Large solutions of elliptic systems of second order and applications to the biharmonic equation
Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Cecilia Yarur
aa r X i v : . [ m a t h . A P ] O c t Large solutions of elliptic systems of second order andapplications to the biharmonic equation ∗ Marie-Fran¸coise BIDAUT-VERON † Marta GARC´IA-HUIDOBRO ‡ Cecilia YARUR § June 6, 2018
Abstract
In this work we study the nonnegative solutions of the elliptic system∆ u = | x | a v δ , ∆ v = | x | b u µ in the superlinear case µδ > , which blow up near the boundary of a domain of R N , orat one isolated point. In the radial case we give the precise behavior of the large solutionsnear the boundary in any dimension N . We also show the existence of infinitely manysolutions blowing up at 0 . Furthermore, we show that there exists a global positive solutionin R N \ { } , large at 0 , and we describe its behavior. We apply the results to the signchanging solutions of the biharmonic equation∆ u = | x | b | u | µ . Our results are based on a new dynamical approach of the radial system by means of aquadratic system of order 4, introduced in [4], combined with the nonradial upper estimatesof [5].
Mathematics Subject Classification: 35J60,35B40,35C20,34A34.Key words: Semilinear elliptic systems, Boundary blow-up, Keller-Osserman estimates, Asymptotic behavior,Biharmonic equation. ∗ The first author was supported by Fondecyt 70100002 and Ecos-Conicyt C08E04, and the second and third authorwere supported by Fondecyt 1070125, Fondecyt 1070951 as well as by Ecos-Conicyt C08E04. † Laboratoire de Math´ematiques et Physique Th´eorique, CNRS UMR 6083, Facult´e des SCiences, 37200 ToursFrance. E-mail address: [email protected] ‡ Departamento de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Casilla 306, Correo 22, Santiagode Chile. E-mail address: [email protected] § Departamento de Matem´atica y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago deChile. E-mail address: [email protected] Introduction
This article is concerned with the nonnegative large solutions of the elliptic system (cid:26) ∆ u = | x | a v δ ∆ v = | x | b u µ , (1.1)in two cases: solutions in a bounded domain Ω in R N , which blow up at the boundary, that islim d ( x,∂ Ω) → u ( x ) = lim d ( x,∂ Ω) → v ( x ) = ∞ , (1.2)where d ( x, ∂ Ω) is the distance from x to ∂ Ω; or solutions in Ω \ { } which blow up at 0 :lim x → u ( x ) = ∞ or lim x → v ( x ) = ∞ . (1.3)We study the superlinear case, where µ, δ >
0, and D = µδ − > , (1.4)and a, b are real numbers such that a, b > max {− , − N } . (1.5)First we recall some well-known results in the scalar case of the Emden-Fowler equation∆ U = U Q (1.6)with Q > . Concerning the boundary blow-up problem, there exists a unique solution U in Ωsuch that lim d ( x,∂ Ω) → U ( x ) = ∞ , and near ∂ Ω U ( x ) = Cd ( x, ∂ Ω) − / ( Q − (1 + o (1)) , where C = C ( Q ). Several researchs on the more general equation∆ U = p ( x ) f ( U )have been done with different assumptions on f and on the weight p , with asymptotic expansionsnear ∂ Ω , see for instance [2], [3], [7], [9], [16], [17], [19], [20], [22]; see also [1], [10] for quasilinearequations. These results rely essentially on the comparison principle valid for this equation,and the construction of supersolutions and subsolutions.The existence and the behavior of solutions of (1.6) in Ω \ { } which blow up at 0:lim x → U ( x ) = ∞ , called large (or singular) at 0, have also been widely investigated during the last decades, seefor example [23], and the references therein. There exists a particular solution in R N \ { } whenever Q < N/ ( N −
2) or N = 1 , , given by U ∗ ( x ) = C ∗ | x | − / ( Q − , with C ∗ = C ∗ ( Q, N ) . Q ≥ N/ ( N − Q 2) or N = 2, any large solution satisfies lim | x |→ | x | / ( Q − U = C ∗ , orlim | x |→ | x | N − U = α > N > , lim | x |→ | ln | x || U = α > , if N = 2 . (1.7)There exist solutions of each type, distinct from U ∗ . Moreover, up to a scaling, there exists aunique positive radial solution in R N \ { } , such that (1.7) holds and lim | x |→∞ | x | / ( Q − U = C ∗ ,see [23] and also [4].In Section 2 we consider the blow up problem of system (1.1) at the boundary.Up to our knowledge all the known results for systems are related with systems for whichsome comparison properties hold, for example (cid:26) ∆ u = u s v δ , ∆ v = u µ v m , where s, m > δ, µ > 0, and δµ ≤ ( s − m − , of competitive type, see [13], or δ, µ < lack of a comparison principle for the system.As a consequence all the methods of supersolutions, subsolutions and comparison, valid for thecase of a single equation fail.Until now the existence of large solutions is an open question in the nonradial case. In theradial case the problem was studied in [15], without weights: a = b = 0. It was shown thatthere are infinitely many nonnegative radial solutions to (1.1) which blow up at the boundaryof a ball provided that (1.4) holds, and no blow up occurs otherwise. In particular, there existsolutions even in the case where either u or v vanishes at 0. This shows the lack of a Harnackinequality, even in the radial case. The precise behavior of the solutions was obtained in [15]for N = 1 , a = b = 0, where system (1.1) is autonomous, with an elaborate proof wich couldnot be extended to higher dimension . Our first main result solves this question in any dimension , with possible weights, andmoreover we give an expansion of order 1 of the solutions: Theorem 1.1 Let ( u, v ) be any radial nonnegative solution of (1.1) defined for r ∈ ( r , R ) , r ≥ , unbounded at r = R . Then lim r → R u ( r ) = lim r → R v ( r ) = ∞ , and u, v admit the followingexpansions near R : u ( r ) = A d ( r ) − γ (1 + O ( d ( r ))) , v ( r ) = B d ( r ) − ξ (1 + O ( d ( r ))) , (1.8) where d ( r ) = R − r is the distance to the boundary, and γ = 2(1 + δ ) D , ξ = 2(1 + µ ) D , (1.9) A = ( γ ( γ + 1)( ξ ( ξ + 1)) δ ) /D , B = ( ξ ( ξ + 1)( γ ( γ + 1)) µ ) /D . (1.10)3ur proof is essentially based on a new dynamical approach of system (1.1), initiated in [4]:we reduce the problem to a quadratic, in general nonautonomous, system of order 4, which,under the assumptions of Theorem 1.1, can be reduced to a nonautonomous perturbation ofa quadratic system of order 2. We then show the convergence of the solution of the originalsystem to a suitable fixed point by using the perturbation arguments of [18].Theorem 1.1 can be applied to sign changing solutions of some elliptic systems, in particularto the biharmonic equation, where δ = 1: Corollary 1.2 Let µ > , b ∈ R . Then any radial solution u of the problem ∆ u = | x | b | u | µ in ( r , R ) , u ( R ) = ∞ , (1.11) satisfies u ( r ) = Ad ( r ) − / ( µ − (1 + O ( d ( r ))) , (1.12) with A µ − = 8( µ + 3)( µ + 1)(3 µ − µ − − . We notice here a case where we find an explicit solution: for N > µ = N +4 N − , equation∆ u = u µ admits the solution in the ball B (0 , ,u ( r ) = C (1 − r ) (4 − N ) / , C / ( N − = N ( N − N − , and v = ∆ u = C ( N − − r ) − N/ ( N − r ) ≥ 0, and (1.8) and (1.12) hold with γ = N − , ξ = N . In Section 3 we consider the problem of large solutions at the origin, that is (1.1)-(1.3).System (1.1) admits a particular radial positive solution ( u ∗ , v ∗ ), given by u ∗ ( r ) = A N r − γ a,b , v ∗ ( r ) = B N r − ξ a,b , r = | x | , (1.13)where γ a,b = (2 + a ) + (2 + b ) δD > , ξ a,b = (2 + b ) + (2 + a ) µD > , (1.14) A DN = γ a,b ( γ a,b − N + 2) ( ξ a,b ( ξ a,b − N + 2)) δ , B DN = ξ a,b ( ξ a,b − N + 2) ( γ a,b ( γ a,b − N + 2)) µ , whenever min { γ a,b , ξ a,b } > N − , or N = 1 , . (1.15)Note that in particular γ , = γ, ξ , = ξ .The problem has been initiated in [26] and [5], see also [27]. Let us recall an importantresult of [5] giving upper estimates for system (1.1) in the nonradial case, stated for N ≥ N ≥ 1. It is not based on supersolutions, but on estimates of themean value of u, v on spheres: Keller-Osserman type estimates [5]. Let Ω be a domain of R N ( N ≥ , containing , and u, v ∈ C (Ω \ { } ) be any nonnegative subsolutions of (1.1), that is, (cid:26) − ∆ u + | x | a v δ ≤ , − ∆ v + | x | b u µ ≤ , ith µ, δ satisfying (1.4). Then there exists C = C ( a, b, δ, µ, N ) such that near x = 0 ,u ( x ) ≤ C | x | − γ a,b , v ( x ) ≤ C | x | − ξ a,b . (1.16)Moreover, one finds in [5] a quite exhaustive study about all the possible behaviors of thesolutions (radial or not) in Ω \ { } .Here we complete those results by proving the existence of local radial solutions large at0 of each of the types described in [5], see Propositions 3.2, 3.4 in Section 3. By using theseresults, we obtain our second main result in this work, which is the following global existencetheorem: Theorem 1.3 Assume that N ≥ and that (1.15) holds. Then there exists a radial positiveglobal solution of system (1.1) in R N \ { } , large near 0, unique up to a scaling, such that lim r →∞ r γ a,b u = A N , lim r →∞ r ξ a,b v = B N ; (1.17) and, for N > , and up to a change of u, µ, a , into v, δ, b , when δ < N + aN − , it satisfies lim r → r N − u = α > , lim r → r N − v = β > , if µ < N + bN − , lim r → r ( N − µ − (2+ b ) v = β > , if µ > N + bN − , lim r → r N − | ln r | − v = β > , if µ = N + bN − , and for N = 2 , lim r → | ln r | − u = α > , lim r → | ln r | − v = β > . Our proof also relies on the dynamical approach of system (1.1) in dimension N by aquadratic autonomous system of order 4, given in [4]. Finally we give an application to thebiharmonic equation: Corollary 1.4 Let N > . Assume that < µ < N +2+ bN − . There exists a positive globalsolution, unique up to a scaling, of equation ∆ u = | x | b u µ in R N \ { } , such that lim r → r N − u = α > , lim r →∞ r (4+ b ) / ( µ − u = C, where C µ − = (4 + b )( N + 2 + b − ( N − µ ) (2 µ + 2 + b )( N + b − ( N − µ ) ( µ − − . This section is devoted to the study of the boundary blow up problem for nonnegative radialsolutions of (1.1). We begin by observing that system (1.1) admits a scaling invariance: if ( u, v )is a solution, then for any θ > r ( θ γ a,b u ( θr ) , θ ξ a,b v ( θr )) , (2.1)where γ a,b , ξ a,b are defined in (1.14), is also a solution.5 .1 Existence and estimates of large solutions We say that a nonnegative solution ( u, v ) of (1.1) defined in (0 , R ) is regular at 0 if u, v ∈ C (0 , R ) ∩ C ([0 , R )). Then u, v ∈ C ([0 , R )) when a, b ≥ − 1, and moreover u ′ (0) = v ′ (0) = 0when a, b > − 1, and u, v ∈ C ([0 , R )) when a, b ≥ Proposition 2.1 Assume (1.5) and only that D = δµ − = 0 . Then for any u , v ≥ , thereexists a unique local regular solution ( u, v ) with initial data ( u , v ) . The result follows from classical fixed point theorem when u , v > 0, by writing the problemin an integral form: u ( r ) = u + Z r τ − N Z τ θ N − a v δ ( θ ) dθ, v ( r ) = v + Z r τ − N Z τ θ N − b u µ ( θ ) dθ. In the case u > v , the existence can be obtained from the Schauder fixed point theorem,and the uniqueness by using monotonicity arguments as in [15]. We give an alternative proofin Section 3, using the dynamical system approach introduced in [4], which can be extended tomore general operators.Next we show that all the nontrivial regular solutions blow up at some finite R > 0, andgive the first upper estimates for any large solution. Our proofs are a direct consequence ofestimates (1.16). Proposition 2.2 (i) Assume (1.4) and (1.5). For any regular nonnegative solution ( u, v ) (0 , , there exists R such that u and v are unbounded near R .