Elliptic equation with critical and negative exponents involving the GJMS operator on compact Riemannian manifolds

Accepted Manuscript Elliptic equation with critical and negative exponents involving the GJMS operator oncompact Riemannian manifolds Mohammed Benalili, Ali Zouaoui

PII: DOI: Reference:

S0393-0440(18)30461-3 https://doi.org/10.1016/j.geomphys.2019.02.003 GEOPHY 3391

To appear in:

Journal of Geometry and Physics

Received date : 18 July 2018 Revised date : 26 January 2019 Accepted date : 10 February 2019 Please cite this article as: M. Benalili and A. Zouaoui, Elliptic equation with critical and negative exponents involving the GJMS operator oncompact Riemannian manifolds, Journal of Geometry and Physics (2019), https://doi.org/10.1016/j.geomphys.2019.02.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ELLIPTIC EQUATION WITH CRITICAL AND NEGATIVE EXPONENTS INVOLVING THE GJMS OPERATOR ON COMPACT RIEMANNIAN MANIFOLDS. MOHAMMED BENALILI AND ALI ZOUAOUI Abstract. In this paper we consider an elliptic equation with critical and negative exponents involving the GJMS (Graham-Jenne-Mason-Sparling) operator of order k on n-dimensional compact Riemannian manifolds with 2k < n. Mutiplicity and nonexistence results are established.

Let (M; g) be an n-dimensional Riemannian manifold. The k-th GJMS operator (Graham-Jenne-Mason-Sparling, see ([5]) Pg is a di¤erential operator de…ned for any integer k if the dimension n is odd, and 2k n otherwise. In the following, we will consider the case 2k n. Pg is of the form k

Pg =

+ lot

where = divg (r) is the Laplacian-Beltrami operator and lot denotes the lower terms. One of the fundamental property of Pg is its behavior with respect to 4 conformal change of metrics: for 2 C 1 (M ) , > 0 and g = n 2k g , with n < 2k , a conformal metric to g , n+2k n 2k

(0.1)

Pgeu = Pg ( u) :

Pg is self-adjoint with respect to the L2 -scalar product. To Pg is associated a conformal invariant scalar function denoted Qg called the Q-curvature. For k = 1, the GJMS operator is ( up to a constant ) the conformal Laplacian and the corresponding Q-curvature function is simply the scalar curvature. For k = 2, the GJMS operator is the Paneitz operator introduced in ([13]). For 2k < n, the Q-curvature is Qg = n 22k Pg (1). A lot of works were devoted the Q-curvature equation in the last two decades (see [2], [3], [4], [5], [7], [9], [13], [17]). Many authors investigated the interactions of conformal methods with mathematical physics which led them to study the Einstein-scalar …elds Lichnerowiz equations (see [6], [8], [12], [14], [15], [16]). These methods have been extended to scalar …elds Einstein-Licherowicz type equation involving the Paneitz operator, (see [9]). In this work we analyze an Einstein-Lichnerowicz scalar …eld equation containing the k-th order GJMS operator on a Riemannian n-dimensional manifold with 2k < n; more precisely we consider the following equation

(0.2)

(

]

Pg (u) = B (x) u2 1 + u>0

A(x) ] u2 +1

+

C(x) up

1991 Mathematics Subject Classi…cation. 58J99-83C05. Key words and phrases. GJMS operator,Critical Sobolev Growth. 1

2

M OHAM M ED BENALILI AND ALI ZOUAOUI

where 2] = n 2n2k and p > 1. In all the sequel of this paper we assume that the operator Pg is coercive which allows us ( see Proposition 2, [17]) to endow Hk2 (M ) with the following appropriated equivalent norm vZ u u (0.3) kuk = t uPg (u)dvg . M

So we deduce from the coercivity of Pg and the continuity of the inclusion Hk2 (M ) ] L2 (M ), the existence of a constant S > 0 such that ]

kuk22]

(0.4)

