L p ESTIMATES AND ASYMPTOTIC BEHAVIOR FOR FINITE ENERGY SOLUTIONS OF EXTREMALS TO HARDY-SOBOLEV INEQUALITIES. 1. Introduction

Transcription

1 L p ESTIMATES AND ASYMPTOTIC BEHAVIOR FOR FINITE ENERGY SOLUTIONS OF EXTREMALS TO HARDY-SOBOLEV INEQUALITIES DIMITER VASSILEV arxiv:math/06066v [math.ap] 3 Jan 006 Abstract. Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp L q regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of R n, which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered L Hardy-Sobolev inequality. [000] Primary 35J65, 35B05. Introduction This paper has three goals. In Section we shall study the L p regularity and asymptotic behavior at infinity of non-linear equations involving critical growth. In its simplest form, our result is the following. For < p < n, we let p = p p and p = np n p be correspondingly the Hölder conjugate and the Sobolev conjugate exponents. Suppose V L p p p (R n ) and u is weak nonnegative solution in R n, i.e. u is of finite energy,, cf. (.6), of the inequality (.) div( u p u) V u p. Then, u L q (R n ) for any p p < q, which in general cannot be improved further, and for any 0 < θ < there exists a positive constant C θ, such that, (.) u(z) C θ + z θ n p p, z R n. We shall actually consider inequalities with a possible singularity on a subspace of R n in the right-hand side, which arise naturally in the study of the extremals of certain Hardy-Sobolev inequalities, see further below and (.5) for the exact setting. Our results are contained in Theorems.,.5,.7 and.8. The asymptotic behavior of solutions to elliptic equations with critical non-linearity has been studied extensively. Generally speaking, the results and the analysis depend on whether one assumes a priori that the solution has finite energy or one is given a smooth solution, see [S], [S], [SW], [E],[GS] and [Li], to name a few. We work under the assumption of finite energy. The paper extends the results of Egnell [E], which correspond to Theorems. and.7 without the possible singularity on a submanifold of R n. In Sections 3 we consider the case p =. The main result here is that, under some natural additional assumption, u has the same asymptotic as the fundamental solution of the laplacian at infinity, see Theorem 3.. The proof exploits the fact that R n has a positive Yamabe invariant and thus the method will be applicable to cases when the ambient space does not have a conformal transformation with the properties of the Kelvin transform. Furthermore, we work Date: January 8, 006.

2 DIMITER VASSILEV in the setting in which V could have a singularity on a subspace, which renders results based on radial symmetry inapplicable. Section 4 contains some calculations which show how one can construct explicit solutions of the Euler-Lagrange equation of certain Hardy-Sobolev inequalities. In the subsequent section we show that these solutions are extremals for which the best constant in the considered Hardy- Sobolev inequalities is achieved. In Section 5, we determine the best constant in the Hardy-Sobolev embedding Theorem involving the distance to a subspace of R n. In order to explain the considered inequality we need a few notations, which shall be used throughout the paper. For p we define the space D,p (R n ) as the closure of Co (Rn ) with respect to the norm ( ) /p (.3) u D,p (R n ) = u p dz. R n Let n 3 and k n. For a point z in R n = R k R n k we shall write z = (x,y), where x R k and y R n k. The following Hardy - Sobolev inequality was proven in Theorem. of [BT]. Theorem. ([BT]). Let n 3, k n, and p, s be real numbers satisfying < p < n, 0 s p, and s < k. There exists a positive constant S p,s = S(s,p,n,k) such that for all u D, p (R n ) we have (.4) ( p(n s) n p u R n x s dz ) n p p(n s) S p,s ( R n u p dz When k = n the above inequality becomes the Caffarelli-Kohn-Nirenberg inequality, see [CKN], for which the optimal constant S p,s was found in [GY]. The case p = was considered earlier in [O] and [GMGT], where the inequality is written in an equivalent, but slightly different form. If we introduce the numbers σ = s(n p) p(n s), hence 0 σ <, and p σ = n p p(n s) the above inequality becomes ( Rn u pσ ) ( /p (.5) x dz σ ) /, C u dz σpσ R n where C is a positive constant. In the case p = and k = n the sharp constant was computed in [GMGT]. When σ < the extremals were found by Lieb in [L], while when σ = we have the classical Hardy inequality, which does not have extremal functions. The main result of Section 5 is the proof of the following Theorem. Theorem.. Suppose n 3 and k n. There exists a positive constant K = K n,k, such that for all u D, (R n ) we have (.6) ( R n k (n ) n u R k x dxdy ) n (n ) ) p. ( K u dz R n Furthermore, K is given in (5.8) and the positive extremals are the functions ( (.7) v = λ (n ) 4 ) n ( (n ) K (n ) ( x + n ) n 4aλ ) + y y o, where λ > 0, y o R n k. ).

