The Paneitz equation in hyperbolic space^{∗}

Hans-Christoph Grunau^{†} Fakultät für Mathematik Otto-von-Guericke-Universität Postfach 4120 D-39016 Magdeburg Mohameden Ould Ahmedou^{‡} Mathematisches Institut Eberhard-Karls-Universität Auf der Morgenstelle 10 D-72076 Tübingen

Wolfgang Reichel^{§} Institut für Mathematik RWTH Aachen Templergraben 55 D-52062 Aachen

Abstract The Paneitz operator is a fourth order differential operator which arises in conformal geometry and satisfies a certain covariance property. Associated to it is a fourth order curvature – the Qcurvature.

We prove the existence of a continuum of conformal radially symmetric complete metrics in hyperbolic space ℍ^{n}, n > 4, all having the same constant Q-curvature.

Moreover, similar results can be shown also for suitable non-constant prescribed Q-curvature functions.

1 Introduction

The fourth order Paneitz operator arises naturally in conformal geometry, when one looks for higher order elliptic operators enjoying some covariance property. We shall be concerned with a corresponding semilinear equation, which comes up when searching conformal metrics with a certain prescribed fourth order curvature invariant – the so called Q-curvature.

Let (M^{n}, g) be a Riemannian manifold of dimension n. The objective of conformal geometry is the following: can one change the original metric g conformally into a new metric h with prescribed properties? This means that one searches for some positive function ρ such that h = ρg and the conformal factor ρ has to satisfy an elliptic boundary value problem.

∗This research was in parts supported by the “Research in Pairs”-program of “Mathematisches Forschungsinstitut Oberwolfach”. This program is financed by the Volkswagen foundation.

†http://www-ian.math.uni-magdeburg.de/home/grunau/

e-mail: Hans-Christoph.Grunau@mathematik.uni-magdeburg.de ‡http://www.mathematik.uni-tuebingen.de/ab/analysis/home.g.html

e-mail: ahmedou@everest.mathematik.uni-tuebingen.de §http://www.instmath.rwth-aachen.de/˜ reichel

e-mail: reichel@instmath.rwth-aachen.de

E.g, for n > 2 let L_{g} := −c_{n}Δ_{g} + R_{g} be the conformal Laplacian, where Δ_{g} is the Laplace Beltrami operator, c_{n} = 4(n − 1)/(n − 2) and R_{g} is the scalar curvature. If one sets the conformal factor ρ = u^{\frac{4}{n−2}} , u > 0 then it is well known that L has the following conformal covariance property:

∀φ ∈ C^{∞}(M) : L_{g}(uφ) = u^{\frac{n+2}{n−2}} L_{h}(φ).

If one prescribes the scalar curvature R_{h} for the metric h then u has to satisfy the second-order equation

L_{g}(u) = u^{\frac{n+2}{n−2}} L_{h}(1) = R_{h}u^{\frac{n+2}{n−2}} .  (1)

In the case R_{h} ≡ const. this is the so called Yamabe problem. In the case R_{h} is a prescribed function it is called the Nirenberg problem.

It turns out that there are many operators beside the conformal Laplacian L_{g} on general Riemannian manifolds of dimension greater than two which enjoy a conformal covariance property. A particularly interesting one is the fourth order operator P_{n} on n-manifolds discovered by Paneitz in 1983, which can be written for n > 4 as:

P_{g} = Δ_{g}^{2} − div_{g}(a_{n}R_{g}g − b_{n} Ric_{g})∇_{g} + \frac{n − 4}{2}Q_{g},

where a_{n} = \frac{(n−2)^{2}+4}{2(n−1)(n−2)}, b_{n} = \frac{4}{n−2}. Here Ric : T M → T M is the (1, 1)-tensor given by Ric_{i}^{j} = g^{jk} Ric_{ki}, the operator ∇_{g} produces the gradient vector-field of a function and div_{g} the divergence of a vector-field. Further, the Q-curvature is given by

Q_{g} = −\frac{2}{(n − 2)^{2}} | Ric_{g} |^{2} + \frac{n^{3} − 4n^{2} + 16n − 16}{8(n − 1)^{2}(n − 2)^{2}}R_{g}^{2} − \frac{1}{2(n − 1)}Δ_{g}R

with | Ric |^{2} := R_{ij}R_{kℓ}g^{ik}g^{jℓ}. In weak form the Paneitz operator may be written

∫_{M}(P_{g}u)φ dv_{g} = ∫_{M} Δ_{g}uΔ_{g}φ + a_{n}R_{g}⟨∇_{g}u, ∇_{g}φ⟩_{g} − b_{n} Ric_{g}(∇_{g}u, ∇_{g}φ) + \frac{n − 4}{2}Q_{g}uφ dv_{g}

for all φ ∈ C_{0}^{∞}(M). In the case n > 4, the conformal factor is usually chosen in the form ρ = u^{4/(n−4)}, u > 0 and the conformal covariance property of the Paneitz operator reads as follows:

∀φ ∈ C^{∞}(M) : P_{g}(uφ) = u^{\frac{n+4}{n−4}} P_{h}(φ).

If one prescribes the Q-curvature for the metric h by a function Q_{h} this leads to the equation

P_{g}(u) = u^{\frac{n+4}{n−4}} P_{h}(1) = \frac{n − 4}{2}Q_{h}u^{\frac{n+4}{n−4}} ,  (2)

which is a fourth-order analogue of (1).

Natural generalizations of problems from second order conformal geometry like the Yamabe problem, the Nirenberg problem or also existence, uniqueness and regularity for equations involving the Paneitz operator or biharmonic mappings are obvious and interesting questions to be studied. We refer to the survey articles of Chang [C1] and Chang, Yang [CY3] and on the lecture notes [C2] for more background information on the Paneitz operator

In the present paper the manifold (M^{n}, g) is the hyperbolic space ℍ^{n} with its standard metric. We focus on finding a complete metric h = U^{\frac{4}{n−4}} g on ℍ^{n} such that h has prescribed Q-curvature. We give conditions on Q (which include the case Q ≡ const.) such that an entire continuum of mutually distinct complete radially symmetric conformal metrics exist all having the same prescribed Q-curvature. In the case where Q ≡ \frac{1}{8}n(n^{2} − 4) this family contains in its “center” the explicitly known standard hyperbolic Poincaré metric, and at least a sub-continuum of these metrics has negative scalar curvature.

We point out that it is surprising to find such highly non-unique solutions. In previous work on the second order Yamabe problem, uniqueness of metrics with constant scalar curvature was found in the case of ℍ^{n} by Loewner-Nirenberg [LN]. In the case of 𝕊^{n} uniqueness (up to isometries) was proved by Obata [O] and later by Caffarelli, Gidas, Spruck [CGS] and Chen, Li [CL]. In the fourth order Paneitz problem, uniqueness (up to isometries) of metrics with constant Q-curvature on 𝕊^{n} was found by Chang, Yang [CY2] for n = 4, by Wei, Xu [WX2] and C.-S. Lin [L] for n > 4 and by Choi, Xu [CX] in the exceptional case n = 3.

In our setting we chose (M, g) to be a non-compact manifold. In contrast to this non-compact case, the literature for the existence of solutions of the prescribed Q-curvature problem on compact manifolds is considerably bigger. We only give a brief survey on results concerning fourth order Paneitz operators. In Chang, Yang [CY1], Wei, Xu [WX1] and Gursky [G] existence results for the constant Q-curvature problem in compact 4-manifolds are given. Recent work of Djadli, Malchiodi [DM] provides further extensions and completions of these works.

