Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, Ferrara, Italy

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1 THE SCALAR CURVATURE EQUATION ON R n AND S n Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, Ferrara, Italy Abstract. We study the existence of positive solutions for the equation u + Kxu n+ n = 0 in R n n 3 which decay to 0 at infinity like x n. Kx is a function which is bounded from above and below by positive constants, and no symmetry assumption on K is made. We find conditions that guarantee existence for a large class of K s. As a consequence one can explicitly show that the set of coefficients for which a solution exists is dense, in C 1 norm, in the space of positive bounded C 1 functions. These conditions also allow us to display a radial K such that the previous problem has non-radial solutions but no radial solution. Some new results for the corresponding problem on S n are also proved. 0. Introduction. In this paper we prove some existence results for the following problem: 0.1 { u + Kxu n+ n = 0 in R n, u > 0 u = O x n as x, for any n 3. Here and throughout the paper K is a function which is bounded from above and below by positive constants. The previous equation is called the scalar curvature equation in R n. The existence of a positive solution for this equation amounts to the existence of a metric g on R n which is conformal to the standard metric g 0 on R n g = u 4 n g0 and whose scalar curvature is K. The decay condition at implies that the metric g gives rise via the stereographic projection to a metric on S n \ {a point} which is equivalent to the standard metric. This implies that the metric g is incomplete. The scalar curvature equation on R n has been studied by several authors, especially after the publication of the paper by W.M. Ni [19], see [0] for a review and many results regarding existence of solutions to this equation, with various asymptotic behaviour at infinity, are available. As far as the author knows all the existence results regarding problem 1 see for instance [16], [17], [6], [8], [5], [14], [15], [] holds under some symmetry assumption on the coefficient K, like radial symmetry, symmetry under a subgroup of the group of rotations or periodicity in one variable. The results of this paper do not assume any symmetry on K Mathematics Subject Classification. 35J0, 35J60, 58G03. 1 Typeset by AMS-TEX

2 GABRIELE BIANCHI Theorem 1. Let K C R n and suppose that K max := max R n K is achieved in at least two isolated points y, z. Suppose moreover that i there exist two constants c > 0 and ρ > n such that Kx K max c x y ρ for x in a neighborhood of y, Kx K max c x z ρ for x in a neighborhood of z, ii if w is a critical point for K with Kw n Kmax, K max then either Kw < 0 and w is a strict local maximum or Kw > 0, iii lim sup x Kx < n Kmax, iv the number of critical points w of K which satisfy Kw n Kmax is finite. Then there exists a solution to problem 0.1. Some remarks about theorem 1: - assumption ii requires that, in a certain range of values, there are neither critical points where K vanish nor saddle points with K < 0; - as n grows n becomes closer to 1 and assumptions ii and iii become weaker; - one can substitute assumption i with a slightly more general one: see the remark following the proof of Lemma.1. Theorem. Let K C R n and let K have only finitely many critical points. Suppose there exists z R n where K achieves a strict local maximum and Kz 0 and a continuous path xt in R n connecting z to some point y, y z, where Ky Kz. Suppose moreover that i min t Kxt > /n K max, ii if w is a critical point for K with Kw [min t Kxt, Kz then either Kw < 0 and w is a strict local maximum or Kw > 0, iii lim sup x Kx < min t Kxt. Then there exists a solution to problem 0.1. The value d := min t Kxt is just the highest value such that the set {x : Kx Kz} \ {z} can be connected to z by some arc in the region {K d}. If Kz = K max then the assumption Kz 0 can be weakened: it suffices requiring z to be an isolated maximum point. Remark. The class of K s which satisfy the assumptions of theorem is dense, in C 1 norm, in the class of positive bounded C 1 functions. The assumption in Theorem 1 about the asymptotic behaviour of K near y and z is automatically satisfied when n = 3. In [4] the author has proved the following result, which implies that when n 4 that assumption is really needed. Theorem A [4]. Let n 4. Take two distinct points y and z in R n and two positive numbers K z, K y. There exists a C positive function K with Kx = K y ε x y n for x in a neighborhood of y, Kx = K z ε x z n for x in a neighborhood of z, for some ε > 0 small enough such that 0.1 has no solution. The value of K on its critical points different from y and z, as well as the value of lim x Kx, can be made arbitrarily close to 0.

3 THE SCALAR CURVATURE EQUATION 3 This theorem extends a result, proved by the author and H. Egnell in [8], regarding the non-existence of radial solutions to 0.1 for some radial K s. We point out that when K y = K z = K max the coefficient K in theorem A satisfies every assumption in theorem except i showing that in that theorem assumption i can not be removed. Theorem together with a non-existence results proved in [6], enables us to prove the following theorem. Theorem 3. There exists a C radial function K such that 0.1 has non-radial solutions but no radial solution. 0. We will explicitly construct the function K in section 3. The scalar curvature problem on S n, that is the existence of a solution to nn u u + Kxu n+ n = 0 in S n, 4 u > 0 u H 1 S n, is closely related to problem 0.1. Problem 0.1 is equivalent via the stereographic projection to this one for a coefficient K which may be discontinuous in the north pole. When K C S n problem 0. has been studied in several papers see [13], [1], [1], [3], [11], [10], [15]. Some strong existence results have been proved for the problem on S 3 and, for any n, when K is close to a constant. Yan-Yan Li has recently proved a generalization to S n, n 4, of some of the results proved for S 3. His result, which is based on a counting index criteria, holds under the assumption that K is very flat near all its critical points. Condition i in theorem 1 of this paper is coherent with his assumption. Our techniques yields some new results also for the problem on S n. Theorems 1 and as well as theorems A and 3 continue to hold, with the obvious change of notations, also for 0.. If K C S n one can also drop the assumption regarding lim sup x K. We will not state the corresponding theorems here and we refer the reader to section 3 theorems 6,7,8 for the precise statements. When n = 3, under the additional assumption that 0.3 K does not vanish at any critical point of K, stronger results can be proved for problem 0.. Theorem 4. Let n = 3, K C S 3 and let 0.3 hold. Suppose there exists z S 3 where K achieves a strict local maximum and and a continuous path xt in S 3 connecting z to some point y, y z, where Ky Kz. Suppose moreover that if w is a critical point for K with Kw [min Kxt, Kz and Kw < 0 then w is a strict local maximum for K. Then there exists a solution to problem 0.. The difference with respect to Theorem is that here it is not necessary to assume min Kxt > /n K max. A simple consequence of this result is the following one. Theorem 5. Let n = 3, K C S 3 and let 0.3 hold. Let us assume that K has at least two local maximum points and that among the critical points of K with

