On the prescribing q curvature problem on s3

On the Prescribing Q-Curvature Problem on S3 CAI, RUILUN (B.S., Shanghai Jiao Tong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2014 ii iii Declaration I hereby declare that the thesis is my original work and it has been composed by myself in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Cai, Ruilun January 19, 2015 iv v To my parents vi Acknowledgement I would like to express my gratitude to my supervisor Professor Xu Xingwang who has brought me to this interesting area. Without his help and support, this thesis will never be finished. I would also like to thank Dr. Sanjiban Santra of University of Sydney for his help and support. During my study at National University of Singapore, I benifit a lot from my academic brothers Dr. Ngo Quoc Anh, Dr. Zhou Jiuru, Dr. Zhang Hong and Mr. Hong Liu. We have a lot of discussion on mathematics and other topics during these years. Cai, Ruilun January 19, 2015 Singapore vii viii ACKNOWLEDGEMENT Contents Acknowledgement vii Contents x Summary xi List of Notations and Conventions xiii Introduction 1.1 Prescribing Scalar Curvature Problem . . . . . . . . . . . . . . . . . . . . 1.2 Prescribing Q-Curvature Problem . . . . . . . . . . . . . . . . . . . . . . 1.3 Result of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background and Preparation 2.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Kazdan-Warner Type Obstruction . . . . . . . . . . . . . . . . . . . . . . Perturbation Result 13 Uniform Bound 21 ix x CONTENTS Proof of Theorem 1.3 33 5.1 A priori estimate for a family of equations . . . . . . . . . . . . . . . . . 33 5.2 Completion of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2.2 Calculation of the Leary Schauder Degree . . . . . . . . . . . . . 38 Conclusions and Further Work 45 34 Chapter 5. Proof of Theorem 1.3 Theorem 5.1. Assume that Q is a positive smooth function defined on S3 satisfying the nondegeneracy condition ( u)2 + |ru|2 > Fix any t0 > 0, ✓ > 0, there exists C > such that for any t [t0 , 1], all solutions of the equation (5.1) satisfy the following inequalities kukC 4,✓ < C, < S3 (Qti j )uj uj ! > > :min uj ! where each tj is in the closed interval [t0 , 1]. Without loss of generality, we may assume that the minimum of each uj is attained at the north pole. We may choose a conformal transformation vj := (uj j j for each j such that )(det(d j )) 1/6 is a solution to the equation PS v j = (Qtj j )vj 5.2. Completion of the Proof 35 which is nomalized in the sense that S3 vj = Thus, since Qt has uniform upper bound and lowerbound for t [t0 , 1]. By Lemma 4.2 and Lemma 4.3, there exists constants C1 and C2 such that C < vj < C for j By choosing a subsequence if necessary, we have vj ! v1 in C 3,↵ (K) for any < ↵ < and K ⇢ S3 \{N }, and tj ! t1 [t0 , 1]. We have v1 C (S3 \{N }) satisfy the equation PS v = Qt1 (N )v17 By a scaling of the solution v1 and a conformal transformation, we have v1 ⌘ 1. The same argument for asymptotic expansion for Q1 = tQ+(1 t)Q0 in the proof of Lemma 4.4 applies. Remark 5.2. Notice that the estimate clearly fails when t0 = 0, as the set of solutions to the standard case PS u + 15 u 16 =0 contains a family of bubbles. 5.2 5.2.1 Completion of the Proof Notations We follow the approach as in Section of Chang-Gursky-Yang [7] and Section of WeiXu [25]. We introduce some notations as there. The equation (5.1) may be written 36 Chapter 5. Proof of Theorem 1.3 as u= 1 P [Qt u ]. S We define a family of (Fredholm) operators Ft [u] = u + PS31 [Qt u ] = u for t [0, 1], where Lt [u] = Lt [u] 1 P [Qt u ]. S Finding a solution to (1.9) is equivalent to finding a zero of Ft for t = 1. To proceed, we may fix any small t0 > and define X = {u C 4,✓ (S3 ) : kukC 4,✓ < C, < u < C}, C where the constant C is given as in Theorem 5.1. Ft is a family of Fredholm operators and continuous in t [t0 , 1]. From Theorem 5.1, we have 62 Ft (@X) for t [t0 , 1]. This guarentees that the Leray-Schauder degree deg(Ft , X, 0) is well defined for t [t0 , 1]. Moreover, by homotopy invariance of