Critical semilinear equations on the Heisenberg group: the eect of the topology of the domain

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1 Nonlinear Analysis 46 (21) Critical semilinear equations on the Heisenberg group: the eect of the topology of the domain G. Citti, F. Uguzzoni Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato 5, 4127 Bologna, Italy Received 28 June 1999; accepted 3 January 2 Keywords: Kohn Laplacian; Semilinear equations; Critical exponent; Homology groups 1. Introduction Let H be the Kohn Laplacian on the Heisenberg group H n and let q =2n +2 be the homogeneous dimension of H n. Precisely, if we denote (x; y; t) the elements of H n = R 2n+1, with x; y R n ;t R, the operator H can be represented as a sum of squares of vector elds n H = (Xj 2 + Yj 2 ); (1.1) j=1 where X j j +2y Y j j 2x We are concerned with the critical semilinear boundary value problem H u = u (q+2)=(q 2) in ; u in ; u = (1.2) where is a connected bounded domain of H n with boundary regular enough. We prove that this problem has a solution if has at least a nontrivial homology group (with Z 2 -coecients) H d () (d N). This result, which is the Kohn Laplacian counterpart of a celebrated theorem by Bahri and Coron [2], completes a research started in the papers [16,28,36]. Supported by University of Bologna. Funds for selected research topics. Corresponding author. addresses: citti@dm.unibo.it (G. Citti), uguzzoni@dm.unibo.it (F. Uguzzoni) X/1/$ - see front matter? 21 Elsevier Science Ltd. All rights reserved. PII: S X()138-3

2 4 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) After the pioneeristic wors by Jerison and Lee [24 26] on the Yamabe problem for CR manifolds, several authors have investigated, with dierent techniques, on semilinear equations for the Kohn Laplacian. Existence and nonexistence results have been established in [22,4,16,9] via variational methods; in [11,31,39] via super and subsolutions methods; in [5 8,14] via blow-up techniques; in [28,29,36 38] via mean value formulas; in [1] via moving planes techniques. We point out that, in spite of the striing similarities of the operator H with the classical Laplacian, the presence of a degenerate direction gives rise to new diculties in our context, principally related to the behavior of solutions near the boundary. Since the exponent (q + 2)=(q 2) is the critical Sobolev exponent, the existence of a solution of (1.2) strictly depends on the geometry of and of the operator H. If is H-starshaped, Garofalo and Lanconelli proved in [22] that such problem has no solutions. In the same paper they gave a rst example of solution to (1.2) on a non-contractible domain ={(x; y; t) r x 2 + y 2 R; t T}, where r; R; T are positive constants. The existence of a solution can be investigated with the concentration compactness principle, exactly as in the critical case of the Laplace operator (see for example [35,3,3], see also [12]). This was rst done in [16]: denoted I the functional naturally associated to Eq. (1.2), its nonnegative Palais Smale sequences can be represented in terms of the solutions to the same problem on dierent open subsets D of H n, called sets at innity (see Denition 3.1). If the structure of these sets is not nown, the representation result shows compactness of (PS) sequences only at some low levels of the functional I and does not apply to the non-perturbed critical equation on general domains. Here we give a complete description of such sets D which can be the whole H n or an open set of the form D = {(x; y; t) a x + b y + ct + q(x; y) }; with a; b R n ;c R and q a quadratic form on R 2n (see Lemma 3.4). The problem of existence or nonexistence of solutions on these sets is, at the authors nowledge, still open. On the other hand, satises a suitable geometrical hypothesis at characteristic points analogous to the dierentiability for the Heisenberg group (see Denition 3.3 of H-at domains), then the sets at innity D are only half-spaces or the whole H n. This geometrical hypothesis is analogous to the H-convexity condition introduced in [8] and seems to be the natural regularity to be required for a domain of H n since it reects the geometrical structure of the operator H. We also refer to [21] for a wea rectiability condition at non-characteristic points. When D is the whole space, the problem at innity (3.1) has been studied in [25]. When D is a half-space, the recent results in [28,36] ensure that problem (3.1) has no solutions. Hence, for an H-at domain we get a complete characterization of the compactness levels of the functional I (Theorem 3.5). As it is well nown this characterization is one of the most important tools in the proof of existence theorems for semilinear equations with critical growth. In the setting of the Laplace operator, Bahri and Coron [2] introduced a powerful topological argument which allows to establish the existence result if the domain has a nontrival homology group. The same technique has been extended to any Riemannian Laplacian

