Open Access

Topological arguments for an elliptic equation involving the fractional Laplacian

  • Mohammed Ali Al-Ghamdi1Email author,
  • Wael Abdelhedi2 and
  • Hichem Chtioui1
Boundary Value Problems20142014:187

https://doi.org/10.1186/s13661-014-0187-2

Received: 21 November 2013

Accepted: 21 July 2014

Published: 26 September 2014

Abstract

In this paper, we consider a fractional Nirenberg-type problem involving σ-exponent of the Laplacian on the standard n-dimensional spheres S n . Using an algebraic topological method and the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the σ-curvature on n-dimensional sphere.

MSC: 35M30, 35Q74, 74A15, 74A60, 74M25.

Keywords

fractional Laplaciancritical exponentcritical points at infinityNirenberg problem

1 Introduction and main results

Let ( S n , g ) be the standard sphere of dimension n, n 2 and K be a positive function on  S n . The classical Nirenberg problem is the following: can one change the original metric g conformally into a new metric g ˜ with prescribed scalar curvature equal to K for n 3 (prescribed Gaussian curvature for n = 2 ). This problem is equivalent to solving the following nonlinear elliptic equations:
Δ g ( u ) + 1 = K e 2 u on  S 2 ,
(1)
L g ( u ) = K u n + 2 n 2 on  S n , n 3 ,
(2)

where L g : = c n Δ g + R g be the conformal Laplacian, Δ g is the Laplace-Beltrami operator, c n = 4 n 1 n 2 and R g is the scalar curvature associated to the metric g.

Recently, several studies have been performed for classical elliptic equations similar to (2) and (1) but with the fractional conformal Laplacians instead of the Laplacian. This operator is introduced first in [1] where Graham, Jenne, Mason, and Sparling constructed a sequence of conformally covariant elliptic operators P k g , on Riemannian manifolds ( M , g ) for all positive integers k if n is odd, and for k { 1 , , n / 2 } if n is even. Moreover, P 1 g is the well-known conformal Laplacian L g used in (2), and P 2 g is the well-known Paneitz operator. Up to positive constants P 1 g ( 1 ) is the scalar curvature R g associated to g and P 2 g ( 1 ) is the Q-curvature. Prescribing scalar curvature and Q-curvature problems on S n was studied extensively in last years, see for example [2]–[10]; see also [11]–[19], and the references therein.

In [20], Chang and Gonzalez generalize the above construction, using the works of Graham-Zworski [21] and Caffarelli-Silvestre [22] for factional Laplacian on the Euclidean space R n , to define conformally invariant operators P σ g of non-integer order σ. These lead naturally to a fractional-order curvature R g : = P σ g ( 1 ) , which will be called here σ-curvature. For the σ-curvatures on general manifolds we refer to [20], [21], [23], and references therein. Corresponding to the Yamabe problem, fractional Yamabe problems for σ-curvatures are studied in [23], [24], and [25] and fractional Yamabe flows on S n are studied in [26].

In this paper, we restrict our attention to the case σ = 1 2 . As in the Nirenberg problem associated to P 1 g and P 2 g , the question of prescribing 1 2 -curvature can be formulated as a Nirenberg-type problem involving the square root of the Laplacian as follows: given a positive function K defined on ( S n , g ) , we ask whether there exists a metric g ˜ conformally equivalent to g such that the 1 2 -curvature is equal to K. This problem can be expressed as finding the solution of the following nonlinear equation with critical exponent,
P 1 2 u = c n K u n + 1 n 1 , u > 0 , on  S n ,
(3)
where
P 1 2 = Γ ( B + 1 ) Γ ( B ) , B = Δ g + ( n 1 2 ) 2 ,
c n = P 1 2 ( 1 ) , Γ is the gamma function and Δ g is the Laplace-Beltrami operator on ( S n , g ) . The operator P 1 2 can be seen more concretely on R n using stereographic projection. The stereographic projection from S n N to R n is the inverse of F : R n S n N defined by
F ( x ) = ( 2 x 1 + | x | 2 , | x | 2 1 | x | 2 + 1 ) ,
where N is the north pole of S n . Then, for all f C ( S n ) , we have
( P 1 2 ( f ) ) F = ( 2 1 + | x | 2 ) ( n + 1 ) 2 ( Δ ) 1 / 2 ( ( 2 1 + | x | 2 ) n + 1 2 ( f F ) ) ,
(4)

where ( Δ ) 1 / 2 is the fractional Laplacian operator (see, e.g., p.117 of [27]).

