Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.


Introduction
The main goal of this paper is to give a fine analysis of blow-up solutions of conformally invariant fully nonlinear second order elliptic equations.
Besides (1)- (5), (σ 1 k k , k ) enjoys other nice and helpful properties, such as concavity and homogeneity properties of σ 1/k k , Newton's inequalities, divergence and variational structures, etc., which we do not assume in this paper. In particular, we would like to note that no concavity or homogeneity assumption on f is being made in the present paper.
For a positive C 2 function u, let A u be the n × n matrix with entries This is sometimes referred to as the conformal Hessian of u. The conformal Hessian A u arises naturally in conformal geometry as follows. Recall that the Riemann curvature Riem g of a Riemannian metric g can be decomposed into traced and traceless parts as where A g = 1 n−2 (Ric g − 1 2(n−1) R g g), Ric g , R g and W g are the Schouten curvature, the Ricci curvature, the scalar curvature and the Weyl curvature of g and denotes the Kulkarni-Nomizu product. While the (1, 3)-valent Weyl curvature remains unchanged under a conformal change of the metric, the Schouten curvature is adjusted by a second order operator of the conformal factor. In particular, if we consider the metric g u := u 4 n−2 g f lat conformal to the flat metric g f lat on R n , then the Schouten curvature A g u of g u is given by the conformal Hessian in the form Consequently, we have where λ(A g u ) denotes the eigenvalues of A g u with respect to the metric g u and λ(A u ) denotes those of the matrix A u .
A u enjoys a conformal invariance property, inherited from the conformal structure of R n , which will be of special importance in our treatment. Recall that a map ϕ : R n ∪ {∞} → R n ∪ {∞} is called a Möbius transformation if it is the composition of finitely many of the following types of transformations: • a translation: x → x +x wherex is a given vector in R n , • a dilation: x → a x where a is a given positive scalar, • a Kelvin transformation: x → x |x| 2 . For a function u and a Möbius transformation ϕ, let where J ϕ is the Jacobian of ϕ. A calculation gives for some orthogonal n × n matrix O ϕ (x). In particular, λ(A u ϕ (x)) = λ(A u (ϕ(x))).
The main result of this paper concerns an analysis on the behavior of a sequence {u k } ∈ C 2 (B 3 (0)) satisfying and where ( f, ) satisfies (1)- (5). Note that no other assumptions on u k is made.
As is known, Eq. (8) is a fully nonlinear elliptic equation. Fully nonlinear elliptic equations involving f (λ(∇ 2 u)) were investigated in the classic paper of Caffarelli et al. [2].
Our paper appears to be the first fine blow-up analysis in this fully nonlinear context. We expect this to serve as a crucial step in the study of the problem on Riemannian manifolds.
To obtain our result on fine analysis of blow-up solutions, we make use of the following Liouville theorems.

below). Then v is radially symmetric about the origin and v(r ) is non-increasing in r .
For −2) and Theorem A was proved by Caffarelli et al. [1]. See also Gidas et al. [5] under some decay assumption of u at infinity. For ( f, ) = (σ 1/2 2 , 2 ) in R 4 and v ∈ C 1,1 loc (R 4 ), the result was proved by Chang et al. [3]. In fact we need a stronger version of Theorem A (see Theorem 1.1) and a variant of Theorem B (see Theorem 1.2). For simplicity, readers are advised that in the main body of the paper all theorems, propositions and lemmas hold under (1)-(5), instead of the stated weaker hypotheses on ( f, ). and Then either v is constant or v is of the form (11) for somex ∈ R n and some positive constants a and b.
and, for some M k → ∞, Then v * is radially symmetric about the origin, i.
2 When ( f, ) satisfies (1)-(4) and an additional hypothesis that f is homogeneous of positive degree, the function v * in Theorem 1.2 is a viscosity solution of (12) and the conclusion follows from Theorem B. However, when f is not homogeneous, v * is not necessarily a viscosity solution of (12).
It is not difficult to see that, under (1)-(4), the function v in Theorem 1.1 is a viscosity solution of (10) (see Remark B.2). We have the following conjecture.
Conjecture Let ( f, ) satisfy (1)- (4), and let 0 < v ∈ C 0 loc (R n ) be a viscosity solution of (10). Then v is of the form (11) for somex ∈ R n and some positive constants a and b.
The notion of viscosity solutions given below is consistent with that in [12].