3 HARDY-SOBOLEV INEQUALITIES 3 Let us note that the results of Section can be applied to the non-negative extremals of the general Hardy-Sobolev inequality of Theorem.. Finally, Section 6 contains some further simple applications, which indicate the direction of some future investigations. The author would like to thank Luca Capogna, Scott Pauls and Jeremy Tyson for organizing the Workshop on Minimal surfaces, Sub-Elliptic PDE s and Geometric Analysis, Dartmouth College, March, 005. The current paper grew out of a conversation with Richard Beals and Yilong Ni on an unrelated question of the author concerning certain equations on Carnot groups, [BY], which lead to Section 4 of this paper. Thanks are also due to Galia Dafni, Nicola Garofalo, Qi Zhang and Congming Li for their interest in the work and valuable conversations.. Regularity and asymptotic of weak solutions The goal of this section is to prove sharp L q regularity for solutions of the considered equations, which would then lead to bounds on the rate of decay at infinity. We start with two definitions. Let p and s be as in Theorem. and denote by p*(s) the Hardy-Sobolev conjugate p(n s) (.) p*(s) = n p and by p the Hölder conjugate p = p. There is another exponent, which will play an p important role. For any s as above we define the exponent r = r(s) to be the Holder conjugate of the exponent r = r (s) defined by (.) r = p p (s) p, thus r = n n p + s. Notice that r, 0 rs p, and furthermore we have the identity (.3) rp = p (rs). However, in general rs could be bigger than k, and thus due to the restriction s < k in the Hardy-Sobolev inequality we shall consider only the case (.4) s(n k) < k(n p), which implies rs < k. Theorem.. Let be an open subset of R n, which is not necessarily bounded, < p < n, 0 s p, s < k and s(n k) < k(n p). Let u D,p () be a non-negative weak solution of the inequality (.5) div ( u p u) V u p x s u in, i.e., (.6) for every 0 φ C o (). u p < u, φ > dz V u p x s uφdz,

4 4 DIMITER VASSILEV a) If V L r (), then u L q ( dz x t ) for any 0 t < min{p,s} and q p (s). b) If V L to () L r () for some t o > r, then u L (). In particular u L q () for every p q <. Remark.. As usual, if we have equality instead of inequality in the above Theorem, the conclusion holds for any weak-solution, without a sign condition. The proof below works with very minor changes. This is the reason we use u rather than u when we are dealing with a non-negative function. Proof. The assumption V L r () together with the Hardy-Sobolev inequality shows that (.6) holds true for any φ D,p (). This can be seen by approximating in the space D,p () by a sequence of test functions φ n C o (), which will allow to put the limit function in the left-hand side of (.6) as u p L p. On the other hand, for φ C o (), using the Hölder and Hardy-Sobolev inequalities we have the estimate (.7) (.8) (.9) (.0) V u p x ( s φdz V r ) /r ( ( V r ) /r [( u r(p ) x rs φ r ) /r u rp (p ) x rs ) rp ( φ pr x rs ) /p ] /r ( u rp (p ) ) S p V L r rp x rs φ L p ( using (.3) ) S p Sp,rs p V Lr u p L φ p L p, which allows to pass to the limit in the right-hand side of (.6). We turn to the proofs of a) and b). a) Let G(x) be a piece-wise smooth, globally Lipschitz function, on the real line, and set (.) F(u) = u 0 G (t) p dt. Clearly, F is a non-negative differentiable function with bounded and continuous derivative. From the chain rule, G(u), F(u) D,p (). In particular, F(u) is a legitimate test function in (.6). We are going to show that if q is a number q p (s) and 0 t < min{p,s}, then u L q ( dz x ) implies u D,p () L κtq ( dz s x ), where κ t t = p (t) p, and for some positive constant C depending on q we have (.) u L κ t q ( dz x t ) C u L q ( dz x s ). Notice that we can apply the Hardy-Sobolev inequality replacing the exponent s with the exponent t. Furthermore, we require t < p as then we have (.3) κ t = p (t) p >. With φ = F(u), taking into account F (u) = G (u) p, the left-hand side of (.5) can be rewritten as (.4) u p < u, F(u) > dh = G(u) p.

5 HARDY-SOBOLEV INEQUALITIES 5 For q p (s), hence q p (p) = p, we define the function G(t) on the real line in the following way, (.5) G(t) = { sign (t) t q p if 0 t l, l q p t if l < t. From the power growth of G, besides the above properties, this function satisfies also (.6) u p F(u) C(q) G(u) p C(q) u q. The constant C(q) depends also on p, but this is a fixed quantity for us. At this moment the value of C(q) is not important, but an easy calculation shows that C(q) C q p with C depending on p. We will use this in part b). Let M > 0 to be fixed in a moment and estimate the integral in the right-hand side of (.6) as follows. (.7) V u p x s F(u)dz = ( V M) V u p x s F(u)dz + ( V >M) V u p x s F(u)dH ( C(q) V G(u) p ( V M) x s dz + C(q) V r ) ( r G(u) pr ) r ( V >M) x sr dz C(q) V u q ( ( V M) x s dz + C(q)Sp p,rs ( V >M) V r ) r ( ) G(u) p dz C(q)M u q L q ( dz x s ) + C(q)Sp p,rs V L r ( V >M) G(u) p L p. At this point we fix once and for all an M sufficiently large, so that C(q)S p p,rs ( ( V >M) V r dh) r, which can be done because V L r. Putting together (.4) and (.7), and using the Hardy- Sobolev inequality, we come to (.8) G(u) p L p (t) ( dz x t ) C(q)M u q L q ( dz x s ) By Fatou s theorem we can let l in the definition of G to infinity and obtain u q C(q)M u q L κ t q ( dz x t ) L q ( dz x s ) The proof of a) is finished. b) Let us observe that the assumption t o > r implies that t o < r and thus 0 < t os < rs. Therefore, for any q p (s) the norm u L qt dz is finite from part (a). We are going to o( ) x st o