On compact manifolds of dimension greater then 4 existence results were given for Einstein manifolds by Djadli, Hebey, Ledoux [DHL] and in the case of invariance of both the manifold and Qcurvature function under a group of isometries by Robert [R]. On the sphere 𝕊^{n} we refer to results of Djadli, Malchiodi, Ould Ahmedou [DMO1], [DMO2] and Felli [FE].

The main results

As a model for hyperbolic space ℍ^{n} we use the Poincaré ball, i.e. ℍ^{n} is represented by the unitball B = B_{1}(0) ⊂ ℝ^{n} with standard co-ordinates x_{1}, …, x_{n} and the Poincaré metric g_{ij} = 4/(1 − |x|^{2})^{2}δ_{ij}. Since ℍ^{n} is conformally flat we may seek the metric h of the form h_{ij} = U^{\frac{4}{n−4}} g_{ij} = u^{\frac{4}{n−4}} δ_{ij} and the corresponding differential equation (2) for u reduces to

Δ^{2}u = \frac{n − 4}{2}Qu^{(n+4)/(n−4)}, u > 0 in B,  u|∂B = ∞.  (3)

The condition u|∂B = ∞ is necessary (and as we shall show also sufficient) for completeness of the metric h. For U = 1 we are at the Poincaré metric. In this case the conformal factor is given explicitly by

u_{0}(x) = (\frac{2}{1 − |x|^{2}} )^{(n−4)/2} .  (4)

The Poincaré metric (u_{0}^{4/(n−4)}δ_{ij})_{ij} with u_{0} as above has constant Q-curvature Q ≡ \frac{1}{8}n(n^{2} − 4).

Infinitely many complete radial conformal metrics with the same constant Q-curvature

Theorem 1. For every α > 0, there exists a radial solution of the prescribed Q-curvature equation (3) in the unit ball with Q ≡ \frac{1}{8}n(n^{2} − 4), infinite boundary values at ∂B and with u(0) = α. Moreover, (i) the conformal metric (u^{4/(n−4)}δ_{ij})_{ij}on B is complete;

(ii) if u(0) > 0 is sufficiently small then the corresponding solution generates a metric with negative scalar curvature.

The existence proof is given in Section 2. Closely related results can be found in a recent and independent work of Diaz, Lazzo, Schmidt [DLS1]. Statement (ii) is discussed in Section 3.

According to forthcoming work [DLS2] of Diaz, Lazzo, Schmidt, one has, for the solutions constructed in Theorem 1, that asymptotically for r ↗ 1

u(r) ∼ C(1 − r^{2})^{(4−n)/2}

where C = C(n) does not depend on the solution. Furthermore, the derivatives of u exhibit a corresponding uniform behavior. This is an even more precise information than just completeness of the conformal metric. However, for the less far reaching statement (i) of completeness, we provide a relatively simple and elementary independent proof in Section A.

The equation (3) is invariant under Moebius transformations of the unit ball. But the only solution which is invariant under all Moebius transformations of the unit ball is the explicit solution (4). Hence, we also have infinitely many distinct nonradial solutions, which is again in striking contrast to the second order analogue of (3). The following is an open problem, which we could not solve in this paper but hope to address in future work:

Find a geometric criterion, which singles out the explicit solution (4) among all other solutions of (3).

One might guess that among all radially symmetric metrics the explicit Poincaré metric is uniquely characterized by a condition of the kind

−C ≤ R_{h} ≤ −\frac{1}{C} < 0

with a suitable constant C. This is however wrong, since it follows from the result of [DLS2] that for every radial solution u of (3) one has that the scalar curvature of the generated metric satisfies lim_{r→1} R_{h} = −n(n − 1). It is however trivially true that the Poincaré metric is the only one with R_{h} ≡ −n(n − 1).

Infinitely many complete radial conformal metrics with the same non-constant Q-curvature

For smooth positive radial functions Q : B → ℝ we give suitable assumptions on Q such that the conformal metric (u^{4/(n−4)}δ_{ij})_{ij} has Q-curvature equal to the given function Q. We can prove a result, which is analogous to Theorem 1.

Theorem 2. Let Q ∈ C^{1}[0, 1] and assume that there are two positive constants Q_{0}, Q_{1} > 0 such that 0 < Q_{0} < Q(r) < Q_{1} on [0, 1]. Suppose further that there exists q ∈ [0, 1) such that rQ′(r) ≥ −qQ(r) on [0, 1], i.e., r^{q}Q(r) is monotonically increasing. Then, for every α > 0, there exists a radial solution of the prescribed Q-curvature equation (3) in the unit ball with infinite boundary values at ∂B and with u(0) = α. Moreover,

(i) the conformal metric (u^{4/(n−4)}δ_{ij})_{ij}on B is complete;

(ii) if u(0) > 0 is sufficiently small, then the corresponding solution generates a metric with negative scalar curvature.

Infinitely many solutions have also been observed by Chang and Chen [CC] in a different conformally covariant fourth order equation in ℝ^{4} with exponential nonlinearity.

R. Mazzeo pointed out that perturbation methods developed by F. Pacard and him [MP] will also apply in the present situation in order to construct neighbourhoods of nonradial solutions close to our radial ones.

2 Shooting method

2.1 Constant Q-curvature

Here we look for radial solutions of (3). By means of a shooting method we shall construct infinitely many distinct solutions. Applying the special Moebius transforms

φ_{a} : B → B,  φ_{a}(x) = \frac{1}{|a|^{2}} (a − (|a|^{2} − 1)\frac{1}{|x − \frac{a}{|a|^{2}} |^{2}} (x − \frac{a}{|a|^{2}} ))  (5)

we even find nonradial solutions by setting

ũ := J_{φ_{a}}^{(n−4)/(2n)} · u ∘ φ_{a},

where J_{φ_{a}} is the Jacobian-determinant of φ_{a}. All these conformal metrics have constant Q-curvature \frac{1}{8}n(n^{2} − 4) and a continuum of them has negative scalar curvature.

In order to construct solutions of (3) with Q ≡ \frac{1}{8}n(n^{2} − 4), we do this for the simplified problem

Δ^{2}u = u^{(n+4)/(n−4)}, u > 0 in B, u|∂B = ∞.

By a simple scaling argument both boundary value problems are equivalent. For radial solutions we study the initial value problem

\{ \begin{matrix}Δ^{2}u(r) = (r^{1−n} \frac{∂}{∂r} (r^{n−1} \frac{∂}{∂r}))^{2} u(r) = u(r)^{(n+4)/(n−4)},  r > 0, \\ u(0) = α,  u′(0) = 0,  Δu(0) = β,  (Δu)′(0) = 0, \end{matrix}  (6)

where α ≥ 0, β ∈ ℝ are given. If necessary, u^{(n+4)/(n−4)} will denote also the odd extension to the negative reals; however, we mainly focus on positive solutions. It is a routine application or modification of the Banach fixed point theorem or the Picard-Lindelöf-result to show that (6) always has unique local C^{4}–solutions.