A sufficient condition was found recently in [28] for the two dimensional sphere, it is an Euler-Hopf-type criterium, namely the authors proved the following existence result.

Theorem 1.1

[28]

Let K be a positive function satisfying the following (nd) condition:

(nd): Assume that K is a smooth function on S 2 having only non-degenerate critical points with
Δ K ( y ) 0 whenever  K ( y ) = 0 .
Let
K = { y S 2 , K ( y ) = 0 } and K + = { y K , Δ K ( y ) > 0 } .
If
( R 1 ) y K + ( 1 ) 2 ind ( K , y ) 1 ,

then (3) has at least one solution. Here, ind ( K , y ) is the Morse index of K at y.

A natural question which arises when looking to the above result is what happens if the total sum in ( R 1 ) is equal to 1, but a partial one is not. Under which condition can one use this partial sum to derive an existence result? Our aim in the first part of this paper is to give a partial answer to this question and to prove new existence criterium which generalizes the index-count formula in ( R 1 ) and recover the previous existence result obtained in [28]. To state our result, we set the following notations. We order the K ( y ) , y K . Without loss of generality, we assume that the elements of K are y 0 , y 1 , , y l with
K ( y 0 ) K ( y 1 ) K ( y l ) .

Let k be an index, 0 k l . We say that k ( A 1 ) if it satisfies the following:

( A 1 ): If there exists an index j > k such that y j K + , we have
1 K ( y j ) > 1 K ( y k ) + 1 K ( y 0 ) .

Now, we are ready to state our first main result.

Theorem 1.2

Let K be a positive function on S 2 and satisfying (nd) condition. If
Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) | 0 ,
then (3) has at least one solution. Moreover, for generic K we have
S Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) | ,

where S denotes the set of solutions of (3).

Observe that the index l ( A 1 ) . It follows that the above mentioned result (Theorem 1.1) is a corollary of our theorem. Actually, we will give in Remark 3.1 a situation where Theorem 1.1 does not give a result, while by Theorem 1.2, we derive the existence of many solutions.

Second, we consider the case n 3 and K close to a constant, i.e. K of the from K = K ε = 1 + ε K 0 , where K 0 C 2 ( S n ) and | ε | small. In order to state our result, we need the following assumption. We say that an integer k ( A 2 ) if it satisfies the following:

( A 2 ): For any y K ε + , we have n ind ( K ε , y ) k + 1 .

We shall prove the following result.

Theorem 1.3

Assume that K ε is a Morse function on S n , n 3 and satisfying (nd) condition. If,
Max k ( A 2 ) | 1 y K ε + n ind ( K ε , y ) k ( 1 ) n ind ( K ε , y ) | 0 ,

then, for | ε | sufficiently small, there exists a solution to the problem (3).

Observe that n ( A 2 ) . We therefore have the following corollary.

Corollary 1.4

Let K ε be a Morse function with (nd) condition. If
y K ε + ( 1 ) n ind ( K ε , y ) 1 ,

then for | ε | sufficiently small, (3) has at least one solution.

The rest of this paper is organized as follows: we first recall some known facts about the variational structure of problem (3) and the associated critical points at infinity. In Section 3, we give the proofs of our main results and in Section 4, we give a more general result for σ ( 0 , 1 ) and 2 n < 2 + 2 σ .

2 Preliminary results

2.1 The variational structure

First, we recall that u H 1 2 ( S n ) is a solution of (3) if the identity
S n P 1 2 u φ d x = c n S n K u n + 1 n 1 φ d x ,
(5)
holds for all φ H 1 2 ( S n ) , where H 1 2 ( S n ) denotes the closure of C ( S n ) under the norm
u H 1 2 ( S n ) = ( S n P 1 2 u u ) 1 / 2 .
(6)
Observe also that for u H 1 2 ( S n ) , we have u n + 1 n 1 L 2 n n + 1 ( S n ) H 1 2 ( S n ) . We associate to problem (3), the functional
I ( u ) = 1 2 S n u P 1 2 u n 1 2 n S n K u 2 n n 1 ,
(7)

defined in H 1 2 ( S n ) .