6 6 DIMITER VASSILEV prove that the L q ( dz x t s) norms of u are uniformly bounded by the o Lqo ( dz x t os) norm of u, where q o = t o p (s). We shall do this by iteration and find a sequence q k which approaches infinity as k. Let q p (s). We use again the function F(u) from part (a) in the weak form (.6) of our equation. The left-hand side is estimated from below as before, see (.4). This time, though, we use Hölder s inequality to estimate from above the right-hand side, (.9) V u p uf(u)dh V L to u p F(u) L t o With the estimate from below we come to V to C(q) G(u) p x s t o C(q) V L to u q L qt o( dz ). x st o G(u) p L p C(q) V L to u q L qt o( dz ). x st o Using the Hardy-Sobolev inequality and then letting l we obtain (.0) u q L q p (st o ) q C C(q) V Lto u p dz ( x st ) L qt dz o( ), x st o o where C is independent of q. Let δ = p (st o ) p t. A small calculation shows that δ > exactly when t o > r. With this notation o we can rewrite (.0) as (.) u q L δqt o( dz x st ) o [ ] q C C(q) q V L to u q L qt o( dz ). x st o Recall that C(q) Cq p. At this point we define q o = p (s)t o and q k = δ k q o, and after a simple induction we obtain (.) u L q k( dz x st o ) k [ C q p ] q j V j=0 Let us observe that the right-hand side is finite, (.3) q j=0 j = q o j= thanks to δ >. Letting j we obtain j k L to j=0 q j δ j < and j= u L qo( dz ). x st o log q j q j <, u C u L qo( dz ). x st o Remark.3. We should keep in mind that the local version of Theorem. is also valid. In other words, we can replace all the spaces in the statement of the Theorem with their local version. The proof is accomplished in a very similar fashion by introducing a local cut-off function.

7 HARDY-SOBOLEV INEQUALITIES 7 With the above Theorem we turn to the equation, in fact slightly more general equation, satisfied by the extremals of the Hardy-Sobolev inequality. Theorem.4. Let < p < n and R n be an open set. If R L and u D,p () is a weak non-negative solution to equation then u L (). div ( u p u) R(z) u p (s) u x s u in, Proof. We define V = R u p (s) p. From the Hardy-Sobolev inequality we have u L p (s) () and thus V L p (s) p (s) p(). Since r = p (s) p (s) p, part (a) of Theorem. shows that u Lq () for p q <. Therefore V L q p p() for any such q and thus by part (b) of the same Theorem we conclude u L (). In the next Theorem we show that one can lower the exponent p in the L q regularity of u. A similar result in the case p =, s = 0 was achieved in [LU]. Our proof is more general and simpler as we do not employ the Lax-Milgram Theorem and thus it works also for equations involving the p-laplacian and s 0. In addition, the estimate we give, even in the case p =, is different and this is crucial for the p-laplacian singular case. Theorem.5. Suppose V L r solution of the inequality and V o L L r. If u is non-negative locally bounded weak (.4) p u V u p x s u + V o, then u L q for every p p < q p. Remark.6. The condition that u is locally bounded holds for example when V o = 0 and V L r L to, for some t o > r by Theorem.. Proof. Let 0 < θ < /p and define the function f = u θ. Our task then is reduced to showing that for any fixed θ the function f L p. Taking into account the Sobolev embedding theorem, it is enough to prove f L p <. Let η m (t) be the following function t θp, t > /m (.5) η m (t) = m θp (t m ), m < t < m 0, t < Notice that η m is a piece-wise smooth function, which is globally Lipschitz, with η m particular the function 0 φ m = η m (u) D,p (). Observe also that m. (.6) 0 η m (t) θp t θp C p,θ t θp > 0. In Indeed, for t > /m we have η m (t) = t θp < + θp t θp as θp > 0, while for m < t < m we have η m (t) m θp ( m m ) = mθp θp t θp < θp t θp.

8 8 DIMITER VASSILEV Using φ m as a test function in (.4) we find u p u φ m dz = { m <u< m } u p u φ m dz + { m <u} u p u φ m dz = m θp { m <u< m } u p dz + { m <u} u p u φ m dz On the other hand we have = m θp { m <u< m } u p dz + θp ( θ) p f p dz. {u>/m} V u p x s φ m dz = {u>/m} V u p x s φ m dz + V u p { m <u< m } x s φ m dz C p,θ {u>/m} C p,θ V L r C p,θ V L r V u p + θp x s {u>/m} {u>/m} dz + m θp V u p { m <u< m } x s dz f p dz + m θp V L r ( u pr x rs dz ) /r f p dz + S p p,rsm θp V L r { m <u< m } u p dz, using pr = p (rs). The other term on the right-hand side can be estimated in the following way (.7) V o φ m dz { m <u< m } V o φ m dz + { m <u} V o φ m dz where C p,θ { m <u< m } V o u θp dz + C p,θ C p,θ m θp (.8) q = as pθ > 0. ( p pθ { m <u} V o u θp dz V o dz + C p,θ V o L q { m <u< m } (u>/m) u p /q L p (u>/m), ) = p p ( pθ) >,

