning radial solutions of the quasilinear problem (S p,q )inr. Lemma 6.1. Let r, >mand w C 1 ([r, + [) C 2 ([r, + [) is a positive supersolution of (6.1) Assume (r 1 w (r) m 2 w (r)) on [r, + [. w(r) > and w (r) < r [r, + [.

11 Positive radial solutions 277 Then there exists a nonnegative number C> such that r m m 1 w(r) >C. Proof. Since u satisfies (6.1) and w (r) <, we deduce that r 1 w (r) is an increasing function on [r, [. Hence there exists a non negative number C such that (6.2) r 1 w (r) m 1 >C r [r, + [. Thus, from Lemma 2.1, there exists a nonnegative number C,m such that (6.3) w(r) C,m r w (r) r [r, + [. Consequently, multiplying (6.3) by r m m 1 we obtain (6.4) r m m 1 u(r) C,m r 1 m 1 w (r) r [r, + [. Then, from (6.2)and (6.4), we deduce that r m m 1 w(r) C,m r 1 m 1 w (r) C,m C 1 m 1 r [r, + [. Hence the proof of the lemma. Our main result is the following: Theorem 6.1. Let u, v C 1 (R ) C 2 (R \) be nonnegative radial solutions of { Δp u b 1 v β, Δ q v c 1 u γ, where b 1 > and c 1 >. Assume (H5) max{p, q} <, β>, and γ>, (H6) Then u = v =. 1 β + 1 γ > p (p 1) + q (q 1). Proof. Since (u, v) is supposed to be radial positive solution, then (u, v) satisfies (r 1 u (r) p 2 u (r)) r 1 b 1 v(r) β, (6.5) (r 1 v (r) q 2 v (r)) r 1 c 1 u(r) γ, u () = v ()=. Integrating (6.5) on (,r) and taking into account that u <, v <, we get ( r ) 1 u [ (r) b1 v β (r) ] 1 (6.6), r > (6.7) v (r) ( r ) 1 1 [c 1 u γ (r)], r >. Thus, from Lemma 2.1, we have (6.8) u(r) C,p r u (r) C,p ( 1 ) 1 p [ r b1 v β (r) ] 1, r > (6.9) ( ) 1 1 v(r) C,p r v q (r) C,p r [c1 u γ 1 (r)], r >.

12 278 A. Ahammou Then, from (6.8) and (6.9), we deduce (6.1) u(r) Cr p b 1 v β (r), r > (6.11) Hence, easily we obtain v(r) Cr q c 1 u γ (r), r >. r β + q r p β (6.12) u(r) Cr q v(r), r > (6.13) r γ + p Multiplying (6.12) by (6.13), we get (6.14) r β + q r q γ v(r) Cr p u(r), r >. γ + p γ q+1 r q C γ v(r) r p β p+1 β u(r), r >. Consequently, from (H5) and Lemma 6.1, there exists a number C> such that for all r>r > wehave r β + q γ + p C. Then, from (H6), we obtain a contradiction. This concludes the proof of the Theorem 6.1. Theorem 6.2. We make the following assumptions: (j) max(p, q) <. { p 1 α, q 1 δ or (jj) (p 1)(q 1) βγ. (jjj) a, b, c, d :[, + [ [, + [ are continuous functions such that inf (a(s),b(s),c(s)d(s)) >. s [,+ [ Under these assumptions, the problem { Δ (S p,q ) p u a(x)u u α 1 + b(x)v v β 1 in R, Δ q v c(x)u u γ 1 + d(x)v v δ 1 in R, has no radial positive solutions in C 1 (R ) C 2 (R \). Proof. By contradiction, let (u, v) be radial positive solution of (S p,q ). Then (u, v) satisfies (6.15) (r 1 u (r) p 2 u (r)) r [ 1 a(r) u(r) α + b(r) v(r) β], (r 1 v (r) q 2 v (r)) r [ 1 c(r) u(r) γ + d(r) v(r) δ], u () = v ()=. Arguing as in proof of Theorem 6.1, we deduce from (jjj) that there exits a nonnegative number C such that (6.16) u(r) Cr p [ a 1 u α (r)+b 1 v β (r) ], r > (6.17) v(r) Cr q [ c 1 u γ (r)+d 1 v δ (r) ], r >.

