CLASSIFICATION OF RADIAL SOLUTIONS TO LIOUVILLE SYSTEMS WITH SINGULARITIES. Chang-Shou Lin. Lei Zhang

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1 Manuscript submitted to AIMS Journals Volume X, Number X, XX 2X doi:1.3934/xx.xx.xx.xx pp. X XX CLASSIFICATION OF RADIAL SOLUTIONS TO LIOUVILLE SYSTEMS WITH SINGULARITIES Chang-Shou Lin Taida Institute of Mathematical Sciences and Center for Advanced Study in Theoretical Sciences National Taiwan University Taipei 16, Taiwan Lei Zhang Department of Mathematics University of Florida 358 Little Hall, P.O.Box Gainesville, Florida , USA Abstract. Let A = (a i ) n n be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: u i + n =1 a i x β e u (x) =, R 2, i = 1,..., n R 2 x β ie ui(x) dx <, i = 1,..., n where β 1,..., β n are constants greater than 2. If all β i s are negative we prove that all solutions are radial and the linearized system is non-degenerate. 1. Introduction. In this article we consider the following singular Liouville system u i + n =1 a i x β e u(x) =, R 2, i I := {1,..., n}, (1) x βi e ui(x) dx <, i I. R 2 where β 1,.., β n are constants greater than 2, A = (a i ) n n is a constant matrix that satisfies (H1) : A is symmetric, nonnegative, irreducible and invertible. A is irreducible means there is no disoint partition of I into I 1 and I 2 such that a i = for all i I 1 and I 2. For the system (1), the irreducibility of A means (1) can not be written as two independent subsystems. If n = 1 and a 11 = 1, the singular Liouville system is reduced to the following single Liouville equation: u + x β e u =, R 2, x β e u <. (2) R 2 21 Mathematics Subect Classification. Primary: 35J47; Secondary: 35J6. Key words and phrases. Liouville system, classification of solutions, non-degeneracy, radial symmetry, singularity. 1

2 2 CHANG-SHOU LIN AND LEI ZHANG Praapat-Tarantello [28] classified all the solutions to (2) and proved, on one hand, that if β/2 N, all solutions are radial and can be written as µ u(x) = log µ (1 + 8(β/2+1) x β+2 ) 2, µ >. 2 On the other hand, a solution may not be symmetric around any point if β/2 N. The proof of Praapat-Tarantello uses properties of integrable system. However, the Liouville system is not integrable and we have to apply new methods. The purpose of this paper is to prove a classification theorem for all the radial solutions to (1). Let u = (u 1,..., u n ) be a solution to (1) we use σ = (σ 1,..., σ n ) to denote its energy: and we set Λ I (σ) as σ i = 1 2π R 2 x βi e ui(x) dx, i I := {1,..., n} (3) Λ I (σ) = 2 i I (β i + 2)σ i a i σ i σ. i, I For J I, Λ J (σ) is understood similarly. The main theorem of this article is Theorem 1.1. Let A satisfy (H1), β 1,..., β n > 2 be constants, 1. If u = (u 1,..., u n ) is a radial solution to (1), then Λ I (σ) =, Λ J (σ) > J I. (4) 2. For each σ = (σ 1,..., σ n ) satisfying (4), there exists a global radial solution u whose energy is σ. 3. If u and v are both radial solutions to (1) with r βi+1 e ui(r) dr = r βi+1 e vi(r) dr, i I. Then u i (r) = v i (δr) + (2 + β i ) log δ for some δ > and all i I. System (1) is reduced to the following form if β 1 =... = β n =, u i + I a ie u =, R 2, R 2 e ui <, R 2. Under the assumption (H1) on A, a standard moving-plane argument shows that all u 1,.., u n are radially symmetric with respect to a common point (see [12] for the proof). The classification of all solutions to (5) has been completed through the works of Chipot-Shafrir-Wolansky [12, 13] and the authors [24]. Among other things Chipot et. al. prove that Theorem A: (Chipot-Shafrir-Wolansky) Suppose A satisfies (H1), for any solution u = (u 1,..., u n ) to (5), its energy σ = (σ 1,..., σ n ) belongs to the hypersurface Γ := {σ = (σ 1,..., σ n ); Λ I (σ) =, Λ J (σ) >, J I. } On the other hand, for any σ Γ, there is a solution u of (5) whose energy is σ. It can be readily verified that the energy of a solution of (5) is invariant under rigid translations and appropriate scalings: Let u be a global solution to (5), then v = (v 1,..., v n ) defined by v i (y) = u i (δy + x ) + 2 log δ, i I (6) (5)