Skuk2

]

where 2] = n 2n2k . Our work is organized as follows: in a …rst section we show the existence of a solution to equation (0.2) obtained by means of the mountain-pass theorem: more precisely we establish the following theorem. Theorem 1. Let (M; g) be a compact Riemannian manifold with dimension n > 2k, A > 0; B > 0, C > 0 are smooth functions on M . Suppose moreover that the operator Pg is coercive and have a positive Green function. If there exists a constant C (n; p; k) > 0 depending only on n; p , k such that k'k2 2]

(0.5)

]

Z

A(x) dvg '2 \

Z

C(x) dvg 'p 1

M

2+2] 2 2]

C (n; p; k) SmaxB(x) x2M

and k'kp 1 p 1

(0.6)

M

p+1 2 2]

C (n; p; k) SmaxB(x)

;

x2M

for some smooth function ' > 0: Then equation (0.2) admits a positive smooth solution. In the second section we prove, by means of the Ekeland’s lemma, the existence of n

a second solution to equation (0.2). In particular, by setting t0 =

1 S max B(x)

2k 4k

,

x2M

1

a=

(2(n k))

2\ 2

and choosing

such that

2a 3

1 2]

<

1 2]

< a , we obtain the following

theorem: Theorem 2. Let (M; g) be a compact Riemannian manifold of dimension n > 2k; with k 2. Suppose the operator Pg is coercive, has a Green positive function and there is a constant C(n; p; k) > 0 which depends only on n; p; k such that: (0.7)

k'k2 2]

]

Z

M

A(x) dvg '2 \

2+2] 2 2]

C (n; p; k) SmaxB(x) x2M

and (0.8)

k'kp 1 p 1

Z

M

C(x) dvg 'p 1

p+1 2 2]

C (n; p; k) SmaxB(x) x2M

for some smooth function ' > 0 on M . If moreover the following conditions occur

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

0 12 Z 1 @ p A(x)dvg A 2]

(0.9)

k aS 4n

3

2+2] 2 2]

SmaxB(x) x2M

M

1

(0.10)

p

1

Z

M

p

2

C(x)dvg

2]

(2

a)

p+1 2]

(aS)

p 1 2]

with p such that 1 < p < 2] + 1 close enough to 2] + 1 and 1 a1 < A (x) 1 a2 , c1 < C (x) 2k ]

where a1 = p

(p 1)k 2nV

2] akSt2+2 0 4nV (M )2

2] k 2 St2+2 0 2nV (M )2

, c1 =

SmaxB(x) x2M

1 2k

1 p

]

]

, a2 =

p+1 2 2]

k 4n

c2

1

(p 1)(2 a) 2] (aS) 4nV (M )2

p 1 S 2] tp+1 o (M )2 1

; V (M ) is the volume of M and 2] = (0.2) admits a second positive smooth solution.

2n n 2k .

p 1 2]

ktp+1 o

, c2 =

Then the equation

In the last section we give a nonexistence result of solution. Mainly we show the following result: Theorem 3. Given a compact Riemannian manifold (M; g) of dimension n > 2k; (k 2 N? ) and A; B; C are positive smooth functions on M and 2 < p < 2] + 1. Assume that there exist two constants R and S1 such that 12 2] ] 0R p p 1+2 B:Cdvg Z C BM R Bdvg > (S1 R)2 (0.11) C(n; p; k) @ A Bdvg M

M

where

2]

2] + p 1 p 1 2] +p 1 . C(n; p; k) = p 1 2] Then the equation (0.2) has no smooth positive solution u with energy kukHk2 (M ) R. 1. Existence of a first solution In this section we show Theorem 1. Before starting the proof, we …rst give an example of manifolds where the GJMS operator Pg has a positive Green function. Proposition 1. Suppose that the metric g is Einstein with positive scalar curvature of dimension n > 2k, then the GJMS operator Pg admits a positive Green function. Proof. On n-dimensional Einstein manifold, the GJMS operator of order k is given by (see [5]) k Y Pg = ( cl Sc) l=1

(n+2l 2)(n 2l) and 4n(n 1)

where cl = Sc is the scalar curvature. If the scalar curvature is positive it is well known that the operator cl Sc has a positive Green function.