3 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) by Bahri and Brezis [1] (similar ideas are also developed in [34,15]). In particular they have pointed out that such technique is based on a few properties of the functional I: precisely on the structure of its (PS) sequences, on a parameterization theorem for a neighborhood of the set of critical points at innity and on an estimate of I on suitable linear combinations of solutions to the problems at innity. Once these estimates are also established in our context, the existence theorem follows. Theorem 1.1. Let be an H-at connected bounded domain of H n. If there exists a positive integer d such that the homology group (with Z 2 -coecients) H d () is nontrivial; then problem (1:2) has a solution in. As in the Euclidean case, the hypothesis on H d () is only a sucient condition for the existence of a solution; indeed we give here an example of a contractible domain such that (1.2) has a solution (Theorem 3.9). The paper is organized as follows. In Section 2 we x the notation. In Section 3 we study the Palais Smale sequences of the functional I. Sections 4 and 5 are devoted to the parameterization of the set of critical points at innity and to the required estimates of the functional I, respectively. In Section 6 we conclude our main result. 2. Notation The Heisenberg group H n is the homogeneous Lie group whose underlying manifold is R 2n+1 and whose group law is dened by ()= =(x + x ;y+ y ;t+ t +2(x y x y)); for every =(x; y; t); =(x ;y ;t ) H n (here denotes the inner product in R n ). The H-dilations are given by : H n H n ; (x; y; t)=(x; y; 2 t) for. The Jacobian determinant of is 2n+2, so that the homogeneous dimension of the group becomes q =2n +2: The homogeneous norm of the space is d (x; y; t)=(( x 2 + y 2 ) 2 + t 2 ) 1=4 (2.1) and the natural distance is accordingly dened d(; )=d ( 1 ): We shall denote by B d (; r) the d-ball of center and radius r. By the denition of d we have (B d (;r)) = B d (; r); r (B d (; 1)) = B d (;r):

4 42 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Hence, if denotes the Lebesgue measure on R 2n+1, from (2.1) we deduce B d (; r) = r q B d (; 1) : As a consequence, for every a b and for every measurable function f :[a; b] R, we have b f(d ()) d = q B d (; 1) f(r)r q 1 dr B d (;b)\b d (;a) if at least one of the integrals exists. The Kohn Laplacian on H n is the second-order operator H dened in (1.1), which is invariant with respect to the left translations of H n and homogeneous of degree two with respect to the dilations. We call subelliptic gradient D H =(X; Y )=(X 1 ;:::;X n ;Y 1 ;:::;Y n ): A remarable property of the Kohn Laplacian is that a fundamental solution of H with pole at zero is given by c q ()= d () q 2 ; (2.2) where c q is a suitable positive constant (see [19]). A basic role in the functional analysis on the Heisenberg group is played by the following Sobolev-type inequality where 2 q c D H 2 2 C (H n ); q := 2q (2.3) q 2 and c is a positive constant. This inequality ensures in particular that for every domain the function = D H 2 is a norm on C (). We denote by S1 () the closure of C () with respect to this norm; S 1 () becomes a Hilbert space with the inner product u; v S 1 = D H u; D H v : Thus there exists a natural orthogonal projection P : S 1 (H n ) S 1 (): (2.4) The exponent q in (2.3) is the critical Sobolev exponent for H since the embedding S 1(), Lq () is continuous but not compact even if is bounded. A variational solution of (1.2) can be found as a critical point of the functional I : S 1 () R; I(u)= 1 2 a D H u 2 q 2 2q u 2q=(q 2) : (2.5)