9 HARDY-SOBOLEV INEQUALITIES 9 Putting the above three estimates together we have shown that the gradient of f satisfies the inequality (.9) m θp u p dz + f p dz { m <u< m } u>/m S p p,rs mθp V L r { m <u< m } u p dz + C p,θ V L r {u>/m} f p dz + C p,θ m θp V o L ( m <u< m ) + C p,θ V o L q (u>/m) u p /q L p. With the above estimates at hand, we can conclude by removing the first terms on each side of the inequality, moving the second term from the right side to the left, and then letting m, provided we have i) S p p,rs V L r > 0 and C p,θ V L r, i.e., V L r ii) for every θ as above we have { m <u< m } V o q dz C(θ) <, where the last constant is independent of m. With this in mind, we fix a sufficiently large R o > 0 and a sufficiently small δ > 0, which we shall chose in a moment, and let α C be a function with supp α { x < δ} {R n \ B Ro } and α = on α { x < δ} {R n \ B Ro }. Then we re-write our inequality in the form (.30) div( u p u) αv u p x s u + g, with g = ( α)v u p x s u + V o g o + V o. Notice that due to the choice of the support of α we have supp g o { x > δ} {B Ro }. Recall we are assuming only V L r and V o L L r. The definition (.8) of q shows that q < (p ), while (.3) p (p) = p p (s) p (0) = p and thus (.3) r = Therefore we have p p (s) p p p p > q < r p p = (p ). and taking into account the definition of g, in particular, the facts that V L r, u L loc and the compact support of g o stays away from x = 0, we conclude that g L q loc for every q < r. So far we have concerned ourselves with the global properties of solutions. Having done this we can obtain the asymptotic behavior of solutions at infinity. The first result concern the local behavior on a ball away from some finite point, see [E] and [Z] for related results. Theorem.7. Suppose s, p, k, and n satisfy the conditions of Theorem.. Let u be a nonnegative solution to the inequality (.5), with V L r. We assume that u has been extended with zero outside. Suppose that q o p is an exponent such that u L qo (). There exist

10 0 DIMITER VASSILEV constants C = C(R n,p, u D,p (), u qo ) > 0 and 0 < R o = R o ( V L r ), such that, for every z R n and R = z / R o we have ( ) qo (.33) max u C u qo dx. B(z,R/) B(z, R) B(z,R) Furthermore, u has the following decay at infinity C (.34) u(z) z. n/qo Proof. Given a function α C o (Rn ), α 0 we consider the function α p F(u) D,p (), see (.) and (.5). We recall (.6), and observe in addition (.35) F(u) G (u) p. We use the fact q/p > in order to see that G is a piece-wise smooth and globally Lipschitz function. Using α p F(u) as a test function in the weak formulation (.6) we have u p < u, (α p F(u)) > dz V u p x s uα p F(u)dz Let us consider the left-hand side (LHS) of the above inequality, which is easily seen to equal (.36) LHS = For any ǫ > 0 we have ab ǫ ap α p G p dz + p p + ǫ p /p b p p, and hence u p α p F(u) u α dz. (.37) u p α p F(u) u α ǫ p αp u p u F(u) + ǫ p /p α p F(u)u p/p p ǫ p αp G p + Cǫ q p α p G p, after using (.35) and (.6) in the last inequality. Inserting (.37) in (.36) we find LHS ( ǫ) α p G p dz Cǫ p /p q p α p G p dz. We shall use the above inequality with a fixed sufficiently small ǫ so that ǫ > 0. With the help of (a + b) p p (a p + b p ), and using once more (.6) and the paragraph after it, we come to (.38) (αg) p dz C V αp G p x s dz + Cq p α p G p dz, where C is a constant independent of q. Therefore, using the Hölder and the Hardy-Sobolev inequalities ( pr = p (rs)! ), we have (.39) (αg) p L p C V L r (supp α) (αg) p L p + Cq p α p G p dz Since V L r (.40) it follows that if 0 z R n and R = z / R o then V r dz 0 as R o. B R (z) Therefore, C V L r (supp α) / for all α with (.4) α Co ( BR (z) ) with R = z /, z R o,

11 HARDY-SOBOLEV INEQUALITIES where R o depends on V and C, and shall be fixed for the rest of the proof. Using the Sobolev inequality, we have shown that for any such α we have (.4) αg L p (αg) L p Cq(p )/p ( α)g L p Therefore, if u L q we can apply Fatou s theorem when l to get (.43) ( α p u δq) /p C q (p )/p( α p u q) /p, where δ = p /p >. In particular, for any 0 < ρ < r < R and α Co ( ) B(z,r) with α on B(z,ρ) and α r ρ, we have (.44) ( ) u δq δq dz B(z, ρ) B(z,ρ) Cp/q q (p )/q (r ρ) p/q ( ) u q q dz. B(z, r) B(z,r) We can define the sequences q j = q o δ j and r j = R ( ) + for j = 0,,... with which Moser s j iteration procedure gives inequality (.33). Let us observe that j=0 q j < and j=0 < thanks to δ >, cf. (.3). The decay property follows immediately taking into account that the volume of B R (z) is proportional to R n, i.e., z n. Combining Theorems.,.5 and (.34), we can assert the following decay of solutions to (.5). Theorem.8. If V L r L to, for some t o > r and u is a non-negative solution of (.5) then there exists a C = C(R n,p, u D,p ()) > 0, such that, u has the following decay at infinity (.45) u(z) C z q u D,p (), for any q < n p p. Remark.9. Let us observe that the fundamental solution of the p-laplacian on R n equals C z n p p, where C is a constant. ln q j q j 3. Asymptotics for the scalar curvature equation In this section we restrict our considerations to the case p = and furthermore we require V = R(z)u with R L, i.e., we consider a non-negative weak solution u of (3.) u R(z)u (s) x s in. From Theorem.8 we know that for any 0 < θ < there exists a constant C θ > 0, such that, (3.) u(z) C θ + z θ(n ) u D,p (). The next result shows that the decay is at least as that of the fundamental solution. Furthermore, if u is a non-negative solution, rather than a subsolution then u has the same decay as the fundamental solution.