4 4 GABRIELE BIANCHI K < 0 the one where K has the lowest value is a strict local maximum. Then there exists a solution to problem 0.. To prove the existence of a critical point for the functional F associated to our problem we will use a min-max method. We look for a min-max over the class of continuous paths connecting two critical points at infinity see section 1 for the definition. A key point in this study is that the Palais-Smale condition does not hold. To overcome this lack of compactness one can go back to the basic elements of the variational theory studying the deformation along the flow lines of F. This has been done by A. Bahri and J. M. Coron in [] and [3]. Their study allows us to regain compactness, along gradient flow, at some values of F. These values are related to the value of the coefficient K in some of its critical points. In our context, with respect to the study in [] and [3], we have further restricted the set of critical points of K which may cause loss of compactness see lemma.3 and proposition 3.. The paper is organized as follows: section 1 contains the basic definitions and recalls some of the results concerning the loss of compactness of the functional F; in section the min-max scheme is defined and it is studied how certain assumptions on K reflect on connection properties of some of the sublevel sets of F; section 3 contains the proofs of the existence results. 1. Loss of compactness on flow-lines. To unify the treatment of the problem on S n and the problem on R n we study the existence of a solution for problem 0. for a coefficient K which is not necessarily continuous in every point of S n. For d 0 let A d = {x S n : Kx d} We will say that K satisfies assumption H1d if H1d K C A d, A d is closed, and K has only finitely many critical points in A d. For the results regarding exclusively S 3 we will assume that H K C S 3, has only a finite number of critical points and K does not vanish at any critical point of K. Let Lu = u nn 4 u be the conformal Laplacian on S n and, for any u H 1 S n, let u L = u + nn S n 4 u dv here dv is the volume element of S n with respect to the standard metric and let <, > L be the corresponding scalar product. Finally let M + = { u H 1 S n : u L = 1 ; u 0 }; We will study the functional F : M + R defined by Fu = 1 S n K u dv,

5 THE SCALAR CURVATURE EQUATION 5 where = n/n is the Sobolev critical exponent. Its gradient is Fu = Fu n /4 u Fu L 1 K u 4/n u. Any critical point u M + can thus be rescaled, by multiplying it by an appropriate positive constant, so that the resulting function is a weak solution of 0.. In a series of paper Brezis, Coron, and Bahri introduced the notion of critical point at infinity for the functional F. Roughly speaking they are the limiting point of a Palais Smale sequence for F that converges weakly to 0 but does not converge strongly. We recall that a Palais Smale sequence for F is a sequence u m such that Fu m converges and Fu m 0 in the dual of H 1 S n. For a S n and λ > 0 let λ 1.1 δ a,λ x = c 0 λ λ cos da, x n Here c 0 is a constant, depending only on the dimension n, such that δ a,λ L = 1 and d, is the geodesic distance in S n with respect to the standard metric. The functions δ a,λ are the only extremal functions for the Sobolev inequality 1. v L S v 0, here v H 1 S n and S is the optimal constant and, after multiplication by a suitable constant, they are the only positive solutions of Lu + u 1 = 0. We observe that as λ becomes large the functions δ a,λ concentrate around the point a. For a positive integer p and for ε > 0, a neighborhood W p, ε of a critical point at infinity is defined as follows see for instance [3]: { W p, ε = u M + s.t. λ 1,..., λ p > 0 : a 1,..., a p S n with u c p λ i > 1/ε i; ε n ij 1 Ka i n 4 δai,λ i < ε, where c = p 1. Ka i n 1 = λ i /λ j + λ j /λ i + λ i λ j da i, a j > 1/ε i j For each u W p, ε we optimize the approximation of u by p 1 α iδ ai,λ i introducing the minimization problem M p M u W p, ε min a i S n,α i >0,λ i >0 u α i δ ai,λ i. Lemma 1.1. For any positive integer p there exists ε 0 > 0 such that for any u W p, ε with ε < ε 0 the minimization problem M has a unique solution up to permutations. This minimization problem has already been considered in various papers, see for instance [3] and [6], and the proof of Lemma 1.1 is available there. 1 }.

6 6 GABRIELE BIANCHI Lemma 1.. Assume 1.1 has no solution in H 1 S n and let u m be a P.S. sequence for F. Let K C 0 S n. Then there exists a positive integer p, a sequence ε m converging to 0 and an extracted subsequence of u m, that we still denote u m, such that u m W p, ε m. There exist p points a i, i = 1,..., p, in S n such that i a i m a i and 1 n. p 1.3 Fu m S 1 Ka i n Here the a i m, i = 1,..., p, are the points associated to u m via the minimization problem M. Lemma 1.3. Assume 1.1 has no solution in H 1 S n and let u m be a P.S. sequence for F. Let us now suppose that K C 0 A d for some d < S lim Fu m 1, that A d is closed and that lim Fu m < /n S Kmax. 1 Then there exists a sequence ε m converging to 0 and an extracted subsequence of u m, that we still denote u m such that u m W 1, ε m. There exist a 1 in A d such that, as m a 1 m a 1 and Fu m S Ka 1. Here a 1 m is the point associated to u m via the minimization problem M. Lemma 1.3 is by now classical, see [1] or [9]. The proof of Lemma 1.4 is similar to that of Lemma 1.3 and it is contained in the appendix. A. Bahri and J.M. Coron have gone further in the study of the loss of compactness for F studying the behaviour of the flow lines for the negative gradient of F near the critical points at infinity. Lemma Let u 0 H 1 S n and let ut be the solution of the following problem u 1.4 t = Fu; u0 = u 0. Let us suppose that ut stays definitively in W p, ε for any ε small enough and some p 1. Let n = 3 and let K satisfy H. Then p = 1 and if at denotes the point associated to ut via the minimization problem M, then at converges to some point a S 3 with Ka = 0 and Ka < 0. Let n 3 and let K satisfy H1d for some d < S lim Fu m 1. Let us assume that lim t Fut < S /n Kmax. 1 Then p = 1 and if at denotes the point associated to ut via the minimization problem M, then at converges to some point a S n with Ka = 0 and Ka 0. Note that the negative gradient flow generated by F leaves M + invariant see for instance [6, Lemma.1] and therefore ut M + for each t. The assertion regarding S 3 has been proved in [3], the proof of the rest of the statement is given in the appendix. Finally we recall another result proved in [3].