Definition 1.1 A positive continuous function v in an open set
⊂ R n is a viscosity supersolution (respectively, subsolution) of and v − ϕ ≤ 0 near x 0 , then either λ(A ϕ (x 0 )) ∈ R n \ or f (λ(A ϕ (x 0 ))) ≤ 1). We say that v is a viscosity solution if it is both a viscosity supersolution and a viscosity subsolution.

Definition 1.2 A positive continuous function v in an open set
⊂ R n is a viscosity supersolution (respectively, subsolution) of We say that v is a viscosity solution if it is both a viscosity supersolution and a viscosity subsolution.
It is clear that for C 2 functions the notions of viscosity solutions and classical solutions coincide. Also, viscosity super-and sub-solutions are stable under uniform convergence, see "Appendix B".

A calculation gives
With the normalization (21), U satisfies Forx ∈ R n and μ > 0, let Note that, in the sense of (6), . Hence, by the conformal invariance (7), for anyx ∈ R n and μ > 0,  1 Note that,m and K are independent of .
To see this, let x * be a point inB 1 In view of (v) and the stated condition on sup B 1 (0) u, x * belongs to some ball B δ * (x i 0 ). By (iv), we then have which implies the assertion.
The following is a quantitative version of Theorem A, and is related to Theorems 1.1 and 1.3.  (5) and the normalization condition (21), and let γ, r 1 > 0 be constants. Then, for every ∈ (0, 1/2], there exist some constants δ * > 0, R * > 0, depending only on ( f, ), γ, r 1 and , and there holds and Then, for some constant C depending only on ( f, ) and b, , the result was proved by Guan and Wang [6]. When ( f, ) satisfies (1)-(4) and is homogeneous of positive degree, Theorem 1.6 was proved in [12].
The rest of the paper is organized as follows. We start in Sect. 2 with the proof of Theorems 1.1 and 1.2. We then prove Theorem 1.6 in Sect. 3. In Sect. 4, we first establish an intermediate quantitative Liouville result and then use it to prove Theorem 1.3. In Sect. 5, we prove Theorem 1.5 as an application of Theorem 1.3. In "Appendix A", we present a lemma about super-harmonic functions which is used in the body of the paper. In "Appendix B", we include a relevant remark on the limit of viscosity solutions of elliptic PDE. Finally we collect in "Appendix C" some relevant calculus lemmas.

Non-quantitative Liouville theorems
In this section, we prove Theorems 1.1 and 1.2. We use the method of moving spheres and establish along the way, as a tool, a gradient estimate which is in a sense weaker than that in Theorem 1.6 but suffices for the moment. (Note that the proof of Theorem 1.6 relies on Theorem 1.2.) (15) and

A gradient estimate
and Then, for some constant C depending only on n and θ , This type of gradient estimate was established and used in various work of the first named author and his collaborators under less general hypothesis on ( f, ). It turns out that the same proof works in the current situation. We give a detailed sketch here for completeness.
We use the method of moving spheres as in [7,8,14,15]. For a function w defined on a subset of R n , we define wherever the expression makes sense. We will use w λ to denote w 0,λ . We start with a simple result.