LeraySchauder degree, deg(Ft , X, 0) is independent of t whenever t0  t  1. In order to prove Theorem 1.3, it suffices to show that deg(Ft0 , X, 0) 6= (5.2) To so, we will show that deg(Ft0 , X, 0) is completely determined by deg(C, B, 0). Recall that in Chapter 3, we have studied the existence of the solution up to the equation 5.2. Completion of the Proof PS u p = provided ✏ = kQ 15 k 37 (Q p )up + (Cp · x)up . (5.3) is small enough. The solution we have constructed is of the form up = T p (1 + w), (5.4) in which the perturbation term w satisfies ˆ S3 wxj = for j = 1, 2, 3, To this end, we define B = {up : p B} ⇢ S = u C (S3 ) : u > 0, and ˆ S3 ⇢ S0 = u S : u xj = for j = 1, 2, 3, u =1 . S3 B and S corresponds to two posible degree of freedom of the family of solutions in (5.3). By the results of Chapter 3, we also have |Cp | = O(✏), where ✏= Q 15 Moreover, deg(C, B, 0) = deg ✓ | | ◆ 6= 0. holds for small enough ✏. Also note that, by homotopy invariance of mapping degree and connectivity of the set of conformal transformations in consideration, deg( | | ) 6= remain unchanged if we change Q into any Q p where p B. 38 Chapter 5. Proof of Theorem 1.3 5.2.2 Calculation of the Leary Schauder Degree First of all, by the continuity of mapping degree under small perturbation, without loss of generality, we may assume that both C and Ft0 have only isolated non-degenerate zeros. In that case, their degrees can be calculated as sums of local degrees of zeroes. Now let u0 be an isolated zero of Ft0 , which comes from the perturbation result. Take a small neighbourhood N of u0 such that @N \ Ft0 [0] = ; and that u0 is the only zero of Ft0 in N . We try to calculate the local degree deg(Ft0 , N, 0) as follows. Take a sequence of compact mappings Ak approximating Lt such that the following properties holds 1. kAk [v] Lt [v]kC 4,✓ (S3 ) ! ¯ as k ! 1; uniformly for all v N 2. the image of each Ak is contained in a finite-dimensional subspace of C 4,✓ (S3 ); 3. Ft,k does not admit a zero on @N , where Ft,k [u] := u Ak [u] Then the local degree deg(Ft , N, 0) = deg(Ft,k , N \ Y, 0) for large enough k. We take Y = Lk i=0 (5.5) Ei as a finite dimensional approximation, where Ei denotes the space of ith -order spherical harmonics. The approximation of each operators are taken as restrictions of the original operators on Y. 5.2. Completion of the Proof 39 To study the local degree of Ft at the solution u0 . For simplicity, we would like to take a conformal transformation such that the zero in question in near by the constant solution u ⌘ to the standard case PS u + 15 u 16 =0 To so, we choose a conformal transformation u˜0 = (u0 0) 0, · (det(d )) let 1/6 . Moreover, for any u N , we define a map T0 by T0 u = (u 0) · (det(d )) 1/6 . Then PS u = ˜=Q where Q 0. Qu if and only if 1˜ Q(T0 u) . PS3 (T0 u) = (5.6) Moreover let F˜t = Ft T0 , then we have ˜ , 0) = deg(Ft , N, 0), deg(F˜t , N ˜ = T0 (N ). where N Thus we may only need to consider the case when ku0 1k1 = o(1). Moreover, by taking one more conformal transformation, without loss of generality we may consider u0 to be in a symmetric class of functions satisfying ˆ S3 xi u0 = for i = 1, 2, 3, 4. 40 Chapter 5. Proof of Theorem 1.3 Hence if u0 satisfies u0 + PS31 [Lt0 (u0 )] = 0, the linearization map of Ft0 around u0 is given by P [Qt0 u0 w]. S Ft00 (u0 )[w] = w (5.7) When considering the linearization, notice that L2 (S3 ) = span{Tu0 B, Tu0 S}. where Tu0 (B) and Tu0 (S) are the tangent space to B and S respectively. For one part of the linearization, let denotes the constant function u ⌘ 1, since Y \ T1 S = E0 E2 E3 ··· Ek and u0 is close to the constant function u ⌘ 1. We may calculate the matrix elements on the space E0 Moreover, since kQt0 15 k E2 E3 ··· = O(✏) and ku0 Ek . 