5 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Moreover, every variational solution is also a classical solution (see [2,22]). Here we loo for solutions of (1.2) as critical points of I constrained on the manifold M = {u S()\{} di(u)(u)=} 1 (more precisely on M + = {u M u }). Remar 2.1. If u S 1 ()\{} then u is a critical point of I i u is a critical point of I M. Moreover for every u,ifweset ( ) 1=(q u 2 2) (u)= u q q then (u)u M and we have ( ) q u : (2.6) I((u)u)= 1 q u q 3. Structure of (PS) sequences, H-at domains In this section we introduce the denition of H-at domains and we study the behavior of the (PS) sequences of the functional I M : It is well nown that the Palais Smale condition is not satised by the functional I and that the loss of compactness is in general due to the solutions of the so-called problems at innity. Denition 3.1. If is a smooth bounded domain of H n, we call set at innity of problem (1.2) any open set D obtained as limit of a subsequence of the following sequence of sets = ( 1 ()); given any sequence ( ) in and any divergent sequence ( )inr + (the structure of these sets at innity will be studied in Lemma 3.4). We call problem at innity related to (1.2) any problem H! = ;! ;! S(D); 1 (3.1) being D a set at innity of (1.2). Moreover, we will denote by I the functional I : S 1 (D) R whose critical points are the solutions of (3.1). When D = H n, a solution to (3.1) is the following C function: C! (x; y; t)= (t 2 +(1+ x 2 + y 2 ) 2 ; (3.2) )(q 2)=4 where C is a suitable positive constant. Moreover, every solution to (3.1) with D = H n taes the form! ; = (q 2)=2! 1 (3.3) for some and H n (this deep result is due to Jerison and Lee [25]).

6 44 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Let us recall the representation theorem for Palais Smale sequences proved in [16]: Theorem C. Let be a smooth bounded domain of H n and let (u ) be a nonnegative sequence in S 1 () such that I(u ) and di(u ) as + : Then there exist u S 1(); msolutions!i ;:::;! m of the problems at innity (3:1); m sequences ( 1 );:::;( m ) in and m divergent sequences ( 1 );:::;( m ) in R + (m N {}) such that (i) u = u + m i=1 P!i i i + o(1) as + ; ins 1(); where u is a solution of di(u )=;! i i i are obtained by translating and dilating! i accordingly to (3:3) and P is dened in (2:4). Moreover; if we denote by! 1 ;:::;! m1 the solutions on the whole H n and by! m1+1 ;:::;! m the solutions on the other sets at innity; we also have (ii) I(u )=I(u )+m 1 I (! )+ m i=m I 1+1 (! i )+o(1); as + ; (iii) i d( i ;@) + as + ; i =1;:::;m 1 ; (iv) i d( i ;@) has a nite limit as + ; i = m 1 +1;:::;m. The description of the sets at innity D was not nown. Actually it requires a careful study of the boundary of. We rst introduce some further notation and then give such description in Lemma 3.4. Given a smooth function : H n R and a point H n, we denote by q H ( ) the quadratic form associated to the Hessian matrix DH 2 along the vector elds of the subelliptic gradient D H, i.e. 2n (q H ( ))(z)= ((D H ) i (D H ) j )( )z i z j z R 2n ; i;j=1 where (D H ) i denotes the ith component of D H. Let us note explicitly that D 2 H ( )is not symmetrical in general, since [X j ;Y j ]= 4@ t. Remar 3.2. A smooth function has the following Taylor expansion of rst order with initial point in the direction of the vector elds of D H : ()= ( )+u X ( )+v Y ( )+o(d(; )) as ; (3.4) where (u; v; w) are the components of 1 () and denotes the scalar product in R n. The second-order expansion is the following: ()= ( )+u X ( )+v Y ( )+w@ t ( ) q H ( )(u; v)+o(d 2 (; )): (3.5) We explicitly remar that the limits in (3.4) and (3.5) are locally uniform in the variable. Denition 3.3. Let be a smooth bounded domain of H n, and let be a smooth function which describes the boundary of in a neighborhood of