4

M OHAM M ED BENALILI AND ALI ZOUAOUI

Denote by Ll = cl Sc, l = 1; :::; k; from the de…nition of the Green function of Ll we know that for all u 2 C 1 (M ), Z (Ll u) (x) = Gl+1 ((x; y) (Ll+1 Ll u) (y)dvg (y) : M

So

u(x) =

Z

1 V ol(M )

Z

u(x)dvg (x) Z Z Z 1 = Gl (x; z)Gl+1 ((z; y)dvg (z) (Ll+1 Ll u) (y)dvg (y)+ u(x)dvg (x) V ol(M ) M M M and letting Z Gl (x; z) (Ll u) (z) dvg (z) +

M

Gl Gl+1 (x; y) =

M

Gl (x; z)Gl+1 ((z; y)dvg (z):

M

And by induction, we get Z u(x) = G1 ::: Gk (x; y)Pg u(y)dvg (y) + M

Z

1 V ol(M )

u(x)dvg (x):

M

Thus Pg admits a positive Green function.

To show the existence of solutions to equation (0.2), we follow the strategy in the proof of the paper by Hebey-Pacard-Pollack [6]. We consider the following "-approximating equations ( " > 0 ) Pg (u) = B (x) u+

(1.1)

2] 1

A (x) u+

+

2[ +1

2

" + (u+ )

C(x)u+

+

p+1 2

2

" + (u+ )

]

where 2[ = 22 , p > 1, which gives us a sequence (u" )" of solutions to (1.1). The solution of equation (0.2) is then obtained as the limiting of (u" )" when " ! 0. To get rid of negative exponents, we consider the energy functional associated to (1.1) de…ned by, for any " > 0 I" (u) = I (1) (u) + I"(2) (u) where I (1) : Hk2 (M ) ! R is given by Z 1 (1) uPg (u)dvg I (u) = 2 M

1 2]

(2)

and I"

: Hk2 (M ) ! R is Z A (x) 1 I"(2) (u) = ] 2 M 2 " + (u+ )

dvg + 2[

Z

B (x) u+

1

Z

M

C(x) p

"+

It is easy to check the following inequality (1.2)

(kuk)

with (t) = and (t) =

1 2 t 2

I (1) (u)

dvg

M

1 p

2]

(kuk)

] 1 S max B t2 ] M 2

] 1 2 1 t + ] S max B(x) t2 . M 2 2

2 (u+ )

1 2

dvg .

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

The function (1.3)

(t) is increasing on [0; t0 ] and decreasing on ]t0 ; +1[, where 0 1 n 4k2k 1 A t0 = @ SmaxB(x) x2M

and (1.4) Lemma 1. Let

(t0 ) =

0

1 n 2k2k 1 k @ A = t20 . SmaxB(x) n

1 2]

1 2

> 0 such that

1 2\

a 2

x2M

1

<

where

< a 2\ 1

a= (2 (n

2\

k)) 2

and put t1 = t 0 . Then we have the following inequality (1.5)

22

(t1 )

]

2]

+2 2

(t0 ) <

1 2k

(t0 ) .

Proof. In fact (t1 )

= =

1 2 t + SmaxB(x) x2M 2 1

]

t21 2]

1 2 t + S maxB(x) x2M 2 0

2

2] 2 ] 2 t0 2]

Since (1.6)

t20

]

SmaxB(x) x2M

= t20

we get ]

(t1 )

= =

2 2 1 2 2 t0 + t2 2 2] 0 " # 2] 2 2 2 1 + t0 2 2] 2 2 t0

1

!