3 LIOUVILLE SYSTEM WITH SINGULARITY 3 for any x R 2 and any δ > clearly satisfies R 2 e vi = R 2 e ui for all i I. It turns out that for any σ Γ, all the global solutions that have the energy σ are related by a translation and a scaling described in (6): Theorem B: ([24]) Suppose A satisfies (H1). Let u = (u 1,..., u n ) and v = (v 1,..., v n ) be global solutions to (5) such that R 2 e ui = R 2 e vi for all i I, then v and u are related by (6) for some δ > and x R 2. Theorem A and Theorem B together give a classification of all the solutions to (5). One obvious question that Theorem 1.1 raises is, for what β 1,.., β n do all the solutions to (1) have to be radially symmetric? We give an affirmative answer for the case of non-positive β. Theorem 1.2. Let u be a solution to (1), A satisfy (H1). Suppose β i ( 2, ] for i I and are not all equal to. Then all components of u are radial functions. Systems (1) and (5) and their reductions appear in many disciplines of mathematics and have profound background in Physics, Chemistry and Ecology. When (5) is reduced to one equation, it becomes the classical Liouville equation u = e u, which is related to finding a metric with constant Gauss curvature. In Physics, the Liouville equation represents the electric potential induced by the charge carrier in electrolytes theory [29] and is closely related to the abelian model in the Chern- Simons theories [18, 19, 17]. The Liouville systems (5)(1) are used to describe models in the theory of chemotaxis [11, 21], in the physics of charged particle beams [2, 14, 22], and in the theory of semi-conductors [26]. For applications of Liouville systems, see [3, 12, 24, 25] and the references therein. Here we note that Liouville systems with singularities are of special importance in Physics and Geometry. For example, the single equation (2) appeared in [27] as a limiting equation in the blow-up analysis of periodic vortices for the Chern-Simons theory of Jackiw and Weinberg [16] and Hong et. al. [15]. In geometry (1) is related to finding metric with conic singularities [4, 5, 6, 8, 2]. It is well known that classification theorems are closely related to blowup analysis and degree-counting theorems. For many equations the asymptotic behavior of blowup solutions are approximated by global solutions. For example, for the Liouville equation u + V e u =, Ω R 2, if V is a positive smooth function, blowup solutions near a blowup point can be well approximated by global solutions to u + e u =, R 2 see [9, 23, 7, 3]. If V is nonnegative and the blowup point happens to be a zero of V, the profile of blowup solutions is similar to that of the global solutions of (2), see [28, 1, 31]. We expect Theorem 1.1 to be useful in the study of singular Liouville systems defined on Riemann surfaces or domains in R 2. The proof of the uniqueness part of Theorem 1.1 (the third statement) is motivated by the authors previous work [24] on the Liouville system with no singularity. The existence part (the second statement) is based on the uniqueness result and is therefore significantly different from the duality method used by Chipot. et. al. in [12]. The first statement in Theorem 1.1 is similar to the corresponding case in [12].

4 4 CHANG-SHOU LIN AND LEI ZHANG For many applications, especially on the construction of bubbling solutions it is important to study the nondegeneracy of the linearized system. Our next result is concerned with the case when β is non-positive. Theorem 1.3. Let β i ( 2, ) for all i I, u = (u 1,..., u n ) solve (1) corresponding to β = (β 1,.., β n ). Let φ = (φ 1,.., φ n ) be a bounded solution to φ i + I a i y β e u(y) φ (y) =, R 2, i I. Then there exists C R such that φ i (r) = C(ru i (r) β i) for all i I. Remark 1. By Theorem 1.2 u is radial in Theorem 1.3. The organization of this paper is as follows. In section two we list standard tools to be used in the proof of Theorem 1.1. Then in section three we prove the three statements of Theorem 1.1. Theorem 1.2 and Theorem 1.3 are proved in section four and section five, respectively. Finally in the appendix we provide proofs for the tools used in the proof of Theorem Preliminary results. In this section we list a few ODE lemmas to be used in the proof of Theorem 1.1. Since these lemmas are standard we put their proofs in the appendix, in order not to disturb the main part of the paper. Lemma 2.1. Let u = (u 1,..., u n ) be a solution to (1) where A satisfies (H1). Then where u i (x) = m i log x + c i + o( x δ ), i I, x > 1, u i (x) = m i x/ x 2 + O( x δ 1 ), i I, x > 1, m i = c i = u i () + δ is some positive small number. n a i σ > 2 + β i, i I, =1 log r n a i r β+1 e u(r) dr =1 Remark 2. u is not assumed to be radial in Lemma 2.1. The next lemma is on the linearized system of (1) expanded along a radial solution u: (rφ i) + a i r β e u rφ =, i I. (7) Lemma 2.2. Let φ = (φ 1,..., φ n ) satisfy (7) with β i > 2 for all i I, then φ i (r) = O(log r) at infinity for i I. Lemma 2.3. Let A satisfy (H1), β i > 2 for i I, then for any c 1,..., c n R, there is a unique solution to { u i (r) + 1 r u i (r) + n =1 a ir β e u(r) =, i = 1,.., n, (8) u i () = c i, i = 1,.., n that exists for all r >. Remark 3. u may not have finite energy. If we further know that a ii > for all i, then the solution has a finite energy:

5 LIOUVILLE SYSTEM WITH SINGULARITY 5 Lemma 2.4. Let a ii > for all i I, then for all c 1,..., c n R, there exists a solution to u i + 1 r u i (r) + I a ir β e u(r) =, < r <, i I, e ui(r) r β+1 dr <, i I, u i () = c i, i I. Lemma 2.5. Let φ be a solution of (rφ i (r)) + n =1 a ir β+1 e u φ (r) =, < r <, φ i () =, i I. Suppose β i > 2 for all i I, then φ i for all i I. (9) 3. Proof of Theorem 1.1. Proof. (Proof of Theorem 1.1) 3.1. The proof of the first statement of Theorem 1.1. Lemma 3.1. Let u = (u 1,.., u n ) be a radial solution of (1) with A satisfying (H1) and β i > 2 for all i. Then Λ I (σ) =, Λ J (σ) >, J I. Proof of Lemma 3.1: This proof uses the same idea as in [12]. Let ũ i (t) = u i (e t ), then ũ i(t) m i as t. (1) The equation for ũ i (t) is ũ i (t) + n a i e (2+β)t+ũ(t) =, t R, i I. (11) =1 Let z i (t) = n =1 ai ũ (t), then z i (t) σ i as t. (11) can be rewritten as z i (t) = e (2+βi)t+ n =1 aiz, i I. (12) Clearly z i (t) < for all i I and all t R. Let w i(t) = z i (t), then by (12) and (1) w i ( ) =, w i (t) < t, w i ( ) = σ i. In addition we have w i ( ) = w i ( ) =. Using the definition of w i we differentiate (12) to obtain w i (t) = (2 + β i )w i(t) + w i(t) a i w (t). (13) I Taking the summation for i I in (13) we can write (13) as w i (t) (2 + β i )w i(t) = 1 2 a i(w i (t)w (t)). i I i I i, I Integrating t from to we obtain Λ I (σ) =. For J I, summation for i J in (13) leads to w i (t) (2 + β i )w i(t) 1 a i (w i w ) (t) = a i w 2 i(t)w (t). i J i J i, J i J, I\J