13 Positive radial solutions 279 Consequently: Case 1. α p 1 and δ q 1. From (6.16) and (6.17) we obtain (6.18) u() α u(r) α Cr p, r >, (6.19) v() δ v(r) δ Cr q, r >. Since u and v are nonincreasing, (6.18) and (6.19) lead us to a contradiction. Case 2. (p 1)(q 1) >βγ. (6.2) u(r) Cr p b 1 v β (r), r >, (6.21) v(r) Cr q c 1 u γ (r), r >. Thus, from (6.2) and (6.21) (6.22) (v(r)) ()() βγ q()+pγ Cr, r >, (6.23) (u(r)) ()() βγ p()+qβ Cr, r >. By an argument like that in Case 1, (6.22) and (6.23), provide a contradiction. This concludes the proof of Theorem 6.2. Acknowledgment. The author is most grateful to a referee for careful and constructive comments on an earlier version of this paper. References [1] A. Ahammou, On the existence of bounded solutions of nonlinear elliptic systems, to appear in International J. Math. and Math. Science. [2] L. Boccardo, J. Fleckinger and F. de Thelin, Elliptic systems with various growth, Reaction Diffusion Systems, (Trieste 1995), Lecture otes in Pure and Applied Math., no. 194, Marcel Dekker, ew York, 1998, p 59 66, Zbl [3] P. Clement, R. Manásevich and E. Mitidieri, Positive solutions for quasilinear system via blow up, Comm. Partial Differential Equations 18 (1993), , Zbl [4] P. Clement, R. Manásevich and E. Mitidieri, Some existence and non-existence results for a homogeneous quasilinear problem, Asymptotic Anaysis 17 (1998), 13 29, Zbl [5] A. Castro, A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a Ball, Proc. Amer. Math. Soc. 11 (1987), 57 64, Zbl [6] A. El Hachimi and F. de Thelin, Infinité de solutions radiales pour un probleme elliptique superlineaire dans une boule, C. R. Acad. Sci., Paris, 315, no.11, (1992), , Zbl [7] F. de Thelin, and J. Velin, Existence and non existence of nontrivial solutions for some nonlinear elliptic systems, Matematica Univ. Compl. Madrid 6 (1993), , Zbl [8] P. Felmer, R. Manásevich and F. de Thelin, Existence and uniqueness of positive solutions for certain quasilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), , Zbl [9] J. Fleckinger, J. Hernandez and F. de Thelin, On maximum principales and existence of positive solutions for some cooperative elliptic systems, Differ. Integral Equ. 8 (1995), 69 85, Zbl [1] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), , Zbl [11] E. Mitidieri, onexistence of positive solutions of semilinear elliptic systems in R, Differ. Integral Equ. 9 (1996), , Zbl

14 28 A. Ahammou [12] L. Peletier and R. Van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential and Integral Eqs. 5 (1992), , Zbl [13] J. Serrin and H. Zou, on-existence of positive solutions of the Lane-Emden systems, Differential Integral Equations. 9 (1996), , Zbl [14] J. Serrin and H. Zou, Existence of entire positive solutions of elliptic Hamiltonian systems, Comm. Partial Differ. Equations, 23 (1998), , Zbl [15] M. A. S. Souto, Sobre a Existencia de Solucões Positivas para Sistemas Couerativos não Lineare, Ph.D. Thesis, UICAMP (1992). [16] R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation, Dyn. Syst. Appl. 3 (1994), , Zbl Département des Mathématiques et Informatique Faculté des Sciences UCD, El Jadida, BP2, Maroc ahammou@ucd.ac.ma This paper is available via