Conformal metrics on $R^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

We study the solutions u ∈ C∞(R2m) of the problem (−∆)u = Q̄e, where Q̄ = ±(2m− 1)!, V := ∫ R2m edx <∞, (1) particularly when m > 1. Problem (1) corresponds to finding conformal metrics gu := e|dx| on R with constant Q-curvature Q̄ and finite volume V . Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value V and the asymptotic behavior of u(x) as |x| → ∞ can be simultaneously prescribed, under certain restrictions. When Q̄ = (2m− 1)! we need to assume V < vol(S), but surprisingly for Q̄ = −(2m− 1)! the volume V can be chosen arbitrarily.


Introduction
We consider the equation where u ∈ C ∞ (R 2m ) and satisfies Equation (2) has been widely studied because of its geometric meaning. Indeed if u solves (2), then the conformal metric g u := e 2u |dx| 2 on R 2m (here |dx| 2 denotes the Euclidean metric on R 2m ) has constant Q-curvature equal to (2m − 1)!. For a brief discussion of the geometric meaning of (2) and a survey of related previous works we refer to the introduction of [12] and the references therein. Here we only mention some relevant facts, necessary to contextualize the results of our present work. First of all the assumption that u ∈ C ∞ (R 2m ) is not restrictive, since any weak solution u ∈ L 1 loc (R 2m ) of (2) with right-hand side in L 1 loc (R 2m ) is smooth, see e.g. [12,Corollary 8].
Also the particular choice of the constant (2m − 1)! in (2) is not restrictive, since it can be changed by considering u + C for C ∈ R.
Next we recall that Problem (2)-(3) possesses the following explicit radially symmetric solutions u(x) = log 2λ 1 + λ 2 |x − x 0 | 2 , λ > 0, x 0 ∈ R 2m , which are called spherical solutions, since they are obtained (up to a Möbius transformation) by pulling back the round metric of S 2m onto R 2m via the stereographic projection.
While in dimension 2, i.e. for m = 1, such spherical solutions exhaust the set of solutions to (2)-(3), as proven by W. Chen and C. Li [4], in the case m ≥ 2 A. Chang and W. Chen [2] showed that non-spherical solutions do exist. In fact they proved that for any m ≥ 2 and every V ∈ (0, vol(S 2m )) there exists a (non-spherical) solution to (2)- (3). This suggests to investigate the properties of such solutions. Building upon the previous work of A. Chang and P. Yang [2], C-S. Lin for m = 2 and L. Martinazzi for m > 2 proved: Theorem A ( [9], [12]) If u solves (2)- (3), then u has the asymptotic behavior where α = 2V vol(S 2m ) and P is a polynomial of degree at most 2m−2 bounded from below. Moreover P is constant if and only if u is spherical. When m = 2 one has V ∈ (0, vol(S 4 )] and V = vol(S 4 ) if and only if u is spherical.

J. Wei and D. Ye complemented the result of C-S. Lin by showing, among other things:
Theorem B ( [17]) For any V ∈ (0, vol(S 4 )) and P (x) = 4 j=1 a j x 2 j with a j > 0, Problem (2)-(3) has a solution with asymptotic expansion (4) for some C ∈ R.
The first result which we prove here is an extension of the result of J. Wei and D. Ye to the case m > 2. We will prove the existence of solutions to (2)-(3) having the asymptotic behavior (4) where P will be any given polynomial of degree at most 2m − 2 satisfying while α > 0 is determined by V ∈ (0, vol(S 2m )). More precisely, define It is worth noticing that (5) is equivalent to the apparently stronger condition lim inf |x|→∞ P (x) |x| a > 0 and lim inf |x|→∞ x · ∇P (x) |x| a > 0, for some a > 0.
Indeed (5) implies the second inequality of (6) by a subtle result of E. Gorin (see [6,Theorem 3.1]), and the second inequality in (6) implies the first one, since one can write A simple example of polynomial belonging to P m is where a j > 0, i j ∈ {1, 2, . . . , m − 1} for 1 ≤ j ≤ 2m, and p is a polynomial of degree at most 2 min{i j } − 1, but in general P m contains polynomials whose higher degree monomials do not split in such a simple way.
The restriction V < vol(S 2m ) in Theorem 1.1 is necessary when m = 2 because of the result of C-S. Lin (Theorem A), but appears to be only a technical issue when m ≥ 3. In fact for m = 3 L. Martinazzi recently proved that there are solutions to (2)-(3) with V arbitrarily large, see [14]. The crucial step in which we need V to be smaller than vol(S 2m ) is Theorem 4.2 below, a compactness result which follows form the blow-up analysis of sequences of prescribed Q-curvature in open domains of R 2m (Theorem 4.1 below) proven by L. Martinazzi, and inspired by previous works of H. Brézis and F. Merle [1] and F. Robert [16]. This compactness is used to prove the a priori bounds necessary to run the fixed point argument of [17], which we closely follow. For m > 2 it remains open whether one can prescribe P ∈ P m and V ≥ vol(S 2m ) in Theorem 1.1.