Lemma 2.1 Let R > 0 and w be a positive Lipschitz function inB
Proof Write w in polar coordinates w(r, θ). It is easy to see that (30) is equivalent to Estimate (31) is readily seen from the estimates

Lemma 2.1 is established.
Proof of Theorem 2.1 By Lemma 2.1, there exists some r 0 It is easy to see that, for some We then define, for x ∈ B 4/3 (0), By the conformal invariance (7), Using the above two displayed equations, the definition ofλ(x), and using the ellipticity of the equation satisfied by v and v x,λ(x) , we can apply the strong maximum principle and Hopf Lemma to infer that eitherλ(x) = 5/3 − |x| or there exists some In the latter case, (29) implies that In either case, we obtain that It is readily seen that Proof of Theorem 1. 1 We may assume that R k ≥ 5 for all k.
Clearly, for every β > 1, there exists some positive constant C(β), independent of k, such . It follows, after passing to a subsequence, that for every 0 and v is super-harmonic on R n . Using the positivity, the superharmonic of v, and the maximum principle, we can find Passing to a subsequence and shrinking R k and c 0 > 0, if necessary, we may assume that and Proof For |x| ≤ R k 5 , we have, by (33) and (34), for all k that 1 where By (35) and Theorem 2.1, there exists c 2 (x) > 0, independent of k, such that Thus, by Lemma 2.1, we can find 0 < λ 1 (x) < r 1 (x) independent of k such that For 0 < λ < λ 1 (x), we have, using (35), that 1 2 (1 + |y|) < |y − x| and we obtain, using (37) and (34), that When and we obtain, using (37) and (34), that Letting we derive from (38) and (39) that Lemma 2.2 follows from (36) and (40). Define, for x ∈ R n and |x| ≤ R k /5, that By (32), (13)- (15). Then either v is constant or

Lemma 2.3 Assume
Sinceλ(x) < ∞, we have, along a subsequence,λ k (x) →λ(x)-but for simplicity, we still use {λ k (x)}, {v k }, etc to denote the subsequence. By the definition ofλ k (x), we have By the conformal invariance (7), Using (16), (41), (42), the definition ofλ k (x), and using the ellipticity of the equation satisfied by v k and (v k ) x,λ k (x) , we can apply the strong maximum principle and Hopf Lemma to infer the existence of some This implies, in view of (33), that On the other hand, ifŷ i is such that |ŷ i | → ∞ and This gives Step 1 is established.
Step 2. It remains to show that either v is constant or, for every x ∈ R n ,λ(x) < ∞.
To this end, we show that ifλ(x) = ∞ for some x ∈ R n , then v is constant. Indeed, assume thatλ k (x) → ∞ as k → ∞. We easily derive from this and the convergence of The above is equivalent to the property that for every fixed unit vector e, r n−2 In particular, α = lim inf |y|→∞ |y| n−2 v(y) = ∞. This implies, by Step 1, thatλ(x) = ∞ for every x ∈ R n , and therefore (43) holds for every x ∈ R n . This implies that v is a constant, see Corollary C.1. (13)- (15) and (18). Then the function v in Theorem 1.1 cannot be constant.

Lemma 2.4 Assume
Proof Fix some t > 0 for the moment. Set ϕ(x) = v(0) − t |x| 2 and fix some r > 0 such that ϕ > 0 in B r (0) and ϕ < v k on ∂ B r (0) for all sufficiently large k. Let and Noting that there is some C > 0 independent of δ and k such that, for large k, Thus, we can select t and δ such that where t 0 is the constant in (18). Since f (λ(A v k (x k ))) = 1, this contradicts (14), (15) and (18).
and it remains to consider the case that, for If v is in C 2 (R n ), the conclusion of Theorem 1.1 follows from the proof of Theorem 1.3 in [8]. An observation made in [11] easily allows the proof to hold for v ∈ C 0,1 loc (R n ). For readers' convenience, we outline the proof below. Let We know that λ(A v ψ (y)) = λ(A v (ψ(y)). Namely, A v ψ (y) and A v (ψ(y)) differ only by an orthogonal conjugation. Introduce We deduce from the above properties of v that for every x ∈ R n , there exists some δ(x) > 0 such that Namely, for some V ∈ R n , A calculation yields Thus Consequently, for somex ∈ R n and d ∈ R, Since v > 0, we must have We have proved that v is of the form (11) for somex ∈ R n and some positive constants a and b.