7 THE SCALAR CURVATURE EQUATION 7 Lemma 1.5. Let ut be a solution to 1.4 then lim t Fut = 0. Connection of the sublevel sets of F. Let y, z be two distinct point in S n. Let P y,z be the set of all continuous paths u s : 0, M + such that.1 as s 0 u s W 1, ε definitively ε > 0 and the point as associated to u s via problem M converges to y, as s u s W 1, ε definitively ε > 0 and the point as converges to z. Since Ky, Kz > 0 it is easy to see that the set of paths P y,z is non empty see for instance the proof of Lemma.1 below. As s 0 Fu s tends to S Ky 1 while as s Fu s S Kz 1. Let us assume that. γ P y,z max u γ Fu > maxs Ky 1, S Kz 1, and let.3 Γ = inf γ P y,z max u γ Fu;. In the proofs of the next two lemmas it is convenient to transform the problem on R n. Let s denotes the stereographic projection from S n \{North pole} to R n and let D 1, R n be the completion of C0 R n in the norm = dx 1/. To R n u H 1 S n it corresponds vx = us 1 x/1 + x n / D 1, R n. The 1 + λ x a n function corresponding to δ a,λ is U b,µ := c 0 λ n for suitable parameters b R n, µ > 0 about this parameters we just observe that as λ then µ and b sa. If u M + and v D 1, R n is the corresponding function then v = 1 and Fu = 1 R n Ks 1 x v dx. The functions U β,µ are the only minimizers of the Sobolev inequality.4 v S v 0, here v D 1, R n and S is the optimal constant and, for any u D 1, R n they satisfy.5 U 1 R n a,λ u dx = S R n U a,λ u dx.

8 8 GABRIELE BIANCHI Lemma.1. Let y, z S n, y z and let us assume that.6.7 Kx Ky cdx, y ρ Kx Kz cdx, z ρ for x in a neighborhood of y for x in a neighborhood of z for some positive constant c and some ρ > n, then inf γ P y,z max Fu < Ky n + Kz n n S. u γ Proof. We may always suppose that both y and z are different from the North pole. Let us transform the problem on R n. To simplify the notations let us keep calling y the point sy, z the point sz and let us denote Ks 1 x by Kx. Let u s = su y,λ + tu z,λ, with 0 s, t, s + t = 1 and λ > 0 large. The path u s, when properly normalized and transformed back in H 1 S n belongs to P y,z. Actually the behaviour at the endpoints U y,λ and U z,λ is not completely correct, but we can continue the path with the desired property. Bahri in [], estimate F3 page 14, proves that as λ.8 R n U y,λ U z,λ dx = Aλ n y z n + oλ n, for some suitable positive constant A. Let us write R n KxU 1 y,λ U z,λ dx = Ky U 1 y,λ R n U z,λ dx+ R n Kx KyU 1 y,λ U z,λ dx. The first integral on the right side, due to.5, is equal to S Uy,λ U z,λ dx and due to.8 it is equal to S Aλ n y z n + oλ n. The second integral on the right side is therefore equal to oλ n. We can conclude that.9 1 KxU y,λ U z,λ dx = S AKyλ n y 1 y n + oλ n, R n and a similar formula holds for KxU y,λ U 1 z,λ dx. Let us study the asymptotic behaviour of u s as λ. u s = s + t U y,λ + st U y,λ U z,λ dx R n = 1 + staλ n y z n + oλ n, where the last equality follows from.8. Now let us prove that.10 Kxu s dx = Kxs U y,λ + t Uz,λdx+ R n R n s 1 t KxU 1 y,λ U z,λdx + st 1 KxU y,λ U 1 z,λ dx + Rλ, R n R n

9 THE SCALAR CURVATURE EQUATION 9 with Rλ oλ n. Let us suppose n 4, that is, 4]. For each x, y 0 it is x + y x y x 1 y + xy 1 c 1 xy for some positive constant c 1 and as a consequence Rλ c 1 st U y,λ U z,λ dx. To estimate the last integral we split it into three parts + + U y,λ U z,λ dx, By,ε Bz,ε By,ε Bz,ε c where ε < y z. By Bx, ε we denote the sphere centered in x and with radius ε. The third integral is Oλ n by the dominated convergence theorem; posing z = λx y the first integral becomes c o λ n { z <λε} 1 + z n/ λ 1 + y 1 y + zλ 1 n/ dx λ ε y z n c o λ n { z <λε} 1 + z n/ dx = Oλ n logλ; and the second integral can be treated in the same way. This proves.10 for n 4. The case n=3 can be treated similarly, using the inequality x + y 6 x 6 y 6 6x 5 y + xy 5 c 4 x 4 y + x y 4 instead of the previous one. Let us now examine.10. We rewrite this formula in the following form Kxu s dx = s Ky U y,λ dx + t Kz U z,λ dx R n R n R n Kx KyU R n + s 1 t KxU 1 y,λ R n +s y,λ dx + t U z,λdx + st 1 Kx KzU z,λ dx R n KxU y,λ U 1 z,λ dx + Rλ. R n Due to.5 and.9 we get Kxu s dx =S s Ky + t Kz R n + Kx K 1 x s Uy,λ + t Uz,λ dx + Oλ n R n + AS λ n y z n s 1 tky + st 1 Kz + Rλ. Here K 1 x is a function equal to Ky in a neighborhood of y and to Kz elsewhere. To conclude the proof of the lemma we want to show that, for λ large enough, max Fu s < Ky n + Kz n n S. s Due to the previous estimates we have lim Fu u s s = lim λ λ Kxu s dx = S s Ky + t Kz Ky n + Kz n n S,

