A new critical curve for the Lane-Emden system

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.


Introduction
We consider the Lane-Emden system where N ≥ 1 and p ≥ q > 0. Introduced independently by Mitidieri [14] and Van der Vorst [21], the Sobolev critical hyperbola plays a crucial role in the analysis of (1.1). In particular, Mitidieri [15] (see also Serrin and Zou [18]) proved that (1.1) has a nontrivial radially symmetric solution if and only if (p, q) lies on or above the hyperbola i.e. when (1.2) 1 p + 1 The Lane-Emden conjecture states that such a result should continue to hold for any positive solution (not necessarily radially symmetric). See Souplet [19] and the references therein for the progress on this conjecture. In this paper we characterize the stability of radially symmetric solutions of the Lane-Emden system (1.1), the definition of which we recall now.
Let us also recall that if (1.2) holds, then is a weak solution of (1.1) provided (1.4) α = 2(p + 1) pq − 1 , β = 2(q + 1) pq − 1 and a = (ST p ) . Our main result states that the stability of a radial solution of the Lane-Emden system is determined by the position of the exponents (p, q) with respect to a new critical curve, which we christen "Joseph and Lundgren", since the exponent introduced by these authors in [11] is the intersection of the curve with the diagonal p = q .
(i) If N ≥ 11 and (p, q) lies on or above the Joseph-Lundgren critical curve i.e.
then any radially symmetric solution (u, v) of (1.1) is stable and satisfies Remark 1.4.
• The above theorem was first proved by Cowan for 1 ≤ N ≤ 10, p ≥ q ≥ 2 and (u, v) not necessarily radial. See [4]. • In the case p = q, using Remarks 1.1(a) and 2.1(a) of Souplet [19] and Farina's seminal work for the case of a single equation [9], part (ii) of the theorem readily follows. The result continues to hold for possibly nonradial solutions, assumed to be stable only outside a compact set. • In the biharmonic case q = 1, the theorem was first proved by Karageorgis [13] using the asymptotics found by Gazzola and Grunau in [12]. • In all the other cases, only partial results were known. To the authors knowledge, the state of the art for nonradial solutions is contained in the following references: Wei and D. Ye [23], Wei, Xu and Yang [22], Hajlaoui, A. Harrabi and D. Ye [10] for the biharmonic case, and Cowan [4] for the general case. We believe that the methods of the paper [8] by Goubet, Warnault and two of the authors should slightly improve the known results (and coincide with [10] in the biharmonic case). • Our result does not cover the case where one of the exponents is less than 1. • The left hand-side in (1.5) is related to the following Hardy-Rellich inequality : The optimal constant C γ in the class of radially symmetric functions ϕ = ϕ(|x|) is given by and the above infimum is never achieved. See Caldiroli and Musina [2]. We remark that the optimal constant C γ in (1.7) corresponds to the left hand-side in (1.5) with γ = α − β ∈ [0, 2).
As an immediate corollary of Theorem 1.2 and standard blow-up analysis, we obtain the following regularity result.
Then, any extremal solution to the system For the notion of extremal solution for systems, we refer to Montenegro [16]. See also Cowan [3] for partial results on general domains. The proof is a straightforward adaptation of Theorem 1.8 in [5], using the version of the blow-up technique introduced by Polacik, Quittner and Souplet [17], so we skip it.

Preliminary Results
The following three results will serve for the purpose of comparing solutions. In the lemma below, we say that a solution is strictly stable in a bounded region Ω ⊂ R N if the principal eigenvalue of the linearized equation with Dirichlet boundary conditions in Ω is strictly positive. 2 be a stable solution of (1.1). Then, given any bounded domain Ω ⊂ R N , (u, v) is strictly stable in Ω. In particular, the linearized operator satisfies the maximum principle, that is, any pair Proof. Since (u, v) is stable in R N , the linearized equation has a strict supersolution in Ω. As observed by Sweers [20] and Busca-Sirakov [1], this implies in turn that the principal eigenvalue of the linearized operator with Dirichlet boundary conditions in Ω is strictly positive and equivalently that the maximum principle holds.
In the next lemma, we say that a solution is minimal if it lies below any (local) supersolution of the same equation. See e.g. [7] for the notion of minimal solution.
Proof. Assume that (u, v) is a strictly stable solution of (2.1). By the maximum principle, In particular, there exists the minimal solution (u m , v m ) of (2.1) and Since (u, v) is strictly stable, the maximum principle holds and implies that φ, ψ ≤ 0 in Ω. It follows that φ ≡ ψ ≡ 0, that is, u = u m and v = v m .
As an immediate consequence of the two previous lemmas, we obtain We next apply Lemma 2.
is the minimal solution of (2.1) and u < u s , v < v s in B R \ B r . This last inequality together with (2.2) yield the conclusion. Proof. Since the implication (i)⇒(ii) is trivial, we only need to prove the implications Assume first that (ii) holds, that is, the singular solution (u s , v s ) is stable outside of a compact set. Thus, (u s , v s ) is stable in R N \ B r for some r > 0. By scale invariance, (u s , v s ) is stable in R N \ B ρ for all ρ > 0.

