PARTIAL REGULARITY FOR A LIOUVILLE SYSTEM

Let Ω ⊂ Rn be a bounded smooth open set. We prove that the singular set of any extremal solution of the system −∆u = μe , −∆v = λe in Ω, with u = v = 0 on ∂Ω, μ, λ ≥ 0, has Hausdorff dimension at most n− 10.


Introduction
In this article we consider the issue of partial regularity of extremal solutions to the Liouville system in Ω, −∆v = λe u in Ω, with Ω a bounded smooth open subset of R n , and λ, µ nonnegative parameters. This system is a generalization of the equation (2) −∆u = λe u in Ω, where λ denotes a positive parameter. It is well known that there is a maximal parameter λ * > 0 for existence of solutions of (2) and for 0 < λ < λ * there is a minimal solution u λ . As λ → λ * , λ < λ * the solution u λ converges to the socalled extremal solution, which turns out to be smooth for n ≤ 9, see [3,11]. The interested reader may find in the book [7] the developments of the theory for the last six decades, with a particular focus on stable solutions. Recently it was proved by K. Wang [13] that for n ≥ 10 the extremal solution of (2) has a singular set of dimension at most n − 10. F. Da Lio [5] obtained partial regularity for any weak stationary solution in dimension 3 (not necessarily stable). See related results for the Lane-Emden equation in [14,6].
Here we generalize the results of [13] to the system (1). For this system, M. Montenegro [12] proved the existence of a nonempty open set U in the quarter plane λ, µ > 0 such that for a couple of parameters (µ, λ) in U there is a smooth minimal solution (u, v) and no smooth solution exists if the couple is in the complement of U. Minimality means u ≤ũ and v ≤ṽ in Ω for any other smooth solution (ũ,ṽ) for the same (µ, λ).
For each slope m > 0, U intersected with the line µ = mλ is a segment {(mλ, λ) : λ ∈ (0, λ * (m))} and at the extremal point (mλ * (m), λ * (m)) ∈ ∂U there is a solution, called the extremal solution. It is defined as the limit as λ ↑ λ * (m) of the minimal solution with parameters (mλ, λ) and it may be singular. In a recent work [8], L. Dupaigne, A. Farina and B. Sirakov proved that the extremal solutions for the Liouville system (1) are smooth if n ≤ 9. C. Cowan [1] had obtained the same conclusion under the restrictions 3 ≤ n ≤ 9 and n−2 In higher dimensions this fails at least in the radial case and for λ = µ, where (1) reduces to (2).
Let us recall that en extremal solution (u, v) satisfies (1) in the sense that u, v ∈ L 1 (Ω), e u dist(·, ∂Ω), e v dist(·, ∂Ω) ∈ L 1 (Ω), and We define the singular set Σ of an extremal solution (u, v) by x ∈ Σ if there is a neighborhood W of x such that u, v are bounded in W . By elliptic regularity, u, v are then smooth in this neighborhood. Theorem 1.1. Assume n ≥ 10 and let (u, v) be an extremal solution of the Liouville system (1) and Σ be its singular set. Then the Hausdorff dimension of Σ is less or equal than n − 10.
The rest of the article is devoted to the proof of this theorem. We first recall a useful inequality which is valid for stable solutions of the system, obtained in C. Cowan, N. Ghoussoub [2] and L. Dupaigne, A. Farina, B. Sirakov [8]. We then state a comparison result between u and v. Next, we perform a Moser iteration scheme to control the growth of some integrals of e u and e v on balls. The final step is an adaptation of an argument of K. Wang [13] using an ε-regularity result. The result in this paper is also closely related to the work of L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault [9] on stable solutions of ∆ 2 u = e u in a bounded domain or entire space.

Proof of Theorem 1.1
From [12] we know that for (µ, λ) ∈ U, the associated minimal solution (u, v) of (1), which is smooth, is stable in the sense that there exist ϕ, ψ : Ω → R, smooth and positive in Ω, satisfying for some η > 0. C. Cowan, N. Ghoussoub [2] and independently L. Dupaigne, A. Farina, B. Sirakov [8] have showed that this stability condition implies the following estimate.
Lemma 2.1. Let (u, v) be a smooth stable solution of the system (1). For any ϕ in H 1 0 (Ω)

2.1.
Comparison. It will be useful later to have the following inequalities between the components of a solution of (1).
Then for any smooth solution to the Liouville system (1) we have: Then due to the maximum principle w ≤ 0 in Ω. For the first inequality in (4) Then by the maximum principlew ≥ 0 in Ω.

2.2.
Reverse Hölder inequality. The following estimate is similar to the one obtained in [8] and [9], see also [4] for the scalar case. We assume that (u, v) is a smooth stable solution of (1).

Remark 1.
Although the constant C depends on µ, λ it remains bounded as (µ, λ) approaches any extremal couple on ∂U.
Proof. Multiply −∆u = µe v by e αu ϕ 2 and integrate by parts to obtain This reads also A similar equality is valid replacing respectively u by v and µ by λ. Introducing α Ω e αv (|∇ϕ| 2 − ϕ∆ϕ), we then have We combine Hölder's inequality and the stability estimate (3) Analogously, we have the same inequality replacing u by v and µ by λ. Hence we obtain Multiplying these inequalities leads to Set δ = ( 16 α 2 − 1). This implies that either hold. Assuming that (8) is true and combining with (6) we get X ≤ CA. Using Young's inequality in (7) we obtain Y ≤ C(A+B) so that X +Y ≤ C(A+B) holds, which is (5). Assuming the validity of (9) we obtain the same conclusion.
A consequence of the previous lemma is the following. Lemma 2.5. Assume (u, v) is a stable smooth solution of (1) with parameter (µ, λ) of the form µ = mλ for some fixed m > 0. For 1 ≤ α < 5 there is C independent of λ such that Ω e αu + e αv ≤ C.
We note that C in general depends on the slope m. In this lemma we need the inequalities between u and v of Lemma 2.2. For the proof, we refer to [8] where the following was proved.
Lemma 2.6. Assume λ ≥ µ. If (u, v) is a stable smooth solution of (1) with parameter (µ, λ) of the form µ = mλ for some fixed m > 0, then for 1 ≤ α < 5 there is C independent of λ such that Ω e αu ≤ C. Lemma 2.5 follows from Lemmas 2.6 and 2.2 in the case λ ≥ µ. By a symmetric argument we obtain the same conclusion if λ ≤ µ.

ε-regularity.
A crucial step is the following ε-regularity result, whose version for stable solutions in the scalar case is due to K. Wang [13], see also [9] for a biharmonic equation with exponential nonlinearity. Lemma 2.7. Let (u, v) be an extremal solution of (1). Then there is ε 2 > 0 such that if for some r 0 > 0 with B r0 (x) ⊂ Ω one has then there is a neighborhood of x such that u, v are smooth in this neighborhood.
For the proof we need the following key step, which is adapted from [13] in the scalar case.
Proof. Let 1 ≤ α < 5. We claim that Σ ⊂ x ∈ Ω : lim sup Therefore for some r 0 > 0 so that B r0 (x) ⊂ Ω we have where ε 2 > 0 is the constant from Lemma 2.7. Then by the same lemma u, v are bounded in a neighborhood of x and hence x ∈ Σ. Since e αu + e αv ∈ L 1 (Ω) by Lemma 2.5, we obtain that H n−2α (Σ) = 0, see e.g. [7,Theorem 5.3.4]. Letting α ↑ 5 we deduce that the Hausdorff dimension of Σ is less or equal than n − 10.