10 10 GABRIELE BIANCHI with equality if and only if s = s = Kz n /4 Ky n + Kz n, t = t = Ky n /4 Ky n + Kz n. Thus there is a problem only when s = s and t = t. We analyze this situation more carefully. In this case where u s = 1 + aλ n + oλ n, a = A y z n KyKz n 4 Ky n + Kz n Furthermore 1 Kxu s dx = Ky n + Kz n n S 1 aλ n cλ +oλ n,. where cλ = c 5 Kx K 1 x Kz R n U y,λ + Ky n U z,λ dx, n and c 5 is a positive constant. Thus, combining the above estimates yields u s Kxu dx s Hence the lemma follows if Ky n + Kz n n S 1 aλ n cλ + oλ n..11 lim inf λ cλ λ n = 0. Conditions.6 and.7 transformed on R n become.1.13 Kx Ky c x y ρ in a neighborhood of y, Kx Kz c x z ρ in a neighborhood of z. Let ε > 0 be such that ε < y z and.1 and.13 hold respectively in By, ε and Bz, ε. We split the integral in the definition of cλ as follows: + + Kx K 1 x By,ε Bz,ε By,ε Bz,ε c Kz n U y,λ + Ky n U z,λ dx. By the dominated convergence theorem the last integral is equal to Oλ n, and also Uz,λ dx = Oλ n, Uy,λ dx = Oλ n. By,ε Bz,ε

11 THE SCALAR CURVATURE EQUATION 11 Therefore we have only to study the asymptotics of By,ε Kx K 1xU and of Bz,ε Kx K 1xUz,λ can be treated in the same way. Kx K 1 xu By,ε y,λ dx dx. Let us consider the first one the other one y,λ dx c By,ε cλ n and the last integral is oλ n when ρ > n. Remark. One could prove the same claim assuming x y ρ U y,λ dx { z <λε} z ρ 1 + z n dx, Kx Ky cdx, y ρ 1 for x in a neighborhood of y, Kx Kz cdx, z ρ for x in a neighborhood of z, for some positive constant c and any ρ 1, ρ larger than nn n+ and satisfying 1 ρ ρ < n. To prove this it suffices to define u s in a slightly different way, that is u s = sδ y,λ + tδ z,µ, with λ not necessarily equal to µ and to repeat the previous proof, studying the asymptotic behaviour of Fu s as λ, µ. One could prove that Theorem 1 continues to hold when hypothesis i in that theorem is substituted by this more general assumption on the asymptotic behaviour of K near y and z. Lemma.. Let y, z belong to two different connected components of {x S n : Kx = K max }. Then inf max Fu > S. γ P y,z u γ K max Proof. Let Γ = inf γ Py,z max u γ Fu. The inequality Γ S K max Sobolev inequality.4. Let us prove that Γ > S K max. follows from the If not there would be a sequence of paths γ n in P y,z with S K 1 max < Fu < S K 1 max + 1/n for each u γ n. For each n let u n γ n : we will determine how to choose u n later. By the concentration-compactness lemma of P.L. Lions see [18] it follows that a either u n converges strongly to some u D 1, R n with u 0, u = 1 b or u m and u m converge as measures respectively to S δx x 0 and δx x 0, for some x 0 S n such that Kx 0 = K max here δx is the Dirac mass concentrated in 0. If a holds then Fu = S K 1 max : this may happen only if K K max and this contradicts the fact that {K = K max } is not connected. Let us now show that we can choose each u n in γ n in such a way that b does not hold. We will call the baricenter x b of a function u D 1, R n the point x b R n such that x x S n b u dx = 0. This point depends continuously on u and on each path γ P y,z the baricenter moves from y to z as s varies from 0 to. If b occurs then the baricenter of u n converges to x 0. Therefore if we choose u n γ n in such a way that for each n the distance of the baricenter of u n from the set {x R n : Kx = K max } is larger than a fixed positive constant then b does not happen. This concludes the proof.

12 1 GABRIELE BIANCHI Lemma.3. Let z S n be a strict local maximum for K and let us suppose that Kz < 0. Moreover let y be a point of S n different from z such that Ky Kz. Then there exists σ > 0 such that inf max γ P y,z u γ S Fu > Kz + σ. Remark. We claim that the same result holds also assuming Kx Kz cdx, z ρ for any x in some neighborhood of z, some c > 0 and ρ 0, n, instead of assuming Kz < 0. Proof. Let u s P y,z. We recall that, by definition of P y,z, when s u s stays definitively in W 1, ε for any ε > 0. Let α s > 0, a s S n, λ s > 0 be the parameters associated to u s via the minimization problem M: we can write u s x = α s δ as,λ s x + e s x, for some e s x H 1 S n. The definition of u s implies that e s L 0, a s z, λ s, as s. The previous decomposition has already been used in various papers. As already proved in these papers see for instance [3] the fact that α s, a s, λ s are the parameters that optimize the approximation of u s by αδ a,λ implies that.16 < e s, δ as,λ s > L = 0 ; < e s, δ a,λ λ > L = 0 ; < e s, δ a,λ as,λ s a > L = 0. as,λ s The condition u s L = 1 is thus equivalent to.17 α s + e s L = 1. Let us write an expansion of F as s..18 Kxu s dx = αs Kxδ a s,λ s dx + α 1 s Kxδ 1 a s,λ s e s dx S n S n S n + 1 α s Kxδ a s,λ s e s dx + O e s 3 L. S n In Lemmas 4.1 and 4. it is proved that for a s close enough to the critical point z Kxδ a s,λ s dx = Ka s δ Ka s a s,λ s dx + c 1 S n S λ + o 1 n s λ, s Kxδ 1 a s,λ s e s dx c Kas + o 1 e s L, S λ n s λ s where the c i s are suitable positive constants. Since δa s,λ s dx = S and Ka s = o1 we have Kxδ a s,λ s dx = Ka s Ka s + c 1 S S n λ + o 1 s λ, s Kxδ 1 a s,λ s e s dx o 1 e s L. S λ n s