6 6 CHANG-SHOU LIN AND LEI ZHANG Integrating the above for t (, ) we obtain (2 + β i )σ i 1 a i σ i σ = 2 i J i, J i J, I\J a i w i(t)w (t)dt. Since the irreducibility of A means that there exists a i > for i J, I \ J we see the right hand side of the above is strictly positive because w i (t) < for all i I and t R and w i (t) < for all i I and t R. Thus we have obtained Λ J (σ) >. Lemma 3.1 is established The proof of the third statement of Theorem 1.1. Proposition 1. Let u and v both be radial solutions to (1) such that r βi+1 e ui(r) dr = r βi+1 e vi(r) dr, i I. Then u i (r) = v i (δr) + (2 + β i ) log δ for some δ > and all i I. To prove Proposition 1 we first establish a uniqueness result for the linearized system: Lemma 3.2. Let φ = (φ 1,..., φ n ) be a bounded solution of (7), then φ i (r) = C(ru i (r) β i) for all i I. Proof of Lemma 3.2: Let φ = (ru 1(r) β 1,..., ru n(r) β n ). Then by computation φ is a solution to the linearized system. Suppose there exists another bounded solution φ 1 which is not a multiple of φ. Without loss of generality we assume φ 1 1() =, as by Lemma 2.5 one of φ 1 i () must be different from 2 + β i. To derive a contradiction we set S = {α; a bounded solution φ = (φ 1,..., φ n ) such that φ 1 () = 2 + β 1, φ i () = α i 3 + β i ;, i = 2,..., n, α = min{2 + β 1, α 2,..., α n }, e ui(s) φ i (s)s 1+βi ds >, r >, i = 1,..., n. }. First we see that 2 + min{β 1,..., β n } S. Indeed the expression of φ gives s 1+βi e ui(s) φ i (s)ds = r 2+βi e ui(r) >. Next we observe that S is a bounded set. Indeed, suppose α < is in S, let φ be the function corresponding to α, then I such that φ () = α. This leads to s1+β e u(s) φ (s)ds < for r small, a contradiction to the definition of S. Let ᾱ be the infimum of S and let α k = (α1, k..., αn) k S be a sequence in S that tends to ᾱ from above. Suppose φ k = (φ k 1,..., φ k n) is the solution corresponding to α k, then we claim that φ k converges to φ = ( φ 1,..., φ n ), which is also a bounded solution with the strict monotonicity property described in S. Indeed, let ψ m = (ψ1 m,..., ψn m ) be the solution to the linearized system such that ψi m() = δm i. Then by Lemma 2.5 n φ k = αmψ k m. m=1

7 LIOUVILLE SYSTEM WITH SINGULARITY 7 Here we recall that by Lemma 2.2 ψi m(r) = O(log r) for r large. Since ᾱ αk i 3 + β i for i I and all k. Along a subsequence φ k tends to φ over all compact subsets of R. The monotonicity property of φ k implies e ui(s) φi (s)s 1+βi ds, i I, r >. On the other hand, since φ k are all bounded functions, for each φ k i we find r l as l such that r l (φ k i ) (r l ). From the equation for rφ k i we have Since A is invertible = n a i r β+1 e u(r) φ k (r)dr =, i I. =1 e ui(r) φ k i (r)r βi+1 dr = n m=1 α k m e ui(r) ψ m i (r)r βi+1 dr. Since ψi m(r) = O(log r), e ui(r) ψi m (r)r βi+1 dr is well defined, we let α k (ᾱ 1,..., ᾱ n ) to obtain e ui(s) φi (s)s βi+1 ds =, i I. (14) As a consequence of (14), φ is bounded. Indeed, the equation for φ is (r φ i(r)) = a i r β+1 e u(r) φ (r), r >. Using φ i (r) = O(log r), r βi+2 e ui(r) = O(r δ ) for some δ > (Lemma 2.1) and (14) we know e ui(s) φi (s)s βi+1 ds = e ui(s) φi (s)s βi+1 ds = O(r δ/2 ) r for r large. Thus φ i (r) = O(r 1 δ ) for all r large, which implies that φ i is bounded. Since each φ i is a non-increasing function, (14) implies that φ i decreases to a negative constant when r. Indeed, by (14) either φ i or φ i decreases to a negative constant. The first possibility does not exist, because the fact φ 1 () = 2 + β 1 > implies that φ 1 decreases into a negative constant at infinity. Also s1+β1 e u1(s) φ i (s)ds > for all r. Consequently for all i in the set I 1 := {i I; a i1 > }, r φ i(r) a i1 s 1+β1 e u1(s) φ1 (s)ds <, r >. Therefore φ i strictly decreases to a negative constant for all i I 1. We can further define I 2 := {i I; a i > for some I 1. }. By the same reason as above φ i decreases to a negative constant at infinity for all i I 2. By the irreducibility of A all the components of φ decrease to negative constants at infinity.