Motivated by the work of Caffarelli and Silvestre [22], several authors have considered an equivalent definition of the operator P 1 2 by means of an auxiliary variable; see [22] (see also [29]–[32], and [33]). In fact, we realize problem (3), through a localization method introduced by Caffarelli and Silvestre on the Euclidean space R n , through which (3) is connected to a degenerate elliptic differential equation in one dimension higher by a Dirichlet to Neumann map. This provides a good variational structure to the problem. By studying this problem with classical local techniques, we establish existence of positive solutions. Here the Sobolev trace embedding comes into play, and its critical exponent 2 = 2 n n 1 .

Namely, let D n = S n × [ 0 , ) . Given u H 1 2 ( S n ) , we define its harmonic extension U = E σ ( u ) to D n as the solution to the problem
{ Δ U = 0 in  D n , U = u on  S n × { t = 0 } .
(8)
The extension belongs to the space H 1 ( D n ) defined as the completion of C ( D n ) with the norm
U H 1 ( D n ) = ( D n | U | 2 d x d t ) 1 / 2 .
(9)
Observe that this extension is an isometry in the sense that
U H 1 ( D n ) = u H 1 2 ( S n ) , u H 1 2 ( S n ) .
(10)
Moreover, for any φ H 1 ( D n ) , we have the following trace inequality:
φ H 1 ( D n ) φ ( , 0 ) H 1 2 ( S n ) .
(11)
The relevance of the extension function U = E σ ( u ) is that it is related to the fractional Laplacian of the original function u through the formula
lim t 0 + U t ( x , t ) = P 1 2 u ( x ) .
(12)
Thus, we can reformulate (3) as follows:
{ Δ U ( x , t ) = 0 and U > 0 in  D n , lim t 0 + U t ( x , t ) = K U n + 1 n 1 ( x , 0 ) on  S n × { 0 } .
(13)
The functional associated to (13) is given by
I 1 ( U ) = 1 2 D n | U | 2 d x d t n 1 2 n S n K U 2 n n 1 d x ,
(14)

defined in H 1 ( D n ) .

Notice that critical points of I 1 in H 1 ( D n ) correspond to critical points of I in H 1 2 ( S n ) . That is, if U = E σ ( u ) satisfies (13), then u will be a solution of problem (3).

Let also define the functional
J ( U ) = U H 1 ( D n ) 2 ( S n K U 2 n n 1 d x ) n 1 n ,
(15)

defined on Σ the unit sphere of H 1 ( D n ) . We set, Σ + = { U Σ / U 0 } . Problem (3) will be reduced to finding the critical points of J subjected to the constraint U Σ + . The exponent 2 n n 1 is critical for the Sobolev trace embedding H 1 ( D n ) L q ( S n ) . This embedding being continuous and not compact, the functional J does not satisfy the Palais-Smale condition, which leads to the failure of the standard critical point theory. This means that there exist sequences along which J is bounded, its gradient goes to zero and which do not converge. The analysis of sequences failing PS condition can be analyzed along the ideas introduced in [2] and [34]. In order to describe such a characterization in our case, we need to introduce some notations.

For a R + n + 1 and λ > 0 , define the function:
δ ˜ a , λ ( x ) = c ¯ λ n 1 2 ( ( 1 + λ x n + 1 ) 2 + λ 2 | x a | 2 ) n 1 2 ,
where x R + n + 1 , and c ¯ is chosen such that δ ˜ a , λ satisfies the following equation:
{ Δ U = 0 and U > 0 in  R + n + 1 , U x n + 1 = U n + 1 n 1 on  R + n + 1 .
Set
δ a , λ = i 1 ( δ ˜ a , λ ) ,

where i is an isometry from H 1 ( D n ) to D 1 , 2 ( R + n + 1 ) ; the completion of C c ( R + n + 1 ¯ ) , with respect to the Dirichlet norm.

For ε > 0 and p N , we define
V ( p , ε ) = { U Σ  s.t.  a 1 , , a p S n , α 1 , , α p > 0  and  λ 1 , , λ p > ε 1  with  U i = 1 p α i δ a i , λ i < ε , ε i j < ε , i j , and | J ( U ) n n 1 α i 2 n 1 K ( a i ) 1 | < ε , i , j = 1 , , p ,
where
ε i j = ( λ i λ j + λ j λ i + λ i λ j | a i a j | 2 ) 1 n 2 .