1 n 2k + 2 2n k 2 2 t0 : n

And since 2] + 2 n = ] 2 2 k

1

and

(t0 ) =

k 2 t n 0

!

.

5

6

M OHAM M ED BENALILI AND ALI ZOUAOUI

we infer that

] 22 2]

(t1 )

+2 2

(t0 ) <

1 2k

(t0 ) .

Now, we check the Mountain-Pass lemma conditions for the functional I" . Lemma 2. The functional I" satis…es the following condition: there exists an open ball B(u1 ; ) of radius > 0 and of center some u1 in Hk2 (M ) and there are u2 2 = B(u1 ; ) and a real number co such that max (I" (u1 ); I" (u2 )) < co

I" (u)

for all u 2 @B(u1 ; ). Proof. Following the strategy of the proof in the paper by Hebey-Pacard-Pollack [6], we let ' 2 C 1 (M ), ' > 0 on M and without loss of generality we may assume k'k = 1. Put 2] +p

(1.7)

C (n; p; k) = (2k

1)

C1 (n; k) =

4n

The inequality (0.5) becomes Z 1 A (x) (1.8) dvg 2] M (t1 ')2]

2k 1 4k

(2k

2]

1) 4n

.

(t0 ) .

Indeed, we have (1.9)

(to ) =

and by (1.7), we get Z A(x) 1 dvg 2\ M (t1 ')2] =

k 2 t . n o

C1 (n; k) t2o\

2\

2+2] 2 2]

S max B (x) M

2k 1 (to ). 4k

Analogously, by putting (1.10)

C2 (n; p; k) =

we obtain 1

(1.11)

p

1

Z

(2k

1) 4n

C(x)

M

p 1

2k 1 (to ). 4k

p 1 dvg

(t1 ')

By relations (1.5), (1.6), (1.8) and (1.11), we infer that Z Z 1 1 A (x) I" (t1 ') (kt1 'k) + ] dv + g 2[ 2 M p 1 M 2 " + (t1 ') (1.12) (t1 ) +

1 2]

Z

M

A (x) 2

" + (t1 ')

dvg + 2[

1 p

1

Z

M

C(x) p

2

" + (t1 ')

C(x) p

2

" + (t1 ')

1 2

dvg <

1

dvg

2

(t0 ) :

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

Again from (1.5), we deduce that (1.13) Z 1 A (x) I" (t0 ') (t0 ) + ] 2 M 2 " + (t0 ')

1

dvg + 2[

p

1

Z

7

C(x) p

M

1

dvg

2

2

" + (to ')

and since A and C are assumed positive, we obtain (1.14)

I" (t0 ')

(t0 ) .

Finally from (1.12) and (1.14), we get I" (t1 ') Noting that lim I" (t')

t!+1

=

2

61 lim 4 kt'k2 2

t!+1

1

lim

t!+1 p

1

(t0 ) < I" (t0 ') :

1 2]

Z

and since

R

lim t2

]

t!+1 ]

M

A(x)

B 2] @B(x) (t') dvg

M

C(x)

p

1

2

" + (t')

2[

1

dvg

" + (t') Z ] 1 1 B (x) '2 dv (g) 2] M 2t2] 2

B(x)'2 dvg > 0, we obtain lim I" (t') =

t!+1

1:

Consequently there is t2 such that t2 > t0

and

I" (t2 ') < 0:

Now, to have the conditions of Lemma 2 ful…lled, we just put u1 = t1 ', u2 = t2 ', u = t0 ' and we take

= t0

t1 > 0 and c0 =

(t0 ).