12 DIMITER VASSILEV Theorem 3.. Let 0 s < and R L. a) If u is a non-negative solution of (3.) then there exists a constant C > 0, such that, (3.3) 0 u(z) b) If u is a non-negative solution of (3.4) u = R(z)u (s) with R 0 then (3.5) C + z n u(z) C + z n, z. x s in, C + z n, z. Proof. We shall prove first the estimate for u from above. Let us extend u as a function on R n be setting it equal to zero outside of. Let us define f = R(z) u x s. Thanks to Theorem.8 we have for any 0 < θ < (3.6) f(z) C θ R(z) + z θ(n+ s) x s. Consider the function v = Γ f, where Γ is the positive fundamental solution of the laplacian Γ(z) = n(n )ω n z n ( recall n > ), where ω n is the volume of the unit ball in R n, so that Γ = δ. For points z,ζ in R n we shall write z = (x,y) R k R n k and ζ = (ξ,η) R k R n k. With this notation and using (3.6) we have ( C n = n(n )ω n ) f(ζ) v(z) = C n dζ z ζ n = C n R n z ζ z + C n f(ζ) dζ z ζ n z ζ > z f(ζ) dζ z ζ n C θ R + z θ(n+ s) z ζ z z ζ n ξ s dζ + C θ R + z n ( + ζ θ(n+ s) dη dξ ) ξ s R k R n k def C θ R + z θ(n+ s) I + C θ R + z n I C θ R + z n, from Lemmas 3.5 and 3.3 ( see also the Remark following Lemma 3.5 for the case k = n). Going back to the bound from above for u we see that the difference u v is a subharmonic function, which goes to zero at infinity and is equal to zero on. From the weak maximum principle we can conclude u v 0,

13 HARDY-SOBOLEV INEQUALITIES 3 from which (3.3). If u is a solution, we can take a ball B centered at the origin and a constant C, such that, the subharmonic function u C Γ goes to zero at infinity, equals to zero on, and is positive on the (compact) boundary B. From the weak maximal principle u C Γ is positive everywhere. The proof of the Theorem is complete. Remark 3.. The second part of of Theorem 3., i.e., when we assume in addition R 0, can be derived also with the help of the Kelvin transform on R n, which can be seen as follows. We know that the Kelvin transform is an isometry between D, () and D, ( ), cf. [E] and [GV]. A calculation using (Ku) = z n ( u)( z ) shows that the Kelvin transform z Ku of u is a non-negative weak solution of the inequality (Ku)(z) R( z x s z ) (s) (z) R( z z s(ku) z ) (s) x s (Ku) (z) in. Notice that R( z z ) L ( ) when R L (). Furthermore, if is a neighborhood of the infinity then 0 is a removable singularity of Ku, i.e., the above inequality is satisfied on {0} and Ku D, ( {0}), cf. [E] and [GV]. In other words, when 0 R L the function Ku satisfies the same inequality and conditions as u. Recalling that u(z) = z n (Ku)( z ) the z claim follows. The following lemma contains a useful fact concerning the value of a certain integral. Lemma 3.3. Let m > n s and 0 s < k < n. The following formula holds (3.7) R k R n k ( + x + y ) m x s dy dx = σ n k σ k B(n k,m n k ) B( k s,m n s ). Proof. With a = + x we have (3.8) an k R n k ( + x + y dy = ) m a m R n k ( + y ) m dy = σ n k a m (n k) 0 r n k ( + r ) m dr = σ n k a m (n k) 0 t n k ( + t) dt n k n k +(m ) = σ n k k B(n,m n k ). am (n k)

14 4 DIMITER VASSILEV Therefore we have (3.9) R k R n k ( + x + y ) m dy dx x s = σ n k B( n k,m n k ) R k ( + x n k m ) x s dx = σ n k σ k B(n k,m n k ) 0 t k s ( + t) dt k s n k +(m k s ) = σ n k σ k B(n k,m n k ) B( k s,m n s ). Remark 3.4. For future reference, let us notice that the above proof amounts to using twice, with the appropriate choice of the involved parameters, the formula R k ( + x ) a x s dx = σ k B(k s,a k s ), which is valid for any a > 0, k > s and a > k s. We end the section with one more technical lemma, which was used in Theorem 3.. Lemma 3.5. Let k, n 3 and 0 s < k n and s <. There exists a constant C > 0 such that I(z) = z ζ n ξ s dζ C z s. z ζ z Proof. To estimate the integral I we observe that I is homogeneous, I(λz) = λ s I (z) for λ > 0. Therefore, if I is finite on z = we can conclude that (3.0) I(z) C z s. In order to see that I is finite when z = let us notice that it depends only on x and y as the integral is invariant under rotation in R k or R n k. A consequence of this fact is that it is enough to show that on z = the integral I is finite at only three point, namely, x = 0, x =, and x = y ( we write x = for the number one on the real axis considered as a point in R k, etc.). For the rest of the proof we assume z =. The case of x 0 is easier, so we shall consider the last two points first. Without any loss of generality we take x = and we split the integral in two parts I(z) = + { z ζ z z } { x ξ 4 }}. { z ζ z z } { x ξ 4 }} On the domain of integration of the first integral we have that ξ is bounded away from zero and so the integral is finite. In turn, on the domain of integration of the second integral z ζ is bounded away from zero and ξ s is integrable near the origin as k > s, hence this integral is finite again. Let us consider now the case x = 0. Introducing ρξ o = ξ and rη o = y η we put I in the form I = ξ 0 = η o = r +ρ /4 ρ k r n k (ρ + r ) n ρ s dr dρdη o dξ o.