If U is a function in V ( p , ε ) , one can find an optimal representation, following the ideas introduced in [35], namely we have:

Lemma 2.1

For any p N , there is ε p > 0 such that if ε ε p and U V ( p , ε ) , then the minimization problem
min { U i = 1 p α i δ ( a i , λ i ) , α i > 0 , λ i > 0 , a i S n } ,
has a unique solution ( α ¯ , λ ¯ , a ¯ ) . Thus, we can write U as follows:
U = i = 1 p α i δ ( a i , λ i ) + v ,

where v belongs to H 1 ( D n ) and satisfies the following condition:

(V0):: v , φ i = 0 for i = 1 , , p , and φ i = δ i , δ i / λ i , δ i / a i , where δ i = δ a i , λ i and , denote the scalar product defined on H 1 ( D n ) by
U , V = D n U V d x d t .

The failure of the Palais-Smale condition can be characterized taking into account the uniqueness result of the corresponding problem at infinity; see e.g. Li-Zhu [36] following the ideas introduced in [34], [37] as follows.

Proposition 2.2

Assume that (3) has no solution and let ( U k ) Σ + be a sequence satisfying J ( U k ) c , a positive number and J ( U k ) 0 . Then there exist an integer p 1 , a positive sequence ( ε k ) k ( ε k 0 ) and an extracted subsequence of ( U k ) , again denoted U k such that U k V ( p , ε k ) .

Following Bahri [35], [38], we set the following definition and notation.

Definition 2.3

A critical point at infinity of J on Σ + is a limit of a flow line U ( s ) of the equation:
{ U s = J ( U ) , U ( 0 ) = U 0 ,

such that U ( s ) remains in V ( p , ε ( s ) ) for s s 0 .

Using Lemma 2.1, U ( s ) can be written as
U ( s ) = i = 1 p α i ( s ) δ ( a i ( s ) , λ i ( s ) ) + v ( s ) .
Denoting a i : = lim s a i ( s ) and α i = lim s α i ( s ) , we denote by
( a 1 , , a p ) or i = 1 p α i δ ( a i , )

such a critical point at infinity.

The following propositions characterize the critical points at infinity of the associated variational problem.

Proposition 2.4

Assume that n = 2 . Under the (nd) condition, the only critical points at infinity of the functional J are:
( y ) : = 1 K ( y ) 1 2 δ ˜ ( y , ) , y K + .

Such a critical point at infinity has a Morse index equal to i ( y ) : = 2 ind ( K , y ) . Its level is c ( y ) : = S ˜ 2 1 K ( y ) 1 2 , where S ˜ 2 is the best constant of Sobolev.

Proof

See Corollary 4.4 of [28]. □

Arguing as in [28], we have the following.

Proposition 2.5

Assume that n 3 and assume that J has no critical point in Σ + . Then the only critical points at infinity of J in V ( 1 , ε ) are
( y ) : = 1 K ( y ) n 1 n δ ˜ ( y , ) , y K + .

The Morse index of ( y ) is equal to i ( y ) : = n ind ( K , y ) . Its level is c ( y ) : = S ˜ n 1 K ( y ) n 1 n . Here S ˜ n is the best constant of Sobolev.

2.2 The unstable manifolds of critical points at infinity

At the beginning of this subsection, we give some basic definitions with will allow us to describe the unstable manifolds of the critical points at infinity in V ( 1 , ε ) .

Definition 2.6

Let K : S n R be a C 2 Morse function and let K the set of critical points of K. If y K , let W s ( y ) designate its stable manifold and W u ( y ) designate its unstable manifold. We have
dim W u ( y ) = ind ( K , y ) and dim W s ( y ) = n ind ( K , y ) .

It is convenient to specify that the notations of stable or unstable manifolds, of flow lines, all are relative to the C 1 vector field ( K ) , with respect to the standard Riemannian structure on S n . Recall the following generic hypothesis:

All stable and unstable manifolds intersect transversely and all such intersections are smooth regularly embedded submanifolds of S n .