13 THE SCALAR CURVATURE EQUATION 13 Regarding the third integral in the right-hand side of.18, due to Lemma 4.1,.19 Kxδ a s,λ s e s dx = Ka s S n δ S n a s,λ s e s dx + o e s L. Putting all the pieces together and writing α s in terms of e s L as in.17 one gets the following expansion, valid as s,.0 Kxu s dx = Ka s Ka s + c 1 S S n λ s Ka s e s L 1S S a s,λ s e s dx + o e s L, 1. λ s S n δ In [7, Lemma 1] and also in [3, Lemma A.] it is proved that for any e s H 1 S n that satisfies.16 e s L δ S n a s,λ s e s dx c 3 > 1S, or, equivalently, e s L 1S δ S n a s,λ s e s dx > c 4 e s L.0 becomes.1 Kxu s S n dx Kz S > 0. Expansion 1 + Ka s Kz S Ka s + c 1 Kz Kzλ s e s Ka s L Kz c 4 + o e s L, 1 λ s Since z is a point of strict local maximum for K and Ka s = Kz + o1 < 0 then there exists σ > 0 such that inf s>0 Kxu S n s dx < KzS 1 σ. Hence sup Fu s > Kz 1 S 1 σ 1. s>0 This estimate holds uniformly with respect to the path in P y,z, since the error term o e s L, 1 λ s in.1 tends to zero uniformly with respect to the function e s. Lemma.4. Let y, z S n and let us assume that. holds. Let xt, t [0, +, be a continuous path in S n such that x0 = z, lim t xt = y. Then inf max Fu S γ P y,z u γ min t [0,+ Kxt. Proof. For a fixed λ > 0 let γt = δ xt,λ : this path is in P y,z. Actually the behaviour at the endpoints is not exactly correct, but we can continue the path with the desired property. Let ε > 0: we claim that there exists λ such that for any λ > λ it is max Fu < S u γt min t [0,+ Kxt + ε..

14 14 GABRIELE BIANCHI We have to study the asymptotic behaviour of Kxδ S n xt,λ dx as λ. Let σ > 0 be small enough. We split the above integral into three parts Kxδ xt,λ dx = Kxt S n + δ S n Bxt,σ xt,λ dx+ Kx + Kxt δ xt,λ dx Bxt,σ C Bxt, σ is the geodesic ball centered in xt with radius σ. The first integral in the right hand side is equal to S Kxt; the last one is bounded in absolute value by c 1 λ n σ n, for some positive constant c 1 that does not depend on xt. Let us now consider the second integral. We can assume that K is bounded in a neighborhood of the curve {xt} of radius σ: let D be such a bound. Bxt,σ Kx Kxt δ xt,λ dx D Bxt,σ dx, xtδ xt,λ dx c λ, for some positive constant c that does not depend on xt. As a conclusion one can write. Kxδ xt,λ dx Kxt c S S λ c 1 λ n σ n n and from this the result follows immediately. 3. Existence results. In the following flow line will mean flow line for F, that is a solution to problem 1.4 for some initial value u 0. Lemma 3.1. Let us assume that F has no critical points. Let P y,z be as in.1, Γ = inf γ Py,z max u γ Fu and let us suppose that Γ > S maxky 1, Kz 1. Let ε > 0 and γs P y,z such that max u γ Fu < Γ + ε. Then there exists ū γ such that if ut denotes the flow line with initial data ū then Fut Γ, Γ + ε, for each t 0. Proof. Let σ > 0 be such that Γ σ > S maxky 1, Kz 1. Let A = {u M : Fu < Γ σ}, B = {u M : Fu Γ σ} and for each u M let gu = du, A/ du, A + du, B here d, denotes the distance in H 1 S n. For each positive s and t let γs, t be the solution of the following problem u t = gu Fu; u0 = γs. Note that for each fixed t γ, t P y,z. Since lim s 0 Fγs = lim s Fγs < Γ σ then there exist an interval [s 1, s ] such that Fγs, t < Γ σ for each s [0, s 1 s, and for each t. Let us prove that there exists s such that Fγ s, t > Γ for each t > 0. If this is not true, for each s [s 1, s ] let τs = inf{t 0 : Fγs, t Γ}.

15 THE SCALAR CURVATURE EQUATION 15 The function τs can not be bounded in [s 1, s ], since otherwise, if T is an upper bound, it is Fγs, T Γ for each s. Choose δ > 0: for each s Fγs, T + δ = Fγs, T T +δ T gγs, t Fγs, t dt < Γ and this contradicts the definition of Γ. We may assume that τs is unbounded in any neighborhood of some point s [s 1, s ]. Due to the continuous dependence of γs, t on t it is Fγ s, τ s Γ. Since F has no critical point then there exists δ > 0 such that Fγ s, τ s+δ < Γ. This inequality is true also for s in some neighborhood of s and this contradicts the definition of s. To conclude the proof it suffices to choose ū = γ s. Proposition 3.. Let P y,z and Γ be as in.1,.3 and let K satisfy H1d for some d < S Γ 1. Let us assume that S maxky 1, Kz 1 < Γ < n S K 1 max, and that there exists an ε > 0 such that if w is a critical point of K and Kw [S Γ 1, S Γ 1 + ε, then either Kw < 0 and w is a strict local maximum or Kw > 0. Then there exists a solution to problem 0.. Proof. Let C = {w S n : Kw = S Γ 1, Kw = 0, Kw 0}. By assumption each point w C is a strict local maximum for K. By Lemma.3 there exists a positive constant σ w > 0 such that 3.1 max u γ Fu > S Kw 1 + σ w for each γ P y,w. Let σ = min w C σ w let σ = 1 if C =. Let δ > 0 be such that δ < minσ, ε, n S Kmax 1 Γ, and moreover let us choose δ in such a way that there is no critical point w of K with Kw S Γ 1, S Γ 1 + δ. Let as argue by contradiction and let us suppose that there is no solution to problem 0.. Let γs P y,z be a path such that Fγs < Γ + δ: Lemma 3.1 states that there exists s such that denoting by ut the flow line for F with initial condition γ s, then Fut Γ, Γ + δ. Due to Lemma 1.5 each subsequence of ut is a Palais-Smale sequence. Since lim Fut Γ + δ < n S K 1 max Lemma 1.3 implies that ut stays definitively in W 1, µ for any positive µ. Let at be the point associated to ut via problem M. Lemma 1.4 says that the point at converges to some point a S n with Ka = 0 and Ka 0. Furthermore lim Fut = S Ka 1. Since there is no critical point w for K with Kw S Γ 1, S Γ 1 + δ it should be S Ka 1 = Γ that is a C. The path which is the union of ut, t 0 and of γs for s 0, s] belongs to P y,a and the value of max F on this path is below Γ+δ. This contradicts 3.1. We recall that hypothesis H1d has been defined at the beginning of section 1.