It is a simple but very useful observation that the initial value problem enjoys a comparison principle, see [MKR]:

Lemma 1. Let u, v ∈ C^{4}([0, R)) and Q̃ ∈ C([0, R)), Q̃ ≥ 0 be such that

\{ \begin{matrix}∀r ∈ [0, R) :  Δ^{2}u(r) − Q̃(r)u(r)^{(n+4)/(n−4)} ≥ Δ^{2}v(r) − Q̃(r)v(r)^{(n+4)/(n−4)}, \\ u(0) ≥ v(0), u′(0) = v′(0) = 0, Δu(0) ≥ Δv(0), (Δu)′(0) = (Δv)′(0) = 0. \end{matrix} (7)

Then we have

∀r ∈ [0, R) :  u(r) ≥ v(r), u′(r) ≥ v′(r), Δu(r) ≥ Δv(r), (Δu)′(r) ≥ (Δv)′(r). (8)

Moreover, (i) the initial point 0 can be replaced by any initial point ρ > 0 if all four initial data at ρ are

weakly ordered,

(ii) a strict inequality in one of the initial data at ρ ≥ 0 or in the differential inequality on (ρ, R) implies a strict ordering of u, u′, Δu, Δu′ and v, v′, Δv, Δv′ on (ρ, R).

The problem (6) has the following entire solutions

U_{α}(r) = α\frac{[n(n^{2} − 4)(n − 4)]^{\frac{n−4}{4}}}{(\sqrt{n(n^{2} − 4)(n − 4)} + (α^{2/(n−4)}r)^{2})^{\frac{n−4}{2}}}  (9)

of (6) with α > 0 and suitably chosen β_{0} := β_{0}(α) := ΔU_{α}(0). It is known that these solutions

are the only positive entire solutions of (6), cf. [L], [WX1]. The metric h = Uα^{\frac{4}{n−4}} δ_{ij} arises as the pullback of the standard metric of the sphere 𝕊^{n} under a stereographic projection to ℝ^{n} .

For our purposes it is enough to show the following result: the solution U_{α} is a separatrix in the r-u-plane, i.e., if we fix α > 0 and consider β as a varying parameter then U_{α} separates the blow-up solutions from the solutions with one sign-change, which lie below U_{α}.

Lemma 2. Let α > 0 be fixed. Then, for β > β_{0}, the solution u = u_{α,β} blows up on a finite interval, which we denote by [0, R(α, β)). The blow-up-radius R(α, β) is monotonically decreasing in β.

Proof. It is useful to have the explicit solutions

V_{α}(r) = α (1 − (\frac{r}{λ_{α}} )^{2})^{−(n−4)/2} ,  (10)

of (6) at hand, where λ_{α} = α^{−2/(n−4)} [n(n^{2} − 4)(n − 4)]^{1/4}. We fix any α > 0, some β > β_{0}(α) and look at the corresponding solution u = u_{α,β} of (6). In order to see that u′(r) − U′_{α}(r) is strictly increasing, note first by Lemma 1 that Δu(r) − ΔU_{α}(r) is positive and strictly increasing. Since u′(r) − U′_{α}(r) = ∫_{0}^{1} rt^{n−1}(Δu − ΔU_{α})(rt) dt it follows that u′(r) − U′_{α}(r) is also strictly increasing. So u(r) cannot converge to 0 and hence has to become unbounded as r → ∞. By integrating successively the differential equation of u we find R large enough such that

u(R) > 0,  u′(R) > 0,  Δu(R) > 0,  (Δu)′(R) > 0.

Since lim_{α̃→0} Vα̃(r) = 0 locally uniformly in C^{4}, we can find a sufficiently small α̃ > 0 such that

u(R) > Vα̃(R),  u′(R) > V ′ ; α̃(R),  Δu(R) > ΔVα̃(R),  (Δu)′(R) > (ΔVα̃)′(R).

But then, the comparison principle Lemma 1 shows that ∀r > R : u(r) > Vα̃(r) and hence, blow up of u at some finite radius R(α, β). The monotonicity of R(α, β) is also a direct consequence of Lemma 1.

Lemma 3. Let α > 0 be fixed. The blow-up radius R(α, β) is a continuous function of β ∈ (β_{0}, ∞).

Proof. Let β > β_{0} be arbitrary but fixed and let denote u = u_{α,β} the corresponding solution of (6). The continuity from the right

β_{k} ↘ β ⇒ R(α, β_{k}) → R(α, β)

follows directly from the monotonicity of R(α, β) in β and continuous dependence on initial data. Only continuity from the left has to be proved.

First we show that for r close enough to R = R(α, β) the functions u, u′, Δu and (Δu)′ are finally strictly increasing. For u, r^{n−1}u′, Δu and r^{n−1} (Δu)′, this follows from successive integration of the differential equation, since the relevant quantities become – at least finally – positive. It remains to consider u′(R − 0) and (Δu)′ (R − 0).

We observe that

∞ = R^{n−1}u′(R − 0) = ∫_{0}^{R} r^{n−1}Δu dr;  (11)

∞ = R^{n−1}(Δu)′(R − 0) = ∫_{0}^{R} r^{n−1}Δ^{2}u dr = ∫_{0}^{R} r^{n−1}u^{(n+4)/(n−4)} dr.  (12)

From this we conclude for r ↗ R:

(Δu)′(r) = ∫_{0}^{r} (\frac{s}{r})^{n−1} u^{(n+4)/(n−4)}(s) ds = r ∫_{0}^{1}(u(rt))^{(n+4)/(n−4)}t^{n−1} dt,
(Δu)′′(r) = ∫_{0}^{1}(u(rt))^{(n+4)/(n−4)}t^{n−1} dt + r\frac{n + 4}{n − 4} ∫_{0}^{1} (u(rt))^{8/(n−4)} u′(rt)t^{n} dt
→ +∞ by (12);
u′(r) = ∫_{0}^{r} (\frac{s}{r})^{n−1} Δu(s) ds = r ∫_{0}^{1} t^{n−1}Δu(rt) dt,
u′′(r) = ∫_{0}^{1} t^{n−1}Δu(rt) dt + r ∫_{0}^{1} t^{n}(Δu)′(rt) dt
→ +∞ by (11).

Moreover, for later purposes we note that for r ↗ R

u′′′(r) = 2 ∫_{0}^{1} t^{n}(Δu)′(rt) dt + r ∫_{0}^{1} t^{n+1}(Δu)′′(rt) dt
≥ \frac{2}{r^{n+1}} ∫_{0}^{r} s^{n}(Δu)′(s) ds − C ≥ \frac{1}{C} Δu(r) − C → +∞.

Here, C denotes a constant which depends on the solution u.

Now, we consider a sequence β_{k} ↗ β. By monotonicity we have R(α, β_{k}) ≥ R(α, β). For t_{k} > 1, which will be adequately chosen below, we define the function

v_{k}(r) := t_{k}^{(4−n)/2}u_{α,β} (\frac{r}{t_{k}} ) ,  (13)

which solves the same differential equation as u_{α,β}. We find values r_{0} − δ < r_{0} < R(α, β) such that

u_{α,β}(r_{0}) > 0, u′_{α,β}(r_{0}) > 0, Δu_{α,β}(r_{0}) > 0, (Δu_{α,β})′(r_{0}) > 0,

and all these quantities are strictly increasing on (r_{0} − δ, R(α, β)). By continuous dependence on data, for β_{k} close enough to β we also have

u_{α,β_{k}}(r_{0}) > 0, u′_{α,β_{k}}(r_{0}) > 0, Δu_{α,β_{k}}(r_{0}) > 0, (Δu_{α,β_{k}})′(r_{0}) > 0.