Definition 2.7

Let y , z K . z is said to be dominated by y, if
W u ( y ) W s ( z ) ,
then there exists (at least) a flow line of ( K ) descending from y to z. Using the dimension argument and the fact that both of W u ( y ) and W s ( z ) are invariant under the action of the flow generated by ( K ) , it is easy to see that
if  W u ( y ) W s ( z ) , then  ind ( K , y ) ind ( K , z ) + 1 .
(16)

Definition 2.8

Let y K + = { y K , Δ K ( y ) > 0 } . In V ( 1 , ε ) , the unstable manifold at infinity W u ( y ) for the critical point at infinity ( y ) , along the flow lines of J is defined and identified by the unstable manifold W ˜ u ( y ) of the critical point y of the function 1 K , along the flow lines of ( 1 K ) multiplied by a factor corresponding to the concentration λ. Precisely in V ( 1 , ε ) , the W u ( y ) have the following description:
W u ( y ) V ( 1 , ε ) = { 1 S ˜ n δ ˜ ( a , λ ) , a W ˜ u ( y ) } ,

where λ is a fixed constant large enough.

Remark 2.9

Observe that W ˜ u ( y ) correspond to W s ( y ) , the stable manifold of the critical point y along the flow lines of ( K ) . Therefore it is easy to see that if W u ( y ) V ( 1 , ε ) , then W u ( y ) behaves as W s ( y ) .

The following lemma gives a sufficient condition to ensure that W u ( y ) is included in V ( 1 , ε ) .

Lemma 2.10

Let y K + . If
W s ( y ) W u ( z ) = , z K K + , then  W u ( y )  diffeomorphic to  W s ( y ) .

Proof

It follows from [28]. The idea is that a flow line in W u ( y ) cannot go out from V ( 1 , ε ) unless the concentration point a ( s ) of the flow line nearby a critical point z of K with Δ K ( z ) < 0 (see proof of Proposition 4.3 of [28]), therefore it is the case when the critical point y is dominated by z K K + . Hence under the condition of the lemma such a situation cannot occur, it follows that every flow line in W u ( y ) is indeed in V ( 1 , ε ) and we then conclude to the result of the lemma using Remark 2.9. □

3 Proof of results

This section is devoted to the proofs of the main results of this paper. Our proofs uses algebraic topological arguments and the tools of the theory of critical points at infinity; see [38]. In our case, the space of variation is contractible and has no topology. However, due to the non-compactness of the problem, there are critical points at infinity whose topological contribution can be computed; see [28]. The main idea is to use the difference of topology of the critical points at infinity between the level sets of the associated Euler-Lagrange functional J, and the main issue is under our conditions on K, there remains some difference of topology which is not due to the critical points at infinity but due to the existence of solution of (3).

3.1 Proof of Theorem 1.2

We denote by y 0 , y 1 , , y l the elements of K . We order the K ( y i ) ’s, y i K . Without loss of generality, we assume that
K ( y 0 ) K ( y 1 ) K ( y l ) .
Recall that, see Proposition 2.4, the critical points at infinity are in one to one correspondence with the elements y i of the sets K + . Like a usual critical point, it is associated to a critical point at infinity ( y i ) , stable and unstable manifolds, W s ( y i ) and W u ( y i ) . Let 0 k 0 l be an index such that
Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) |
is achieved. Let
c 1 = 1 K ( y 0 ) + 1 K ( y k 0 ) .

Since k 0 satisfies ( A 1 ), we can find ε > 0 satisfying the following:

For any j > k 0 such that y j K + , we have
c ( y j ) = 1 K ( y j ) > c 1 + ε .
Therefore the only critical points at infinity under the level c 1 + ε are:
( y j ) , y j K +  such that  0 j k 0 .
Let
X k 0 : = y j K + , 0 j k 0 W u ( y j ) .
Given c R , we set
J c : = { u Σ + , J ( u ) c } .
We claim that
X k 0  is contractible in  J c 1 + ε .
(17)
Indeed, let
X k 0 : = y j K + , 0 j k 0 W s ( y j ) ,
where W s ( y j ) is the stable manifold of y j along the flow lines of ( K ) . Observe that for any y K + , we have ind ( K , y ) = 1  or  2 , thus from (16), y cannot be dominated through the flow lines of ( K ) only by critical points y of K such that ind ( K , y ) = 2 , thus it satisfies Δ K ( y ) > 0 . Therefore we obtain
W s ( y ) W u ( z ) = , z K K + .
Using now Lemma 2.10, we derive that
X k 0  is diffeomorphic to  X k 0 .
More precisely,
X k 0 = { 1 S ˜ 2 δ ˜ ( x , λ ) , x X k 0 } ,

where λ is a fixed constant large enough and S ˜ 2 is the best constant of Sobolev.