Lemma 1 allows us to apply the Mountain-Pass Lemma to the functional I" . Let C" = inf maxI" (u) 2 u2

where denotes the set of paths in Hk2 (M ) joining the functions u1 = t1 ' and u2 = t2 '. So C" is a critical value of I" and moreover C" >

(t0 )

and by putting (t) = t', for t 2 [t1 ; t2 ], we see that C" is uniformly bounded when " goes to 0, so we get (1.15)

0<

(t0 ) < C"

C

for " su¢ ciently small and C > 0 not depending on ": Consequently there exists a sequence (um )m of functions in Hk2 (M ) such that (1.16)

I" (um )

!

m!+1

C"

and

DI" (um )

!

m!+1

0

3

C 7 A dvg 5

2

2

M

=

0

Z

8

M OHAM M ED BENALILI AND ALI ZOUAOUI

By Lemma 2 the sequence (um )m of Hk2 (M ) is a Palais-Smale sequence (P-S) for the functional I" . Theorem 4. The Palais-Smale sequence (um )m is bounded in Hk2 (M ) and converges weakly to nontrivial smooth solution u" of equation (1.1). Proof. By (1.16) we get for any ' 2 Hk2 (M )

DI" (um ) ' = o (k'k)

i.e. for any ' 2 (1.17) +

Hk2

(M ) one has Z Z 'Pg um dvg = M

Z

2[ +1

2

" + u+ m

dvg +

Z

M

Z

1 2 +

1 2

2]

dvg =

Z

M

Z

p 2 +1

dvg + o (kum k)

2

A(x) (u+ m) " + u+ m

[ 2 2 +1

dvg

2

2

" + (um )

p 2 +1

um Pg um dvg +

M 2

A(x) (u+ m) "+

Z

2

M

C(x) (u+ m)

M

(1.18)

" + u+ m

M

M

or

'dvg

C(x)u+ m'

in particular, for ' = um we have Z Z um Pg um dvg B(x) u+ m +

2] 1

M

A (x) u+ m'

M

B (x) u+ m

2 u+ m

dvg + 2[ +1

1 2

1 2 Z

dvg + o (kum k) .

Z

B(x) u+ m

2

dvg

M

M

2

C(x) (u+ m) " + u+ m

2

p 2 +1

dvg = o (kum k) .

On the other hand it comes from (1.16) that Z Z 1 1 2] um Pg um dvg B(x) u+ dvg m ] 2 M 2 M (1.19) Z Z 1 A(x) C(x) 1 + ] dv + g p 1 dvg = C" + o (kum k) . [ 2 2 M p 1 M 2 2 + 2 " + u+ " + u m m So by adding (1.18) and (1.19) we get (1.20) Z Z Z 2 k 1 A(x) (u+ 1 A(x) 2] m) B(x) u+ dv + dv + dvg g g m [ +1 ] 2 2[ n M 2 M 2 M 2 + 2 " + u+ " + u m m Z Z 2 1 C(x) (u+ 1 C(x) m) + dvg + p p 1 dvg = C" + o (kum k) . +1 2 M p 1 M 2 + 2 2 + 2 " + um " + um For su¢ ciently large m we deduce that Z k 2] B(x) u+ dvg m n M

2C" + o (kum k)

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

9

or

Z n 1 2] dvg B(x) u+ C" + o (kum k) m 2] M 2] and plugging this last inequality in (1.19) we obtain Z 1 n um Pg um dv (g) C" + ] C" + o (kum k) 2 M 2 n (n 2k) C" + o (kum k) 2nC" + o (kum k) . 2n Hence for m large enough, we have Z um Pg um dv (g) 4nC" + o (1) 4nC" + 1 nC" +

M

i.e.