For suitably chosen t_{k} we conclude that

v_{k}(r_{0}) = t_{k}^{(4−n)/2}u_{α,β} (\frac{r_{0}}{t_{k}} ) ≤ t_{k}^{(4−n)/2}u_{α,β}(r_{0}) < u_{α,β_{k}}(r_{0}),
v′_{k}(r_{0}) = t_{k}^{(2−n)/2}u′_{α,β} (\frac{r_{0}}{t_{k}} ) ≤ t_{k}^{(2−n)/2}u′_{α,β}(r_{0}) < u′_{α,β_{k}}(r_{0}),
Δv_{k}(r_{0}) = t_{k}^{−n/2}Δu_{α,β} (\frac{r_{0}}{t_{k}} ) ≤ t_{k}^{−n/2}Δu_{α,β}(r_{0}) < Δu_{α,β_{k}}(r_{0}),
(Δv_{k})′(r_{0}) = t_{k}^{(−n−2)/2}(Δu_{α,β})′ (\frac{r_{0}}{t_{k}} ) ≤ t_{k}^{(−n−2)/2}(Δu_{α,β})′(r_{0}) < (Δu_{α,β_{k}})′(r_{0}).

By continuous dependence on data, we may achieve

t_{k} ↘ 1 (k → ∞).

The comparison result of Lemma 1 yields for r ≥ r_{0}:

u_{α,β_{k}}(r) ≥ v_{k}(r).

This gives finally

R(α, β) ≤ R(α, β_{k}) ≤ R(v_{k}) = R(α, β) · t_{k} → R(α, β) as k → ∞,

where R(v_{k}) denotes the blow-up-radius of v_{k}. The proof is complete.

Lemma 4. Let α > 0 be fixed. Then, for the limits of the blow-up radius R(α, β), one has:

lim_{β↘β_{0}} R(α, β) = ∞,  lim_{β↗∞} R(α, β) = 0.  (14)

Proof. The first claim is just a consequence of the global existence of the solution for β = β_{0} and continuous dependence of solutions on the initial data. The proof of the second statement relies upon some rescaling arguments. First we note that the same argument as in the proof of Lemma 2 shows that R(0, 1) < ∞. By the comparison result from Lemma 1 we conclude that

∀α′ > 0 :  R(α′, 1) ≤ R(0, 1) < ∞.  (15)

For β > 0 we find the relation

u_{α,β}(r) = (\frac{α}{α′} ) u_{α′,1} ((\frac{α}{α′} )^{2/(n−4)} r) ,  (16)

where α′ is chosen such that

β = (\frac{α}{α′} )^{n/(n−4)} , i.e. α′ = α · β^{(4−n)/n}.

Obviously, α′ ↘ 0 for β ↗ ∞. We read from (16) and (15) that

R(α, β) = R(α′, 1) (\frac{α′}{α} )^{2/(n−4)} ≤ R(0, 1) (\frac{α′}{α} )^{2/(n−4)} = R(0, 1)β^{−2/n},

which tends to 0 as β → ∞.

Theorem 3. For every α > 0 there exists a radial solution of (6) with u(0) = α which blows up at r = 1. Moreover,

(i) if u, ũ are two such solutions with u(0) < ũ(0) then Δu(0) > Δũ(0),

(ii) if 0 < u(0) ≤ [n(n^{2} − 4)(n − 4)]^{\frac{n−4}{8}} then the corresponding solution generates a metric with negative scalar curvature.

Proof. Let α > 0 be fixed, and denote u_{α,β} the solution of (6). According to Lemmas 3 and 4, we find a suitable β > β_{0}(α) such that for the blow-up-radius, we have precisely R(α, β) = 1. Property (i) is a consequence of Lemma 1. To see property (ii) note that under the hypothesis 0 < u(0) < V_{α_{0}}(0) with α_{0} = [n(n^{2} − 4)(n − 4)]^{\frac{n−4}{8}} we find by (i) that Δu(0) > ΔV_{α_{0}}(0) > 0 and hence Δu > 0 on [0, 1). Thus by Lemma 8 below the solution u generates a metric with negative scalar curvature.

In order to complete the proof of Theorem 1, it remains to prove the completeness of the induced metrics. Indeed, these metrics are complete, see Section A.

2.2 Nonconstant Q-curvature

To obtain radial solutions of (3) for a prescribed smooth radial Q-curvature function Q : B → ℝ we also use the shooting method. For simplicity let Q̃ := \frac{n−4}{2}Q. We then study the problem

Δ^{2}u = Q̃ u^{(n+4)/(n−4)}, u > 0 in B,  u|∂B = ∞,  (17)

such that the conformal metric (u^{4/(n−4)}δ_{ij})_{ij} has Q-curvature equal to the given function Q. In all of our discussion we make the following assumptions on the function Q̃:

(Q1) there are two positive constants Q_{0}, Q_{1} such that 0 < Q_{0} < Q̃(r) < Q_{1} on [0, 1], Q̃ ∈ C^{1}[0, 1],

(Q2) there exists q ∈ [0, 1) such that rQ̃′(r) ≥ −qQ̃(r) on [0, 1], i.e., r^{q}Q̃(r) is increasing.

We extend Q̃ as a C^{1}-function to [0, ∞) which is bounded on [1, ∞) and satisfies (Q1), (Q2) on [0, ∞).

Theorem 4. Let Q̃ satisfy (Q1), Q(2). For every α > 0, there exists a radial solution of the prescribed Q-curvature equation (17) in the unit ball with infinite boundary values and with u(0) = α. Moreover,

(i) if u, ũ are two solutions with u(0) < ũ(0) then Δu(0) > Δũ(0),

(ii) if u(0) > 0 is sufficiently small then the corresponding solution generates a metric with negative scalar curvature.

The initial value problem for (17) takes the form

\{ \begin{matrix}Δ^{2}u(r) = (r^{1−n} \frac{∂}{∂r} (r^{n−1} \frac{∂}{∂r}))^{2} u(r) = Q̃(r)u(r)^{(n+4)/(n−4)},  r > 0, \\ u(0) = α,  u′(0) = 0,  Δu(0) = β,  (Δu)′(0) = 0, \end{matrix}  (18)

where α > 0, β ∈ ℝ are given. Existence and uniqueness of local C^{4}-solutions denoted by u_{α,β} is standard.

We recall from (9) the definition of β_{0} = β_{0}(α) = ΔU_{α}(0) < 0.

Lemma 5. Let α > 0 be fixed. Then there exists a value β^{∗} ∈ [\sqrt{Q_{1}}β_{0}, \sqrt{Q_{0}}β_{0}] with the following properties:

(i) For −∞ < β < β^{∗} the solution u_{α,β} is decreasing and has a finite first zero.

(ii) For β > β^{∗} the solution u_{α,β} blows up on a finite interval [0, R(α, β)). For fixed α, the blowup-radius is decreasing in β.

(iii) For β = β^{∗} the solution u_{α,β^{∗}} exists on [0, ∞) and converges to 0 at ∞.

Proof. For simplicity we assume that 1 ≤ Q̃(r) ≤ 2 for r ∈ [0, ∞). As in the proof of Lemma 2 we find with the help of the same subsolution Vα̃(r) (α̃ > 0 small enough) that for β > β_{0} the solution u_{α,β} must blow up at a finite value R(α, β). Likewise, we can use the functions Ū_{α}(r) := U_{α}(\sqrt[4]{2}r)

solving Δ^{2}Ū_{α} = 2Ūα^{\frac{n+4}{n−4}} on [0, ∞) as supersolutions to see that for β < \sqrt{2}β_{0} the solutions u_{α,β} have a finite first zero. Hence we can define

β^{∗} = sup{β ∈ ℝ : u_{α,β} has a finite first zero}
= inf{β ∈ ℝ : u_{α,β} blows up at a finite value},

where it is easy to see that the two numbers coincide. Moreover, β^{∗} ∈ [\sqrt{2}β_{0}, β_{0}]. Finally, the solution u_{α,β^{∗}} must exist on [0, ∞) and can therefore only decay to 0 at ∞.