7 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) (i.e. : B d ( ;R) R, ()= B d ( ;R), () i B d ( ;R) and the Euclidean gradient of is always dierent from ). The point is called characteristic if D H ( )=. If is characteristic, we shall say that is H-at at if q H ( )=: (3.6) We shall call H-at if it is H-at at any characteristic point of its boundary. According to (3.4) and (3.5), the condition q H ( ) = implies that the nontrivial Taylor polynomial of least order is a plane. Hence the boundary of an H-at domain has at every point a tangent plane, in the sense of the Heisenberg distance. This condition seems to be natural, since it is satised by the balls of the metric. Lemma 3.4. Let be a smooth bounded domain of H n ; let ( ) be a sequence in and let ( ) be a divergent sequence in R +. We set = ( 1 ()): Then (taing a subsequence if necessary) D as ; where (i) if d( ;@) is unbounded; then D is the whole space H n ; (ii) if d( ;@) is bounded; then there exist a; b R n ;c Rand a quadratic form q on R 2n such that D = {(x; y; t) a x + b y + ct + q(x; y) } (up to a left translation); (iii) if d( ;@) is bounded and is H-at; then D is a half-space of H n. Proof. (i) We may assume d( ;@) +. Since B d (; d( ;@))=( 1 )(B d ( ;d( ;@))) we get H n as. (ii) and (iii) Since d( ;@) is bounded, + and is bounded, then the sequence tends to a Let be a function that denes the boundary of in a neighborhood B d ( ;r )of. We now x =(x ;y ;t ) H n. By denition ; where := ( 1). Note that we can assume B d ( ;r ) for every suciently large. Indeed d( ; ) d( ; )+d( ; ( 1)) = d( ; )+d(; 1)=d( ; )+ 1 d(;) and this last term tends to as. Then by the choice of, ( ) : Let now be a point such that d( ;@)=d( ; ). If we consider the Taylor expansion of with initial point stated in (3.4) we get ( )=u X ( )+v Y ( )+o(d( ; )) as ;

8 46 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) where (u; v; w) are the components of 1 ( )= 1 ( 1), given by u = x x + 1 x ; v= y y + 1 y ; w = t t 2(x y x y )+2 1 (x (y y ) y (x x )) + 2 t : Hence (for any large ) ( ) ( )=( (x x )+x ) X ( ) +( (y y )+y ) Y ( )+o(1) as : We rst assume that is not characteristic. Moreover, since d( ; ) is bounded we can assume that (x x ) and (y y ) have limits c 1 and c 2, respectively. Hence (c 1 + x ) X ( )+(c 2 + y ) Y ( )+o(1) as : This proves that tends to the half-space D = {(x; y; t) (c 1 + x) X ( )+(c 2 + y) Y ( ) }: We now assume that is characteristic. In this case we have to use the second-order Taylor expansion recalled in (3.5): ( )=((x x )+ 1 x ) X ( )+((y y )+ 1 y ) Y ( ) +(t t 2(x y x y )+2 1 (x (y y ) y (x x )) + 2 t )@ t ( )+ 1 2 q H ( )(x x + 1 x ;y y + 1 y )+o(d( ; ) 2 ) as : We get (for any large ) ( ) 2 ( ) =( (x x )+x ) X ( )+( (y y )+y ) Y ( ) +( 2 (t t 2(x y x y ))+2(x (y y ) y (x x )) + t )@ t ( )+ 1 2 q H ( )( (x x ) + x ; (y y )+y )+o(1) as : As before, we can assume that the sequences (x x ), (y y ) and 2(t t 2(x y x y )) converge to c 1, c 2 and c 3, respectively. If % := D H ( ) is unbounded, we can also assume that % +, % 1 D H ( ) (; ) and we get (dividing all by % ) (c 1 + x ) +(c 2 + y ) + o(1) as :