15 HARDY-SOBOLEV INEQUALITIES 5 Letting r = t cos φ, ρ = t sin φ we come to iff s < and k > s. / I = σ n k σ k t s dt 0 π 0 (sin φ) k (cos φ) n k sin φ s dφ <, 4. A non-linear equation in R n related to the Yamabe equation on groups of Heisenberg type Suppose a and b are two natural numbers, λ > 0, and for x,y R + = (0,+ ), define the function φ = λ [ (x + α) + (y + β) ], where α, β R. Proposition 4.. The function φ satisfies the following equation in the plane (4.) φ a + b + φ φ + a x φ x + b y φ y = aλ α x + bλ β, xy 0. y Proof. Set ξ = λ(x + α), η = λ(y + β) and define φ(ξ,η) = φ(x,y). Then we have Thus we have, (4.) Since φ = ξ + η we have x = λ ξ Σ def = φ a + b + φ φ and y = λ η. + a x φ x + b y φ y = λ φ a + b + λ φ φ + aλ x φ ξ + bλ y φ η. Σ = 4λ n + λ 4(ξ + η ) ξ + η + a x λξ + b y λη (4.3) = nλ + aλ x bλ (λx + λα) + (λy + λβ) y = nλ + aλ + bλ + aλ α x Hence, taking into account a + b = n, we proved Σ = aλ α x + bλ β. y + bλ β. y

16 6 DIMITER VASSILEV Noting that φ + a x φ x + b y φ y is the laplacian in R n R a+ R b+, n = a + b +, acting on functions with cylindrical symmetry, i.e., depending on x and y only, we are lead to ask the following question. Question 4.. Given two real numbers p o and q o find all positive solutions of the equation u n u u = p o x + q o y, (x,y) Rn R a+ R b+, which have at most a quadratic growth condition at infinity, u C( x + y ). As usual a simple transformation allows to remove the appearance of the gradient in the above equation. For a function F we have F(u) = F (u) u + F (u) u and thus u τ = τ(τ )u τ u + τu τ u = τ uτ (u u + (τ ) u ). Therefore we choose τ such that (τ ) = n, i.e., τ = n and then rewrite the equation for u as u n = n u n (u u n u ) = (n ) ( u n p o x + q ) o. y This is the equation which we will study. As a consequence of the above calculations we can write a three parameter family of explicit solutions. Proposition 4.3. Let λ > 0. The function v(x,y) defined in R n R a+ R b+ by the formula (4.4) v(x,y) = λ n ( ( x + α) + ( y + β) ) n, (x,y) R n R a+ R b+, satisfies the equation (4.5) v = v n n where ( p x + q ), y (4.6) p = α (n )λ a, q = β (n )λ b. Let us observe that the above equation is invariant under rotations in the x or y variables. Also if v is a solution then a simple calculations shows that for any t 0 the function v t (x,y) = t (n )/ v(tx,ty) is also a solution. Another observation is that the same principle works if we split R n in more than two subspaces. For example, if we take three subspaces we can consider the equation v = v n p n f( x, y, z ), f(x,y,z) = + q + r x y z and ask the question of finding all positive solutions with the same behavior at infinity as the fundamental solution. Clearly the function v = λ n ( ( x + α) + ( y + β) + ( z + γ) ) n, with the obvious choice of α, β and γ is a solution.

17 HARDY-SOBOLEV INEQUALITIES 7 5. The best constant and extremals of the Hardy-Sobolev inequality In this Section we give the proof of Theorem.. It was proven in [SSW] that there are extremals with cylindrical symmetry, i.e., functions depending only on x and y for which the inequality becomes equality. On the other hand, it was shown in [MS] that all extremals of inequality (.5) have cylindrical symmetry after a suitable translation in the y variable, see also [CW] and [LW] for some related results. Theorem 5. ([MS]). If u D, (R n ) is a function for which equality holds in (.5) then i) for any y R n k the function u(.,y) is a radially symmetric decreasing function in R k ; ii) there exists a y o R n k such that for all x R k the function u(x,. + y o ) is a radially symmetric decreasing function on R n k. We turn to the proof of Theorem., in which we find the extremals and the best constant in (.5) in the case σp σ =, i.e., s = in Theorem.. Proof. (of Theorem.) By Theorem. and Theorem.5 of [BT] there is a constant K for which (.6) holds and this constant is achieved, i.e., the equality is achieved. A small argument shows that a non-negative extremal u of the naturally associated variational problem inf R n u dz subject to the constraint (5.) R n k satisfies the Euler-Lagrange equation (n ) n u R k x dxdy = (5.) u = Λ x u n n, u D, (R n ), where Λ = K (n ) n. From Theorem.4 and standard elliptic regularity results we can see that v is a C function on x 0. Furthermore, u L loc (Rn ) and u is C smooth in the y variables. In particular u C 0,α loc (Rn ) for any 0 < α <. In order to see these claims let v = u yj for some j. Hardy s inequality shows v x L q loc for any < q < k. From elliptic regularity v W,q loc (Rn, δq ) for any < q < k and hence the Sobolev embedding gives v Wloc (Rn ), where δ = n, n q >. After finitely many iterations we see that v Wloc (Rn ) from which we can invoke Remark.3 to conclude v L loc (Rn ). The same argument can be done for the higher order derivatives in the y variables. For the x derivatives we argue similarly. We consider ( v = u xi ) v = V x v V o x. We note that V, V o L loc (Rn ) and it is not hard to see that the proof of Theorem.4, cf. also Remarks. and Remark.3, allows to conclude v L loc (Rn ), and hence x u L loc (Rn ) as claimed. From Theorem. of [MS] we can assume that u is an extremal with a cylindrical function after a suitable translation in the y variable. Thus we can assume that u has cylindrical symmetry. Introducing ρ = x, r = y we have that u is a function of ρ and r. We define U(ρ,r) = u by restricting u to two lines through the origin-one in R k, the other in R n k. From the regularity of u it follows that U is a smooth function of r for any fixed ρ. For any fixed r it is a smooth function of ρ when ρ 0, and Lipschitz for any ρ. Furthermore, in the first quadrant ρ > 0,r > 0