Now, let
U : [ 0 , 1 ] × X k 0 Σ + , ( t , 1 S ˜ 2 δ ˜ ( x , λ ) ) t H ( y 0 ) δ ˜ ( y 0 , λ ) + ( 1 t ) K ( x ) δ ˜ ( x , λ ) t K ( y 0 ) δ ˜ ( y 0 , λ ) + ( 1 t ) K ( x ) δ ˜ ( x , λ ) ,
U is continuous and satisfies U ( 0 , 1 S ˜ 2 δ ˜ ( x , λ ) ) = 1 S ˜ 2 δ ˜ ( x , λ ) X k 0 , and U ( 1 , δ ˜ ( x , λ ) ) = 1 S ˜ 2 δ ˜ ( y 0 , λ ) , a fixed point of X k 0 . Furthermore, arguing as in [3], we derive that
J ( U ( t , 1 S ˜ 2 δ ˜ ( x , λ ) ) ) S ˜ 2 ( 1 K ( y 0 ) + 1 K ( x ) ) 1 2 ( 1 + o ( x , λ ) )
for each ( t , 1 S ˜ 2 δ ˜ ( x , λ ) ) [ 0 , 1 ] × X k 0 . Here o ( x , λ ) tends to zero when λ tends to +∞. Taking λ large enough, since x X k 0 we have K ( x ) K ( y k 0 ) , we get
J ( U ( t , 1 S ˜ 2 δ ˜ ( x , λ ) ) ) c 1 + ε , ( t , 1 S ˜ 2 δ ˜ ( x , λ ) ) [ 0 , 1 ] × X k 0 .
Therefore the contraction U is performed under the level c 1 + ε and X k 0 is then contractible in J c 1 + ε . Hence claim (17) follows. Let
θ ( X k 0 ) : = U ( [ 0 , 1 ] × X k 0 ) .
Now, we use the gradient flow of ( J ) to deform θ ( X k 0 ) . Since θ ( X k 0 ) J c 1 + ε and the only critical points at infinity of J in J c 1 + ε are ( y j ) , 0 j k 0 , by using a deformation lemma, see Proposition 7.24 and Theorem 8.2 of [39], we get
θ ( X k 0 ) X k 0 w < θ ( X k 0 ) W u ( w ) ,
(18)

where w is a solution of (3) dominated by θ ( X k 0 ) . Here denotes retracts by deformation.

Now observe that it follows from the above deformation retract that problem (3) has necessarily a solution. Otherwise it follows from (18) that
θ ( X k 0 ) X k 0 .
(19)
Let M be a cw complex in dimension k, it is well known (see [40]) that the Euler-Poincaré characteristic of M, χ ( M ) is given by the following:
χ ( M ) = r = 0 k ( 1 ) r n ( r ) ,
(20)
where n ( r ) is the number of cells of dimension r in M. We apply this to our situation. By construction X k 0 is a finite cw complex, where the cells of dimension an integer r in X k 0 are given by the unstable manifolds of the critical points at infinity ( y ) such that i ( y ) = r . According to (20), we derive that
χ ( X k 0 ) = y j K + , 0 j k 0 ( 1 ) i ( y j ) .
From another part, it is easy to see that χ ( θ ( X k 0 ) ) = 1 , since θ ( X k 0 ) is a contractible set. Thus, we derive from (19), taking the Euler characteristic of both sides that
1 = y j K + , 0 j k 0 ( 1 ) i ( y j ) .

Such an equality contradicts the assumption of Theorem 1.2.

Now, for generic K, it follows from the Sard-Smale theorem that all solutions of (3) are non-degenerate solutions; see [10]. We apply the Euler-Poincaré characteristic argument, we derive from (18) that
1 = y j K + , 0 j k 0 ( 1 ) i ( y j ) + w < θ ( X k 0 ) ( 1 ) i ( w ) ,
where i ( w ) denotes the Morse index of the solution w. It follows then that
| 1 y j K + , 0 j k 0 ( 1 ) i ( y j ) | S ,

where S denotes the set of solutions of (3).