8 8 CHANG-SHOU LIN AND LEI ZHANG Now we claim that ᾱ ɛ S for ɛ > small. To see this, consider φ + tφ 1 for t sufficiently small. Recall that φ 1 1() =, thus φ 1 () + tφ 1 1() = 2 + β 1. Clearly φ + tφ 1 solves (7). By choosing t positive or negative with t small we can make min i I Since φ + tφ 1 is bounded we have φ i () + tφ 1 i () = ᾱ ɛ >. e ui ( φ i + tφ 1 i )s βi+1 ds =, i = 1,..., n. Since φ i (r) tends to a negative constant as r and φ 1 is bounded, we know for r large and t small Consequently r e ui ( φ i (s) + tφ 1 i (s))s βi+1 ds <. e ui(s) ( φ i (s) + tφ 1 i (s))s βi+1 ds > r >. Thus ᾱ ɛ S for some ɛ > small, a contradiction to the definition of ᾱ. Lemma 3.2 is established. Proof of Proposition 1: We shall consider u i (r) + 1 r u i (r) + I a ir β e u(r) =, < r <, e ui(r) r βi+1 dr <, i I, Let u i () = c i, i = 1,..., n 1, u n () =. Π 2 := {σ = (σ 1,.., σ n ); Λ I (σ) =, Λ J (σ) >, J I. }. Π 1 := {C = (c 1,.., c n 1 ); (15) has a solution. }. Note that by Lemma 2.4 Π 1 = R n 1 if a ii > for all i. We claim that the mapping from Π 1 to Π 2 is locally one to one. Indeed, let M be the following matrix: c1 σ 1... cn 1 σ 1 M = c1 σ n 1... cn 1 σ n 1 We claim that M is nonsingular. We prove this claim by contradiction. Suppose there exists a non-zero vector D = (d 1,..., d n 1 ) T such that MD =. Then by setting γ = d 1 c d n 1 c n 1 we have For Π 2, Λ I (σ) = reads (15) γ σ 1 = γ σ 2 =... = γ σ n 1 =. (16) i, I a i σ i σ = 2 i I (2 + β i )σ i. By differentiating both sides with respect to γ we have ( a i σ 2 β i ) γ σ i =. i

9 LIOUVILLE SYSTEM WITH SINGULARITY 9 Since Λ J (σ) > implies a iσ > 2 + β i, (16) implies γ σ n =. Set φ i = γ u i (i I), then φ = (φ 1,..., φ n ) satisfies (7) and From γ σ i = (i I) we have φ i () = d i, i = 1,..., n 1, φ n () =. e ui φ i (s)s 1+βi ds =, i I. (17) As a consequence of (17), φ is bounded. Indeed, integrating (7) from to r rφ i(r) = = r a i s 1+β e u(s) φ (s)ds a i s 1+β e u(s) φ (s)ds = O(r δ ) for some δ >. Therefore φ (r) = O(r 1 δ ), which proves that φ i is bounded. By Lemma 3.2 φ i = c(ru i β), then we see immediately that c = because φ n () =, this is not possible because not all d i s are zero. Therefore we have proved that M is nonsingular for all C = (c 1,..., c n 1 ) Π 1. We further assert that there is one-to-one correspondence between Π 1 and Π 2. This is proved in two steps as follows. Case 1: a ii >, i I. In this case, by Lemma 2.4 Π 1 = R n 1. The mapping from Π 1 to Π 2 is proper and locally one to one. Here we claim that Π 2 is simply connected. Assuming this, since R n 1 and Π 2 are simply connected, there is one to one correspondence between them. Let u = (u 1,..., u n ) and v = (v 1,.., v n ) be two radial solutions such that u n () = v n () =, x βi e ui = x βi e vi (i I). Then u R 2 R 2 i () = v i () for i = 1,..., n 1. By Lemma 2.3 u i v i for all i I. Now we prove that Π 2 is simply connected. Indeed, using m i = a iσ, Λ I (σ) = can be written as a i (2 + β i )(2 + β ) = a i (m i 2 β i )(m 2 β ). (18) i, I i, I Therefore Π 2 is part of a quadratic surface, the boundary of which is restricted by Λ Ji (σ) = where J i is I with the index i removed. Λ Ji (σ) > reads m i 2 β i > a ii 2 σ i. In another word in the coordinate system represented by m i, we use n coordinate planes to bound the quadratic hypersurface described in (18). Other restrictions Λ J >, when J is obtained from I with at least two indices removed, do not affect the topological information of Λ I (σ) =. Thus Π 2 is a part of the quadratic hypersurface in the first quadrant and is therefore simply connected. Proposition 1 is proved in this case. Case 2: There exists i such that a i,i =. We prove this case by a contradiction. Suppose c k = (c k 1,..., c k n 1) (k = 1, 2) are two distinct points on Π 1 that correspond to the same energy: let u 1, u 2 be two solutions corresponding to c 1 and c 2 respectively such that e u1 i (r) r 1+βi dr = e u2 i (r) r 1+βi dr = σ i, i I.

10 1 CHANG-SHOU LIN AND LEI ZHANG ( ) σ Since the matrix M (n 1) (n 1) is nonsingular at c 1 and c 2, there is a oneto-one mapping between a neighborhood of c k to a neighborhood of σ in Π 2. Since c c 1 c 2, we choose the neighborhoods around them to be disoint. Now consider a perturbation system u i (r) + 1 r u i (r) + I (a i + ɛδ i )r β e u =, r >, i I, r βi+1 e ui dr <, i I, (19) u 1 () = c 1,...u n 1 () = c n 1, u n () =. Let u k,ɛ be the solution to (19) that corresponds to the initial condition c k = (c k 1,..., c k n 1, ) (k = 1, 2). Let σ k,ɛ = (σ k,ɛ,.., σk,ɛ n ) be defined as We claim that σ k,ɛ i = 1 r βi+1 e uk,ɛ i (r) dr, i = 1,.., n. σ k,ɛ = (σ 1,.., σ n ) + (1), k = 1, 2. (2) and σ k,ɛ i = σ i + (1), i = 1,.., n, = 1,.., n 1, k = 1, 2. (21) c c Assuming (2) and (21) for the moment. Now the matrix c1 σ k,ɛ 1... cn 1 σ k,ɛ 1... c1 σ k,ɛ n 1... cn 1 σ k,ɛ n 1 is non-singular at c k (k = 1, 2) for ɛ small. On the other hand, σ 1,ɛ and σ 2,ɛ both satisfy i, I (a i + ɛδ i )σ k,ɛ i σ k,ɛ = (22) Λ ɛ J >, J I. Λ ɛ I (σk,ɛ ) := i I 2(2 + β i)σ k,ɛ i We use Π ɛ to represent the hyper-surface described as above. For σ 2,ɛ = (σ 2,ɛ 1 Π ɛ, we can find c 1,ɛ = (c 1,ɛ 1,.., c1,ɛ n 1 ) such that c 1,ɛ = c 1 + (1), = 1, 2,.., n 1 and a solution ū 1,ɛ of (19) with the initial condition (c 1,ɛ After using Λ ɛ I (σ2,ɛ ) = in (22) we have β+1 r eū1,ɛ dr = σ 2,ɛ, = 1, 2,.., n 1. βn+1 r eū1,ɛ n dr = σ 2,ɛ n. 1,.., c1,ɛ n 1, ) such that,.., σ2,ɛ n ) Then the difference between c 1 and c 2 implies c 1,ɛ c 2 for ɛ small. A contradiction to the uniqueness property satisfied by the system (19). To finish the proof we now verify (2) and (21). Here we require ɛ (, δ ) where δ is so small that the matrix (a i + ɛδ i ) n n is non-singular for all ɛ (, δ ).