Lemma 6. Let α > 0 be fixed. Then, the blow-up radius R(α, β) is a continuous function of β ∈ (β^{∗}, ∞).

Proof. Let β > β^{∗} be fixed. Continuity of the blow-up radius from the right follows as before. For the continuity from the left one shows first that for r close enough to R = R(α, β) the functions u, u′, Δu and (Δu)′ are finally strictly increasing. For u, r^{n−1}u′, Δu and r^{n−1} (Δu)′, this follows from successive integration of the differential equation. To see the strict monotonicity of u′, (Δu)′ near the blow-up point one finds as before

∞ = R^{n−1}u′(R − 0) = ∫_{0}^{R} r^{n−1}Δu dr;  (19)

∞ = R^{n−1}(Δu)′(R − 0) = ∫_{0}^{R} r^{n−1}Δ^{2}u dr = ∫_{0}^{R} r^{n−1}Q̃(r)u^{(n+4)/(n−4)} dr.  (20)

From this we conclude for r ↗ R:

(Δu)′(r) = ∫_{0}^{r} (\frac{s}{r})^{n−1} Q̃(s)u^{\frac{n+4}{n−4}} (s) ds = r ∫_{0}^{1} Q̃(rt)u^{\frac{n+4}{n−4}} (rt)t^{n−1} dt,
(Δu)′′(r) = ∫_{0}^{1} Q̃(rt)u^{\frac{n+4}{n−4}} (rt)t^{n−1} dt + r\frac{n + 4}{n − 4} ∫_{0}^{1} Q̃(rt)u^{\frac{8}{n−4}} (rt)u′(rt)t^{n} dt
+r ∫_{0}^{1} Q̃′(rt)u^{\frac{n+4}{n−4}} (rt)t^{n} dt
→ +∞ by (20),

where we have used hypothesis (Q2). The same proof as in Lemma 3 shows that u′′(r), u′′′(r) → ∞ as r ↗ R. The actual continuity proof of Lemma 3 was based on finding a subsolution

v_{k}(r) := t_{k}^{−γ}u_{α,β} (\frac{r}{t_{k}} ) .

with γ = \frac{n−4}{2} and suitable t_{k} > 1. For non-constant Q we need to choose a different positive γ, since the condition for v_{k} being a subsolution is given by

Δ^{2}v_{k} = t_{k}^{−γ−4+γ \frac{n+4}{n−4}} Q̃(r/t_{k})vk^{\frac{n+4}{n−4}} ≤ Q̃(r)vk^{\frac{n+4}{n−4}} .

To achieve this we use hypothesis (Q2). Hence we need to choose γ > 0 such that −γ − 4 + γ \frac{n+4}{n−4} ≤ −q. Since q ∈ [0, 1) one possible choice is γ = 3(n − 4)/8. Then the rest of the proof of Lemma 3 goes through.

Lemma 7. Let α > 0 be fixed. Then, for the limits of the blow-up radius R(α, β), one has:

lim_{β↘β^{∗}} R(α, β) = ∞,  lim_{β↗∞} R(α, β) = 0.  (21)

Proof. For β = β^{∗} there exists a global solution tending to 0 at ∞. By continuous dependence on the initial data the first statement follows. The proof of the second statement is adapted from Lemma 4. Let v_{α,β} be the solution of Δ^{2}v = Q_{0}v^{\frac{n+4}{n−4}} with v(0) = α, v′(0) = 0, Δv(0) = β, (Δv)′(0) = 0. The argument of Lemma 5 shows that v_{0,1} blows up at the finite point S(0, 1). For α′ > 0 let us denote the blow-up point of v_{α′,1} by S(α′, 1). Then S(α′, 1) ≤ S(0, 1) < ∞. For positive β we find the relation

v_{α,β}(r) = (\frac{α}{α′} ) v_{α′,1} ((\frac{α}{α′} )^{2/(n−4)} r) ,  (22)

where α′ is chosen such that

β = (\frac{α}{α′} )^{n/(n−4)} , i.e. α′ = α · β^{(4−n)/n}.

We see that v_{α,β} is a subsolution to u_{α,β}. The blow-up positions therefore satisfy

R(α, β) ≤ S(α, β) = S(α′, 1) (\frac{α′}{α} )^{2/(n−4)} ≤ S(0, 1) (\frac{α′}{α} )^{2/(n−4)} = S(0, 1)β^{−2/n},

which tends to 0 as β → ∞.

Proof of Theorem 4. The proof follows from Lemmas 6 and 7. Let us prove property (ii). If we define V := Q1^{\frac{4−n}{8}} V_{α_{0}} with α_{0} = [n(n^{2} − 4)(n − 4)]^{\frac{n−4}{8}} then Δ^{2}V = Q_{1}V ^{\frac{n+4}{n−4}} . Therefore, if 0 < u(0) < V (0) then Δu(0) > ΔV (0) > 0 by an argument similar to (i), and hence Δu > 0 on [0, 1). Thus by Lemma 8 below the solution u generates a metric with negative scalar curvature. □

In order to finish the proof of Theorem 2, it remains to show the completeness of the induced metrics. See Section A.

3 Subharmonicity and negative scalar curvature

Let us recall that we consider conformal metrics of the form

h_{ij} = u^{4/(n−4)}δ_{ij}.  (23)

In order to compute the scalar curvature it is more convenient to write the conformal factor as

h_{ij} = v^{4/(n−2)}δ_{ij},

i.e. we set v := u^{(n−2)/(n−4)}, u = v^{(n−4)/(n−2)}. The scalar curvature R_{u} of the metric (h_{ij})_{ij} is then given by

R_{u} = −\frac{4(n − 1)}{(n − 2)}v^{−(n+2)/(n−2)} Δv = −\frac{4(n − 1)}{(n − 2)}u^{−(n+2)/(n−4)} Δu^{(n−2)/(n−4)}
= −\frac{4(n − 1)}{(n − 4)}u^{−(n+2)/(n−4)} (u^{2/(n−4)}Δu + \frac{2}{(n − 4)}u^{(6−n)/(n−4)}|∇u|^{2}) .  (24)

The following lemma is an immediate consequence of this formula:

Lemma 8. Let u : B → (0, ∞) be a C^{4}-function such that −Δu ≤ 0 in B. Then the conformal metric h given by (23) satisfies

R_{u} ≤ 0 in B.

For radially symmetric solutions, also the converse is true:

Proposition 1. Let u : B → (0, ∞) be an unbounded smooth radially symmetric solution of the perturbed Paneitz equation (17) for the hyperbolic ball with Q > 0. Assume further that R_{u} ≤ 0 in B. Then

−Δu ≤ 0 in B.

Proof. Since Δ^{2}u > 0, the function −Δu is superharmonic. So, if we assume that −Δu > 0 somewhere, then in particular

−Δu(0) > 0.

Since u is assumed to be radially symmetric, we also have

∇u(0) = 0.

Now, formula (24) would give R_{u}(0) > 0, a contradiction.

A Completeness of the conformal metric

Completeness of the metric h = u^{\frac{4}{n−4}} δ_{ij} on B means that every maximally extended geodesic curve has infinite length. However, the following lemma reduces this to a property, which is simpler to check.