(ii) Any solution ( u, v ) which is nonnegative in an interval ( r , R ) and unbounded at R ,satisfies lim r → R u = lim r → R v = lim r → R u ′ = lim r → R v ′ = ∞ . (2.2) and there exists C = C ( N, δ, µ ) > such that near r = R , u ( r ) ≤ C ( R − r ) − γ , v ( r ) ≤ C ( R − r ) − ξ . (2.3) Proof. (i) Let ( u, v ) be any nontrivial regular solution. Suppose first that v > 0. Then from(1.1), r N − u ′ is positive for small r , and nondecreasing, hence u is increasing. If the solution isentire, then it satisfies (1.16) near ∞ : indeed by the Kelvin transform, the functions u ( x ) = | x | − N u ( x/ | x | ) , v ( x ) = | x | − N v ( x/ | x | ) , satisfy in B (0 , \ { } the system ( − ∆ u + | x | a v δ = 0 , − ∆ v + | x | b u µ = 0 , a = ( N − δ − ( N + 2 + a ) , b = ( N − µ − ( N + 2 + b ), and γ a,b , ξ a,b are replaced by N − − γ a,b , N − − ξ a,b . Then the estimate (1.16) for ( u, v ) implies the one for ( u, v ) and thus u tends to 0 at ∞ , which is contradictory. Furthermore, from u ≤ u + r a (2 + a )( N + a ) v δ , v ≤ v + r b (2 + b )( N + b ) u µ ,u and v blow up at the same point R > r N − u ′ is increasing, it has a limit as r → R . If this limit is finite, then u ′ isbounded, implying that u has a finite limit; this contradicts our assumption. Thus (2.2) holds.By (2.1) we can assume R = 1 and make the transformation r = Ψ( s ) = (cid:26) (1 + ( N − s ) − / ( N − , if N = 2 ,e − s , if N = 2 , (2.4)(in particular r = 1 − s if N = 1), so that s describes an interval (0 , s ], s > 0, and we get thesystem (cid:26) u ss = F ( s ) v δ v ss = G ( s ) u µ (2.5)with F ( s ) = r N − a , G ( s ) = r N − b ; (2.6)hence lim s → F = lim s → G = 1. Then (cid:26) − u ss + v δ ≤ − v ss + u µ ≤ , s ], thus from the Keller-Osserman estimates (1.16), there exists C = C ( N, δ, µ ) > u ( s ) ≤ Cs − γ , v ( s ) ≤ Cs − ξ , near s = 0 and (2.3) follows. In this section we prove Theorem 1.1. Consider a solution blowing up at R = 1. In the case of dimension N = 1, and a = b = 0, wehave that F ≡ G ≡ (cid:26) u ss = v δ v ss = u µ . (2.7)Following the ideas of [4], we are led to make the substitution X ( t ) = − su s u , Y ( t ) = − sv s v , Z ( t ) = sv δ u s , W ( t ) = su µ v s , t = ln s , t describes ( −∞ , t ], and we obtain the autonomous system X t = X [ X + 1 + Z ] ,Y t = Y [ Y + 1 + W ] ,Z t = Z [1 − δY − Z ] ,W t = W [1 − µX − W ] . (2.8)We study the solutions in the region where X, Y ≥ Z, W ≤ 0. In this region system (2.8)admits two fixed points O = (0 , , , , M , = ( γ, ξ, − − γ, − − ξ ) (2.9)where γ and ξ are defined in (1.9). We intend to show that trajectories associated to the largesolutions converge to M , . Observe that system (2.7) has a first integral, which is a crucialpoint in what follows: u s v s − u µ +1 µ + 1 − v δ +1 δ + 1 = C, equivalently e − t uv ( XY + XZδ + 1 + Y Wµ + 1 ) = C. Since any large solution at r = 1 satisfies lim r → u = lim r → v = ∞ , we obtain XY + XZδ + 1 + Y Wµ + 1 = o ( e t )as t → −∞ . Thus, eliminating W , we get the nonautonomous system of order 3 X t = X [ X + 1 + Z ] ,Y t = Y [ Y + 1] − ( µ + 1) X ( Y + Zδ +1 ) + o ( e t ) ,Z t = Z [1 − δY − Z ] . (2.10)which appears as a perturbation of system X t = X [ X + 1 + Z ] ,Y t = Y [ Y + 1] − ( µ + 1) X ( Y + Zδ +1 ) ,Z t = Z [1 − δY − Z ] . (2.11)Moreover, by using a suitable change of variables, system (2.10) reduces to a nonautonomoussystem of order 2 , and we can show that the last system behaves like an autonomous one. Thenwe come back to the initial system and deduce the convergence.In the case N ≥ a, b not necessarily equal to 0, we first reduce the problem to a systemsimilar to (2.8), but nonautonomous, and we prove that it is a perturbation of (2.8). Moreoverwe produce an identity that plays the role of a first integral, allowing us to reduce to a doubleperturbation of (2.11). We manage with the two perturbations in order to conclude.8 .2.2 Steps of the proof Our proof relies strongly in a result due to Logemann and Ryan, see [18]. We state it below forthe convenience of the reader. Theorem 2.3 [18, Corollary 4.1] Let h : R + × R M → R M be of Carath´eodory class. Assumethat there exists a locally Lipschitz continuous function h ∗ : R M → R M such that for all compact C ⊂ R M and all ε > , there exists T ≥ such that sup c ∈ C ess sup τ ≥ T || h ( τ, x ) − h ∗ ( x ) || < ε Assume that x is a bounded solution of equation x τ = h ( τ, x ) on R + such that x (0) = x . Then the ω -limit set of x is non empty, compact and connected, and invariant under the flowgenerated by h ∗ . The proof of Theorem 1.1 requires some important lemmas. By scaling we still assume that R = 1. Lemma 2.4 Let ( u, v ) be any fixed solution of system (1.1) in [ r , , unbounded at . Let usset t = log s , where s = Ψ − ( r ) is defined in (2.4). Let F, G be defined by (2.6). Then thefunctions X ( t ) = − su s u > , Y ( t ) = − sv s v > , Z ( t ) = sF ( s ) v δ u s < , W ( t ) = sG ( s ) u µ v s < , (2.12) satisfy the (in general nonautonomous) system X t = X [ X + 1 + Z ] ,Y t = Y [ Y + 1 + W ] ,Z t = Z [1 − δY − Z − α ( t )] ,W t = W [1 − µX − W − β ( t )] , (2.13) where α ( t ) = 2 N − a N − e