Proof of Theorem 1.2
Proof of Theorem 1. 2 We start with some preparation as in the proof of Theorem 1.1. We may assume that R k ≥ 5 for all k.
In case (54), we obtain that v * is radially symmetric about the origin thanks to Lemma C.1. To finish the proof, we assume in the rest of the argument that (55) holds and derive a contradiction. We first collect some properties ofλ(x). We start with an analogue of Lemma 2.3. By (46), let α := lim inf |y|→∞ |y| n−2 v * (y) ∈ (0, ∞].

Lemma 2.6
Under the hypotheses of Theorem 1.2, ifλ(x) < |x| for some x ∈ R n \{0}, then Proof We adapt Step 1 in the proof of Lemma 2.3. Assume thatλ(x) < |x| and (without loss of generality) thatλ k (x) →λ(x). Arguing as before but using the strong maximum principle for solutions with isolated singularities [10, Theorem 1.6] instead of the standard strong maximum principle, this leads to the existence of some y k ∈ ∂ B R k (0) such that This implies, in view of (47), that On the other hand, as in the proof of Lemma 2.3, we can use The conclusion is readily seen.
As in the proof of Lemma 2.6, there exists y k ∈ ∂ B R k (0) such that We know that In the computation below, we use o(1) to denote quantities such that Fix some δ > 0 and consider |x − x 0 | < |x 0 |/2. We note that (1)).

It follows that
Recalling (56), we arrive at Thus, in view of (48), we can find small¯ > 0 depending only on δ, c,λ(x 0 ) and the function m(·) such that, for all |x − x 0 | <¯ and for large k, This implies that (cf. (56)), that The conclusion follows.
We now return to drawing a contradiction from (55). By Lemma 2.7, we infer from (55) that there exists some r 0 > 0 such thatλ(x) < |x| for all x ∈ B r 0 (x 0 ). We can then argue as in the proof of Theorem 1.1, using Lemma 2.6 instead of Lemma 2.3 to obtain for somex ∈ R n and some a, b > 0. For small δ > 0, let

Local gradient estimates
In this section, we adapt the argument in [12] to prove Theorem 1.6.
We can now apply Theorem 2.1 to obtain Passing to a subsequence and recalling (58) and (61), we see thatv i converges in C 0,α (α < α < 1) on compact subsets of R n to some positive, locally Lipschitz function v * .
On the other hand, if we definē then by the conformal invariance (7), we have .
This contradicts (59), in view of (62) and the convergence ofv i to v * . We have proved (57).
From (57), we can find some universal constant C > 1 such that Applying Theorem 2.1 again we obtain the required gradient estimate in B 1/4 (0).

A quantitative centered Liouville-type result
In this subsection, we establish: Then for every > 0, there exists a constant δ 0 > 0, depending only on ( f, ) and , such that, for all sufficiently large k,

Proof of the equivalence between Propositions 4.1 and 4.2 It is clear that Proposition 4.2 implies Proposition 4.1.
Consider the converse. Let δ 0 = δ 0 ( ) be as in Proposition 4.1. Arguing by contradiction, we assume that there are some > 0 and a sequence of R k and u k ∈ C 2 (B R k (0)) such that k but the last estimate in Proposition 4.2 fails for each k. Defineū Returning to the original sequence u k we arrive at a contradiction.