11 LIOUVILLE SYSTEM WITH SINGULARITY 11 For u k, there exists R large such that for r > R and some δ >, (u k i ) (r)r 2 β i 2δ, i = 1,..., n, k = 1, 2. For δ small we have u k,ɛ i converges uniformly to u k i over r R. For r = R we have (u k,ɛ (r)) r (2 + β + δ) at r = R, ɛ δ. Then by the super-harmonicity of u k,ɛ Thus, C > and R 1 R such that Hence for k = 1, 2, σ ɛ = it is easy to show (u k,ɛ (r)) r (2 + β + δ) for r R. e uk,ɛ (r) r β+1 dr = (2) is verified. To show (21) u k,ɛ c σ ɛ i c = r β e uk,ɛ (r) Cr (2+δ) for r R 1 (23) e uk (r) r β+1 dr + o(1) = σ + (1), = 1,.., n. r βi+1 e uk,ɛ i (r) uk,ɛ i (r)dr, i = 1,.., n, k = 1, 2. (24) c satisfies the following linearized equation: ( uk,ɛ i ) = c l By Lemma =1 (a i + ɛδ i )r β e uk,ɛ u k,ɛ c l, i = 1,.., n, l = 1,..., n 1. uk,ɛ i (r) C ln r, r 2, i = 1,.., n, l = 1,.., n 1 (25) c l where the constant C is independent of ɛ (, δ ). Moreover, for any fixed R >, u 1,ɛ i c l (r) converges uniformly to u1 i c l (r) over < r < R with respect to ɛ. Using the decay estimates (23) and (25) in (24) we obtain (21) by elementary analysis. Proposition 1 is proved in all cases The proof of the second statement of Theorem 1.1. Our proof is based on the uniqueness result and is completely different from the method employed in [12]. We divide the proof into two cases according to the diagonal entries of A. Case one: a ii > for all i I. In this case, by Lemma 2.4, for any c 1,..., c n 1 R, there exists a unique finite energy solution u = (u 1,..u n ) such that u i () = c i for i = 1,.., n 1 and u n () =. By Proposition 1 there is a biection between the initial condition (c 1,.., c n 1, ) and Π 2 (see the notation in the proof of Proposition 1). Thus Theorem 1.1 is proved in this case. Case two: There exists i I such that a i,i =.

12 12 CHANG-SHOU LIN AND LEI ZHANG Let σ Π 2, then for ɛ > we consider u ɛ i = I (a i + ɛδ i ) x β e uɛ (x), R 2, u ɛ i () = cɛ i, i = 1,..., n 1, uɛ n() =, r βi+1 e uɛ i (r) dr = σ ɛ i, i I where σ ɛ = (σ ɛ 1,..., σ ɛ n) is a point on the hyper-surface Π ɛ 2 := {σ = (σ 1,..., σ n ); σ i >, i I, Λ ɛ I(σ) =, Λ ɛ J(σ) >, J I} such that σ ɛ σ as ɛ. Here we recall that Λ ɛ I (σ) is defined as Λ ɛ I(σ) := 2 (2 + β i )σ i (a i + ɛδ i )σ i σ. i I i, I The vector (c ɛ 1,..., c ɛ n 1, ) is the initial condition corresponding to σ ɛ. Now we claim that max c ɛ i C i = 1,..., n 1 (26) for some C > independent of ɛ. Indeed, if this is not the case, without loss of generality we assume c ɛ 1 is the largest among c ɛ i and tends to infinity. Re-scale uɛ according to c ɛ 1 to make the maximum of all components at equal to. The re-scaled system has to converge in Cloc 2 (R2 ) norm to a partial system. Indeed, the first component converges because all the components are bounded. The n th component tends to because the initial condition is before the scaling and all components are non-increasing. Therefore for the limit function v = (v 1,..., v n ) without loss of generality we assume v m+1 =...v n = for some 1 < m < n. For i = 1,..., m we easily observe that σ i := The reason is for each fixed R > we have R r βi+1 e vi(r) dr σ i, i = 1,..., m. (27) r βi+1 e vi(r) dr σ ɛ i + (1), i = 1,..., m. Clearly (v 1,..., v m ) satisfies v i + m =1 a ir β e v =, i = 1,..., m, By Lemma 2.1 r βi+1 e vi(r) dr σ i, i = 1,.., m. m a i σ > 2 + β i, i = 1,...m. (28) =1 We claim that σ = ( σ 1,..., σ n ) with σ m+1 =... = σ n = satisfies Λ I ( σ) =. Indeed, let v m+1 =... = v n and H i = 1 if i = 1,..., m and H i = for i = m + 1,..., n. Then the system for v can be written as n v i + a i r β H e v =, i = 1,..., n. =1 Apply the standard method to obtain the Pohozaev identity to the system above we have Λ I ( σ) =. Let J = {1,...m} we have Λ J ( σ) =. Let z i = σ i σ i. From

