Large solutions of elliptic systems of second order and applications to the biharmonic equation

In this work we study the nonnegative solutions of the elliptic system \Delta u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu \delta>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation \Delta^2 u=|x|^{b}|u|^{\mu}. Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, combined with nonradial upper estimates.


Introduction
This article is concerned with the nonnegative large solutions of the elliptic system ∆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 is where C = C(Q). Several researchs on the more general equation have been done with different assumptions on f and on the weight p, with asymptotic expansions near ∂Ω , see for instance [2], [3], [7], [9], [16], [17], [19], [20], [22]; see also [1], [10] for quasilinear equations. 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\ {0} which blow up at 0: called large (or singular) at 0, have also been widely investigated during the last decades, see for example [23], and the references therein. There exists a particular solution in R N \ {0} whenever Q < N/(N − 2) or N = 1, 2, given by U * (x) = C * |x| −2/(Q−1) , with C * = C * (Q, N ). There exist solutions of each type, distinct from U * . Moreover, up to a scaling, there exists a unique positive radial solution in R N \ {0}, such that (1.7) holds and lim |x|→∞ |x| 2/(Q−1) U = C * , see [23] and also [4].
In Section 2 we consider the blow up problem of system (1.1) at the boundary.
On the contrary the problem (1.1)-(1.2) has been the object of very few works, because it brings many difficulties. The main one is the lack of a comparison principle for the system. As a consequence all the methods of supersolutions, subsolutions and comparison, valid for the case of a single equation fail.
Until now the existence of large solutions is an open question in the nonradial case. In the radial case the problem was studied in [15], without weights: a = b = 0. It was shown that there are infinitely many nonnegative radial solutions to (1.1) which blow up at the boundary of a ball provided that (1.4) holds, and no blow up occurs otherwise. In particular, there exist solutions even in the case where either u or v vanishes at 0. This shows the lack of a Harnack inequality, 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 could not be extended to higher dimension.
Our first main result solves this question in any dimension, with possible weights, and moreover 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 0 , R), r 0 ≥ 0, unbounded at r = R. Then lim r→R u(r) = lim r→R v(r) = ∞, and u, v admit the following expansions near R : where d(r) = R − r is the distance to the boundary, and

9)
A 1 = (γ(γ + 1)(ξ(ξ + 1)) δ ) 1/D , B 1 = (ξ(ξ + 1)(γ(γ + 1)) µ ) 1/D . (1.10) Our 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 of a quadratic system of order 2. We then show the convergence of the solution of the original system 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 particular to the biharmonic equation, where δ = 1: Then any radial solution u of the problem We notice here a case where we find an explicit solution: for N > 4 and µ = N +4 N −4 , equation ∆ 2 u = u µ admits the solution in the ball B(0, 1), In Section 3 we consider the problem of large solutions at the origin, that is (1.1)-(1.3).
The problem has been initiated in [26] and [5], see also [27]. Let us recall an important result of [5] giving upper estimates for system (1.1) in the nonradial case, stated for N ≥ 3, but its proof is valid for any N ≥ 1. It is not based on supersolutions, but on estimates of the mean value of u, v on spheres: Keller-Osserman type estimates [5]. Let Ω be a domain of R N (N ≥ 1), containing 0, and u, v ∈ C 2 (Ω\ {0}) be any nonnegative subsolutions of (1.1), that is, with µ, δ satisfying (1.4). Then there exists C = C(a, b, δ, µ, N ) such that near x = 0, Moreover, one finds in [5] a quite exhaustive study about all the possible behaviors of the solutions (radial or not) in Ω\ {0}.
Here we complete those results by proving the existence of local radial solutions large at 0 of each of the types described in [5], see Propositions 3.2, 3.4 in Section 3. By using these results, we obtain our second main result in this work, which is the following global existence theorem: and, for N > 2, and up to a change of u, µ, a, into v, δ, b, when δ < N +a and for N = 2, lim r→0 |ln r| −1 u = α > 0, lim Our proof also relies on the dynamical approach of system (1.1) in dimension N by a quadratic autonomous system of order 4, given in [4]. Finally we give an application to the biharmonic equation: There exists a positive global solution, unique up to a scaling, of equation

Large solutions at the boundary
This section is devoted to the study of the boundary blow up problem for nonnegative radial solutions 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 > 0, r → (θ γ a,b u(θr), θ ξ a,b v(θr)), where γ a,b , ξ a,b are defined in (1.14), is also a solution.