From the work of Brézis-Merle we also borrow a simple but fundamental critical estimate, whose generalization is Lemma A.2 below, which is used in Lemma 3.6 below.
As we shall now show, things go differently when the prescribed Q-curvature is negative. Consider the equation whose solutions give rise to metrics g u = e 2u |dx| 2 of Q-curvature −(2m − 1)! in R 2m . One can easily verify that under the assumption (3) Equation (7) has no solutions when m = 1, see e.g. [11,Proposition 6]. On the other hand, when m ≥ 2 we have: Theorem C ( [11]) For every m ≥ 2 there is some V > 0 such that Problem (7)-(3) has a radially symmetric solution. Every solution to (7)-(3) (a priori not necessarily radially symmetric) has the asymptotic behavior given by (4) where α = − 2V vol(S 2m ) and P is a non-constant polynomial of degree at most 2m − 2 bounded from below.
Notice that, contrary to Chang-Chen's result [2], the existence part of Theorem C does not allow to prescribe V . Moreover its proof is based on an ODE argument which only produces radially symmetric solutions. It is then natural to address the following question: For which values of V and which polynomials P does Problem (7)-(3) have a solution with asymptotic behavior (4) (with α = − 2V vol(S 2m ) )? In analogy with Theorem 1.1 we will show: Theorem 1.2 For any integer m ≥ 2, given P ∈ P m and V > 0, there exists a solution of (7)-(3) having the asymptotic behavior (4) for α = − 2V vol(S 2m ) .
The remarkable fact which allows for large values of V in Theorem 1.2 (but not in Theorem 1.1) is that, as shown in [13], when the Q-curvature is negative, compactness is obtained even for large volumes, compare Theorems 4.1 and 4.2 below. This in turn depends on Theorem C above, and in particular on the fact that the polynomial in the expansion (4) of a solution to (7)-(3) is necessarily non-constant.
About the assumption that P ∈ P m in Theorems 1.1 and 1.2, we do not claim nor believe that it is optimal, but it is technically convenient in the crucial Lemma 3.5 below, where it is needed in (22). Since a solution to (2)-(3) or (7)-(3) must satisfy (4) for α = ± 2V vol(S 2m ) , a necessary condition on P and V is but it is unknown whether this condition is also sufficient to guarantee the existence of a solution to (2)-(3) or (7)-(3) with asymptotic expansion (4), at least in the negative case, or for V < vol(S 2m ) in the positive case. Also replacing (5) with the weaker assumption (which implies the first inequality in (6), hence (8)) creates problems, since (9) does not imply (5) when deg P ≥ 4, see e.g. Proposition A.4 in the appendix, and as already noticed (5) is crucial in Lemma 3.5 below.
Finally, we remark that new difficulties arise when recasting the above problems in odd dimension. For instance in dimension 3 T. Jin, A. Maalaoui, J. Xiong and the second author studied in [8] the non-local problem proving the existence of some non-spherical solutions with asymptotic behavior as in (4). Whether also in this case one can show an analog to Theorems 1.1 and 1.2 above is an open question.
Notation In the following C will denote a generic positive constant, whose dependence will be specified when necessary, and whose value can change from line to line. We will also write 2 Strategy of the proof of Theorems 1.1 and 1.2 where γ m is defined by Let V , α = ± 2V vol(S 2m ) and P ∈ P m be given as in Theorem 1.1 or 1.2. We would like to find a solution to (2) or (7) of the form for a suitable choice of C ∈ R and of a smooth function v( and notice that (5) implies for some C 1 , C 2 > 0. Now if we assume (3), then the constant C in (11) is determined by the function v. Indeed An easy computation shows that u given by (11) satisfies and (3) if and only if C = c v and Then we will use a fixed point method in the spirit of [17] to find a solution v to (15) in the Banach space and of course v will also be smooth by elliptic estimates. In order to run the fixed-point argument we introduce the following weighted Sobolev spaces.
For v ∈ C 0 (R 2m ) and c v as in (14) we have This follows easily from (13) and dominated convergence.
The following Lemma will be proven in Section A.2 below.
given by T v =v wherev is the only solution to and compactness follows from the continuity of S and ((−∆) m ) −1 and the compactness of E.
If v is a fixed point of T , then it solves (15) and u = v + c v − P − αu 0 is a solution of (2) or (7) (depending on the sign of K in (12)) and (3), with asymptotic expansion (4). Then in order to prove Theorems 1.1 and 1.2 it remains to prove that T has a fixed point, and we shall do that using the following fixed-point theorem.