(1.21)

kum k

2

4nC" + 1

Thus the sequence (um )m is bounded in Hk2 (M ); so we can extract a subsequence, still denoted (um )m , which veri…es: 1. um ! u" weakly in Hk2 (M ) : 2. um ! u" strongly in Lp (M ), 8p < n 2n2k 3. um ! u" a.e. in M: 2] 1

2]

]

4. (um ) ! u2" 1 weakly in L 2] Furthermore, noting that 8k 2 N:

u+ m

2

1

q

+"

<"

(M ). q

and "

q

2 Lp (M ) 8p

1

where " > 0 and q > 0; we get by Lebesgue’s dominated convergence theorem that q

2

q

2

(u+ ! (u+ strongly in Lp (M ) for any p m) + " " ) +" above property (2) we infer that u+ m 2 u+ m

q

+"

!

u+ " " + u+ "

1 and with the

2 q

strongly in L2 (M ). So if we let m go to +1 in (1.17) we obtain that u" is weak a solution of the equation (1.22)

Pg u" = B (x) u+ "

2] 1

+

A (x) u+ " 2[ +1

+

C(x)u+ "

" + u+ "

2

dv (g)

C" + o(kum k).

" + u+ "

\

where 2b = 22 and p > 1. Now, by (1.20) we have Z 1 2] M

A (x) "+

[ 2 2 u+ m

Letting m ! +1 and taking in mind (1.15), we infer that Z 1 A (x) dv (g) C (1.23) [ ] 2 M 2 2 " + u+ "

2

p+1 2

10

M OHAM M ED BENALILI AND ALI ZOUAOUI

where C is the upper bound of C" . Now, if for a sequence of su¢ ciently small positive "j the u"j were to be 0, then we would infer by (1.23) that Z 1 (1.24) A (x) dv (g) C [ 2] "2j M which contradicts the assumption A > 0: Therefore, for " su¢ ciently small, u" is a non trivial solution of equation (1.1). Next, we will show the regularity of u" . First we write the equation (1.22) in the form Pg u" = b(x; u" )u" where b(x; u" ) = B (x) u+ "

(1.25)

2] 2

A (x)

+ "+

A

Since

2 u+ "

2[ +1

+

C 2 u+ "

[ 2 2 +1 u+ "

C(x)

+

" + u+ "

2 L1 (M ) and u" 2 Hk2 (M )

p+1 2

2

p+1 2

.

]

L2 (M ), we

"+( ) "+( ) n deduce that b 2 L 2k (M ). By the work of S. Mazumdar ( see the proof of theorem 5 page 28 in [10] ) we obtain that u" 2 Lp (M ) for any 0 < p < +1. According to ([1]), we get u" 2 Hkp (M ) for all 1 < p < +1. By the same arguments as in the proof of proposition 8.3 in ([1]) we conclude that u" 2 C 2k; (M ) with 2 (0; 1). Now, R since the operator Pg has a positive Green function G that is to say u" (x) = M G(x; y)b(y; u" (y))u" (y) dvg (y) with b(x; u" )u" 0 is given by (1.25), we deduce that u" 0. Thus u" is solution to the "-approximating equation

Pg (u" ) = B (x) u2"

(1.26)

]

1

A (x) u"

+

(" +

2[ +1 u2" )

+

C(x)u"

(" + u2" )

p+1 2

.

Now, we are in position to prove Theorem 1. Proof. From what precedes u" is a C 2k (M )-nontrivial solution to equation (1.1), moreover u" is a weak limit of the sequence (um )m given by (1.21) which allows us by the lower semicontinuity of the norm to write ku" k

lim inf kum k .

m!1

From the inequalities (1.15), (1.21) we deduce that the sequence (u" )" of the "approximating solutions is bounded in Hk2 (M ) for su¢ ciently small " > 0 i.e. 2

(1.27)

ku" k

4nC + 1.

Let ("m )m be sequence of positive numbers tending to 0 as m goes to 1 such that the relations (1.27) and (1.23) are true with " = "m . Thus by (1.27) we can extract a subsequence still labelled (u"m )m ful…lling: (i) u"m ! u weakly in Hk2 (M ) (ii) u"m ! u strongly in Lp (M ) for p < 2] (iii) u"m ! u a.e. in M . ]

(vi) u2"m

1

! u2

]

1

2]

weakly in L 2]

1

:

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

11

Thus by (iii) and applying Fatou’s lemma to relation (1.23) with " = "m , we obtain Z 1 A (x) (1.28) dv (g) C. 2] M u2] Furthermore the sequence (u"m )m is uniformly bounded from below i.e. there exists > 0 such that u"m . Since if it is not the case, there exist a sequence (x"m )m in M such that u"m (x"m ) ! 0 as m ! 1. By the compactness of M , there is a subsequence of (x"m )m still labelled (x"m )m such that x"m ! xo as m ! 1. Now by assumption the operator Pg admits a positive Green function, then we can write 1 0 Z C B A (y) u"m (y) C(y)u"m (y) 2] 1 C + + u"m (x"m ) = G (x"m ; y) B p+1 A dvg : @B (y) u"m (y) 2[ +1 2 2 M 2 " + u"m (y) " + u"m (y) Since the last two terms of the inequality are positive we drop them Z 2] 1 u"m (x"m ) G (x"m ; y) B (y) u"m (y) dvg : M

Passing to the liminf, provides Z u(xo )

2] 1

G (xo ; y) B (y) u (y)

dvg

M

which implies that u = 0 and leads to a contradiction with (1.28).Thus, there exists > 0, such that u"m . We can once again use Lebesgue’s dominated convergence theorem to get 1 2

q

"m + (u"m )

!

1 strongly in Lp (M ) , 8p u2q

1, 8q > 0.

Since for m large enough u"m > 0 there is ~" > 0 such that 1 "m + u2"m

q

1 with q > 0. ~"q

Thus by the Lebesgue’s dominated convergence theorem, we get that 1 "m + u2"m

q

!

1 strongly in Lp (M ) , 8p u2q

1; 8q > 0:

Finally, with (ii), it follows that u"m 2

"m + (u"m )

2[ +1

!

1 strongly in L2 (M ) . u2] +1

with u > 0. Letting "m ! 0 in (1.22) as m ! +1;we get that u is a weak positive solution of equation (0.2) . By the same arguments as those of the regularity of the solution u" of the equation (1.22) we infer that u 2 C 2k; (M ) with 2 (0; 1). Since u > 0, the right-hand-side of (0.2) has the same regularity as u and by successive iterations we obtain that u 2 C 1 (M ).

12

M OHAM M ED BENALILI AND ALI ZOUAOUI

2. Existence of a second solution According to the previous section our functional satis…es the following inequalities (2.1)

I" (t1 ') < (t0 ) < I" (t0 ')

C"

where t0 , t1 are real numbers such that 0 < t1 < t0 , ' 2 C 1 (M ) with ' > 0 and k'k = 1. On the other hand and as I" (t') tends to 1 as t goes to +1, there is t2 >> t0 such that I" (t2 ') < 0, which means by the previous section that I" admits C" as a local maximum. Now if we let t and " tend both to 0+ , the functional I" goes to +1. Indeed, h i (2) lim+ lim+ I" (t:') = lim+ I (1) (t:') + I0 (t:') t!0 "!0 t!0 Z ] 2 B(x)(t:')2 dv(g) = lim+ (t:')Pg (t:') ] 2 t!0 M 3 2 Z Z A(x) C(x) 1 1 5 dv(g) + + lim+ 4 ] p 1 dv(g): 2] 2 p 1 t!0 (t:') (t:') M

M

= +1:

and it follows that for " small enough, there is near 0 a real number 0 < t0 << t1 such that I" (t0 ') > (t0 ) > I" (t1 '). What let us see, from this fact, that our functional has a local lower bound. We will give the necessary conditions for this lower bound to exist, then we show by Ekeland’s lemma that this lower bound is reached. Before beginning the proof of theorem 2, we will establish some preliminary lemmas. Lemma 3. Let

> 0 such that 2 2]

a 2

(2.2)

<

where

2

2

< a 2]

1

a=

2]

(2 (n

k)) 2

and put a 2

t3 =

1 2]

t0

then we have the following inequality a (t0 ): 2 Proof. Since (t1 being de…ned as in Lemma 1 )