18 8 DIMITER VASSILEV of the ρr-plane it satisfies the equation (5.3) U = Λ ρ U n n. Using the equation and the smoothness of U in r it is not hard to see that U has bounded first and second order derivatives on ((0, ) (0, )), cf. Lemma 5.. Let φ(ρ,r) = U n. The calculations of Section 4 show that φ satisfies the following equation in the plane (5.4) φ n φ φ + a ρ φ ρ + b r φ r Λ n ρ = 0, where a = k, b = n k. Let µ > 0 and consider φ = µ φ. Clearly φ is a solution of φ n Let us choose µ such that φ φ Λ µ(n ) = n, i.e., + a ρ φ ρ + b r φ r µ = 4Λ (n ). Λ µ(n ) ρ = 0. With this choice of µ we see that φ satisfies equation (4.) in [GV]. Furthermore, a small argument using the homogeneity of the Kelvin transform, see Remark 3., shows it satisfies the asymptotic behavior (4.37) of [GV], except the inequality for the derivatives hold only on x 0. We can apply (4.40) of [GV] by noticing that the integrals on the ρ and r axis vanish as U has bounded first and second order derivatives in the punctured neighborhood of any point from the closed first quadrant, a fact which we observed above. Hence (4.43) of [GV] after setting A = λ gives φ = λ [ (r + n 4aλ ) + s ], Recalling that φ = µ φ and the value of µ we come to φ = λ 4Λ [ n (r + (n ) 4aλ ) + s ]. This shows that v must equal ( v = λ (n ) 4 ) n [ (n ) Λ n ( x + n ] n 4aλ ) + y ( = λ (n ) n ) [ n K (n ) ( x + n ] n 4aλ ) + y. The value of K is determined by (5.) after fixing λ arbitrarily, say λ =, since the value of the integral in (5.) is independent of λ. With this goal in mind we set p = n 4a and compute the integral (5.5) = [ ( n ) n R n k R k x K n [ ( x + p) + y ] n = K (n ) n ( n ) (n ) R n k R k ](n ) n dxdy [ x ( x + p) + y ] n dxdy

19 Let a = x + p. Then we compute (5.6) HARDY-SOBOLEV INEQUALITIES 9 R n k (a + y ) n dy = a n+k R n k ( + y dy ) n = σ n k k B(n, n + k ), an+k where σ n k is the volume of the unit n k dimensional sphere and B(.,.) is the beta function. On the other hand after a simple computation we find σ k r k dx = R k x ( x + p) n+k p n+k+ dr 0 (r + ) n+k (5.7) Plugging in (5.5) come to ( K (n ) n n = (5.8) ) (n )σn k = B( n k, n + k σ k pn+k+b(k,n ). σ k ) B(k,n + k ) pn+k+ = k+3 (n ) n k 3 (k ) n+k+ σ n k σ k B( n k, n + k )B(k,n ). The proof is complete taking into account the allowed translations in the y variable. In the above proof we used the following simple ODE lemma, which can be proved by integrating the equation. Lemma 5.. Suppose f is a smooth function on R \ {0}, which is also locally Lipschitz on R, i.e., on any compact interval there is a constant L, such that, f(t ) f(t ) L t t for any two points t,t on this interval. If f satisfies the equation f + k t f = a t + b, t > 0, where k is a constant k > and a, b are L loc order derivatives near the origin. functions, then f has bounded first and second 6. Some applications Let us consider the prescribed scalar curvature equation on R n (6.) u = R(z) u, where R is a bounded function and u is a non-negative function. As usual we say that u is of finite energy if u D,p (R n ) is finite. Clearly if g = u 4/(n ) g o is a metric conformal to the Euclidean metric g o on R n then the finite energy condition is equivalent to g having finite volume. The results of Section can be extended to the case of the scalar curvature equation of many non-compact manifold with positive Yamabe invariant, which among other things will be done in [VZ], but the following result, which follows from the fast decay Theorem 3. (a) of