Lemma 9. Let u be a radial solution of (17). The induced metric u^{\frac{4}{n−4}} δ_{ij} on ℍ^{n} is complete if and only if

∫_{0}^{1} u(r)^{2/(n−4)} dr = ∞.

Proof. To see necessity of the above condition note that for fixed z ∈ ℝ^{n}\{0} the curve γ(r) = rz/|z| for r ∈ (−1, 1) is a maximally extended geodesic and its length is given by

2 ∫_{0}^{1}⟨γ′(r), γ′(r)⟩_{h}^{1/2} dr = 2 ∫_{0}^{1} u^{\frac{2}{n−4}} dr.

Next we prove sufficiency. Let γ be a maximally extended geodesic in (B, h) parameterized over ℝ. Then lim_{t→±∞} |γ(t)| = 1. Clearly γ has infinite length if δ(t) = dist_{h}(γ(t), 0) becomes unbounded for t → ±∞. Since

δ(t) = ∫_{0}^{|γ(t)|} u^{\frac{2}{n−4}} (r) dr

the claim follows.

We recall that according to forthcoming work [DLS2] of Diaz, Lazzo, Schmidt, one has, for the solutions with constant Q-curvature constructed in Theorem 1, that asymptotically for r ↗ 1

u(r) ∼ C(1 − r^{2})^{(4−n)/2},

where C = C(n) does not depend on the solution. This gives in particular that

∫ ^{1} u(r)^{2/(n−4)} dr = ∞

and so, the completeness of the conformal metric. This work covers a very general situation, is quite involved and relies on deep work of Mallet-Paret and Smith [MPS] on Poincaré-Bendixson results for monotone cyclic feedback systems. Moreover, we expect all these solutions to oscillate infinitely many times around the explicit solution (4) and around each other.

In what follows we give an independent and relatively simple and elementary proof of the statement of completeness by means of a suitable transformation and energy considerations. The proof applies in the same way both to the case of constant and non-constant Q-curvature functions. The final statement of completeness is given in Theorem 5 in Section A.6 below.

Estimates from above and a first nonoptimal estimate from below are deduced in the original setting of equation (17). For the final conclusion that ∫^{1} u(r)^{2/(n−4)} dr = ∞ we have to perform a change of variables such that r ↗ 1 is replaced by s → ∞ so that elementary qualitative theory of dynamical systems becomes applicable. This procedure is somehow motivated by techniques recently developed for fourth order equations in [GG, FG].

A.1 Pohožaev’s identity for solutions of (17)

The following is true for every r ∈ (0, 1), cf. [PO], [PS]:

−\frac{n − 4}{2n} ∫_{B_{r}(0)} x · ∇Q̃(x)u^{\frac{2n}{n−4}} dx

= ∫_{S_{r}(0)} ∇Δu · ν(x · ∇u + \frac{n − 4}{2}u) + \frac{2 − n}{2}Δu∇u · ν dσ

− ∫_{S_{r}(0)} Δu(x^{T} D^{2}u ν) − \frac{1}{2}(x · ν)(Δu)^{2} + \frac{n − 4}{2n}(x · ν)Q̃(x)u^{\frac{2n}{n−4}} dσ.

For radial solutions this implies

− \frac{n − 4}{2n} ∫_{0}^{r} Q̃′(s)u^{\frac{2n}{n−4}} s^{n} ds

= r^{n−1}(Δu)′(ru′ + \frac{n − 4}{2}u) + \frac{n}{2}r^{n−1}u′Δu − r^{n}(\frac{1}{2}(Δu)^{2} + \frac{n − 4}{2n}Q̃(r)u^{\frac{2n}{n−4}} ). (25)

A corresponding equality holds for radial solutions on [ρ, r], where the integration on the left-hand side is from ρ to r and on the right-hand side the corresponding term evaluated at ρ is subtracted.

A.2 Maximal blow-up rate for radial solutions of (17)

Proposition 2. Let u : B → [0, ∞) be an unbounded smooth radial solutions of the perturbed Paneitz equation (17) on the unit ball with 1 ≤ Q̃(r) ≤ 2. Then there exists a constant C = C(u) such that

u(r) ≤ C(\frac{1}{1 − r^{2}} )^{\frac{n−4}{2}} .

Proof. As was shown in the proof of Lemma 3, we may choose ρ ∈ (0, 1) such that

u, u′, u′′, Δu, (Δu)′ > 0 are increasing in (ρ, 1).

By C we denote a constant depending on u. By using the analogue of Pohožaev’s identity (25) on the interval [ρ, r] we obtain for all r ∈ (ρ, 1)

− \frac{n − 4}{2n} ∫_{ρ}^{r} Q̃′(s)u^{\frac{2n}{n−4}} (s)s^{n} ds + \frac{n − 4}{2n}r^{n}Q̃(r)u^{\frac{2n}{n−4}} (r) + \frac{r^{n}}{2} (Δu(r))^{2}

= r^{n−1}(Δu)′(ru′ + \frac{n − 4}{2}u) + \frac{n}{2}r^{n−1}u′Δu + C. (26)

We estimate the two sides of the equality separately. Right-hand side: The following estimates for r > ρ obtained by integration

u(r) = u(ρ) + ∫_{ρ}^{r} u′(s) ds ≤ u′(r) + C,
Δu(r) ≤ (Δu)′ (r) + C.

Hence the entire right-hand side of (26) can be estimated by C_{1}u′(r) (Δu)′ (r) + C_{2} and since u′(r), (Δu)′ (r) → ∞ for r → 1 we find that Cu′(r) (Δu)′ (r) for ρ < r < 1 is an upper estimate for the right-hand side of (26). Left-hand side: After dropping the last term in the left-hand side of (26) a lower bound is given by

− \frac{n − 4}{2n} ∫_{ρ}^{r} Q̃′(s)u^{\frac{2n}{n−4}} (s)s^{n} ds + (1 − ε)\frac{n − 4}{2n} ∫_{ρ}^{r} \frac{d}{ds}(s^{n}Q̃(s)u^{\frac{2n}{n−4}} (s)) ds

+ ε\frac{n − 4}{2n}r^{n}Q̃(r)u^{\frac{2n}{n−4}} (r), (27)

where ε ∈ (0, 1) is chosen later. The two integrals add up to

∫_{ρ}^{r} ( − εQ̃′(s) + n(1 − ε)Q̃(s)s^{−1})\frac{n − 4}{2n}u^{\frac{2n}{n−4}} (s)s^{n} + (1 − ε)Q̃(s)s^{n}u^{\frac{n+4}{n−4}} (s)u′(s) ds,

which is positive provided ε = ε_{0} is chosen sufficiently small. Hence, for finding a lower bound for (27) the two integrals can be dropped. Moreover, by using 1 ≤ Q̃ ≤ 2 we obtain finally that

ε_{0} \frac{n − 4}{2n}r^{n}u^{\frac{2n}{n−4}} (r)

is lower bound for the right-hand side of (26).

Hence, (26) yields the existence of a constant C = C(u, ρ, ε) such that

u^{\frac{2n}{n−4}} ≤ Cu′ (Δu)′ on [ρ, 1).

Multiplication with u′ leads to

(u^{\frac{3n−4}{n−4}} )′ ≤ Cu′^{2} (Δu)′ = C (u′^{2}Δu)′ − 2Cu′u′′Δu ≤ C (u′^{2}Δu)′ on [ρ, 1),

and integration shows

u^{\frac{3n−4}{n−4}} ≤ C_{1}u′^{2}Δu + C_{2} ≤ Cu′^{2}Δu on [ρ, 1).