t e t , β ( t ) = 2 N − b N − e t e t . (2.14) Moreover we recover u, v by the relations u = s − γ F − D G − δD ( | Z | X ) D ( | W | Y ) δD , v = s − ξ F − µD G − D ( | W | Y ) D ( | Z | X ) µD . (2.15) Proof. Since ( u, v ) is unbounded, (2.2) holds. We make the substitution (2.4), which leadsto system (2.5), with F, G given by (2.6). Clearly we can assume that u s < v s < , s ], lim s → | u s | = lim s → | v s | = lim s → u = lim s → v = ∞ . Then we can define X, Y, Z, W by (2.12) and we obtain system (2.13) with α ( t ) = − s F ′ ( s ) F ( s ) , β ( t ) = − s G ′ ( s ) G ( s ) ;then (2.14) follows, and we deduce (2.15) by straight computation.Next we prove that system (2.13) is a perturbation of the corresponding autonomous system(2.8): 9 emma 2.5 Let N ≥ . Under the assumptions of Theorem 1.1, there exist k > and ¯ t < t such that /k ≤ X, Y, | Z | , | W | ≤ k for t ≤ ¯ t. (2.16) Moreover, setting XY + XZδ + 1 + Y Wµ + 1 = ̟ ( t ) µ + 1 , (2.17) we have ̟ ( t ) = O ( e t ) as t → −∞ . Proof. We establish some integral inequalities, playing the role of a first integral, then we usethem to prove (2.16), and finally we deduce the behavior of ̟ .(i) Integral inequalities. Let σ , θ ∈ R and set H σ,θ ( s ) = r − N (cid:18) u s v s − F ( s ) v δ +1 δ + 1 − G ( s ) u µ +1 µ + 1 − σvu s + θuv s N − s (cid:19) = r − N uve − t (cid:18) XY + X ( Z + ¯ α ( t )) δ + 1 + Y ( W + ¯ β ( t )) µ + 1 (cid:19) , where ¯ α ( t ) = σ ( δ + 1) s N − s and ¯ β ( t ) = θ ( µ + 1) s N − s . It can be easily verified that H ′ σ,θ ( s ) = ( N − − σ − θ ) u s v s + F ( s ) v δ +1 δ + 1 ( N + a − σ ( δ +1))+ G ( s ) u µ +1 µ + 1 ( N + b − θ ( µ +1)) . (2.18)By choosing first the constants σ = σ > θ = θ > H ′ σ ,θ ( s ) < H σ ,θ ( s ) ≥ − C for some t C > 0; next choosing σ = σ < θ = θ < H ′ σ ,θ ( s ) > H σ ,θ ( s ) ≤ C for some C > 0. Hence, there exists functions ¯ α i ( t ) , ¯ β i ( t ) , i = 1 , , which are O ( e t ) as t → −∞ , and such that XY + X ( Z + ¯ α ( t )) δ + 1 + Y ( W + ¯ β ( t )) µ + 1 ≥ − C r N − e t uv (2.19) XY + X ( Z + ¯ α ( t )) δ + 1 + Y ( W + ¯ β ( t )) µ + 1 ≤ C r N − e t uv . (2.20)(ii) Estimates from below in (2.16). Using that u ss ≤ v δ and multiplying by 2 u s < u s ) s ≥ v δ u s = (2 v δ u ) s − δv δ − uv s > (2 v δ u ) s since v s < , s ], hence u s − v δ u ≤ C = ( u s − v δ u )( s ); since lim s → v δ u = ∞ , it followsthat u s ≤ (5 / v δ u on (0 , s ] , for sufficiently small s . Using the same method for the secondequation, we obtain from (2.12) that X ( t ) ≤ | Z ( t ) | , Y ( t ) ≤ | W ( t ) | , on ( −∞ , t ] . (2.21)10lso, from the generalized L’Hˆopital’s rule,lim s → | Z | F = lim s → sv δ − u s ≤ lim s → δsv δ − v s + v δ − u ss = lim s → (cid:18) δY − F (cid:19) , and by symmetry δ lim s → Y ≥ s → | Z | , µ lim s → X ≥ s → | W | . (2.22)Suppose now that lim t →−∞ X = 0. From (2.22), lim t →−∞ X ≥ /µ , hence there is a sequence { t n } → −∞ of local minima of X such that lim n →∞ X ( t n ) = 0, and from the definition of X in (2.12), X ( t n ) > n sufficiently large. At each t n we have that X t ( t n ) = 0 and X tt ( t n ) ≥ 0. From (2.13), using that X ( t n ) = 0, we have that X ( t n ) + 1 = | Z ( t n ) | and hence | Z ( t n ) | > 1. Since X tt ( t n ) = X ( t n ) Z t ( t n ), it follows that Z t ( t n ) ≥ 0, and thus, from the thirdequation in (2.13), 1 − δY ( t n ) + | Z ( t n ) | ≤ 0, implying1 ≤ | Z ( t n ) | ≤ δY ( t n ) + α ( t n ) . (2.23)From (2.19) and (2.21), we deduce Y ≤ Y ( ¯ β ( t ) + ( µ + 1) X ) + 3( µ + 1) X ¯ α ( t ) δ + 1 + O ( e t ) , hence lim n →∞ Y ( t n ) = 0, which contradicts (2.23). We conclude that lim t →−∞ X > 0, andsimilarly for Y , thus X, Y, | Z | , | W | are bounded from below.(iii) Estimates from above. From (2.3), s γ u and s ξ v are bounded as s → , thus from (2.15) and(2.21), X Y δ is bounded as t → ∞ . Since X, Y are bounded from below, they are boundedfrom above, and then also | Z | and | W | , from (2.22), hence (2.16) holds.(iv) Conclusion. From (2.19), (2.20), since X, Y are bounded and | ¯ α i ( t ) | , | ¯ β i ( t ) | ≤ Ce t , XY ≥ X | Z | δ + 1 + Y | W | µ + 1 − C e t and XY ≤ X | Z | δ + 1 + Y | W | µ + 1 + C e t , (2.24)for some C , C > 0. Then we deduce (2.17).Next we show that a convenient combination of our solution ( X, Y, Z, W ) satisfies a systemof order 2. We have Lemma 2.6 Under the assumptions of Theorem 1.1, and with the above notations, let x ( τ ) = − X ( t ) Z ( t ) , y = − Y ( t ) Z ( t ) , τ = − Z ¯ tt Z ( σ ) dσ. (2.25) Then ( x, y ) lies in the region R := { ( x, y ) | /k ≤ x ≤ k , δ + 1 + 12( µ + 1) k ≤ y ≤ k } for τ ≥ ˜ τ > , and satisfies (cid:26) x τ = x ( − x − δy + 2) + ̟ ( τ ) y τ = ( δ +1 − y )(( δ + 1) y − ( µ + 1) x ) + ̟ ( τ ) , (2.26) where ̟ ( τ ) = O ( e − Kτ ) and ̟ ( τ ) = O ( e − Kτ ) for some K > , as τ → ∞ . roof. We first reduce system (2.13) to a system of order 3: from relation (2.17) we eliminate W in the system (2.13) and obtain X t = X [ X + 1 + Z ] ,Y t = Y [ Y + 1] − ( µ + 1) X ( Y + Zδ +1 ) + ̟ ( t ) ,Z t = Z [1 − δY − Z − α ( t )] , which is a perturbation of system (2.11). Next, defining x = − XZ , y = − YZ , we get the system ( x t = Z [ x (2 − x − δy ) + ̟ ] y t = Z h ( δ +1 − y )(( δ + 1) y − ( µ + 1) x ) + ̟ i with ̟ = − α ( t ) XZ = O ( e t ) , ̟ = ̟ ( t ) − α ( t ) YZ = O ( e t ) , (2.27)from Lemma 2.5, and then Z t = Z (1 + Z ( δy − 1) + α ( t )) . (2.28)and τ ( t ) defined by (2.25) for t ≤ ¯ t describes [0 , ∞ ) as t describes ( −∞ , ¯ t ], and τ / k ≤ | t | ≤ kτ for t ≤ ¯ t . Hence we deduce (2.26), and the