Existence and estimates of large solutions
We say that a nonnegative solution (u, v) of (1.
We first give an existence and uniqueness result for regular solutions: The result follows from classical fixed point theorem when u 0 , v 0 > 0, by writing the problem in an integral form: In the case u 0 > 0 = v 0 , 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 proof in Section 3, using the dynamical system approach introduced in [4], which can be extended to more general operators.
Next we show that all the nontrivial regular solutions blow up at some finite R > 0, and give the first upper estimates for any large solution. Our proofs are a direct consequence of estimates (1.16).  and there exists C = C(N, δ, µ) > 0 such that near r = R, Proof. (i) Let (u, v) be any nontrivial regular solution. Suppose first that v 0 > 0. Then from (1.1), r N −1 u ′ is positive for small r, and nondecreasing, hence u is increasing. If the solution is entire, then it satisfies (1.16) near ∞: indeed by the Kelvin transform, the functions 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 and v blow up at the same point R > 0.
(ii) Since r N −1 u ′ is increasing, it has a limit as r → R. If this limit is finite, then u ′ is bounded, 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 , so that s describes an interval (0, s 0 ], s 0 > 0, and we get the system

The precise behavior near the boundary
In this section we prove Theorem 1.1.

Scheme of the proof
Consider a solution blowing up at R = 1. In the case of dimension N = 1, and a = b = 0, we have that F ≡ G ≡ 1 in (2.6), and we are concerned with the system Following the ideas of [4], we are led to make the substitution where t = ln s, t describes (−∞, t 0 ], and we obtain the autonomous system (2.8) We study the solutions in the region where X, Y ≥ 0 and Z, W ≤ 0. In this region system (2.8) admits two fixed points where γ and ξ are defined in (1.9). We intend to show that trajectories associated to the large solutions converge to M 0,1 . Observe that system (2.7) has a first integral, which is a crucial point in what follows: Since any large solution at r = 1 satisfies lim r→1 u = lim r→1 v = ∞, we obtain as t → −∞. Thus, eliminating W , we get the nonautonomous system of order 3 (2.10) which appears as a perturbation of system (2.11) Moreover, by using a suitable change of variables, system (2.10) reduces to a nonautonomous system of order 2, and we can show that the last system behaves like an autonomous one. Then we come back to the initial system and deduce the convergence.
In the case N ≥ 1 or a, b not necessarily equal to 0, we first reduce the problem to a system similar to (2.8), but nonautonomous, and we prove that it is a perturbation of (2.8). Moreover we produce an identity that plays the role of a first integral, allowing us to reduce to a double perturbation of (2.11). We manage with the two perturbations in order to conclude.

Steps of the proof
Our proof relies strongly in a result due to Logemann and Ryan, see [18]. We state it below for the convenience of the reader.
Then the ω-limit set of x is non empty, compact and connected, and invariant under the flow generated by h * .
The proof of Theorem 1.1 requires some important lemmas. By scaling we still assume that R = 1. (2.4). Let F, G be defined by (2.6). Then the functions satisfy the (in general nonautonomous) system (2.14) Moreover we recover u, v by the relations 2) holds. We make the substitution (2.4), which leads to system (2.5), with F, G given by (2.6). Clearly we can assume that u s < 0 and v s < 0 on (0, s 0 ], lim s→0 |u s | = lim s→0 |v s | = lim s→0 u = lim s→0 v = ∞. Then we can define X, Y, Z, W by (2.12) and we obtain system (2.13) with 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): Proof. We establish some integral inequalities, playing the role of a first integral, then we use them to prove (2.16), and finally we deduce the behavior of ̟.
Next we show that a convenient combination of our solution (X, Y, Z, W ) satisfies a system of order 2. We have Lemma 2.6 Under the assumptions of Theorem 1.1, and with the above notations, let Then (x, y) lies in the region for τ ≥τ > 0, and satisfies
Proof. 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 which is a perturbation of system (2.11). Next, defining x = − X Z , y = − Y Z , we get the system from Lemma 2.5, and then Z t = Z(1 + Z(δy − 1) + α(t)).
Hence system (2.26) appears as an exponential perturbation of an autonomous system that we study now: Lemma 2.7 Consider the system (2.29) The fixed points of system (2.29) are O = (0, 0), and and m 0 is a sink. Any solution of the system (2.29) which stays in the region R 0 converges to the fixed point m 0 as τ → ∞.
Hence, by the Bendixson-Dulac Theorem, system (2.29) has no limit cycle. From the Poincaré-Bendixon Theorem, the ω-limit set Γ of any solution of (2.29) lying in R 0 is fixed point, of a union of fixed points and connecting orbits. But m 0 is the unique fixed point in R 0 . Then any solution in R 0 converges to m 0 as τ → ∞.

Remark 2.8
It is easy to prove that there exists a connecting orbit joining the two points ℓ 0 and m 0 , but it is not located in R 0 .
We can now conclude.