Lemma 4.1 Under the hypotheses of Proposition 4.1 except for
Moreover, for every > 0, there exists k 0 ≥ 1 such that Proof We first prove (65). Since v k satisfies (63), we deduce from Theorem 1.6 that where C is independent of k. This yields (65) in view of Theorem 1.1. We now prove (66). Suppose the contrary, then there exists some > 0 and sequences of Because of (65), r i → ∞.
As in the proof of Lemma 2.2, there exists λ i > 0 such that By the explicit expression of U , there exists some small δ > 0 independent of i such that, for large i, By the uniform convergence of v k i to U on compact subsets of R n , we have, for large i, As in the proof of Lemma 2.2, the moving sphere procedure does not stop before reaching λ = 1 + δ, namely we have, for large i, Sending i to ∞ leads to A contradiction-since we see from the explicit expression of U that U 1+δ (y) > U (y) for all 1 < 1 + δ < |y| ≤ 2.  1, depending only on ( f, ) and , such that, for all sufficiently large k, and Proof Assume without loss of generality that ∈ (0, 1/2). Since v k → U in C 0 loc (R n ), there exist r 2 > 1 and k 1 , depending on , such that for all By (5), Using the superharmonicity of v k and the maximum principle, we obtain Thus, for any δ 2 ∈ (0, 2 n−2 ), we have for all sufficiently large k that Now if δ 1 < δ 2 , (69) is readily seen from (71) and (74). Letv Then Enlarging k 1 if necessary, we can apply Corollary A.3 in "Appendix A" to get where here and below C is some positive constant depending only on n. On the other hand, by Lemma 4.1, we have (after enlarging k 1 if necessary) It now follows from (75) and (76) that where c 1 depends only on n. (70) is then established for ≤ 1 c 1 with r 1 = 2r 2 and δ 1 = δ 2 /8. The conclusion for > 1/c 1 also follows.
If ( f, ) satisfies in addition the conditions (18), (19) and the normalization condition (21), then δ 3 can be chosen to be any constant smaller than R n U 2n n−2 dx.
Proof We adapt the proof of [7,Lemma 6.4]. Arguing by contradiction, we can find a sequence of 0 < u j ∈ C 2 (B 2 ) such that f (λ(A u j )) = 1 in B 2 (0), where y j ∈ B 3/2 (0) and d(y) = 3/2 − |y|. Let Then by the conformal invariance property (7) By Theorem 1.6, there is a constant C independent of j such that Thus, after passing to a subsequence, we can assume that v j converges in C 0 loc (R n ) to some positive function v (as v j (0) = 1). This contradicts (77).
The above argument can be adapted to prove the last assertion of the lemma: Eq. (77) is replaced by On the other hand, by Theorem 1.1, we have v j → U in C 0 loc (R n ). This gives a contradiction.

It follows from Lemma 4.4 that
for some universal constant C. Sinceṽ k also satisfies f (λ(Aṽ k )) = 1, we can apply Theorem 1.6 to obtain where C is universal. The conclusion then follows from Lemma 4.1.
Proof of Proposition 4.1 Fix > 0. In view of Lemma 4.2 (cf. (69)), we only need to prove that there exist δ 0 > 0 such that, for all sufficiently large k, Suppose the contrary of the above, then, after passing to a subsequence and renaming the subsequence still as {v k } and {R k }, there exist In view of the convergence of v k to U , |y k | → ∞ as k → ∞. Consider the following two rescalings of v k : for some constant C independent of k.
The above violates the radial symmetry ofv * . Proposition 4.1 is established.

Detailed blow-up landscape
The proof of Theorem 1.3 uses the following consequence of the Harnack-type inequality for conformally invariant equations, see [4,7,16].