Now, as above, we can estimate

Δu(r) = u′′(r) + \frac{n − 1}{r}u′(ρ) + \frac{n − 1}{r} ∫_{ρ}^{r} u′′(s) ds ≤ Cu′′(r) + C ≤ Cu′′(r)

and we may proceed to the inequality

u^{\frac{3n−4}{n−4}} ≤ C(u′)^{2}Δu ≤ C(u′)^{2}u′′.

In a similar way, multiplication with u′ and integration leads to

u^{\frac{4n−8}{n−4}} ≤ Cu′^{4} on [ρ, 1), u^{\frac{n−2}{n−4}} ≤ Cu′ on [ρ, 1).

Solutions of Cv′ = v^{\frac{n−2}{n−4}} on some interval [ρ, δ) are given by

v_{δ}(r) = (\frac{n − 4}{2}C)^{\frac{n−4}{2}} (δ − r)^{\frac{4−n}{2}} , δ ≤ 1.

If for some value of r_{0} ∈ [ρ, 1) we would have u(r_{0}) > v_{1}(r_{0}) then u(r_{0}) > v_{δ}(r_{0}) for some δ ∈ (0, 1). Then u stays strictly above v_{δ} and hence u blows up somewhere in the interval (ρ, δ), i.e., strictly before the point 1. This contradiction shows that u(r) ≤ v_{1}(r) for all r ∈ [ρ, 1). This establishes the claim.

A.3 A first estimate from below for the blow-up rate of radial solutions to (17)

Let u = u(r) solve Δ^{2}u = Q̃(r)u^{\frac{n+4}{n−4}} on [0, 1), u(1) = ∞ with 1 ≤ Q̃(r) ≤ 2. Then, for r ≥ r_{0} we may assume that u(r) is increasing and (Δu)′(r) ≥ 0. Thus

(Δu)′(r) = (\frac{r_{0}}{r} )^{n−1}(Δu)′(r_{0}) + ∫_{r_{0}}^{r} (\frac{s}{r})^{n−1}Q̃(r)u^{\frac{n+4}{n−4}} (s) ds ≤ (Δu)′(r_{0}) + 2u^{\frac{n+4}{n−4}} (r),

and hence

Δu(r) ≤ Δu(r_{0}) + (Δu)′(r_{0}) + 2u^{\frac{n+4}{n−4}} (r) = K + 2u^{\frac{n+4}{n−4}} (r)

with suitably chosen K = K(u) > 0. Now let v be the unique radial solution of

Δv = K + 2v^{\frac{n+4}{n−4}} for r_{0} < r < 1, v(r_{0}) = u(r_{0}), v(1) = ∞.

Then v is a subsolution for u and

u(r) ≥ v(r) ≥ C(\frac{1}{1 − r^{2}} )^{\frac{n−4}{4}} on [r_{0}, 1)

where C = C(r_{0}; u). Hence we have proved the following result:

Proposition 3. Let u : B → [0, ∞) be an unbounded smooth radial solution of the perturbed Paneitz equation (17) on B with 1 ≤ Q̃(r) ≤ 2. Then there exists a constant C = C(u) such that

u(r) ≥ C(\frac{1}{1 − r^{2}} )^{\frac{n−4}{4}} on [1/2, 1).

A.4 A transformation: Moving the boundary r = 1 to ∞

The equation (17) reads in radial coordinates

u^{(4)}(r) + \frac{2(n − 1)}{r}u′′′(r) + \frac{(n − 1)(n − 3)}{r^{2}}u′′(r) − \frac{(n − 1)(n − 3)}{r^{3}}u′(r) = Q̃(r)u^{\frac{n+4}{n−4}} (r).

With the transformation

u(r) = (1 − r^{2})^{\frac{4−n}{2}} v(− log(1 − r^{2})), v(t) = e^{(4−n)t/2}u(\sqrt{1 − e^{−t}}), t ∈ (0, ∞)

we get

K_{4}(t)v^{(4)}(t) + K_{3}(t)v′′′(t) + K_{2}(t)v′′(t) + K_{1}(t)v′(t) + K_{0}v(t) = \frac{1}{16}q(t)v^{\frac{n+4}{n−4}} (t)  (28)

with

K_{0} = \frac{1}{16}(n^{4} − 4n^{3} − 4n^{2} + 16n),
K_{1}(t) = \frac{1}{16}((1 − e^{−t})^{2}(−4n^{2} + 24n − 32) + (1 − e^{−t})(4n^{3} − 16n^{2} − 16n + 64)
+ 4n^{3} − 4n^{2} − 24n),
K_{2}(t) = \frac{1}{16}((1 − e^{−t})^{2}(4n^{2} − 40n + 80) + (1 − e^{−t})(16n^{2} − 16n − 96) + 4n^{2} + 8n),
K_{3}(t) = (1 − e^{−t})^{2}(n − 4) + (1 − e^{−t})(n + 2),
K_{4}(t) = (1 − e^{−t})^{2},
q(t) = Q̃ (\sqrt{1 − e^{−t}}) .

Eventually, it will be useful to have the values K_{i}^{∞} = lim_{t→∞} K_{i}(t), i.e,

K_{0}^{∞} = \frac{1}{16}n(n − 2)(n + 2)(n − 4),  K_{1}^{∞} = \frac{1}{2}(n − 1)(n^{2} − 2n − 4),
K_{2}^{∞} = \frac{3}{2}n^{2} − 3n − 1,  K_{3}^{∞} = 2n − 2,  K_{4}^{∞} = 1.

In view of the differentiability properties assumed on Q̃ it is enough to consider

q(t) = 1 + αe^{−t}

as a prototype.

Note that (28) has always the constant solution v_{0} ≡ 0. Moreover, in the case of constant Q, i.e. α = 0, it has a second constant solution v_{1} ≡ (16K_{0})^{\frac{n−4}{8}} .

Motivated by the observation that

u′(r) = 0 ⇔ v′(t) + \frac{n − 4}{2}v(t) = 0

we transform (28) into a system for w(t) = (w_{1}(t), w_{2}(t), w_{3}(t), w_{4}(t))^{T} by setting

w_{1}(t) = v(t), w_{2}(t) = v′(t) + \frac{n − 4}{2}v(t), w_{3}(t) = v′′(t) + \frac{n − 4}{2}v′(t), w_{4}(t) = v′′′(t) + \frac{n − 4}{2}v′′(t).

The resulting system is

\{ \begin{matrix}w′_{1}(t) & = & −\frac{n−4}{2}w_{1}(t) + w_{2}(t) \\ w′_{2}(t) & = & w_{3}(t) \\ w′_{3}(t) & = & w_{4}(t) \\ K_{4}(t)w′_{4}(t) & = & C_{2}(t)w_{2}(t) + C_{3}(t)w_{3}(t) + C_{4}(t)w_{4}(t) + \frac{1}{16}q(t)w_{1}(t)^{\frac{n+4}{n−4}} , \end{matrix}  (29)

where

C_{m}(t) = − ∑_{k=m−1}^{4} K_{k}(t)(\frac{4 − n}{2})^{k+1−m}.