9 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Hence, in this case, tends to a half-space. If % is bounded, we can assume that D H ( ) has limit (; ) and get that tends to the set D = {(x ;y ;t ): (c 1 + x ) +(c 2 + y ) +(c 3 +2(x c 2 y c 1 )+t )@ t ( )+ 1 2 q H ( )(c 1 + x ;c 2 + y ) } = 1 c ({(x; y; t): x + y + t@ t ( )+ 1 2 q H ( )(x; y) }): Moreover if is H-at, then q H ( )= and D is again a half-space. The recent nonexistence results on half-spaces contained in [28] and [36] ensure that (3.1) has no solutions on a half-space. Then the representation theorem can be improved as follows: Theorem 3.5. Let be an H-at domain of H n and let (u ) be a nonnegative sequence in S 1 () such that I(u ) and di(u ) as + : Then there exist u S 1(); m sequences ( 1);:::;( m ) in and m divergent sequences ( 1 );:::;( m ) in R + (m N {}) such that (i) u = u + m i=1 P! i i + o(1) as + ; in S 1(), where u is a solution of di(u )= and! i i are the well-nown functions dened by (3:2) and (3:3); (ii) I(u )=I(u )+mi (! )+o(1); as + ; (iii) i d( i ;@) + ; as + ; (iv) i j + j i + i j d 2 ( i ; j ) + ; as + ; i j. Proof. Since is H-at, the sets at innity can be only half-spaces or the whole H n (see Lemma 3.4). On the other hand the problem at innity (3.1) has no solutions when D is a half-space by means of [28,36]. Hence (i) (iii) follow directly from Theorem C. Assertion (iv) can be proved exactly as the analogous assertion in [12]. Proposition 3.6. An analogous result holds for the constrained functional I M. Indeed the following relations between Palais Smale sequences of Iand I M hold: (i) If (u ) is a sequence in S 1() such that I(u ) and di(u ) ; then (u ) 1; I((u )u ) and di M ((u )u ) as. (ii) If (u ) is a sequence in M such that I(u ) and di M (u ) ; then di(u ) as. As a consequence we can give a complete characterization of the compactness levels of (u ), exactly as in the Euclidean case. Remar 3.7. In the hypothesis of Theorem 3.5, if R + \{mi (! )} m N, then there exists u S 1()\{} such that di(u ) =, i.e. u is a variational solution to the equation in (1.2). Moreover, if ]I (! ); 2I (! )[, then u u strongly in S 1().

10 48 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) We remar that in Theorem 3.5, cannot belong to ];I (! )[ since di(u )= and u imply I(u ) I (! ). We now want to mae a few comments on the notion of H-at domains. Roughly speaing, they are C 2 -domains which are enough at at characteristic points. Let us start with a property of the characteristic points: Remar 3.8. If is a characteristic point then the straight line through orthogonal and incident to the t-axis is tangent at. Indeed if =(z ;t )= (x ;y ;t ) is characteristic and = is a local equation at then D H ( )= implies = D H ( );z n = (x ;j (@ xj ( )+2y t ( )) + y ;j (@ yj ( ) 2x t ( ))) j=1 = z ( );z ; i.e. (z ; ) ( ) where ( ) generates the orthogonal space at. Here we have denoted by ( )=( z ( );@ t ( )) the Euclidean gradient of. This simple condition immediately provides some examples: the domain A obtained by rounding o the edges of the annular region A = {(z; t)=(x; y; t) H n 1 z 2; t 1}, has no characteristic points. Hence it is H-at. The d-balls B d (; r) are remarable examples of H-at domains which exhibit characteristic points. As an application of Theorem 3.5 we can prove the following result. Theorem 3.9. There exists a contractible bounded domain of H n on which problem (1:2) admits a solution. Proof. In the papers [17,18,32,33] analogous results are obtained, with dierent proofs, in the setting of the Laplace operator. Not all these proofs can be extended directly to our context. For example in adapting the proof of [18] a delicate question arises when we try to modify the domain near the characteristic points in order to mae it H-at and eep the required symmetry properties. On the other hand, the proof in [33], for its generality, can be extended to our setting. It is based on a deformation argument and on the notion of Newtonian capacity. The topological argument can be repeated with no diculties for the Heisenberg operator; the notion of capacity has been extended to the H-capacity in our context (see for example [27,23,13] for its properties). Then, by means of Theorem 3.5 and the arguments of [33], problem (1.2) has a solution on any contractible H-at domain obtained by removing a small H-capacity set from a non-contractible domain. A simple example is given below. Let us set =(B d (; 1)\B d (; 1 2 ))\{(z; t) = (x; y; t) H n t ; z } for a small and dene by smoothing and eeping the symmetry in z. In light of Remar 3.8, it is immediate to recognize that has exactly two characteristic points in (; 1 4 ) and (; 1) and it is H-at.