The set of initial data for blow up
Here we suppose a = b = 0. By scaling, for any ρ > 0 there exists solutions which blow up at ρ. Let us call ρ(u 0 , v 0 ) the blow-up radius of a regular solution with initial data (u 0 , v 0 ). From (2.1), we find Then for any (u 0 , v 0 ) ∈ S 1 there is a unique λ such that ρ(λ γ u 0 , λ ξ v 0 ) = 1. Thus there exist infinitely many solutions blowing up at R = 1, including in particular two unique solutions with respective initial data (ū 0 , 0) and (0,v 0 ). Using monotonicity properties, it was shown in [15] that the set Next we give some properties of S extending some results of [15] to higher dimensions.
Proof. We claim that the mapping 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 the fixed point theorem of a strict contraction, since min{δ, µ} ≥ 1, then we have local continuous dependence of the initial conditions, even if u 0 = 0 or v 0 = 0, and the result follows classically.
(ii) The function ρ is upper semi-continuous. We can start from a point r 0 > 0 instead of 0. We prove that for any positive (ũ 0 ,ṽ 0 ), considering any solution (ũ,ṽ) equal to (ũ 0 ,ṽ 0 ) at r 0 , with blow-up pointρ, for anyr >ρ, any solution (u, v) starting from r 0 with data sufficiently close to (ũ 0 ,ṽ 0 ), blows up beforer : suppose that it is false, then there exists a sequence of positive solutions (u n , v n ), with data (ũ n ,ṽ n ) at r 0 , tending to (ũ 0 ,ṽ 0 ), increasing, and blowing up at ρ n ≥r. We can assumer = 1. Making the change of variables (2.4) we get solutions of system (2.5) with u and v decreasing. In fact estimates (1.16) hold with a universal constant, in any B(0, k)\ {0} ⊂ 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 0 , N, δ, µ) such that Passing to the limit we find that u, v are bounded at the pointρ < 1, which is contradictory. Then the claim is proved. Thus S is a curve with (u 0 , v 0 ) = ρ γ (cos θ, sin θ) cos θ, ρ ξ (cos θ, sin θ) sin θ , θ ∈ [0, π/2] , as a parametric representation.
3 Behavior of system (1.1) near the origin

Formulation as a dynamical system
In [4] the authors study general quasilinear elliptic systems, and in particular the system where ε 1 = ±1, ε 2 = ±1. Near any point r where u(r) = 0, u ′ (r) = 0 and v(r) = 0, v ′ (r) = 0, they define with t = ln r, so system (3.1) becomes One recovers u and v by the formulas 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 to study system (3.1) in a unified way. In our case ε 1 = ε 2 = −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 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 of system (3.1), and they are called nonadmissible. System (3.3) has sixteen fixed points, including O = (0, 0, 0, 0). The main one is 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, and M 0 are linked to the large solutions near 0. Notice that P 0 ∈ R for 2+b N −2 < µ < N +b N −2 and Q 0 ∈ R for 2+a N −2 < δ < N +a N −2 . We are not concerned by the other fixed points

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 , if and only the corresponding solution (X, Y, Z, W ) converges to N 0 (resp. R 0 , resp. S 0 ) as t → −∞. For any u 0 , v 0 ≥ 0, not both 0, there exists a unique local regular solution (u, v) with initial data (u 0 , v 0 ).
Proof. The proof in the case u 0 , v 0 > 0 is done in [4,Proposition 4.4]. Suppose u 0 > 0 = v 0 , and consider any regular solution (u, v) with initial data (u 0 , 0). We find then from (3.2) the corresponding trajectory (X, Y, Z, W ) converges to R 0 as t → −∞. Next we show that there exists a unique trajectory converging to R 0 . We write Under our assumptions it lies in R. Setting Y =Ȳ +Ỹ , Z =Z +Z, W =W +W , the linearization at R 0 gives the eigenvalues are The unstable manifold V u has dimension 1 and V u ∩ {X = 0} = ∅, hence there exist precisely one admissible trajectory such that X < 0 and Z > 0. Moreover it satisfies Then from (3.4) u has a positive limit u 0 , and v=O(e 2t ), thus v tends to 0; then (u, v) is regular with initial data (u 0 , 0). By (2.1) we obtain existence for any (u 0 , 0) and the uniqueness still holds. Similarly the solutions with initial data (0, v 0 ) correspond to S 0 .

Local existence of large solutions near 0
Next we prove the existence of different types of local solutions large at 0, by linearization around the fixed points A 0 , G 0 , H 0 , P 0 , Q 0 . 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 of logarithmic type. All the following results extend by symmetry, after exchanging u, δ, a, γ a,b and v, µ, b, ξ a,b .
Proof. (i) We study the behaviour of the solutions of (3 with eigenvalues If δ < N +a N −2 and µ < N +b N −2 , then we have λ 3 , λ 4 > 0; the unstable manifold V u has dimension 4, then there exists an infinity of trajectories converging to A 0 as t → −∞, interior to R, then admissible, with Z, W < 0. The solutions satisfy lim Hence from (3.4), the corresponding solutions (u, v) of (1.1) satisfy (3.5). If δ > N +a N −2 or µ > N +b N −2 , then λ 3 < 0 or λ 4 < 0, respectively, and V u has at most dimension 3, and it satisfies Z = 0 or W = 0 respectively. Therefore there is no admissible trajectory converging at −∞.
(ii) Here we study the behaviour near P 0 .

Remark 3.3 If µ > N +b
N −2 , in (ii) the two functions u, v are large near 0. If µ < 2+b N −2 , then u is large near 0 and v tends to 0.
Next we study the behavior near M 0 , which is the most interesting one. N ≥ 1 and (1.15). Then (up to a scaling) there exist infinitely many solutions defined near r = 0 such that