estimates of ̟ , ̟ . Notice that 1 /k ≤ x, y ≤ k for any τ ≥ τ ≥ ˜ τ > ,y − δ + 1 = 1 XZ ( Y Wµ + 1 + o (1)) ≥ µ + 1) k , ending the proof.Hence system (2.26) appears as an exponential perturbation of an autonomous system thatwe study now: Lemma 2.7 Consider the system (cid:26) x τ = x (2 − x − δy ) y τ = ( y − δ +1 )(( µ + 1) x − ( δ + 1) y ) . (2.29) The fixed points of system (2.29) are O = (0 , , and j = (cid:18) , δ + 1 (cid:19) , ℓ = (cid:18) δ + 2 δ + 1 , δ + 1 (cid:19) , m = ( x , y ) = (cid:18) δ + 1) µδ + 2 δ + 1 , µ + 1) µδ + 2 δ + 1 (cid:19) , and m is a sink. Any solution of the system (2.29) which stays in the region R converges tothe fixed point m as τ → ∞ . Proof. The point m is a sink: the eigenvalues of the linearized system of (2.29) at m are theroots ℓ , ℓ of equation ℓ + δµ + 3 + 2 µ + 2 δµδ + 2 δ + 1 ℓ + 2 µδ + 2 µ + 1 µδ + 2 δ + 1 = 0 , equivalently ( γ + 1) ℓ + ( γ + ξ + 1) ℓ + 2( ξ + 1) = 0 , (2.30)12nd they have negative real part. Next we show that (2.29) has no limit cycle in (0 , ∞ ) × (1 / ( δ + 1) , ∞ ). Let B = x p ( y − δ +1 ) − q , where p, q are parameters. Writing (2.29) under theform x τ = F ( x, y ), y t = G ( x, y ), we obtain ∇ · ( B ( F , G )) = B x x τ + B η y τ + B ( F x + G y ) := M B , where M = ( µ − − p − q ( µ + 1)) x − ( pδ − q ( δ + 1) + 3 δ + 2) ( y − δ + 1 ) + p ( δ + 2) + q ( δ + 1) + 1 δ + 1 . Choosing q = µδ +2 δ +2 µδ +2 δ +1 and p = µ − − q ( µ + 1), we find that( δ + 1) M = − ( µδ + 2 µ + 1 µδ + 2 δ + 1 + δ + 2) < . Hence, by the Bendixson-Dulac Theorem, system (2.29) has no limit cycle. From the Poincar´e-Bendixon Theorem, the ω -limit set Γ of any solution of (2.29) lying in R is fixed point, of aunion of fixed points and connecting orbits. But m is the unique fixed point in R . Then anysolution in R converges to m as τ → ∞ . Remark 2.8 It is easy to prove that there exists a connecting orbit joining the two points ℓ and m , but it is not located in R . We can now conclude. Proof of Theorem 1.1. (i) Convergence for system (2.26). From Proposition 2.3, the ω -limitset Σ of our solution ( x, y ) of (2.26) is nonempty, compact, connected and contained in R , andΣ = S ℓ ∈ Σ ,τ ≥ ˜ τ ϕ ( τ, ℓ ) , where ϕ ( τ, ℓ ) denotes the trajectory of (2.29) such that ϕ (˜ τ , ℓ ) = ℓ . Sincelim τ →∞ ϕ ( τ, ℓ ) = m , there holds m ∈ Σ. Since m is a sink of (2.29), then from the standardstability theory, see for example [6, Theorem 3.1, page 327], ( x, y ) converges to m .(ii) Convergence for system (2.13) . By setting g ( t ) = 1 /Z , we find from (2.28) that g ′ +(1 − α ) g =1 − δy, hence by L’Hˆopital’s rule,lim t →−∞ Z = lim t →−∞ ( e R t ¯ t (1 − α ) ) ′ ( ge R t ¯ t (1 − α ) ) ′ = lim t →−∞ − α − δy = − µδ + 2 δ + 1 µδ − − (1 + γ ) = Z . Hencelim t →−∞ X = − lim t →−∞ xZ = 2( δ + 1) µδ − γ = X , lim t →−∞ Y = lim t →−∞ yZ = 2( µ + 1) µδ − ξ = Y . Finally, from (2.17), we obtain lim t →−∞ W = − (1 + ξ ) = W . That means ( X, Y, Z, W )converges to M , defined at (2.9). Then from (2.15) we deduce the estimates u ( r ) = A d − γ (1 + o (1)) , v ( r ) = B d − ξ (1 + o (1))where A , B are given by and (1.10).(iii) Expansion of u and v. We first consider system (2.26). Setting x = x + ˜ x, y + ˜ y, we finda system of the form (˜ x τ , ˜ y τ ) = A (˜ x, ˜ y ) + Q (˜ x, ˜ y ) + ( ̟ , ̟ )13here (˜ x, ˜ y ) → (0 , , the eigenvalues ℓ , ℓ of A satisfy max(Re( ℓ , ℓ )) = − m < − / ( γ +1) , and Q is quadratic and ̟ ( τ ) , ̟ ( τ ) = O ( e − Kτ ) . There exists an euclidian structure with a scalarproduct where hA (˜ x, ˜ y ) , (˜ x, ˜ y ) i ≤ − m k (˜ x, ˜ y ) k . Then the function τ η ( τ ) = k (˜ x, ˜ y ) k ( τ )satisfies an inequality of the type η τ ≤ − ( m − ε ) η + Ce − Kτ for any ε > τ large enough.Then η ( τ ) = O ( e − Kτ ) + O ( e − ( m − ε ) τ ) . (2.31)Then the convergence of ( x, y ) to ( x , y ) is exponential. From (2.28), the convergence of Z to Z is exponential. Writing τ under the form τ = c + Z t + Z ∞ t ( Z − Z ) , we deduce that τ = c + Z t + O ( e kt ) for some k > . From (2.27) we obtain that ̟ , ̟ = O ( e − K τ ) with K = 1 / | Z | ; taking K = K = 1 / ( γ + 1) in (2.31), we find that η ( τ ) = O ( e − K τ ) = O ( e t ) , because m > K . Then from (2.28) we deduce that | Z − Z | = O ( e t ) , andthen from (2.25), | X − X | + | Y − Y | = O ( e t ) , and in turn | W − W | = O ( e t ) from (2.17).Finally we come back to u and v by means of (2.15): recalling that s = e t and r = 1 + O ( s ) as s → , we deduce that u ( r ) = A s − γ (1 + O ( s )) , v ( r ) = B s − ξ (1 + O ( s ))and the expansion (1.8) follows from (2.4). Proof of Corollary 1.2. Let u be a radial solution of (1.11). Then u and v = ∆ u satisfy (cid:26) ∆ u = v ∆ v = | x | b | u | µ and then u ( r ) > r , R ) and u ( R ) = ∞ . Integrating twice the second equation in thissystem, we have that lim r → R v ( r ) = ∞ and Theorem 1.1 applies. Here we suppose a = b = 0. By scaling, for any ρ > ρ . Let us call ρ ( u , v ) the blow-up radius of a regular solution with initial data ( u , v ). From(2.1), we find ρ ( λ γ u , λ ξ v ) = λ − ρ ( u , v ) . Then for any ( u , v ) ∈ S there is a unique λ such that ρ ( λ γ u , λ ξ v ) = 1 . Thus there existinfinitely many solutions blowing up at R = 1, including in particular two unique solutions withrespective initial data (¯ u , 0) and (0 , ¯ v ). Using monotonicity properties, it was shown in [15]that the set S = n ( u , v ) ∈ [0 , ∞ ) × [0 , ∞ ) : lim r → u = lim r → v = ∞ o is contained in [0 , ¯ u ] × [0 , ¯ v ]. Next we give some properties of S extending some results of [15]to higher dimensions. Proposition 2.9 Let N ≥ . If min { δ, µ } ≥ , then S is a simple curve joining the two points (¯ u , and (0 , ¯ v ) . roof. We claim that the mapping ( u , v ) ∈ [0 , ∞ ) × [0 , ∞ ) \ { , } 7−→ ρ ( u , v ) is continuous.As in [11] this will follow from our global estimates.