11 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Parameterization of the set of quasi-critical points In this section we give a parameterization of the set V (m; ) which contains the Palais Smale sequences of the functional I. By the representation given in Theorem 3.5, it is natural to introduce (for every m N) the function % = % m :(; + ) m m (; + ) m S(); 1 m %(; ; )= i P! i; i ; (4.1) i=1 where! i; i are dened by (3.2) and (3.3) and P by (2.4). We also set, for every, { B = (; ; ) i d( i ;@) 1 ; 1 2 i 2 i; i + } j + i j d 2 ( i ; j ) 1 i j (4.2) j i and V (m; )={u M + : (e; ; ) B such that u %(e; ; ) }; (4.3) where M + is dened in Remar 2.1 and e =(1;:::;1) (; + ) m. Lemma 4.1. For every m N there exists such that for every u V (m; ) the problem min{ u %(; ; ) : (; ; ) B 4 } has a unique solution (up to permutations) and denes a continuous function :V (m; ) m = m ; where m = m is the quotient of m with respect to the group of permutations of {1;:::;m}. Proof. The proof follows the lines of the analogous one in [2]. In particular the variational argument is the same and we omit it, but we focus on a technical estimate, which depends on the geometrical properties of the operator and requires some care in our setting. The existence of a minimum can be proved as in [2]. In order to prove that it is unique we assume by contradiction that there exists a sequence ( )inr + such that as +, and for every N there exists u V (m; ) such that the function (; ; ) u %(; ; ) attains its minimum at two distinct points ( ; ; ) and ( ; ; ). One can prove that (up to permutations) b i := i = i 1= 1 o(1); i := i ( i i )=o(1); i := i i = o(1), as +. For simplicity of notation, in the sequel we will omit the index and we will denote by o(1) a sequence which goes to zero as :

12 41 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Arguing as in [2, (A26)], we get that m i = o(1) ( b j + j + j ) C i j=1 i (! i! i ); (4.4) where we have set! i =! i; i,! i =! i; i. Let us dene i = H n i (! i! i ). Using the denition of! i,weget i = (q+2)=2 i H n ( (q 2)=2 i ( i ( 1 i ))! ( i ( 1 )) i (q 2)=2 i! ( i ( 1 i ))) d (if we set = i ( 1 i )) = ()(! () (1 + b i ) (q 2)=2! ( i (1+bi))) d: H n Now, we tae the Taylor expansion (in the usual sense) of order one of the function!(b i ; i )()=! () (1 + b i ) (q 2)=2! ( i (1+bi)) with respect to the variables b i and i in a neighborhood of. If we call i =(x i ;y i ;t i ) and =(x; y; t), it is easy to prove that d!=dt i (; )() is odd as a function of t, while! is even with respect to the same variable. Hence d! ()t i (; )()d =: H dt n i Arguing in the same way with all the components of i,weget i = Now setting n ( Z = we have d! ()b i (; )()d + O( b i + i ) 2 : H db n i i=1 x + y i = b i ( Z! ()+ q 2 H 2!2q=(q 2) ()) d + O( b i + i ) 2 : n Let us estimate I R := ( Z! ()+ q 2 ()) d B d (;R) = q 2 2q 2!2q=(q 2) (Z! 2q=(q 2) ()+q! 2q=(q 2) B d (;R) ()) d (4.5)