Lemma 4.6
Let ( f, ) satisfy (13)- (15) and (5). There exists a constant C 6 , depending only Proof We give the proof here for completeness. By (5), Thus, by Corollary A.2 in "Appendix A" as well as the maximum principle, It follows that The constantm in the result can be selected to be the least integer satisfyinḡ (Clearly, this is an obvious upper bound for m if the x i 's satisfies (iii).) Let δ 3 and C 3 be the constants in Lemma 4.4. Fix some N 0 > C 1 δ 3 . Then there is some r 0 ∈ (3/2, 2) such that By Lemma 4.4, this implies that Let C 0 and δ 0 be as in Proposition 4.2 (corresponding to = 0 ). We can assume without loss of generality that C 0 > 2 and δ 0 < 1. (92) We now declare This choice of C * will become clear momentarily. (93), C * ≥ 2C 7 , and so, by (91), (93) gives Hence, an application of Proposition 4.2 to u on the ball B R 1 (x 1 ) leads to In particular, where we have used (93) in the last estimate.
we stop. Otherwise, in view of (94), there is some We then let R 2 = δ 0 R 1 2 so that (93) implies Hence, by Proposition 4.2, We then repeat the above process to define U 3 , V 3 , and to decide if a local max x 3 can be selected in V 3 , etc. As explain above, the number m of times this process can be repeated cannot exceedm. We have obtained the set of local maximum points {x 1 , . . . , x m } of u and have verified (i) and (iv) for (vi) is readily seen as (ii) is also clear for From construction, we have By Theorem 1.6, this implies that Also, note that, for and so It is now clear that (iii) and (v) hold for K sufficiently large. The proof is complete.

A quantitative Liouville theorem
Proof of Theorem 1.5 Assume by contradiction that, for some k v k (R k y) for |y| ≤ 3.
Then f (λ(A u k )) = 1 in B 3 (0) and Lemma A.1 For any 0 < 2ρ < ρ 0 < ρ 1 < ρ 2 < 1, there exists some constants C, C > 1, depending only on n, ρ 0 , ρ 1 , ρ 2 , such that the Green's function G for B 1 \B ρ satisfies Consequently, Proof In the following we use C 1 , C 2 , . . . to denote positive constants depending only on ρ 0 , ρ 1 , ρ 2 and n. For a fixed x satisfying ρ 1 ≤ |x| ≤ ρ 2 , it follows from the maximum principle that for some positive constant C 1 , It follows that for some positive constants C 2 and C 3 , Since G(x, y) is a positive harmonic function of y in (B 1 \B ρ )\{x}. We can apply the Harnack inequality to obtain, for some C 4 , By the maximum principle, It follows that for some K , Lemma A.1 is established.

Corollary A.2 For any
Then, for some constants C depending only on n, ρ 0 , ρ 1 , ρ 2 , Proof This follows from Corollary A.1 by sending ρ → 0.

Appendix B: A remark on viscosity solutions
In this section we consider the convergence of viscosity solutions in a slightly more general context. Let R n×n , Sym n×n , Sym n×n + denote the set of n × n matrices, symmetric matrices, and positive definite symmetric matrices, respectively. Let M = R n×n or M = Sym n×n . Let ⊂ R n , U ⊂ M be open, F ∈ C(U ), A ∈ C( × R × R n × Sym n×n ; M) and consider partial differential equations for the form F(A(x, u, ∇u, ∇ 2 u)) = 0.
To keep the notation simple, we will abbreviate A[u] = A(·, u, ∇u, ∇ 2 u), and whenever we write F(M), we implicitly assume that M ∈ U .
In applications, it is frequently assumed that M + N ∈ U for M ∈ U and N ∈ Sym n×n + , The following definition is "consistent" with the assumptions (99), (100) and (101) and with Definition 1.1. Proof The proof is standard and we include it here for readers' convenience. We will only show (b). The proof of (a) is similar. Fix x 0 ∈ and assume that ϕ ∈ C 2 ( ) such that (v − ϕ)(x 0 ) = 0 and v −ϕ ≥ 0 in some small ball B ρ (x 0 ). We need to show that F(A[ϕ](x 0 )) ≥ 0.
Thus, as min B δ (x 0 ) (v − ϕ δ ) = 0, there is some x k ∈ B δ (x 0 ) such that Also, since we have that x k → x 0 as k → ∞.