(i) The function ρ is lower semi-continuous. Indeed the local existence is obtained by thefixed point theorem of a strict contraction, since min { δ, µ } ≥ 1, then we have local continuousdependence of the initial conditions, even if u = 0 or v = 0, and the result follows classically.(ii) The function ρ is upper semi-continuous. We can start from a point r > u , ˜ v ), considering any solution (˜ u, ˜ v ) equal to (˜ u , ˜ v ) at r ,with blow-up point ˜ ρ , for any ˜ r > ˜ ρ , any solution ( u, v ) starting from r with data sufficientlyclose to (˜ u , ˜ v ), blows up before ˜ r : suppose that it is false, then there exists a sequence ofpositive solutions ( u n , v n ), with data (˜ u n , ˜ v n ) at r , tending to (˜ u , ˜ v ), increasing, and blowingup at ρ n ≥ ˜ r . We can assume ˜ r = 1. Making the change of variables (2.4) we get solutions ofsystem (2.5) in (0 , s ], satisfying C = C ( r , N, a, b ) (cid:26) − u ss + C v δ ≤ − v ss + C u µ ≤ u and v decreasing. In fact estimates (1.16) hold with a universal constant, in any B (0 , k ) \ { } ⊂ Ω such that the mean values of u and v on ∂B (0 , r ) are strictly monotone.Then there exists a constant C = C ( C , N, δ, µ ) such that u n ( s ) ≤ Cs − γ , v n ( s ) ≤ Cs − ξ for s ≤ s , that means u n ( r ) ≤ C ( r − N − − γ , v n ( r ) ≤ C ( r − N − − ξ for r ∈ [ r , . Passing to the limit we find that u, v are bounded at the point ˜ ρ < 1, which is contradictory.Then the claim is proved. Thus S is a curve with( u , v ) = h ρ γ (cos θ, sin θ ) cos θ, ρ ξ (cos θ, sin θ ) sin θ i , θ ∈ [0 , π/ , as a parametric representation. In [4] the authors study general quasilinear elliptic systems, and in particular the system (cid:26) − ∆ u = − ( u rr + N − r u r ) = ε r a v δ , − ∆ v = − ( v rr + N − r v r ) = ε r b u µ , (3.1)where ε = ± ε = ± 1. Near any point r where u ( r ) = 0 , u ′ ( r ) = 0 and v ( r ) = 0, v ′ ( r ) = 0 , they define X ( t ) = − ru r u , Y ( t ) = − rv r v , Z ( t ) = − ε r a v δ u r , W ( t ) = − ε r b u µ v r , (3.2)15ith t = ln r , so system (3.1) becomes X t = X [ X − ( N − 2) + Z ] ,Y t = Y [ Y − ( N − 2) + W ] ,Z t = Z [ N + a − δY − Z ] ,W t = W [ N + b − µX − W ] . (3.3)One recovers u and v by the formulas u = r − γ a,b ( | ZX | ) /D ( | W Y | ) δ/D , v = r − ξ a,b | W Y | ) /D | ZX | µ/D , (3.4)and we notice the relations γ a,b + 2 + a = δξ a,b and ξ a,b + 2 + b = µγ a,b . As mentioned in [4], system (3.3) is independent of ε i , i = 1 , 2, and thus it allows tostudy system (3.1) in a unified way. In our case ε = ε = − 1, then XZ = − r a +2 v δ /u and Y W = r b +2 u µ /v, thus we are led to study (3.3) in the region R = { ( X, Y, Z, W ) | XZ ≤ , Y W ≤ } . This system is quadratic , and it admits four invariant hyperplanes: X = 0 , Y = 0 , Z =0 , W = 0. The trajectories located on these hyperplanes do not correspond to a solution ofsystem (3.1), and they are called nonadmissible . System (3.3) has sixteen fixed points, including O = (0 , , , M = ( X , Y , Z , W ) = ( γ a,b , ξ a,b , N − − γ a,b , N − − ξ a,b ) , which is interior to R whenever (1.15) holds; it corresponds to the particular solution ( u ∗ , v ∗ )given in (1.13). Among the other fixed points, as we see below, N = (0 , , N + a, N + b ) ,R = (0 , − (2 + b ) , N + a + (2 + b ) δ, N + b ) , S = ( − (2 + a ) , , N + a, N + b + (2 + a ) µ ) , are linked to the regular solutions, and A = ( N − , N − , , , G = ( N − , , , N + b − ( N − µ ) , H = (0 , N − , N + a − ( N − δ, ,P = ( N − , ( N − µ − − b, , ( N + b − ( N − µ )) ,Q = (( N − δ − − a, N − , N + a − ( N − δ, , and M are linked to the large solutions near 0 . Notice that P 6∈ R for bN − < µ < N + bN − and Q 6∈ R for aN − < δ < N + aN − . We are not concerned by the other fixed points I = ( N − , , , , J = (0 , N − , , , K = (0 , , N + a, , L = (0 , , , N + b ) , which correspond to non admissible solutions, from [4], and C = (0 , − (2 + b ) , , N + b ) , D = ( − (2 + a ) , , N + a, , which can be shown as non admissible as t → −∞ . .2 Regular solutions First we give an alternative proof of Proposition 2.1. Proposition 3.1 Assume (1.5) and D = 0 . Then a solution ( u, v ) is regular with initial data ( u , v ) , u , v > (resp. ( u , , u > , resp. (0 , v ) , v > ), if and only the correspondingsolution ( X, Y, Z, W ) converges to N (resp. R , resp. S ) as t → −∞ . For any u , v ≥ , notboth , there exists a unique local regular solution ( u, v ) with initial data ( u , v ) . Proof. The proof in the case u , v > u > v , and consider any regular solution ( u, v ) with initial data ( u , . We find v ′ = u µ N + b r b (1 + o (1)) , v = u µ ( N + b )(2 + b ) r b (1 + o (1)) , ( r N − u ′ ) ′ = u µ δ r N − a +(2+ b ) δ (( N + b )(2 + b )) δ (1 + o (1)) , u ′ = u δµ r a +(2+ b ) δ (( N + b )(2 + b )) δ ( N + a + (2 + b ) δ ) (1 + o (1));then from (3.2) the corresponding trajectory ( X, Y, Z, W ) converges to R as t → −∞ . Nextwe show that there exists a unique trajectory converging to R . We write R = (cid:0) , ¯ Y , ¯ Z, ¯ W (cid:1) = (0 , − (2 + b ) , N + a + (2 + b ) δ, N + b ) . Under our assumptions it lies in R . Setting Y = ¯ Y + ˜ Y , Z = ¯ Z + ˜ Z, W = ¯ W + ˜ W , thelinearization at R gives X t = λ X, ˜ Y t = ¯ Y h ˜ Y + ˜ W i , Z t = ¯ Z h − δ ˜ Y − ˜ Z i , W t = ¯ W h − µX − ˜ W i ;the eigenvalues are λ = 2 + a + δ (2 + b ) > , λ = − (2 + b ) < , λ = − ¯ Z < , λ = − ( N + b ) < . The unstable manifold V u has dimension 1 and V u ∩ { X = 0 } = ∅ , hence there exist preciselyone admissible trajectory such that X < Z > 0. Moreover it satisfieslim t →−∞ e − λ t X = C > , lim t →−∞ Y = ¯ Y , lim t →−∞ Z = ¯ Z, lim t →−∞ W = ¯ W . Then from (3.4) u has a positive limit u , and v = O ( e t ) , thus v tends to 0; then ( u, v ) is regularwith initial data ( u , u , 0) and the uniqueness stillholds. Similarly the solutions with initial data (0 , v ) correspond to S . Next we prove the existence of different types of local solutions large at 0, by linearizationaround the fixed points A , G , H , P , Q . For simplicity we do not consider the limit cases,where one of the eigenvalues of the linearization is 0, corresponding to behaviors of u, v oflogarithmic type. All the following results extend by symmetry, after exchanging u, δ, a, γ a,b and v, µ, b, ξ a,b . 17 roposition 3.2 Assume N > .(i) If δ < N + aN − and µ < N + bN − , then there exist solutions ( u, v ) to (1.1) such that lim r → r N − u = α > , lim r → r N − v = β > . (3.5) If δ > N + aN − or µ > N + bN − , there exist no such solutions.(ii) Let γ a,b > N − and let µ < bN − or µ > N + bN − . Then there exist solutions ( u, v ) of (1.1)such that lim r → r N − u = α > , lim r → r ( N − µ − (2+ b ) v = β ( α ) > , (3.6) with β ( α ) = α µ / (( N − µ − N − b )(( N − µ − − b ) . If γ a,b < N − , there exist no suchsolutions.(iii) If µ < bN − then there exist solutions ( u, v ) of (1.1) such that lim r → r N − u = α > , lim r → v = β > . (3.7) If µ > bN − there exist no such solutions. Proof. (i) We study the behaviour of the solutions of (3.3) near A as t → −∞ . The lineariza-tion at A gives, with X = N − X, Y = N − Y , ˜ X t = ( N − h ˜ X + Z i , ˜ Y t = ( N − h ˜ Y + W i , Z t = λ Z, W t = λ W, with eigenvalues λ = λ = ( N − > , λ = N + a − ( N − δ, λ = N + b − ( N − µ. If δ < N + aN − and µ < N + bN − , then we have λ , λ > 0; the unstable manifold V u has dimension 4 , then there exists an infinity of trajectories converging to A as t → −∞ , interior to R , thenadmissible, with Z, W < 0. The solutions satisfy lim t →−∞ e − λ t Z = Z < t →−∞ e − λ t W = W < 0, with lim t →−∞ X = lim t →−∞ Y = N − . Hence from (3.4), the corresponding solutions ( u, v )of (1.1) satisfy (3.5). If δ > N + aN − or µ > N + bN − , then λ < λ < 0, respectively, and V u has at most dimension 3, and it satisfies Z = 0 or W = 0 respectively. Therefore there is noadmissible trajectory converging at −∞ .(ii) Here we study the behaviour near P . Setting P = ( N − , Y ∗ , , W ∗ ), with Y ∗ = ( N − µ − − b, W ∗ = N + b − ( N − µ, the linearization at P gives, with X = N − X, Y = Y ∗ + ˜ Y , W = W ∗ + ˜ W ,˜ X t = ( N − h ˜ X + Z i , ˜ Y t = Y ∗ h ˜ Y + ˜ W i , Z t = λ Z, ˜ W t = W ∗ h − µ ˜ X − ˜ W i . By direct computation we obtain that the eigenvalues are λ = N − > , λ = Y ∗ , λ = N + a − δY ∗ = D ( γ a,b − ( N − , λ = − W ∗ . γ a,b > N − 2. Then λ > 0. If µ > N + bN − , then also λ , λ > V u hasdimension 4, then there exist an infinity of admissible trajectories, with Z < , converging as t → −∞ . If µ < bN − , then λ , λ < 0, thus V u has dimension 2, and V u ∩{ Z = 0 } has dimension1, thus there also exist an infinity of admissible trajectories with Z < t → −∞ . Then lim t →−∞ e − λ t Z = C < 0, lim t →−∞ X = N − 2, lim t →−∞ Y = Y ∗ and lim t →−∞ W = W ∗ ,thus (3.4), ( u, v ) satisfy (3.6). If γ a,b < N − 2, then λ < V u = V u ∩ { Z = 0 } and thereis no admissible trajectory converging when t → −∞ .(iii) We consider the behaviour near G . The linearization at G gives, with X = N − X, W = N + b − ( N − µ + ˜ W ,˜ X t = ( N − h ˜ X + Z i , Y t = (2 + b − ( N − µ ) Y,Z t = ( N + a ) Z, W t = ( N + b − ( N − µ ) h − µ ˜ X − ˜ W i , and the eigenvalues are λ = N − > , λ = 2 + b − ( N − µ, λ = N + a > , λ = ( N − µ − N − b. If µ < bN − , then λ , λ < 0. Then V u has dimension 3, and V u ∩ { Y = 0 } and V u ∩ { Z = 0 } have dimension 2. This implies that V u must contain admissible trajectories such that X > N − > Y < Z < W > N + b − ( N − µ > t →−∞ X = N − t →−∞ W = N + b − ( N − µ > 0. Moreover, lim t →−∞ e − λ t Y = C < t →−∞ e − λ t Z = C < 0, thus (3.7) follows from (3.4). Let now µ > bN − , so that λ < 0. If µ < N + bN − , then λ < V u has dimension 2, and also V u ∩ { Y = 0 } , hence V u = V u ∩ { Y = 0 } ,and there exists no admissible trajectory. If µ > N + bN − , then λ > V u has dimension 3 andalso V u ∩ { Y = 0 } , there is no admissible trajectory. Remark 3.3 If µ > N + bN − , in (ii) the two functions u, v are large near . If µ < bN − , then u islarge near and v tends to . Next we study the behavior near M , which is the most interesting one. Proposition 3.4 Assume N ≥ and (1.15). Then (up to a scaling) there exist infinitely manysolutions defined near r = 0 such that lim r → r γ a,b u = A N , lim r → r ξ a,b v = B N . Proof. Setting X = X + ˜ X, Y = Y + ˜ Y , Z = Z + ˜ Z, W = W + ˜ W , the linearized system is ˜ X t = X ( ˜ X + ˜ Z ) , ˜ Y t = Y ( ˜ Y + ˜ W ) , ˜ Z t = Z ( − δ ˜ Y − ˜ Z ) , ˜ W t = W ( − µ ˜ X − ˜ W ) . 