13 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) (if we denote by = d = d the outer unit normal to B d (;R), by dh q 2 Hausdor (q 2)-dimensional measure and we notice that q = div Z) = q 2! 2q=(q 2) Z dh q 2 d (;R) (since Z = Zd = d = d = d and setting for brevity g =((q 2)=2q)! 2q=(q 2) ( L 1 )) g = R d dh q d (;R) the By Federer s coarea formula, since g L 1 (H n ), we have + ( ) d (;R) d dh q 2 dr = g + : H n Hence, there exists a divergent sequence of radii (R ) N such that I R as : This proves that i = O( b i + i ) 2 : Recalling (4.4), we obtain i = o(1) m ( b j + j + j ): j=1 With essentially the same arguments, it is possible to show that i + b i = o(1) m ( b j + j + j ): j=1 This implies that i = b i = i = for every i =1;:::;m, which is a contradiction. Hence the minimum is unique and is well dened. With similar arguments one can prove that is also continuous. 5. Estimates of the functional I In this section, we give some estimates of the functional I M, on a suitable nite dimensional set. The main idea is to nd an expression of the functional I in terms of the Green function G of related to H. In what follows we denote by H ()= H(; )= (; ) G(; ) the regular part of G (we recall that (; )=c q d(; ) 2 q, see (2.2)).

14 412 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Proposition 5.1. Let K be a compact subset of and let m 1 = { [; 1] m m i=1 i =1} be the standard simplex. Let m 1 ; K m ;. We set d = min i j d( i ; j ); 2 = m i=1 2 i and 2q=(q 2) = m i=1 2q=(q 2) i. If d 1; then we have I((%(; ; ))%(; ; )) ( ) ( q = I (! ) 1+ C m ( ) ) 2q=(q 2) i q i 2 H( i ; i ) i j ( 2 (q+2)=(q 2) i j 2q=(q 2) i j 2 i=1 ) ( G( i ; j ) + O 1 (d) q 1 where %(; ; ) =%( 1 ;:::; m ; 1 ;:::; m ;;:::;) has been dened in (4:1) and in Remar 2:1. Proof. We only consider the part of the proof which needs to be handled with a little of care in the setting of the operator H (the rest of the proof is analogous to the one in [2]). Recalling (2.6), we only need to estimate m j=1 jp! j and m j=1 jp! j q, where! j =! ;j. Setting h i =! i P! i, we have D H P! j D H P! i = D H! j D H P! i = j P! i = j (! i h i ) = j H n! i H n \ j! i ) ; j h i : (5.1) Let us consider the case when i = j. The delicate point is to estimate the integral in the far right-hand side. For this purpose we notice that h i is H -harmonic in and compare it with H i. Indeed, for we have h C i() (q 2)=2 d(; i ) q 2 C (q+2)=2 ; where C is the constant introduced in (3.2). Then, by the maximum principle for H, we get h C i() c q (q 2)=2 H i () C (q+2)=2 : (5.2) The Taylor expansion of second order of the function H i elds of D H can be written (see (3.5)) in the direction of the vector H i ()=H i ( i )+( 1 i ()) 1 XH i ( i )+( 1 i ()) 2 YH i ( i )+O(d 2 (; i )):

15 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Let l = By the symmetry of B d (;l) with respect to the rst 2n variables, we get B d ( i;l) i H(; i )d = H( i ; i ) = H( i ; i ) B d ( i;l) ( + O B d ( i;l) (with some simple computations) ( ) H( i ; i ) 1 =C 2 (q 2)=2 + O q=2 ; where C 2 = H n. Now, using (5.2), we nd ( i d + O i H( i ; i ) H n i d(; i ) 2 d ) B d ( i;l) H n \B d ( i;l) i h i = C ( ) C 2 H( i ; i ) 1 c q q 2 + O q 1 : From this and (5.1), with some easy computations, we get D H P! i D H P! i = qi (! ) C C 2 H( i ; i ) c q q 2 + O ) i d(; i ) 2 d i ( ) 1 q 1 : The case i j is more technical but does not present new diculties with respect to the proof in [2] (as well as the estimate in norm q ). Setting for brevity % = %(; ; ), we nally get ( ) q=2 D H % 2 =(qi (! ) 2 ) q=2 C C c q I (! ) q 2 i j q m i 2 2 ( ) 2 H( i; i ) 1 + O () q 1 : i=1 Analogously we can prove that ( ) (q 2)=2 % 2q=(q 2) =(qi (! )) (q 2)=2 q C C 2 m 1+ c q I (! ) q 2 i j (q+2)=(q 2) i j 2q=(q 2) G( i ; j ) + O i=1 ( 1 ) () q 1 i j 2 G( i; j ) ( ) 2q=(q 2) i H( i ; i )