1k1 = o(1), (5.7) reduces to 105 P [w] + (O(✏) + o(1))kPS31 [w]k. 16 S Ft00 (u0 )[w] = w (5.8) Now we calculate the matrix element of the linearization map Ft0 (u0 ) in the subspace E0 (i) dim(Ei ) We choose a basis {⇠j }j=1 E2 E3 ··· Ek . for each subspace Ei (i = 0, 2, 3, · · · , k). We need to calculate each (i ) (i ) < ⇠j11 , Ft0 (u0 )[⇠j22 ] >L2 (S3 ) 5.2. Completion of the Proof 41 Let denotes the ith eigenvalue of the Paneitz operator PS3 . We have (i ) (i ) < ⇠j11 , Ft0 (u0 )[⇠j22 ] >L2 (S3 ) 105 (i2 ) (i ) (i ) = < ⇠j11 , ⇠j22 PS3 [⇠j2 ] >L2 (S3 ) +o(1) 16 ✓ ◆ 105 (i ) (i ) = < ⇠j11 , ⇠j22 >L2 (S3 ) +o(1) 16ai2 ✓ ◆ 105 = as ✏ ! i1 i2 j1 j2 + o(1) 16ai2 The linearization map Ft0 (u0 ) restricted in the subspace E0 E2 E3 ··· Ek is represented by (a perturbation of) the matrix 0 B8 B 105 B0 16a2 B B 105 B0 16a B B. B. . . B. @ 0 ··· ··· ··· . ··· 105 16ak C C C C C C C C C C A On the other hand, as in [7] we can compute the linearization of Ft0 in the direction of Tu0 (B). Notice that L2 (S3 ) = T1 (B) T1 (B) \ Y = E1 , T1 (S) and T1 (S) \ Y = E0 E2 ··· Ek Since u0 is a perturbation to the constant function u ⌘ 1, Tu0 (B) is transversal to the space E0 E2 ··· Ek . Hence we can find a basis for Tu0 (B), consisting of 42 Chapter 5. Proof of Theorem 1.3 { i , i = 1, 2, 3, 4}. They are given by i = xi + e i + ✏i , where ei E0 ··· E2 Ek remain bounded and ✏i ! as ✏ ! 0. Now we calculate the linearization of Ft0 , in the direction Tu0 (B). In the view of the equation, PS3 up + (Qt0 p )up = (C(p) · x)up (5.9) Assume that the solution u0 corresponds to the point p0 B, clearly C(p0 ) = 0. We consider the solution up of the equation (5.9), with p = p0 + s⇠i B, where ⇠i is the ith -coordinate in R4 and s > 0. Notice that Ft0 (up ) = PS31 [C(p) · xup ] To calculate the matrix element, we consider < j , Ft0 (up ) >L2 (S3 ) = ˆ S3 (xj + ej + ✏j )PS31 [C(p) · xup ] Since C(p0 ) = and up is close to constant u ⌘ 1. The only term that contains the 1st order of s is in ˆ S3 x j PS 16s [C(p) · xup ] = (C (p0 ))ij 105 ˆ S3 x2j + higher order terms in s We denote the positive coefficient before s(C0 (p0 ))ij by a. As a result, we can express the L L L derivative of (5.7) with respect to the natural decomposition Y = E0 E1 · · · Ek 5.2. Completion of the Proof 43 as a perturbation of the following matrix: x 0 B B B0 aC0 0 B B 105 B0 x B 16a2 B B0 x 105 16a B B B . . . B. @ x 0 ··· ··· ··· ··· . ··· 105 16ak C C C C C C C C C C C C C A where al is the lth -eigenvalue of the operator PS3 . The sign of the determinant of the above matrix is the same as the sign of the determinant of C0 . Thus, the local degree deg(Ft0 , N, 0) and deg(C, Up0 , 0) equals, where Up0 is a small neighbourhood of p0 . By summing up all the zeros, we have deg(Ft0 , X, 0) = deg(C, B, 0) = deg This finishes of the proof of Theorem 1.3. ✓ | | ,S ◆ 6= 44 Chapter 5. Proof of Theorem 1.3 Chapter Conclusions and Further Work The prescribing Q-curvature problem on S3 is studied in this thesis. We proved the existence of a solution to the prescribing Q-curvature equation (1.9) by applying LeraySchauder degree theory under some assumptions on the prescribed function. Inspired by the work of X. Chen and X. Xu [5] on prescribing scalar problem on Sn . It is interesting to try a flow approach for the prescribing Q-curvature problem (for n = or n cases). However, the difficulties is that maximum principle for parabolic equation is not applicable to such Q-curvature flow, because the Paneitz operator is a 4th order operator. To this end, there are difficulty in proving long time existence of a Q-curvature flow constructed similar to the scalar curvature flow in [5]. 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Based on the previous work, we study the prescribing Q- curvature problem on 3-dimensional sphere S3 The equation we need to solve is 1 PS3 u + Qu 2 7 =0 (1.9) u>0 where PS3 u = 2 S3 u + 1 2 S3 u 15 u 16 For such a prescribed function Q, we associate it with a vector field : S 3 ! R4 defined by (x) = S3 Q( x)x + rQ(x), where x is the position vector in R4 In the dimensional three case, the prescribing Q- curvature. .. In this thesis, we consider the prescribing Q- curvature problem on 3-dimensional sphere S3 In view of partial di↵erential equations, we established an existence result for a positive solution to the prescribing Q- curvature equation 2 S3 u + 1 2 S3 u 15 1 