19s described in [4], the eigenvalues are the roots λ , λ , λ , λ , of the characteristic polynomial f ( λ ) = det X − λ X Y − λ Y δ | Z | | Z | − λ µ | W | | W | − λ = ( λ − X )( λ + Z )( λ − Y )( λ + W ) − δµX Y Z W , (3.8)where we recall that X , Y > Z , W < 0. We write f in the form f ( λ ) = λ + E λ + F λ + G λ − H , with E = Z − X + W − Y ,F = ( Z − X )( W − Y ) − X Z − Y W ,G = − Y W ( Z − X ) − X Z ( W − Y ) ,H = DX Y Z W . We note that E < , F > G = − E [ Y Z + X W ] < 0. From (1.4) we have H > λ λ λ λ < 0. Hence there exist two real roots λ < < λ , with λ > max( { X , Y , | Z | , | W |} from (3.8), and two roots λ , λ , which may be real or complex. From the form of f ( λ ) in(3.8), we also see easily that if the roots λ , λ are real, they are positive. Next we claim thatRe λ > 0. Suppose Re λ = 0. Then f ( i Im λ ) = 0 , then G = E F G + E H , and thus,dividing by E ,0 = G − E F G + E H = E (cid:16) [ Y Z + X W ] + 2 [ Y Z + X W ] F − H (cid:17) , hence [ Y Z + X W + F ] = F + 4 H > F ; but Y Z + X W + F = ( X − W )( Y − Z ) ∈ (0 , F )which is a contradiction. Since Re λ is a continuous function of ( δ, µ ), it is sufficient to find avalue ( µ, δ ) satisfying (1.15) for which it is positive. Taking δ = µ , the equation in λ reducesto two equations of order 2: f ( λ ) = ( λ − X ) ( λ − | Z | ) − δ X Z = (cid:2) λ − ( X + | Z | ) λ − ( δ − X | Z | (cid:3) (cid:2) λ − ( X + | Z | ) λ + (1 + δ ) X | Z | (cid:3) , and X + | Z | > 0, thus the claim is proved. Then V u has dimension 3 and V s has dimension1. Hence the result follows. Remark 3.5 In the case N = 1 , two roots are explicit: λ = − , λ = 2 + γ + ξ , and λ , λ are the roots of equation λ − (1 + γ + ξ ) λ + 2(1 + γ )(1 + ξ ) = 0 . (3.9)20 he 4 roots are real if (1 + γ + ξ ) − γ )(1 + ξ ) ≥ , that means ( δµ + 3 + 2 µ + 2 δ ) − µδ + 2 δ + 1)( µδ + 2 µ + 1) ≥ , which is not true for δ = µ , but is true for example when δ/µ is large enough. The roots ofequation (3.9) and the roots of equation (2.30) relative to the linearization of system (2.29) at m are linked by the relations ℓ = λ / | Z | , ℓ = λ / | Z | . Indeed M = M , defined at (2.9)satisfies relation (2.17) with ̟ = 0 , thus ( X , Y , Z ) is a fixed point of system (2.11) and thelinearization of (2.11) at this point gives the eigenvalues − , λ , λ . The point m is the imageof ( X , Y , Z ) by the transformation (2.25), which divides the eigenvalues by | Z | , due to thechange in time t τ . Here we prove our second main result. Proof of Theorem 1.3. From the proof of Proposition 3.4, the linearization at M admitsa unique real eigenvalue λ < 0. From (3.8) a generating eigenvector ( u , u , u , u ) satisfies u u < u u < 0, and hence it is of the form ~u = ( − α , − β , σ , ρ ), or − ~u . Thereexist precisely two trajectories T ~u and T − ~u converging to M as t → ∞ and the convergenceof X, Y, Z, W is monotone near t = ∞ ; from (3.4), the corresponding solutions ( u, v ) of system(1.1) satisfy (1.17).We consider the trajectory T ~u corresponding to ~u. Let us show that the convergence ismonotone in all R . Notice that neither of the components can vanish, since system (1.1) is ofKolmogorov type. Near t = ∞ , X and Y are increasing, and Z, W are decreasing. Supposethat there exists a greatest value t such that X has a minimum local at t , hence X tt ( t ) = X ( t ) Z t ( t ) ≥ , Z ( t ) = N − − X ( t ) , thus Z t ( t ) ≥ t ≥ t such that Z t ( t ) = 0, and Z tt ( t ) = − δZ ( t ) Y t ( t ) ≤ , Z ( t ) = N + a − δY ( t ) , then Y t ( t ) ≤ 0. There exists t ≥ t such that Y t ( t ) = 0, and Y tt ( t ) = Y ( t ) W t ( t ) ≥ , Y ( t ) = N − − W ( t ) . There exists t ≥ t such that W t ( t ) = 0 and W tt ( t ) = − W ( t ) X t ( t ) ≤ 0. From the definitionof t , this implies t = t , and then all the conditions above imply that ( X, Y, Z, W )( t ) = M , which is impossible. Hence X stays strictly monotone, and similarly Y, Z, W also stay strictlymonotone. Since X, Y > 0, and Y, Z < 0, then T ~u is bounded, hence defined on R andconverges to some fixed point L = ( l , l , l , l ) of the system as t → −∞ and necessarily l < X , l < Y , l > Z , l > W . • Case N > 2. First we note that along T ~u we always have X, Y > N − 2. Indeed, if atsome point t we have X ( t ) = N − 2, then X t ( t ) = ( N − Z ( t ) < 0, which is contradictory.Hence the possible values for L are A , or P when µ ≥ N + bN − , or Q when δ ≥ N + aN − , since I isnonadmissible. By hypothesis, γ a,b > N − , then either µ < N + bN − or δ < N + aN − . We can assume21hat δ < N + aN − . Then Q 6∈ R , then L = A or P . When µ < N + bN − , then L = A . When µ > N + bN − , from Proposition 3.2(i), we have L = A , thus L = P . In the limit case µ = N + bN − ,we find P = A . From the linearization at A we have λ = λ = N − > , λ = N + a − ( N − δ > , λ = 0 . Coming back to the proof of Proposition 3.2(i), we find that the convergence of Z and ˜ X = X − ( N − 2) to 0 are exponential. From the fourth equation in (3.3) we see that W t + W > − /W ≤ C | t | near −∞ . Then, there exists m > W t = W ( − − µW − ˜ X ) = W ( − O ( e mt );integrating over ( t, t ), t < 0, we obtain that W ( t ) = t − + O ( t − ). In turn we estimate Y ;setting Y = ˜ Y + W , then Y t = ( N − Y + Y ( Y − W ) + W ( − µ ˜ X − W ) , and thus Y t = (( N − 2) + ε ( t )) Y + O ( t − ) , implying Y = O ( t − ) and thus Y = N − − t − + O ( t − ) . Next we find that Z t /Z = λ + t − + O ( t − ) , which yields lim t →−∞ e − λ t | t | − δ | Z | = C > . Finally, by replacing in (3.4), and deducethe behavior of u and v as claimed:lim r → r N − u = C > r → r N − | log( r ) | − v = C > . • Case N = 2. Then necessarily L = O = (0 , , , , , a, b . Since Z t = Z (2 + a − δY − Z ) and Y and Z tend to 0as t tends to −∞ , Z converges exponentially to 0, and similarly W . Since X t ≤ X , it followsthat X ≥ C | t | − near −∞ . 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