Polyhomogeneous expansions at the boundary for some blowing-up solutions of a class of semi-linear

elliptic equations

Piotr T. Chruściel^{∗} Département de Mathématiques Faculté des Sciences Parc de Grandmont F-37200 Tours, France

Tours preprint 136/96, July 1996

Let g be mooth Riemannian metric on a smooth compact manifold with boundary \overline{M}, consider the problem

Δ_{g}φ =| φ |^{α−1} φ ,  α > 1 ,  (0.1)
lim_{p→∂M} φ(p) = ∞ ,  (0.2)

Δ_{g} -the Laplace-Beltrami operator on functions. It was shown in [?] that one then necessarily has (cf. also [?])

lim_{p→∂M} [φ(p) x(p)^{\frac{2}{α−1}} ] = C_{α} ≡ (\frac{2(α + 1)}{(α − 1)^{2}})^{\frac{1}{α−1}} .  (0.3)

Here x(p) denotes the Riemannian distance from p to ∂M. It follows from the results of [?] that the behavior of φ near ∂M can be described in a much more precise way. More precisely, we have the following:

Theorem 0.1 Let (\overline{M}, g) be a smooth compact Riemannian manifold with boundary, let α > 1, set M = int\overline{M}. Let φ ∈ W_{2}^{p,loc}(M) satisfy

Δ_{g}φ =| φ |^{α−1} φ,

lim_{p→∂M} [φ(p) x(p)^{\frac{2}{α−1}} ] = C_{α},

where x(p) = dist(p, ∂M). Define

μ_{+} = \frac{2(α + 1)}{α − 1}.

Let \overline{N} be a compact submanifold of \overline{M} forming a neighborhood of ∂M such that φ |\overline{N}> 0, and x ∈ C^{∞}(\overline{N}). There exist functions φ_{i} ∈ C^{∞}(\overline{N}), i ∈ ℕ, such that the following hold : For any k ∈ ℕ let I ∈ ℕ be such that Iμ_{+} ≥ k. Then 1. if μ_{+} ∉ ℕ, then

∗E–mail: chrusciel@univ-tours.fr

x^{\frac{2}{α−1}} φ − φ_{0} − x^{μ+}φ_{1} − ... − x^{Iμ_{+}}φ_{I} ∈ C^{k}(\overline{N}).

2. If μ_{+} ∈ ℕ, then

x^{\frac{2}{α−1}} φ − φ_{0} − x^{μ+} ln xφ_{1} − ... − x^{Iμ_{+}} ln^{I} xφ_{I} ∈ C^{k}(\overline{N}).

If φ_{1} |_{∂M}= 0, then (in both cases)

x^{\frac{2}{α−1}} φ ∈ C^{∞}(\overline{N}).

Remarks: 1. Similar results can be established using [?] when (M, g) is only of finite degree of differentiability, with some finite number of functions φ_{i} of finite degree of differentiability.

2. For j ≤ Int(μ_{+}) and j < μ_{+} the functions (∂jxφ_{0}) |_{∂M} are uniquely deter-

mined by g_{ij} |_{∂M} and (∂k xg_{ij}) |_{∂M}, for some finite number of transverse derivatives of g_{ij} at ∂M. Similarly when μ_{+} ∈ ℕ the function φ_{1} |_{∂M} is uniquely determined by the boundary values of a finite number of derivatives of the metric in directions transverse to ∂M.

Proof: By elliptic regularity on N = int\overline{N} we have φ ∈ C^{∞}(N). Set

u = \frac{φx^{\frac{2}{α−1}}}{C_{α}} − 1.

u ∈ C^{∞}(N) and satisfies the equation

Lu = ω + F (u) ,
F (u) = C_{α}^{α−1}[(1 + u)^{α} − 1 − αu] , ω = \frac{2xΔ_{g}x}{α−1} ,
Lu = x^{2}Δ_{g}u − \frac{4x}{α−1}g(Du, Dx) − \frac{2}{α−1}[α + 1 + xΔx]u .

L is of the form considered in [?] with indicial exponents

μ_{−} = −1 ,  μ_{+} = \frac{2(α + 1)}{α − 1} .  (0.4)

Let ψ = φ |_{∂N∩M} . By the arguments of proof of [?, Proposition 7.3.1] there exists a solution φ̂ of

Δφ̂ = |φ̂|^{α−1}φ̂ ,
φ̂|_{∂N∩M} = ψ ,
φ̂x^{α−1} − C_{α} = O(x) .

(The function u_{0} required in [?, Proposition 7.3.1] can be taken as u_{0} ≡ 0; cf. also Remark 2 following [?, Proposition 7.3.1]). By the asymptotic maximum principle (cf., e.g., [?, Theorem 3.5]) we have φ ≡ φ̂ which implies

u = O(x) .

Theorem 7.4.1 of [?] gives

u ∈ 𝒜_{phg}(N) ∩ C^{0}(\overline{N}) .

By a closer examination of the structure of Equation (??) one obtains the claimed results. □

Another example of equation to which the results of [?] apply is the equation

Δ_{g}u = e^{αu} ,  α > 0 ,  (0.5)

with the asymptotic condition

lim_{p→∂M} u(p) = ∞ .  (0.6)

It has been shown by Bandle and Marcus [M. Marcus, private communication] that u then necessarily satisfies, for x ≤ ε < 1, for some ε > 0,

|u − \frac{2}{α} ln \frac{1}{x} − \frac{1}{α} ln \frac{2}{α}| ≤ Cx ln \frac{1}{x} .

We have the following:

Theorem 0.2 Let (\overline{M}, g) be a smooth Riemannian manifold with boundary, let α > 0 and let φ ∈ C^{∞}(M) satisfy

Δ_{g}u = e^{αu} .

Suppose that there exists a constant C such that

|u − \frac{2}{α} ln \frac{1}{x}| ≤ C .

Let \overline{N} be a compact submanifold of \overline{M} forming a neighborhood of ∂M such that x ∈ C^{∞}(\overline{M}). There exist functions φ_{i} ∈ C^{∞}(\overline{N}), i ∈ ℕ, such that, for all k ∈ N,

u − \frac{2}{α} ln \frac{1}{x} − \frac{2}{α} ln \frac{2}{α} − φ_{1}x ln x − φ_{2}x^{2} ln^{2} x − ... − φ_{k}x^{k} ln^{k} x ∈ C^{k}(\overline{N}) .

φ_{1}|∂M is proportional to the mean extrinsic curvature of ∂M in \overline{M}. If that mean extrinsic curvature vanishes, then

u − \frac{2}{α} ln \frac{1}{x} ∈ C^{∞}(\overline{N}).

Proof: Write u in the form u = \frac{1}{α}[2 ln \frac{1}{x} + ln \frac{2}{α} + ω]. The function ω solves the equation

x^{2}Δ_{g}ω = 2(e^{ω} − 1) + 2xΔ_{g}x.

The proof follows now that of Theorem ?? (in the present case we have μ_{−} = 0, μ_{+} = 1). □

References

[1] L. Andersson and P.T. Chruściel, On asymptotic behaviour of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”, Dissert. Math. 355 (1996), 1–100.

[2] C. Bandle and M. Marcus, Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 155–171.

[3] C.R. Graham and J.M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186–225.

[4] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, Jour. Anal. Math. 59 (1992), 231–250, Festschrift on the occasion of the 70th birthday of Shmuel Agmon.