u + Qu 16 2 7 =0 on S3 The Schmidt-Lyapunov reduction method is applied to obtain a perturbation result (Q close to constant 15 ) 8 for the solvability... Perturbation Result In this section, we use the Lyapunov-Schmidt’s reduction method to solve the equation (1.9) We’ll prove the following perturbation result ˆ ˆ ˆ Theorem 3.1 Let Q be a smooth function defined on S 3 with kQk1 = 1 If Q satisfies: ˆ ˆ 1 the nondegeracy condition |rQ|2 + | Q| 2 > 0, 2 the degree condition deg( | | ) 6= 0, where (x) = ˆ ˆ Q( x)x + rQ(x), then there exists some ✏0 > 0 such that the. .. important role in the prescribing Q- curvature problem Let be a conformal transformation on S3 and g = u 4 g0 be a conformal metric on S3 , then we have ⇤ g0 = (T u) 4 g0 (2.3) 2.2 Kazdan-Warner Type Obstruction 9 where the conformal factor after the transformation is given by T u = (u )|det d | 1 6 (2.4) After the action of the conformal transformation, the equation (1.9) becomes 1 PS3 (T u) + (Q 2 )(T u)... combining the results of several lemmas below Theorem 4.1 If Q is a smooth positive function defined on S3 satisfying the nondegeneracy condition ( Q) 2 + |rQ|2 > 0, then there exist positive constants C1 and C2 which only depend on M = max Q and m = min Q, such that any positive solution u of 1 Qu 2 PS 3 u = 7 satisfies C1  u  C2 Lemma 4.2 Assume that u is a positive solution of the equation (1.9), then there... Backgrounds The equation (1.9) has a variational structure We may consider the corresponding functional J[u] = ✓ˆ Qu 6 S3 = ✓ˆ ◆1/3 ✓ˆ S3 Qu 6 S3 ◆1/3 ✓ˆ ◆ uPS3 u , ( u) 2 S3 ◆ 15 2 u , 16 1 |ru|2 2 (2.1) u > 0 The equation (1.9) has negative exponent in its nonlinear term In contrast with the case when n 5, the standard Sobolev inequality is no longer applicable to obtain a lower bound for the functional J[u]... for some constants cj (j = 1, 2, 3, 4) Then the equation (3.6) has a solution if and only if ˆ S3 f xi = c i ˆ S3 x2 i for i = 1, 2, 3, 4 (3.7) 16 Chapter 3 Perturbation Result If the above condition (3.7) holds, the solution is smooth and satisfies the following estimate kwkW 2,2 (S3 )  Ckf kL2 (S3 ) Moreover, w is unique within the class of function satisfying the following condition ˆ S3 wxi = 0... Q- curvature equation has a negative exponent in its nonlinear term Similar to the works of X Xu and J Wei in [24] and [25], the Schmidt-Lyapunov reduction method is applied to obtain the following perturbation result ˆ Theorem 1.2 Let Q be a smooth function defined on S3 , which satisfies ˆ ˆ 1 the nondegeneracy condition, |rQ|2 + | Q| 2 > 0; 6 Chapter 1 Introduction 2 the degree condition, deg ˆ 3 kQk1 = 1... obtained a perturbation result of the prescribing Q- curvature problem for n 5, which is similar to that of A Chang and P Yang on the prescribing scalar curvature problem They proved the following result Theorem 1.1 For n 5, there exists ✏n > 0, depending only on the dimension, such that if a positive function f 2 C 1 (Sn ) satisfies the following conditions: 1 f n(n2 4) 8 1  ✏n , 2 f is non-degenerate of... the equation 1 PS3 u + Qu 2 15 8 has a positive smooth solution, where Q = 7 =0 (3.1) ˆ + Q and 0 < ✏ < ✏0 We will prove the theorem in several steps Firstly, for any point p in the unit ball B in R4 , we introduce an operator 7/6 Qp u 7 Sp [u] = P u + 2 ( S3 Qp u 6 )7/6 S3 ´ 13 on S3 (3.2) 14 Chapter 3 Perturbation Result where Qp = Q p, = 15 vol (S3 ) 8 In the definition of Sp [u], the exponent . (1.8)doesnotsatisfiesthePalais-Smale condition. Thus the direct variational method fails. 1.3 Result of This Thesis Based on the previous work, we study the prescribing Q- curvature problem on 3-di m ensi on al sphere. prescribing Q- curvature problem on 3-dimensional sphere S 3 . In view of partial di↵erential equations, we established an existen ce result for a positive solution to the prescribing Q- curvature equation  2 S 3 u. of equations 33 5.2 Completion of the Proof 35 5.2.1 Notations 35 5.2.2 Calculation of the Leary Schauder Degree 38 6 Conclusions and Further Work 45 Summary In this thesis, we consider the prescribing