13 LIOUVILLE SYSTEM WITH SINGULARITY 13 the definition of σ i we know that z i for i = 1,..., m Since 1 < m < n we have Λ J (σ) >, Λ J (σ) Λ J ( σ) > gives ( ( a i σ (2 + β i ))z i + ( ) a i σ (2 + β i ))z i <. (29) i J J J Since J a i σ > 2 + β i for all i J, we also have J a iσ > 2 + β i for all i J because σ i σ i. Clearly (29) is impossible. (26) is proved. Similarly there is a lower bound for c ɛ 1,..., c ɛ n 1. As ɛ, the u ɛ converges to u that corresponds to σ. Theorem 1.1 is proved in both cases. 4. Proof of Theorem 1.2. The proof of Proposition 4.1 in [12] can be readily applied to prove Theorem 1.2. We include it for the convenience of readers. For λ >, let u λ i (x 1, x 2 ) = u i (2λ x 1, x 2 ). Set Σ λ = {x R 2 ; x 1 > λ.} and T λ be the boundary of Σ λ. The equation for u λ = (u λ 1,..., u λ n) is u λ i + I a i x λ e uλ =, i I (3) where x λ = (2λ x 1, x 2 ). Set wi λ = uλ i u i to be defined in Σ λ for λ >. For wi λ we have wi λ + a i x βi e ξλ w λ = a i ( x λ β x β )e uλ where e ξλ i = ewλ i e w i wi λ w i Since β i for all i I, Let f = log log( x + 3), then f(x) = w λ i + = 1 e wi+t(wλ i wi) dt. a i x βi e ξλ w λ. (31) 3 r(r + 3) 2 log(r + 3) 1 (r + 3) 2 log 2 (r + 3). Therefore for any ɛ >, there exists C(ɛ) > such that Let z λ i = wλ i f f 1, r > C(ɛ). (32) r2+ɛ /f, then the following lemma holds. Lemma 4.1. There exists R > independent of λ such that for λ >, if x is a point where a negative minimum of min{z λ 1,..z λ n} is attained, then x B R. Proof. ( Proof of Lemma 4.1) From (31) we obtain z λ i + 2 z λ i f f + zλ i f f + a i e ξλ z λ. (33) Suppose zi λ(x ) = min z λ(x ) < and x is where the negative minimum for zi λ is attained. Here we note that the global minimum of zi λ should be attained. Indeed, by Lemma 2.1, u i (x) = m i log x + c i + O( x δ )

14 14 CHANG-SHOU LIN AND LEI ZHANG when x is large. Thus, for λ >, since x λ < x, w λ i (x) = u λ i (x) u i (x) O( x δ ), x >> 1. Thus lim x zi λ (x). Let J = { I; z (x ) }. Here we observe that the image of the origin is not in J, because of the decay rate of u i. We rewrite (33) as z λ i + 2 z λ i f f + zλ i f f in a small neighborhood of x. Then at x, + J z λ i (x ), 2 z λ i (x ) f(x ) f(x ) a i e ξλ z λ. (34) =. For J, since w λ (x ), we have u λ (x ) u (x ), so if x is large, by Lemma 2.1, e ξλ (x) x 2 δ for x close to x and some δ >. Thus a i e ξλ (x) Czi λ (x ) x 2 δ. J a i e ξλ (x) z λ (x ) z i (x ) On the other hand if x is large z λ i (x ) f(x ) f(x ) > z λ i (x ) x 2 ɛ. Therefore by choosing ɛ < δ/2 we see that (34) can not hold if x is large. Lemma 4.1 is established. By Lemma 4.1 and Lemma 2.1, min{w λ 1,..., w λ n} > in Σ λ for λ sufficiently large. Thus set λ := inf{λ > ; min{w λ 1,..., w λ n} > in Σ λ }. Lemma 4.2. λ =. Proof. (Proof of Lemma 4.2) If λ >, we first prove that w λ i > in Σ λ for all i I. Indeed, let I = {i I; w λ i }. If I is not empty, the irreducibility of A implies all w λ i in Σ λ. However, not all β i are, so for some i I, we have w λ i + I a i x β e ξ λ w λ = a i ( x λ β x β )e u λ <. I A contradiction. Next we derive a contradiction to the definition of λ. Let λ k tend to λ from the left. Thus λ k > for all large k. We can assume that min i I w λ k i < in Σ λk because otherwise, the strong maximum principle implies w λ k i > in Σ λk, a contradiction to the definition of λ. Therefore, let x k be where the minimum of min i I wi λ be attained and there is i k I such that w λ k i k (x k ) = min i I,x Σλk w λ k i <. By Lemma 4.1, x k B R for some R > and all k. Along a subsequence {x k } converges to x Σ λ such that for some i I, w λ i (x ) =. Since we have proved that w λ i > for all i I in Σ λ, x T λ. However, w ik (x k ) = leads to w λ i (x ) =, a contradiction to the Hopf Lemma. Lemma 4.2 is established. Thus we have proved λ =, which leads to u i ( x 1, x 2 ) u i (x 1, x 2 ), x 1, i I. Moving the plane from all possible directions we obtain the symmetry of u i. Theorem 1.2 is established.