Remark 3.1

Here, we want to consider some situation where the result of [28] does not give solution to problem (3), but using Theorem 1.2 we derive that problem (3) admits a solution. For this, let K : S 2 R be a C 2 positive Morse function satisfying the (nd) condition such that K + is reduced to 3 points y 0 , y 1 and y 2 with K ( y 0 ) K ( y 1 ) K ( y 2 ) . y 0  is the absolute maximum of K, so ind ( K , y 0 ) = 2 . Assume also that ind ( K , y 1 ) ind ( K , y 2 ) { 1 , 2 } and
1 K ( y 2 ) > 1 K ( y 0 ) + 1 K ( y 1 ) .
It is easy to see that
y K + ( 1 ) ind ( K , y ) = 1 .
However, we have
Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) | = | 1 y j K + 0 j 1 ( 1 ) ind ( K , y j ) | 0 ,

so by Theorem 1.2 we derive the existence of a solution of problem (3).

3.2 Proof of Theorem 1.3

Let ε R , K ε = 1 + ε K 0 and J ε be the associated variational problem. So, for u Σ + , we have
J ε ( u ) = u 2 ( S n K ε | u | 2 n n 1 d σ g 0 ) n 1 n .
Let us observe that in the case of the unit ball, in view of uniqueness result of Li and Zhu [36], a solution of (3) cannot be achieved as a minimum of J ε , unless for ε = 0 . In this case, the functional J 0 possesses a n-dimensional manifold Z of critical points, given by
Z = { δ ˜ ( a , λ ) , a S n , λ > 0 } .
Let S ˜ n be the best constant of Sobolev associated to the Sobolev trace embedding H 1 ( S n ) L 2 n n 1 ( S n ) , it is easy to see that for every δ ˜ ( a , λ ) Z , we have
J 0 ( δ ˜ ( a , λ ) ) = S ˜ n = inf u Σ + J 0 ( u ) .
Given a , b R , we set
J ε a = { u Σ + , J ε ( u ) a }
and
J ε b a = { u Σ + , b J ε ( u ) a } .

We have the following lemma.

Lemma 3.2

Let η > 0 , for | ε | sufficiently small, we have
J ε S ˜ n + η J 0 S ˜ n + 2 η J ε S ˜ n + 3 η .

Proof

Let u Σ + , we have
J ε ( u ) = 1 ( S n | u | 2 n n 1 d x + ε S n K 0 | u | 2 n n 1 d x ) n 1 n = J 0 ( u ) 1 ( 1 + ε ( S n | u | 2 n n 1 d x ) 1 S n K 0 | u | 2 n n 1 d x ) n 1 n .
Using the fact that K 0 is bounded on S n , we derive that
J ε ( u ) = J 0 ( u ) ( 1 + O ( ε ) ) ,

where O ( ε ) is independent of u, tends to zero when ε tends to zero. Hence the lemma follows. □

Now let ( y 1 , , y q ) a critical point at infinity of q-masses. It is easy to see that the level of J ε at ( y 1 , , y q ) goes to q S ˜ n , when ε tends to zero. Since
c ε ( y 1 , , y q ) = S ˜ n ( i = 1 q 1 K ε ( y i ) n 1 2 ) 2 n .
Now, let η = S ˜ n 4 , we can therefore assume that | ε | is so small that all the critical points at infinity of J ε of two masses or more are above the level S ˜ n + 3 η , and the critical points at infinity of one mass are below S ˜ n + η . Therefore,
the functional  J ε  has no critical points at infinity in  J ε ( S ˜ n + η ) S ˜ n + 3 η .
(21)
Arguing by contradiction and assume that (3) has no solution. It follows from (21) that
J ε S ˜ n + 3 η J ε S ˜ n + η ,
(22)
where denotes retracts by deformation. Using Lemma 3.2, we derive that
J 0 S ˜ n + 2 η J ε S ˜ n + η .
(23)
Now we claim that
J ε S ˜ n + η  is a contractible set .
(24)

Indeed, from (23), it is sufficient to prove that J 0 S ˜ n + 2 η contractible set.