Lemma 2.4 (Theorem 11.3 in [5]) Let T be a compact mapping of a Banach space X into itself, and suppose that there exists a constant M such that for all x ∈ X and t ∈ (0, 1] satisfying tT x = x. Then T has a fixed point. In order to apply Lemma 2.4 to the operator T defined in (16) we will prove in Section 3 the following a priori bound, which completes the proof of Theorems 1.1 and 1.2.
with M independent of v and t.

A priori estimates and proof of Proposition 2.5
Throughout this section let t ∈ (0, 1] and v ∈ C 0 (R 2m ) be fixed and satisfy tT v = v, that is where c v is as in (14). Also definew Proof. Letṽ(x) be defined as the right-hand side of (19). Then for |x| ≥ 1, using (14) we writẽ We first show that lim Let R > 1 be fixed. Then for |x| > 2R, we split and we will show that I i → 0 as |x| → ∞ for 1 ≤ i ≤ 5. For i = 1, since lim |x|→∞ log |x| |x−y| = 0 uniformly with respect to y ∈ B R (0), from the dominated convergence theorem we get From (13) we also have → 0, as |x| → ∞.
Using that 1 2 < |x| |x−y| < 2 on A 4 and that K ∈ L 1 (R 2m ) we find that for every ε > 0 it is possible to choose R so large that Finally, again using that K log(| · |) ∈ L 1 (R 2m ) with the dominated convergence theorem we get Since ε can be chosen arbitrarily small, (20) is proven. Since Then by the Liouville theorem for polyharmonic functions (see e.g. Theorem 5 in [12]) w is a polynomial, and since it vanishes at infinity, it must be identically zero, i.e. v ≡ṽ.
By Lemma 2.2 and (13), we have with C independent of t and v, and together with Lemma 3.3 and Lemma 3.6 below we obtain where C is independent of v and t. Now Proposition 2.5 follows at once from the continuity of the embedding M p 2m,δ (R 2m ) → C 0 (R 2m ) (see Lemma 2.3).
Remark. An alternative way of getting uniform bounds on v C 0 is to get uniform upper bounds ofw and use them in (19).
Using Lemma 3.1 one can prove the following decay estimate for the derivatives of v at infinity.
Proof. Notice that ∇v = ∇w, so it is enough to work with v.
Using (19) for |x| > 1 one can compute Fix ε > 0 and R 1 > 1 such that For |x| > 2R 1 , we split R 2m in to three disjoint domains: Then Since R 1 is fixed, for |x| large enough we have by the mean-value theorem hence with (14) we get Since K goes to zero rapidly at infinity,w is bounded, and |x − y| ≤ |x|/2 on A 2 , we have as |x| → ∞.
On A 3 we have |x − y| ≥ |x|/2, which implies |x| |x−y| ≤ 2 . Hence Since ε is arbitrarily small, the proof is complete. Proof. Since u 0 is a fixed function and locally bounded, it is enough to prove that w :=w − tαu 0 is locally uniformly upper bounded. Now where we used (14) and that |K| is positive and continuous.
In addition in the case when Q > 0 we have Moreover Lemma 3.1 gives and with Fubini's theorem we get Therefore Theorem 4.2 implies that there exists C = C(R) > 0 (independent of w) such that A consequence of the local uniform upper bounds ofw is the following local uniform bound for the derivatives of v: Proof. Let x ∈ B R . Then from (19) and Lemma 3.3, we have where the last integral is bounded using (14). Since u 0 is smooth, α is fixed and t ∈ (0, 1], then the lemma follows.
Now to prove uniform upper bounds forw outside a fixed compact set, first we will need the following result, which relies on a Pohozaev-type identity.
Lemma 3.5 For given ε > 0, there exists R 0 = R 0 (ε) > 0 only depending on K (and not on v or t) such that Proof. Taking R → ∞ in Lemma A.1 and noticing that the first term on the right-hand side of (28) vanishes thanks to (13) and last two terms vanish thanks to Lemma 3.2, we find Thanks to (6) we can find C 1 > 0 and R 1 ≥ 1 such that Then for some R ≥ R 1 to be fixed later we bound where in the equality on the third line we used (21). Now using (14) and (18), we compute (I) = 2mt|α|γ m , and using Lemma 3.4 we bound where Ω := x ∈ R 2m : x · ∇P (x) + α < 0 .