15 LIOUVILLE SYSTEM WITH SINGULARITY Uniqueness theorem on the linearized system. In this section we prove Theorem 1.3. The following lemma describes the proection of u on sin kθ and cos kθ. Lemma 5.1. Let φ i,k (r) satisfy and φ i,k + 1 r φ i,k + a i r β e u(r) φ,k k2 r 2 φ i,k =, < r < (35) I φ i,k (r) Cr k (1 + r) 2k, r >, k 1. (36) If there exists f = (f 1,.., f n ) such that f i + 1 n r f i k2 r 2 f i + a i r β e u f <, r > (37) =1 and f i (r) >, r >, lim f i (r)/r k =, r Then φ ik. lim f i(r)r k =. (38) r Proof. (Proof of Lemma 5.1) We only need to show φ ik. Suppose this is not the case. Then because of the assumptions on the decay rates, without loss of generality we assume w 1 (r ) = φ 1,k(r ) f 1 (r ) = max i,r φ i,k (r) f i (r) >. (39) Note that the maximum can be attained because of the decay assumptions on φ i,k and (38). The equation for w 1, after simple derivation, is w 1 + ( 2f f 1 r )w 1 + w 1 ( f f 1 k2 f 1 r f 1 r 2 + a 11r β1 e u1 ) n + a 1 r β k e φ u k, =. f 1 Near r, w 1 (r) >. Thus in the neighborhood of r, using (37) we have w 1 + ( 2f n f 1 r )w 1 > a 1 r β e f w u 1 φ k,. f 1 =2 The left hand side of the above is non-positive when evaluated at r, while the right hand side is non-negative. A contradiction. Lemma 5.1 is established. =2 Proof. (Proof of Theorem 1.3) Let f i = u i (r). Direct computation shows that f i + 1 r f i 1 r 2 f i + a i r β e u f = a i β r β 1 e u. Since all β i <. f = (f 1,..., f n ) satisfies (37) and (38) for all k 2. Let φ k = (φ k 1,..., φ k n) be the radial part of the proection onto, say, sin kθ. Then φ k satisfies (35). Since φ k is bounded, it is easy to apply standard ODE theorem to obtain that (36) also holds. Thus all the proections on sin kθ and cos kθ are all zero for β.

16 16 CHANG-SHOU LIN AND LEI ZHANG Finally we prove that for the proection on sin θ or cos θ is also zero. Let φ 1 = (φ 1,1,.., φ 1,n ) be the proection of φ on sin θ. Then we have φ 1,i + 1 r φ 1,i 1 r 2 φ 1,i + a i r β e u φ 1, =. Since φ 1 is bounded, the standard ODE theory implies that φ 1,i behaves like O(1/r) at infinity and like O(r) near. We shall use f = ( u 1,..., u n) as the function to maorize φ 1. To apply the same argument as in the proof of Lemma 5.1, The problem is the maximum may tend to or infinity. We first prove that this can not happen at : φ i /f i is strictly increasing near if φ i is positive near. (4) Clearly once (4) is proved, φ 1, thus Theorem 1.3 would be established. Now we prove (4). Let z i = φ 1,i /r and F i = f i /r. Direct computation yields z i + 3 r z i + a i r β e u z =, r >. and F i + 3 r F i + a i r β e u F = a i β r β 2 e u. Since φ 1,i is positive near, z i () > (if z i () =, there is no need to consider this case, as the maximum can not tend to ). Easy to see that near, z i (r) = z i () + O(r β+2 ) and F i (r) = F i () + a i β + 2 rβ e u() + O(r β+1 ). Proving φ 1,i /f i to be increasing near is the same as proving that z i /F i is increasing near. Since β i <, one immediately sees that z i /F i is increasing near. Next we prove that z i /F i is decreasing if z i is positive at infinity. Assume z i (r) = q i /r 2 + O(r 3 ) at infinity. We have known that, for some δ i >, Thus u i (r) = m i log r + c i + O(r δi ) r > 1. e ui(r) = e ci r mi + O(r mi δi ), r >> 1. We obtain, by integration on the equation for z i, that z i(r) = 2q i r 3 + a i e c q m β 2 rβ m 1 + O(r β m 1 δ ). z i (r) = q i r 2 a i e c q (m β 2)(m β ) rβ m + O(r β m δ ). Correspondingly to compute F i, we use the equation for u i to obtain (ru i(r)) = a i r β+1 e u = a i r 1+β m e c + O(r 1+β m δ ).

17 LIOUVILLE SYSTEM WITH SINGULARITY 17 Using ru i (r) m i at infinity, we have ru i(r) = m i + a i e c m β 2 r2+β m + O(r 2+β m δ ). Consequently F i = u i (r) r = m i r 2 a i e c m β 2 rβ m + O(r β m δ ). F i = 2m i r 3 + m a i e c β m β 2 rβ m 1 + O(r β m 1 δ ). Our goal is z if i z i F i < near infinity when z i is positive near infinity. Using the expressions above it is enough to show if the following is negative: a i e c ( q i + m i q )r β m 2. (41) m β By q i /m i = max I q /m, q i > and β i < for all i I, we have (41). Therefore z i /F i is decreasing near infinity. Theorem 1.3 is proved. 6. Appendix. In this appendix, we prove the ODE lemmas stated in section two. Proof. (Proof of Lemma 2.1) The proof is standard ( for example, see [1]). We include it for the convenience of the reader. Let w i (x) = ( 1 1 log x y + R 2π 2π log(1 + y )) a i y β e u(y) dy. (42) 2 Clearly w i is well defined and satisfies w i (x) = a i x β e u(x), R 2 and (u i w i ) =, R 2. By Lemma 4.1 in [24] u i C on R 2. Next we claim that w i (x) lim x log x = m i. To see the above, it is easy to obtain for ɛ >, there exist R(ɛ) >> 1 and R 1 >> R such that for x > R 1 1 log x y + log(1 + y ) a i y β e u m i ɛ. 2π log x Also B R log x y + log(1 + y ) R 2 \B R log x Thus u i w i C log(1 + x ), which leads to for some C i R. Next we claim that a i y β e u dy ɛ. u i = w i + C i (43) m i β i > 2 for all i I. (44)