Let u o J 0 S ˜ n + 2 η , we solve
{ u s = J 0 ( u ) , u ( 0 ) = u o .
Let u ( s , u 0 ) be the solution for s > 0 . Using the results of [38], concerning the Yamabe functional on S n , which is similar to our function J 0 , we know that the Palais-Smale condition is satisfied for the above differential equation, up to s = + . When s tends to +∞, u ( s , u 0 ) converges to a single mass in Z, thus
J 0 S ˜ n + 2 η Z .

We drive then that J 0 S ˜ n + 2 η is contractible since Z is a contractible set. Hence our claim follows.

Now let k 0 be the integer where the
max k ( A 2 ) | 1 y K + , i ( y ) k ( 1 ) i ( y ) |
is achieved. Here i ( y ) = n 1 ind ( K , y ) is the Morse index of ( y ) . Let
X k 0 = y K + , i ( y ) k 0 W u ( y ) .
(25)
X k 0 is a stratified set of dimension at most k 0 in J ε S ˜ n + η , without loss of generality we can assume that X k 0 is of dimension k 0 . Observe that X k 0 is contractible in J ε S ˜ n + η , since J ε S ˜ n + η is a contractible set. More precisely there exists
h : [ 0 , 1 ] × X k 0 J ε S ˜ n + η , ( t , u ) h ( t , u ) ,
(26)

where h ( 0 , u ) = u , h is continuous and h ( 1 , u ) = u ˜ a fixed point in J ε S ˜ n + η .

Let θ ( X k 0 ) = h ( [ 0 , 1 ] × X k 0 ) . θ ( X k 0 ) is a contractible stratified set of dimension k 0 + 1 . Now, we use the gradient flow of ( J ) to deform θ ( X k 0 ) . By transversally arguments we can assume that the deformation avoids all critical points at infinity having their Morse indices greater or equal to k 0 + 2 . Using the fact that θ ( X k 0 ) J ε S ˜ n + η , we get
θ ( X k 0 ) y K + , i ( y ) k 0 + 1 W u ( y ) .
Since k 0 satisfying ( A 2 ), there are no critical point at infinity ( y ) , y K + of index k 0 + 1 , we therefore have
θ ( X k 0 ) X k 0 .
(27)
Thus, we derive from (27), taking the Euler characteristic of both sides that
1 = y K + , i ( y ) k 0 ( 1 ) i ( y ) .

Such equality contradicts the assumption of Theorem 1.3. This completes the proof of Theorem 1.3.

4 A general existence result

In the last part of this paper, we give a generalization of Theorem 1.2. Namely instead of assuming that σ = 1 2 and n = 2 , we assume that σ ( 0 , 1 ) and 2 n < 2 + 2 σ . Following the same scheme as Theorem 1.2, we state here an existence result to the following problem:
P σ u = c n , σ K u n + 2 σ n 2 σ , u > 0 , on  S n ,
(28)
where
P σ = Γ ( B + 1 2 + σ ) Γ ( B + 1 2 σ ) , B = Δ g + ( n 1 2 ) 2 ,

c n , σ = P σ ( 1 ) , Γ is the gamma function and Δ g is the Laplace-Beltrami operator on ( S n , g ) . We are now ready to state the following existence result.

Theorem 4.1

Let K C 1 ( S n ) , be a positive function satisfying (nd) condition with 2 n < 2 + 2 σ and σ ( 0 , 1 ) . If
Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) | 0 ,
then (28) has at least one solution. Here, K + = { y S n , K ( y ) = 0 , Δ K ( y ) > 0 } . Moreover, for generic K we have
S Max k ( A 1 ) | 1 y j K + 0 j k ( 1 ) ind ( K , y j ) | ,

where S denotes the set of solutions of (28).

Proof

The proof goes along with the proof of Theorem 1.2, therefore we will only sketch the differences. Keeping the notation of the proof of Theorem 1.2 and regarding the expansion of the functional J, we observe that the strong interaction of the bubbles in dimensions n < 2 + 2 σ forces all blow up points to be single. Thus, the only critical points at infinity associated to problem (28) correspond to
( y ) : = 1 K ( y ) n 2 σ n δ ( y , ) , y K + .

The Morse index of such critical point at infinity ( y ) is: i ( y ) = n ind ( K , y ) . Now the remainder of the proof is identical to the proof of Theorem 1.2. □

Declarations

Acknowledgements

We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax, Tunisia

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