From (22) we infer that Ω ⊂ B R 1 . Then with Lemma 3.3 we find where C 2 does not depend on t or v. To complete the proof it suffices to take R 0 = R so large that To prove uniform upper bound ofw on the complement of a compact set, we use the Kelvin transform. For R > 1 define Lemma 3.6 There exists ε > 0 sufficiently small such that if R 0 = R 0 (ε) > 1 is as in Lemma 3.5, then ξ( Proof. Using (31) for n = 2m and k = m and recalling that Then with the change of variable y = R 0 x |x| 2 and Lemma 3.5 we obtain for R 0 = R 0 (ε) large enough (and ε > 0 to be fixed later) Iteratively using the maximum principle it is easy to see that Now fix ε > 0 small enough (and consequently R 0 = R 0 (ε) > 0 large enough) so that by Lemma A.2 below, there exists p > 1 such that e 2mξ 1 is bounded in L p (B 1 ). As usual this bound, as well as ε, R 0 are independent of t and v.
Since |∆ k ξ 2 | is uniformly bounded on ∂B 1 for k = 0, 1, 2, ..., m − 1 by Lemma 3.4 andw + is uniformly bounded on ∂B R 0 by Lemma 3.3, so that ξ + is uniformly bounded on ∂B 1 , by the maximum principle we get uniform bounds of ξ 2 in B 1 . Hence, noticing that (13), and using (25), we can bound Consequently by elliptic estimates and Sobolev embedding there exists a conastant C > 0 (independent of v and t) such that and therefore ξ ≤ξ ≤ |ξ 1 | + |ξ 2 | ≤ C in B 1 , with C not depending on v and t. Here we state a slightly simplified version of Theorem 1 from [13] which we will use to prove the uniform upper bound of Theorem 4.2 below. This theorem was originally proved by F. Robert [16] in dimension 4 and under the assumption V k > 0, and is a delicate counterpart to the blowup analysis initiated by H. Brézis and F. Merle [1] in dimension 2. The crucial fact which we shall use is that in order to lose compactness V 0 must be positive somewhere and V k e 2mu k L 1 must approach or go above Λ 1 := (2m − 1)!vol(S 2m ).

Theorem 4.1 ([13])
Let Ω ⊆ R 2m be a connected set. Let (u k ) ⊂ C 2m loc (Ω) be such that Then one of the following is true: (Ω) for some u 0 ∈ C 2m (Ω), or (ii) there is a finite (possibly empty) set S = {x (1) , ...., x (I) } ⊂ Ω such that V 0 (x (i) ) > 0 for 1 ≤ i ≤ I, and up to a subsequence u k → −∞ locally uniformly in Ω \ S, and in the sense of measures in Ω, where In particular, in case (ii) for any open set Ω 0 Ω with S ⊂ Ω 0 we have for a function K ∈ C 0 (B R ) and assume that for given C 1 , C 2 > 0 one has where C only depends on R, C 1 , C 2 , Λ (in case (c 1 ) holds and not (c 2 )) and K.
Proof. Assume that there is a sequence of functions u n ∈ C 2m (B R ) and a sequence of points x n ∈ B R/2 such that u n satisfies the conditions (a), (b), and (c 1 ) or (c 2 ), and assume that Then we can apply Theorem 4.1 with V k = K for every k, and because of (27), we clearly are in case (ii) of the theorem. Assume that S = ∅. Then K > 0 on S, hence condition (c 2 ) does not hold. On the other hand condition (c 1 ) contradicts (26). Then S = ∅, hence u k → −∞ uniformly in B R/2 , contradicting (27).
Proof. Integrating by parts we find and integrating by parts m times the first term on the right-hand side of (29) we find (see e.g. [15,Lemma 14] for the simple proof) and using the divergence theorem we obtain and putting together the above equations we conclude.
The proof of the following lemma can be found in [12] (Theorem 7). It extends to arbitrary dimension Theorem 1 of [1].
Proof. We shall prove the lemma by induction on k ∈ N. Notice that for k = 0 (31) is trivial.
For |β| ≤ 2m, we have From the embedding W 2m,p (A) → C 0 (A) there exists a constant S > 0, such that u C 0 (A) ≤ S u W 2m,p (A) , for all u ∈ W 2m,p (A). Hence ≤ CSR −γ f M p 2m,δ , γ = 2m/p + δ > 0. (32) Since R ≥ 1 is arbitrary (32) and on B 2 we have we conclude that M p 2m,δ (R 2m ) ⊂ C 0 (R 2m ), and actually By (32) and (33), on any compact set Ω R 2m the sequence f n W 2m,p (Ω) is bounded and from the compact embedding W 2m,p (Ω) → C 0 (Ω), we can extract a subsequence converging in C 0 (Ω). Then up to choosing Ω = B n and extracting a diagonal subsequence we have f n → f locally uniformly for a continuous function f , and actually f ∈ C 0 (R 2m ) and the convergence is globally uniform thanks to (34).

A.3 Condition (9) does not imply (5)
Proposition A.4 For n ≥ 2 there exists a polynomial P of degree 4 in R n satisfying (9) but not (5).