16 16 GABRIELE BIANCHI Theorem 6. Let K satisfy H1d with d = n Kmax. Suppose there exists two points y and z of S n belonging to two different connected components of {x S n : Kx = K max }. Moreover suppose that - there exist two constants c > 0 and ρ > n such that Kx K max cdx, y ρ for x in a neighborhood of y, Kx K max cdx, z ρ for x in a neighborhood of z, - if w is a critical point for K and Kw n Kmax, K max, then either Kw < 0 and w is a strict local maximum or Kw > 0. Then there exists a solution to problem 0.. Proof. Let P y,z and Γ be as in.1,.3. Due to Lemmas.1 and. S K 1 max < Γ < n S K 1 max. The theorem follows from proposition 3.. Theorem 7. Suppose there exists z S n where K achieves a strict local maximum and Kz 0 and a continuous path xt in S n connecting z to some point y different from z, where Ky Kz. Suppose moreover that - min t Kxt > /n K max, - if w is a critical point for K and Kw [min t Kxt, Kz, then either Kw < 0 and w is a strict local maximum or Kw > 0. Finally let K satisfy H1d for some d < min t Kxt. Then there exists a solution to problem 0. Proof. Let P y,z and Γ be as in.1,.3. We may choose d > n Kmax. Due to Lemmas.3 and.4 S Kz 1 < Γ < d 1 S. The theorem follows from proposition 3.. Proof of Theorems 1 and. Theorems 1 and follows immediately, via a stereographic projection, from the two previous results. In order to prove Theorem 3 we need to extend slightly lemma 1.4 to include those K s which are symmetric under to the group of rotations around the North pole-south pole axis and which have critical points besides the two poles. Lemma 1.4 does not apply to these coefficients since they have infinitely many critical points. Let x 1, x,..., x n+1 be euclidean coordinates in R n+1 and let S n be embedded in R n+1 in such a way that the North pole has coordinate 0, 0,..., 1 and the South pole 0, 0,..., 1. Lemma 1.4. Let Kx = Kx n+1, and let ut be the solution of problem 1.4 for some initial data u 0 H 1 S n. Let us suppose that ut stays definitively in W 1, ε for any ε small enough. Let us assume that Kr is C, with only a finite number of critical points, in {r [ 1, 1] : Kr d}, for some d < S lim Fut 1.

17 THE SCALAR CURVATURE EQUATION 17 Then, denoting by at the point associated to ut via the minimization problem M, at n+1 converges to some r [ 1, 1] with dk dr r = 0 and d K dr r 0. See the appendix for a proof. Proof of Theorem 3. Let K 0 r be a C positive function which is equal to 1 in [0, 1/4], constant in [1/, + and with dk 0 dr < 0 in 1/4, 1/. Let ϕr be equal to r α r α for some α > n in [0, ] and identically 0 elsewhere. We claim that problem 0.1 with Kx = K 0 x ε 0 ϕ8 x ε ϕ 1 x has a solution, but no radial solution, for any ε 0, ε > 0 with ε 0, ε small. The non-existence of any radial solution follows from Theorem 0.4 in [6]. On the other hand, let z be any point with z = 1/4, let y be the origin and let P y,z and Γ be as in.1,.3. z and y are points where K achieves its absolute maximum and moreover there is a continuous path connecting z and y and on this path min K = 1 ε 0. Therefore, due to lemmas. and.4, Γ S, S1 ε 0 1 ]. For ε 0 small, the only critical points of K with values in the range [1 ε 0, 1 are those on { x = 1/8}, and on those points K > 0. Arguing exactly as in the proof of proposition.3, using Lemma 1.4 instead of Lemma 1.4, one proves the existence of a solution. Remark. In view of theorem 1 in [4] it seems worth mentioning that the coefficient Kr in the previous theorem can be made arbitrarily close in C 1 norm to a radial function which is strictly decreasing in [0, 1] and strictly increasing in [1, ]. Let now pass to prove the results regarding exclusively S 3. Theorem 8. Let n = 3 and let K satisfy H. Let y be a point with Ky = K max and let us suppose that there exists a point z, different from y, where K achieves a local maximum. Furthermore let us suppose that if w is a critical point for K with Kw KyKz Ky+ Kz, Kz and Kw < 0 then w is a strict local maximum for K. Then there exists a solution to problem 0.. Proof. Let P y,z and Γ be as in.1,.3. Since z is a local maximum and we have assumed that K 0 in each critical point of K, it is Kz < 0. Lemma.3 applies to this situation and together with lemma.1 implies that S 6 Kz 1 < Γ < Ky 1 + Kz 1 S 6. One can now conclude repeating the proof of Proposition 3.. Note that in this case to assert that ut stays in W 1, µ for each µ > 0 it is not necessary to assume that Γ < n S Kmax. 1 Proof of Theorem 4. The proof proceeds as that of Theorem 8. Here 1 S 6 Kz 1 < Γ < S 6 min Kx x xt