18 18 CHANG-SHOU LIN AND LEI ZHANG Indeed, if this is not the case, there exists i I such that m i β i = min{m 1 β 1,.., m n β n } 2. By (42) and (43) we have u i (x) = 1 ( log x y + log(1 + y )) 2π R 2 Easy to check thus log x y + log(1 + y ) log(1 + x ), a i y β e u(y) dy C. u i (x) m i log( x + 1) C (2 + β i ) log( x + 1) C a contradiction to x βi R 2 e ui (x) <. (44) is established. Now u i (x) can be written as u i (x) = 1 log x y a i y β e u(y) dy + c i. (45) 2π R 2 c i can be determined as in the statement. Finally we derive the error term O(r δ ). To see this we set E 1 = {y; y x /2.} E 2 = {y; y x x 2.} E 3 = R 2 \ (E 1 E 2 ). Using e ui(y) = O( y mi ) in (45) one obtains 1 2π E 1 log x y Similarly by elementary estimates E 2 E 3 log x y a i y β e u(y) dy = m i log x + O( x δ ). a i y β e u(y) dy = O( x δ ). The gradient estimate for u i is obtained by standard estimates. Lemma 2.1 is established. Proof. (Proof of Lemma 2.2) Let ψ(t) = (ψ 1 (t),..., ψ n (t)) be defined as Then ψ satisfies ψ i (t) + ψ i (t) = φ i (e t ), i I. a i e u(et )+(2+β )t ψ (t) =, < t <, i I. Let ψ n+1 = ψ 1,..., ψ 2n = ψ n and F = (ψ 1,.., ψ 2n ) T, then F satisfies F = MF ( ) I where M =. B is a n n matrix with B B i = a i e u(et )+(2+β )t. For t > 1, the solution for F is F(t) = lim N eɛm(t N )...e ɛm(t) F(). (46) where t,..., t N satisfy t = ɛ, =,.., N, ɛ = t/n. Since u i (e t ) + (2 + β i )t ( m i β i )t when t is large and m i > 2 + β i, we have B e δt for some δ > and t large. With this property we further have M k Ce kδ1t, k = 2, 3,... t > (47)

19 LIOUVILLE SYSTEM WITH SINGULARITY 19 for some δ 1 >. Using (47) in (46) we have Lemma 2.2 is established. F(t) = O(t), t > 1. Proof. (Proof of Lemma 2.3) If a solution u = (u 1,..., u n ) exists, it would satisfy n u i (r) = c i () a i t β+1 (log r log t)e u(t) dt, i = 1,..., n. =1 We first prove the existence of a solution on < r < δ for some small δ > by iteration: Let u () = (,.., ) and n u (k+1) i (r) = c i () a i t β+1 (log r log t)e u(k) (t) dt, i = 1,..., n. =1 For δ > small and r (, δ), since β i + 1 > 1, it is easy to see that such a sequence converges. Therefore the existence of a solution for over (, δ) is proved. The existence for r (δ, ) clearly holds because of the right hand side is a Lipschitz function of u. The proof of the uniqueness of the solution is the same as that in Lemma 2.5 later in the section. Lemma 2.3 is established. Proof. (Proof of Lemma 2.4) By Lemma 2.3 a solution to (8) exists for r >. We ust need to show that e ui(r) r βi+1 dr <, i I. Let v i (t) = u i (e t ) + (2 + β i )t (i I), then v = (v 1,..., v n ) satisfies v i (t) + a i e v(t) =, < t <, i I. From the equation for u we have ru i(r) = a i e u(s) s β+1 ds <, r >, i I. The last inequality is strict because a i and not all equal to. Consequently v i (t) < 2 + β i for t R. Fix t R we have, for t > t, v i(t) = v i(t ) t t a i e v(s) ds v i(t ) a ii t t e vi(s) ds, i I. Since a ii > there exists t > t such that v i (t) <. Choose v i (t 1) = δ < for some δ >, then we see v i (t) v i (t 1 ) δ(t t 1 ), t > t 1 which is equivalent to u i (r) < ( 2 β i δ) log r + C for r > e t1. e ui(r) r βi+1 dr <. Lemma 2.4 is established. Therefore

20 2 CHANG-SHOU LIN AND LEI ZHANG Proof. (Proof of Lemma 2.5) The proof is standard, we include it for the convenience of the reader. Clearly we only need to show that φ i in (, δ) for δ > small. Write φ i (r) as 1 s φ i (r) = ( a i t β+1 e u(t) φ (t)dt)ds s I = a i t β+1 e u(t) φ (t)(log r log t)dt I Let α = min{β 1,..., β n } + 1, since all β i > 2 we have α ɛ > 1 for some ɛ > small. For the ɛ we choose δ > small so that log r log t < t ɛ for r < δ and t r. Thus φ i (r) C t α ɛ φ (t) dt, r < δ for some C. Let φ(r) = i I φ i(r) and F (r) = tα ɛ φ(t) dt, then F (r) Cr α ɛ F (r), F, F () =. Since α ɛ > 1, F. Lemma 2.5 is established. Acknowledgments. Part of the work was finished when the second author was visiting Taida Institute for Mathematics Sciences (TIMS) in March 211. He would like to thank TIMS for the warm hospitality. The second author is also partially supported by a grant from National Science Foundation. REFERENCES [1] D. Bartolucci, C. C. Chen, C. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (24), [2] W. H. Bennet, Magnetically self-focusing streams, Phys. Rev. 45 (1934), [3] S. Chanillo, M. K-H Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type. Geom. Funct. Anal. 5 (1995), no. 6, [4] S. Y. Chang, M. Gursky and P. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. and PDE, 1 (1993), [5] S. Y. Chang and P. Yang, Prescribing Gaussian curvatuare on S2, Acta Math., 159 (1987), [6] C. C. Chen and C. S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom., 49 (1998), [7] C. C. Chen, C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (22), no. 6, [8] C. C. Chen, C. S. Lin, Mean field equations of Liouville type with singular data: Sharper estimates, Discrete and continuous dynamic systems, 28, No 3, 21, [9] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), [1] W. X. Chen, C. M. Li, Qualitative properties of solutions to some nonlinear elliptic equations in R 2. Duke Math. J. 71 (1993), no. 2, [11] S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis, Math. Biosci. 56 (1981), [12] M. Chipot, I. Shafrir, G. Wolansky, On the solutions of Liouville systems. J. Differential Equations 14 (1997), no. 1, [13] M. Chipot, I. Shafrir, G. Wolansky, Erratum: On the solutions of Liouville systems [J. Differential Equations 14 (1997), no. 1, 59 15; MR (98:3553)]. J. Differential Equations 178 (22), no. 2, 63. [14] P. Debye and E. Huckel, Zur Theorie der Electrolyte, Phys. Zft 24 (1923),

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