16 414 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) and we can conclude ( % ) q I((%)%)= 1 q % q = I (! ) ( 2 ( i j ( ) q C C c q I (! ) q 2 ( ) ) 2q=(q 2) i 2 i 2 2 (q+2)=(q 2) i 2q=(q 2) From the previous proposition we can deduce j m H( i ; i ) i=1 C C 2 2c q I (! ) q 2 ) ( ) i j 2 G( i ; j ) 1 + O () q 1 : Lemma 5.2. Let K and m 1 be as in the previous proposition. (a) For every m N; m 2; there exist and such that I((%(; ; ))%(; ; )) mi (! ); (5.3) for every (; ) m 1 K m with i for some i. (b) There exists m N such that for every there exists such that (5:3) holds for every m m ; ; m 1 and K m with min i j d( i ; j ). (c) For every m N; m 2; and for every there exist and such that (5:3) holds for every m 1 [; 1] m ; and K m with min i j d( i ; j ). (d) There exist m N and such that for every (5:3) holds for every m 1 and K m. We omit the proof since in Proposition 5.1 we provide an expression of the functional I which is formally the same as the expression nown for the analogous problem associated to the classical Laplacian. Then Lemma 5.2 follows as in [2]. 6. Proof of the existence result Theorem 1.1 will be proved as an application of the well-nown theory introduced by Bahri and Coron in the Laplacian case and extended by Bahri and Brezis to the Riemannian case. In their paper Bahri and Brezis pointed out that this technique can be applied to loo for critical points of a functional any time the structure of its (PS) sequences is nown. Here we loo for critical points of the functional I constrained on the manifold M and we state their theorem in our particular situation.

17 G. Citti, F. Uguzzoni / Nonlinear Analysis 46 (21) Theorem 6.1. Let be a smooth connected bounded domain of H n. If Theorem 3:5 and Lemmas 4:1; 5:2 are satised and there exists a positive integer d such that the homology group H d () is nontrivial; then problem (1:2) has a solution. This theorem is proved in [1, Section 8] and we do not repeat the proof here but we only want to give an idea of it for the reader s convenience. It is based on the topological properties of the sublevels of the functional I M W m = {u M + I(u) (m +1)I (! )}: If by contradiction (1.2) has no solutions, then the representation Theorem 3.5 implies that is homeomorphic to a retract of W 1. Indeed the set V (1;) dened in (4.3) is a deformation retract of W 1 and, roughly speaing, the function % 1 (e; :; ) : V(1;) induces isomorphisms on homology groups with inverses induced by the function :V (1;) dened in Lemma 4.1. The estimates of the functional I on the image of % provided in Lemma 5.2, ensure that there exists m such that % m is homologically trivial. The main point of the proof of [1] is a topological argument which leads to a commutative diagram which shows that % m is trivial if and only if % 1 is. This gives rise to a contradiction. Proof of Theorem 1.1. Theorem 3.5, Lemmas 4.1 and 5.2 have already been proved in Sections 3, 4 and 5, respectively. Hence Theorem 1.1 directly follows from Theorem 6.1. Acnowledgements We would lie to than M. Mulazzani for some useful discussions on the topological aspects of the problem. References [1] A. Bahri, H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, in: S. Gindiin (Ed.), Topics in Geometry. In Memory of Joseph D Atri. Birhaeuser, Boston; Prog. Nonlinear Dierential Equ. Appl. 2 (1996) 1 1. [2] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the eect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) [3] V. Benci, G. Cerami, Existence of positive solutions of the equation u + a(x)u = u N +2=N 2 in R N, J. Funct. Anal. 88 (199) [4] S. Biagini, Positive solutions for a semilinear equation on the Heisenberg group, Boll. Un. Mat. Ital. (7) 9-B (1995) [5] I. Birindelli, Nonlinear Liouville theorems, Lecture Notes in Pure and Applied Mathematics, Vol. 194, Deer, New Yor, 1997, pp

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