Proof of Theorem 1.2
We start this section with the following simple remark. To see this, we first note that (u, v) satisfies This implies that r −→ r N −1 u ′ (r) and r −→ r Proof. Assume by contradiction that there exists a radially symmetric solution (u, v) of (1.1) for which (3.2) fails to hold and set Since (3.2) is not fulfilled, U ′ and V ′ must change sign in (0, ∞). Indeed, otherwise U ′ < 0 or V ′ < 0 in (0, ∞) which implies (since U (∞) = V (∞) = 0) that u s ≥ u or v s ≥ v in (0, ∞). Now, the maximum principle yields u s ≥ u and v s ≥ v in (0, ∞) and this contradicts our assumption.
Assume next that (1.5) fails to hold. We establish first the following result.
Proposition 3.3. Assume (p, q) does not satisfy (1.5). Then, for any stable solution (u, v) of (1.1) we have . Also u − u s < 0 in a neighborhood of the origin and by Remark 3.1 we have u(x) − u s (x) → 0 as |x| → ∞. By the maximum principle, we deduce u − u s ≤ 0 in R N \ {0} which contradicts our assumption.
Hence u − u s and v − v s change sign on (0, ∞). Denote by r 1 (resp. r 2 ) the first sign-changing zero of u − u s (resp. v − v s ). From Corollary 2.3, u − u s (resp. v − v s ) cannot be zero in a whole neighborhood of r 1 (resp. r 2 ). Without losing generality, we may assume that r 1 ≤ r 2 .
We claim that u − u s has a second sign-changing point r 3 > r 1 . Indeed, otherwise u−u s ≥ 0 in R N \B r 1 which by the maximum principle implies that v − v s ≥ 0 in R N \ B r 2 . Therefore, u ≥ u s , v ≥ v s in R N \ B r 2 which implies that (u s , v s ) is a stable solution of (1.1) in R N \ B r 2 and thus, contradicts Proposition 2.4. Hence, there exists r 3 > r 1 a second sign-changing point of u − u s . Further, we must have r 3 ≥ r 2 for otherwise r 1 < r 3 < r 2 . Then u(r 3 ) = u s (r 3 ) and v(r 3 ) < v s (r 3 ) which by Corollary 2.3 yields u < u s , v < v s in B r 3 \ {0}. But this is impossible since u(r 1 ) = u s (r 1 ). Thus, We next claim that v − v s has a second sign-changing point r 4 > r 2 . As before, if this is not true, then v − v s ≥ 0 in R N \ B r 2 and by the maximum We show next that r 4 ≥ r 3 . Assuming the contrary we have r 2 < r 4 < r 3 . At this stage, two cases may occur: Case 1: v ≤ v s in (r 4 , r 3 ). Remark that u(r 3 ) = u s (r 3 ) and v(r 3 ) ≤ v s (r 3 ). By Corollary 2.3 we deduce u < u s in B r 3 which is impossible since u(r 1 ) = u s (r 1 ). Case 2: v − v s has a third sign-changing point ρ ∈ (r 4 , r 3 ). Then v − v s > 0 on (r 2 , r 4 ) and v − v s < 0 on (r 4 , ρ). On the other hand, and v − v s = 0 on ∂(B ρ \ B r 4 ). The maximum principle yields v − v s > 0 on (r 4 , ρ), a contradiction. We have proved that r 4 ≥ r 3 . We claim that u − u s has a third sign-changing point r 5 > r 3 . Indeed, if this is not true, then u − u s ≤ 0 in R N \ B r 3 and by the maximum principle 4 . This is clearly impossible since u(r 1 ) = u s (r 1 ). Hence, u − u s has a third sign-changing point r 5 > r 3 . If . By the maximum principle we infer that u−u s ≥ 0 in B r 5 \B r 3 which implies u−u s ≥ 0 in B r 5 \B r 1 . This contradicts the fact that r 3 ∈ (r 1 , r 5 ) is a sign-changing point of u − u s .
If r 5 > r 4 then u(r 4 ) ≤ u s (r 4 ) and v(r 4 ) = v s (r 4 ). By Corollary 2.3 we deduce u < u s , v < v s in B r 4 which is again a contradiction.
We are now ready to complete the proof of Theorem 1.2(ii). We adapt an idea introduced in [6]. Assume there exists a positive stable radially symmetric solution (u, v) of (1.1) and set By the strong maximum principle, (u, v) cannot touch (u s , v s ), so there exists a sequence {R k } converging to +∞ such that (3.7) lim k→∞ u(R k ) u s (R k ) = 1.
Define u k (r) = R α k u(R k r) , v k (r) = R β k v(R k r) r ≥ 0. By scale invariance we have (3.8) 0 < u k < u s , 0 < v k < v s in R N \ {0} and (u k , v k ) solves the Lane-Emden system (1.1) in R N \ {0}. By elliptic regularity, {(u k , v k )} converges uniformly in C 2 loc (R N \ {0}) to a solution ( u, v) of (1.1) which, in view of (3.8), also satisfies On the other hand, By the strong maximum principle we deduce that u ≡ u s in R N \ {0}. This is impossible, since u is a stable solution by construction while u s is unstable when (1.5) fails.