20 0 DIMITER VASSILEV u, i.e., at least as fast as the fundamental solution, is indicative of what is to be expected, see also [Le]. Theorem 6.. Suppose R L and g o is the Euclidean metric on R n. Let u be a positive solution to (6.). If R n with the conformal metric g = u 4/(n ) g o has finite volume then u has fast decay and the metric g is incomplete. The second application concerns the original motivation of Badiale and Tarantello [BT] to consider the Hardy-Sobolev inequality. The following equation has been proposed, cf. [Ch], [B] and [R] for further details, as a model to study elliptic galaxies. (6.) u = φ( x )u q, 0 < u D, (R 3 ) ( z = (x,y) R 3!). It is also required that u is of finite mass, i.e., φu q dz <. Using the results of this paper we can show the following Theorem. Theorem 6.. Suppose ( + x ) γ φ L (R 3 ) for some 0 < γ <. If (γ) < q < 6, then any solution of (6.) is also of finite mass. Proof. Since the dimension of the ambient space is three we have (0) = 6 and < (γ) < 6. Given any q satisfying (γ) < q < 6 we can find an s < γ, such that, q = (s) and u satisfies the equation u = φ( x )u q = V u (s) x s with V = x s φ ( + x ) s φ ( + x ) γ φ L (R). Theorem 3. (a) implies that u decays at least as fast as the fundamental solution of the laplacian in R 3 Since q > (γ) > it follows u(z) C + z. φu q dz <, which shows that every finite energy solution is also of finite mass. References [B] Bertin, G., Dynamics of Galaxies, Cambridge University Press 000 [BS] Badiale, M. & Serra, E. Critical nonlinear elliptic equations with singularities and cylindrical symmetry. Rev. Mat. Iberoamericana 0 (004), no., [BT] Badiale, M. & Tarantello, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 63 (00), no. 4, [BY] Beals, R., & Ni, Y., Private Communication March 005 [C] Ciotti, L., Lecture Notes on Stellar Dynamics, Scuola Normale Superiore di Pisa editore, 00 [Ch] Chandrasekhar, S., Principles of Stellar Dynamics, enl. ed. New York, Dover, 960. [CKN] Caffarelli, L., Kohn R., & Nirenberg L., First Order Interpolation Inequality with Weights, Compositio Math. 53 (984) [CW] Catrina, F., & Wang, Z-Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. 54 (00), no., 9 58

21 HARDY-SOBOLEV INEQUALITIES [CGS] Caffarelli, L., Gidas, B., & Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 4 (989), no. 3, 7 97 [E] Egnell, H., Asymptotic results for finite energy solutions of semilinear elliptic equations. J. Differential Equations 98 (99), no., [FMT] Filippas, S., Maz ya, V. G. & Tertikas, A., Sharp Hardy-Sobolev inequalities, C. R. Math. Acad. Sci. Paris 339 (004), no. 7, [GMGT] Glaser, V., Martin, A., Grosse, H., & Thirring, W., A family of optimal conditions for absence of bound states in a potential, In: Studies in Mathematical Physics. Lieb, E.H., Simon, B., Wightman, A.S. (eds.), pp Princeton University Press 976 [GS] Gidas, B. & Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (98), no. 4, [GV] Garofalo, N. & Vassilev, D., Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-laplacians, Math. Ann. 38 (000), no. 3, [GV], Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type, Duke Math J, 06 (00) 3, [GY] Ghoussoub, N. & Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 35 (000), no., [L] Lieb, E.H., Sharp constants in the Hardy Littlewood-Sobolev and related inequalities, Ann. of Math. () 8 (983), no., [Le] Leung, M. C., Asymptotic behavior of positive solutions of the equation gu + Ku p = 0 in a complete Riemannian manifold and positive scalar curvature, Comm. Partial Differential Equations 4 (999), no. 3-4, [Li] Lin, C-S, Estimates of the scalar curvature equation via the method of moving planes. III. Comm. Pure Appl. Math. 53 (000), no. 5, [LU] Lanconelli, E., & Uguzzoni, F., Asymptotic behavior and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group, Boll. Un. Mat. Ital., (8) -B (998), [LW] Lin, C-S, & Wang, Z-Q., Symmetry of extremal functions for the Caffarrelli-Kohn-Nirenberg inequalities. Proc. Amer. Math. Soc. 3 (004), no. 6, [M] Maz ya, V.G., Sobolev Spaces, Springer, 985 [MS] Mancini, G. & Sandeep, K., Cylindrical symmetry of extremals of a Hardy Sobolev inequality, Ann. Mat. Pura Appl. (4) 83 (004), no., 65 7 [O] Okikiolu, G. O., Aspects of the theory of bounded integral operators in L p -spaces, Academic Press, London, 97 [R] Rein, G., Stationary and static stellar dynamic models with axial symmetry, Nonlinear Anal. 4 (000), no. 3-4, Ser. A: Theory Methods, [S] Serrin, J., Local behavior of solutions of quasi-linear equations. Acta Math [S], Singularities of solutions of nonlinear equations. 965 Proc. Sympos. Appl. Math., Vol. XVII pp [SSW] Secchi, S., Smets, D. & Willem, M., Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris 336 (003), no. 0, 8 85 [SW] Serrin, J., & Weinberger, H. F., Isolated singularities of solutions of linear elliptic equations, Amer. J. Math [V] Vassilev, D., Existence of solutions and regularity near the characteristic boundary for sub-laplacian equations on Carnot groups, to appear in Pacific J. Math. [V], Ph. D. Dissertation, Purdue University 000. [VZ] Vassilev, D. N. & Zhang, Q., Finite energy solutions of the Yamabe equation on non-compact manifolds, work in progress [Z] Zhang, Q., A Liouville type theorem for some critical semilinear elliptic equations on noncompact manifolds, Indiana Univ. Math. J., 50 (00), (Dimiter Vassilev) University of California, Riverside, Department of Mathematics, Riverside, CA 95 address: dvassilev@math.ucr.edu