18 18 GABRIELE BIANCHI due to Lemmas.3 and.4. Proof of Theorem 5. Let z be a strict local maximum point for K such that Kz = min{kw : Kw = 0, Kw < 0} and let y be a point where Ky = K max : with this choice of z and y Theorem 5 is just a corollary of Theorem Appendix Proof of Lemma 1.3. We follow [1], with the necessary modifications. Given a function v defined on S n, y S n and a constant r > 0 we define now a rescaled function v y,r on S n in the following way: Let h 1, h,..., h n be a geodesic coordinate system on S n, centered in y, we define v y,r h 1, h,..., h n = r n vrh1, rh,..., rh n. Furthermore given y S n we denote by y S n the antipodal point. For u H 1 S n let Eu = 1 u nn + u dv 1 S 4 K u + dv. n S n If u m is a Palais-Smale sequence of positive functions for the functional F and, for each m v m 1 = u m Fu m n 4 then v m 1 is a P.-S. sequence for E. We can write 4.1 v 1 m L = u m L Fu m n 4 = Fu m n 4, Ev 1 m 0 strongly in H 1 S n. Since v m 1 L is bounded, we may assume that v m 1 v 0 weakly in H 1 S n. To be more precise this is true only for a subsequence: we will often extract subsequences without explicitly mentioning this fact. If v 0 is not identically 0, then it is a solution of problem 0. and therefore we may assume u 0 0. As proved in [1], there exist sequences rm 1 of positive constants tending to 0, ym 1 of points of S n such that v m 1 y 1 m,rm 1 v1 0 weakly in H 1 S n as m. By weak continuity, v 1 is a weak solution of 4. Lu + K 1 u + u = 0 in S n, where K 1 is a weak and a.e. pointwise limit of the sequence of functions rm 1 n K y 1 m,rm 1. Note that if u is a weak solution of 4. then u 0 and 0 = u L K 1 u dv u L sup K 1 u dv S n S n S n u L 1 sup K 1 S u 4 n L, S n that is, if u 0, 4.3 S 4 n u sup S n K 1 L.

19 THE SCALAR CURVATURE EQUATION 19 We may assume that y 1 m converges to some point a 1 S n. Let us now prove that a 1 A d. If this is not the case, since A d is closed, lim sup x a1 Kx < d. Moreover, by definition of K 1, sup K 1 d. Therefore, by v 1 n L S d. On the other hand, by 4.1 and since the norm L is invariant with respect to rescaling, v m 1 y 1 m,rm 1 L = v m 1 L = Fu m n 4. Combining this with 4.4 one gets lim Fu m = lim v m 1 y 1 m,rm 1 4 n L v1 4 n L S d, which contradicts the assumptions of this Lemma. Therefore Ka 1 d and moreover, since K is continuous in a 1 K 1 Ka 1. But in the case K 1 constant, all the positive solutions of the equation 4. are known and we can write for some λ > 0, b S n. Let Note that 4.5 v 1 = S n Ka1 n 4 δb,λ v m = v m 1 v 1 y 1 m,1/rm 1 v m L = v 1 m L v 1 L + o1, Ev m 0 strongly in H 1 S n. Let us now show that v m 0 strongly in H 1 S n. Let as argue by contradiction and let us suppose that this is not the case. As observed above, since problem 0.1 has no solution, then v m 0 weakly. Arguing as above there exist sequences rm of positive constants tending to 0, ym of points of S n such that, as m, v m y m,r m v 0 weakly in H 1 S n and v is a weak solution of Lu + K u + u = 0 in S n, where K is a weak limit of the sequence of functions rm n K y m,rm. One can argue as in the case of 4.3 to prove that 4.6 S sup S n K v 4 n L. By and 4.6, one gets the following chain of inequalities lim Fu m n = lim v 1 m L = lim v m L + v 1 L v L + v 1 L S n + sup K 1 x S n, K max S sup K x n

20 0 GABRIELE BIANCHI which contradicts the assumptions. This concludes the proof that v m 0 in H 1 S n, that is v m 1 v 1 y 1 m,1/rm 1 0. The proof is easily concluded writing this in terms of u m and keeping in mind that that u m L = 1, δ b,λ L = 1. Let u W 1, ε for some ε > 0 small enough. Let α > 0, a S n, λ > 0 be the parameters associated to u via the minimization problem M: we can write ux = αδ a,λ x + ex, for some ex H 1 S n. The fact that α, a, λ are the parameters that optimize the approximation of u by αδ a,λ implies see [3] for a proof that V 0 < e δ a,λ > L = 0; < e δ a,λ a > L= 0; < e δ a,λ λ > L= 0. In what follows we always assume that K satisfies H1d. Lemma 4.1. Let a A d+ε for some ε > 0 and let e satisfies V 0. Then Kxδ 1 a,λ e dv Ka c 1 + o 1 λ λ e L ; Kx Kaδ a,λ e dv O e L ; λ Kxδ 1 1 δ a,λ a,λ λ a dv = c Ka + O 1 λ λ ; Kxδ 1 a,λ λ δ a,λ λ dv = c Ka 3 λ + o 1 λ. Lemma 4.. Let z A d+ε for some ε > 0 and let z be a critical point for K with Kz < 0. Let ρ be small enough so that Bz, ρ A d+ε and Kz Kx for any x Bz, ρ. Let a Bz, ρ and let e satisfies V 0. Then the following expansion holds : Kx Kaδ a,λ dv = c 4 Ka λ + o 1 λ Proof of Lemmas 4.1 and 4.. Let K C S n coincide with K in A d. Assuming K C S n the previous estimates have been proved in [3] Lemmas A.6, A.7, A.8 for n = 3 and generalized to any n in [] the proofs given for n = 3 can be repeated for any n just adapting the notations. Therefore the previous estimates with K replaced by K are true. Since the distance from A d+ε to A d c is positive, by dominated convergence theorem, for any β > 0 there is a positive constant d 1 > 0 such that, as λ, K K β δ a,λ dv d 1 λ n.