By explicit calculations we get C_{1}(t) ≡ 0 and

C_{2}(t) = −\frac{1}{8}n^{3} + \frac{1}{2}n
C_{3}(t) = 1 − \frac{3}{4}n^{2} + e^{−t}(\frac{1}{2}n^{2} − n) + e^{−2t}(\frac{1}{2}n − 1)
C_{4}(t) = −\frac{3}{2}n + e^{−t}(2n − 2) + e^{−2t}(2 − \frac{1}{2}n)

To get an idea about the behavior of the almost-autonomous system (29) we replace the functions C_{i}(t) by their limit C_{i}^{∞} = lim_{t→∞} C_{i}(t), i = 2, 3, 4 and t ↦ q(t) by the constant 1. In other words we put for the moment α = 0 and study the resulting autonomous system

\{ \begin{matrix}w′_{1}(t) & = & −\frac{n−4}{2}w_{1}(t) + w_{2}(t) \\ w′_{2}(t) & = & w_{3}(t) \\ w′_{3}(t) & = & w_{4}(t) \\ w′_{4}(t) & = & C_{2}^{∞}w_{2}(t) + C_{3}^{∞}w_{3}(t) + C_{4}^{∞}w_{4}(t) + \frac{1}{16}w_{1}(t)^{\frac{n+4}{n−4}} , \end{matrix}  (30)

where

C_{2}^{∞} = −\frac{1}{8}n^{3} + \frac{1}{2}n, C_{3}^{∞} = 1 − \frac{3}{4}n^{2}, C_{4}^{∞} = −\frac{3}{2}n.

The autonomous system has the steady-states

O = (0, 0, 0, 0) and P = ((16K_{0})^{\frac{n−4}{8}} , \frac{n − 4}{2}(16K_{0})^{\frac{n−4}{8}} , 0, 0);

note that O is also a steady state for the almost autonomous system (29). At the point O the system (30) has the linearized stability matrix

M_{O} = (\begin{matrix}\frac{4−n}{2} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & C_{2}^{∞} & C_{3}^{∞} & C_{4}^{∞} \end{matrix})

with four negative eigenvalues

λ_{1} = 2 − \frac{n}{2} > λ_{2} = 1 − \frac{n}{2} > λ_{3} = −\frac{n}{2} > λ_{4} = −1 − \frac{n}{2}

and corresponding eigenvectors

ϕ_{1} = (1, 0, 0, 0),  ϕ_{2} = (1, −1, −1 + \frac{n}{2}, −\frac{(n − 2)^{2}}{4} ),
ϕ_{3} = (1, −2, n, −\frac{n^{2}}{2} ),  ϕ_{4} = (1, −3, 3 + \frac{3n}{2}, −\frac{3(n + 2)^{2}}{4} ).

Thus O is asymptotically stable for (30). At the point P the linearized stability matrix is

M_{P} = (\begin{matrix}\frac{4−n}{2} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{n+4}{n−4}K_{0} & C_{2}^{∞} & C_{3}^{∞} & C_{4}^{∞} \end{matrix})

with the eigenvalues

μ_{1} = 1, μ_{2} = −n, μ_{3} = \frac{1 − n}{2} − \frac{i}{2}\sqrt{n^{2} + 2n − 9}, μ_{4} = \frac{1 − n}{2} + \frac{i}{2}\sqrt{n^{2} + 2n − 9}.

Thus P has a three-dimensional stable manifold and a one-dimensional unstable manifold.

A.5 Stability of O in the nonautonomous equation (28)

Lemma 10. The origin O is an asymptotically stable steady state of the system (29). Moreover the following holds

(i) if w is a solution to the system (29) such that for a sequence t_{k} → ∞, one has that w(t_{k}) → O, then for any ε > 0 one has that eventually

|w(t)| ≤ exp ((\frac{4 − n}{2} + ε) t);

(ii) the corresponding solution u(r) = (1 − r^{2})^{\frac{4−n}{2}} w_{1}(− log(1 − r^{2})) of the original equation (17) is bounded near r = 1.

Proof. System (29) has the form

w′(t) = M_{O}w(t) + G(t, w(t));

G(t, w) = (\frac{1}{16} + O(e^{−t})) (0, 0, 0, w_{1}^{(n+4)/(n−4)})^{T} + e^{−t}Bw + e^{−2t}Cw

with constant 4 × 4–matrices B and C. In particular

lim_{t→∞,w→O} \frac{G(t, w)}{|w|} = 0,

i.e. Condition [H, (8.11)] is satisfied. Since all eigenvalues of M_{O} are below μ := (4 − n)/2, the corollary of [H, Theorem 8.1] shows asymptotic stability of the origin O. Moreover, for a solution w with w(t_{k}) → O, it follows from this corollary that

limsup_{t→∞} \frac{log |w(t)|}{t} ≤ μ = \frac{4 − n}{2}.

Hence, for any ε > 0, one has that eventually

|w(t)| ≤ exp ((\frac{4 − n}{2} + ε) t) .

For the solution u of the original equation (17) this means that for r < 1 close enough to 1

u(r) ≤ (1 − r^{2})^{−ε} .

In view of the minimal blow up rate for unbounded solutions proved in Proposition 3, this shows that r ↦ u(r) has to remain bounded near r = 1.

A.6 Energy considerations

Theorem 5. Let u : B → [0, ∞) be an unbounded smooth radial solutions of the perturbed Paneitz equation (17) on the unit ball with 1 ≤ Q̃(r) ≤ 2. Then

∫ ^{1} u(r)^{2/(n−4)}dr = ∞.

Proof. First we take from Proposition 2 that in the transformed coordinates, v is bounded. Then, as in [FG, Lemma 2], we see that also v′, …, v^{(4)} are bounded.

Let us assume for contradiction that

∫ ^{1} u(r)^{2/(n−4)}dr < ∞,

which gives that

∫_{0}^{∞} v(s)^{2} ds ≤ C ∫_{0}^{∞} v(s)^{2/(n−4)} ds < ∞.

Testing the differential equation (28) once with v and once with v′ gives that for t → ∞

∫_{0}^{t} v′′(s)^{2} ds − K_{2}^{∞} ∫_{0}^{t} v′(s)^{2} ds = O(1);

K_{3}^{∞} ∫_{0}^{t} v′′(s)^{2} ds − K_{1}^{∞} ∫_{0}^{t} v′(s)^{2} ds = O(1).

Observe that only the terms with constant coefficients are relevant since all other terms contain a factor e^{−t} and produce finite integrals.

Combining the two equations above gives

(K_{2}^{∞}K_{3}^{∞} − K_{1}^{∞}) ∫_{0}^{t} v′′(s)^{2} ds = O(1).

Since

(K_{2}^{∞}K_{3}^{∞} − K_{1}^{∞}) > 0,

this shows first

∫_{0}^{∞} v′′(s)^{2} ds < ∞

and then

∫_{0}^{∞} v′(s)^{2} ds < ∞.

Testing the differential equation (28) with v′′′ finally gives

∫_{0}^{∞} v′′′(s)^{2} ds < ∞

so that

∫_{0}^{∞} (w_{1}(s)^{2} + w_{2}(s)^{2} + w_{3}(s)^{2} + w_{4}(s)^{2}) ds < ∞.

Consequently there is a sequence t_{k} ↗ ∞ such that

lim_{k→∞} (w_{1}, w_{2}, w_{3}, w_{4}) (t_{k}) = 0.

Since O = (0, 0, 0, 0) is stable, this shows that

lim_{t→∞} (w_{1}, w_{2}, w_{3}, w_{4}) (t) = 0.

From Lemma 10 we conclude that u(r) remains bounded near r = 1, contradicting the assumption on u.

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