21 THE SCALAR CURVATURE EQUATION 1 This proves the expansion in lemma 4.. Since λ δ a,λ λ, 1 δ a,λ λ a d δ a,λ we have also K Kδ 1 1 δ a,λ a,λ λ a dv d 3λ n ; K Kδ 1 a,λ λ δ a,λ λ dv d 4λ n. Similarly, by Hölder and Sobolev inequalities K Kδ 1 a,λ e dv d 5λ n+ e L ; K Kδ a,λ e dv d 6λ e L. These estimates conclude the proof. Proof of Lemma 1.4. For each t let αt, λt be the constants associated together with at to ut via problem M. We write ut = αtδ at,λt + e t with e t H 1 S n. Since ut W 1, ε definitively for any ε > 0, λt, e t L 0 as t. Let us start proving that for t large enough at {Kx d + σ} for some σ > 0 independent on t. As in.17 αt = 1 e t L. Formula.18, valid for e t L 0, can be rewritten as Kxut dx = Kxδ at,λt dx + O e t L, S n S n moreover see. as λ, Therefore, as t, Kxδ at,λt dx Kat + O 1 S S λt. n S n Kxut dx S Kat + O 1 λt + O e t L. Since we assumed that d < S lim Fut 1, this expansion concludes the proof that at {Kx d + σ}. In [3] Lemma 6, for n = 3, and in [] see pages 91, 9, for general n, a system of differential equations for at, λt is derived from the condition that ut is a flow line. The authors prove this under the assumption K C S n. The same result can be proved under our weaker assumptions on K provided that at A d+σ. For instance, one can follow [3] where the proof is more concise repeating the proof of Lemma and then of Lemma 6. One has just to replace the estimates needed in those lemmas, and proved in [3] when K is regular, with those proved in Lemmas 4.1 and 4. of this paper. We will not write here the details.

22 GABRIELE BIANCHI The functions at and λt satisfy the following differential equations: dλ λ dt = d Kat 11 + o1 Kat n+ 4 λ + O Fut + Ka 1 λ + o λ, 4.8 da dt = d Ka Fut 1 + o1 Ka n+ 4 λ + O + 1 λ λ 3, for some positive constants d 1, d. Let us perform a stereographic projection in order to be able to work linearly. Let 4.9 a t = at t s 0 O Fus λ ds, where O Fut /λ is the one appearing in 4.8 and s 0 will be chosen later. By mean value theorem 4.8 becomes 4.10 da dt = d Ka 1 + o1 Ka n+ 4 λ + O Multiplying 4.7 by λ one gets 4.11 dλ dt λ λ O Fut d 3. 1 t Fus λ ds + 1 s 0 λ λ 3. t Since on any flow line F is bounded from above lim t 0 Fus ds = lim t Fut Fu 0 is bounded. This and 4.11 imply that λs d 4 s + 1. Let us now introduce the new variable τt = t 1 0 λ 1 sds. As t τt becomes 4.1 da dτ = d 1 + o1 Ka tτ Fus + O ds + 1. Ka n+ 4 s 0 λ λ Let y 1, y,..., y m be the critical points of K in {K d+σ}. If Ky i Ky j there is no trajectory of the flow associated to K leading from y i to y j and the same is true also for some neighborhood of y i and y j. Let ρ > 0 be such that whenever Ky i Ky j there is no trajectory of the flow associated to K leading from some point in By i, ρ to some point in By j, ρ. Formula 4.1 implies that 1 d dτ Ka 1 = d 1 + o1 Ka Ka n+ 4 + O tτ Fus s 0 λ ds + 1 λ Since λ when τ and K is bounded, the previous estimates implies that a cannot stay definitively outside of m i=1 By i, ρ, if s 0 is large enough. Therefore a has to enter some of the By i, ρ..

23 THE SCALAR CURVATURE EQUATION 3 We claim that, for τ large enough, if a leaves By i, ρ and enters By k, ρ then Ky k > Ky i. This is true for a trajectory of the flow associated to K and therefore, by continuity, also for a, when s 0, λ and τ are large enough. Due to this a can leave one of the By i, ρ to reenter another one just a finite number of times and a, after a certain time, rests in one of these neighborhoods, let us say By i0, ρ. Choosing s 0 large enough, the same is true for a. It remains only to prove that Ky i0 0. Arguing by contradiction let us suppose that Ky i0 > 0. When at is very close to y i0 4.7 can be rewritten as 1 dλ λ dt = d Ky i o1 Ky i0 n+ 4 λ + O Fut Therefore if Ky i0 > 0 then λt stays bounded, which is a contradiction. Proof of lemma 1.4. Let r 1, r,..., r m be the critical points of Kr in {r [ 1, 1] : Kr d}. Let U i ρ be a ρ neighborhood of {x S n : x n+1 = r i }. Let now choose ρ > 0 such that there is no trajectory of the flow associated to K leading from some point in U i ρ to some point in U j ρ whenever Kr i Kr j. The proof works as the previous one, with this new definition of the neighborhoods of the critical points. References [1] T. Aubin, Meileures constantes dans le théorème d inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal , [] A. Bahri, Critical points at infinity in some variational problems, Pitman research notes in mathematics 18, Longman Pitman, London, [3] A. Bahri, J.M. Coron, The scalar curvature problem on the standard three-dimensional sphere, J. Funct. Anal , [4] G. Bianchi, Non-existence and symmetry of solutions to the scalar curvature equation, Comm. P.D.E. to appear. [5] G. Bianchi, J. Chabrowski, A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. T. M. A , [6] G. Bianchi, H. Egnell, A variational approach to the equation u + Ku n+ n = 0 in R n, Arch. Rational Mech. Anal , [7], A note on the Sobolev inequality, J. Funct. Anal , [8], An ODE approach to the equation u + Ku n+ n = 0 in R n, Math. Zeit , [9] H. Brezis, J.M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal , [10] S. Y. A. Chang, M.J. Gursky, P. C. Yang, The scalar curvature equation on - and 3-sphere, Calc. Var , [11] S. Y. A. Chang, P. C. Yang, A perturbation result in prescribing scalar curvature on S n, Duke Math. J , [1] J. Escobar, R. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Mat , [13] J. Kazdan, F. Warner, Existence and conformal deformations of metrics with prescribed Gaussian and scalar curvature, Ann. of Math , [14] Y.Y. Li, On u = Kxu 5 in R 3, Comm. Pure Appl. Math , [15], On Nirenberg problem and related topics, Topological methods in Nonlinear Anal. to appear. [16] Y. Li, W. M. Ni, On conformal scalar curvature equation in R n, Duke Math. J ,

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