(2.3)

(t3 ) >

t3 =

1 2]

a 2

t0 < t0 = t1

and (2.4)

a 2

2 2]

>

a : 2

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

13

by (2.3), we get ]

1 2 t2 t3 S: max B(x) 3] M 2 2 2 1 1 a 2] 1 2a t = t0 2 2 2] 0 2 n 1 a 22] a n 2k k 2 = t : k 2 2 2 2n n 0

(t3 ) =

Knowing by (1.4) that k 2 t = n 0

(t0 )

we deduce a 22] a a n + 2k 2 2 2 a > (t0 ) 2 where we used the inequality (2.4) in the last line.

(t0 )

(t3 ) =

Lemma 4. Given a Riemannian compact manifold (M; g) of dimension n > 2k; k 2 N? , 1 < p < 2] + 1 and ' 2 C 1 (M ); ' > 0 in M with k'k = 1: Then there is a constant ? > 0 such that: 8" 2 ]0; ? [ the following inequalities take place 0 12 Z Z p A(x) 2 a@ (2.5) dvg A(x)dvg A 2] 2] aSt 2 o (" + (t :') ) 2 3

M

M

and

(2.6)

Z

M

C(x) (" + (t3 :')2 )

p

1

2

dvg

a aS

2

0 12 Z 1 @ p C(x)dvg A p 1

p 1 2]

to

where t3 ; a are chosen as in Lemma 3 .

M

Proof. Let ?

(2.7)

=

"

2

#

2 2]

2

1

a

2

V (M ) 2]

where V (M ) denotes the volume of M and =

aS 2

By Hölder’s inequality, we get 0 12 Z p @ (2.8) A(x)dvg A M

2 2]

t20 :

2]

Ik" + (t3 :')2 k 22] 2

14

M OHAM M ED BENALILI AND ALI ZOUAOUI

where

Z

I=

A(x) 2]

dvg :

(" + (t3 :')2 ) 2

M

Independently, the Minkowski’s inequality can be written k" + (t3 :')2 k 2]

k"k 2] + t23 k'2 k 2]

2

2

2

consequently 2]

k" + (t3 ')2 k 22]

(2.9)

2] 2

k"k 2] + t23 k'2 k 2] 2

2

:

2

Notice that

2

k"k 2] = " [V (M )] 2] 2

and k'2 k 2] = k'k22] 2

by inequality (0.4) we have 2

k'k22]

Since k'k = 1 we get

S 2] k'k2 : 2

k'k22]

(2.10)

S 2]

and therefore (2.9) becomes 2

k" + (t3 ') k

2] 2 2] 2

"V (M

2 2]

+

2] 2

2 t23 S 2]

:

Taking account of a 2

t3 =

1 2]

t0

(2.8) is written as 0 12 Z p @ A(x)dvg A

I

"V (M )

2 2]

2 2]

aS 2

+

M

?

And letting " such that 0 < " < 0 12 Z p @ A(x)dvg A

"

M

2

2

1

a

#

+

2] 2

a

] aSt20

2

2 2]

2

2

I

a

.

Finally we deduce I

0

12 Z p @ A(x)dvg A M

.

, we get

I

I

! 22]

t20 :

2 a ] aSt20

:

aS 2

2 2]

t20

! 22]

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

15

In the same fashion, we get Z

M

C(x) (" + (t3 :')2 )

p

1

2

dvg

aS

2

12 0 Z 1 @ p C(x)dvg A : p 1

p 1 2]

a

to

M

Now we are able to prove the existence of a second solution to equation (0.2), that is to say the proof of Theorem 2. Proof. The proof will be done in four steps 1 st -step. The functional I" has a local lower bound. This consists to …nd a strictly positive real number has the following double inequality (2.11)

I" (t3 ')

?

such that 8" 2 ]0;

?

[ one

8' 2 C 1 (M ); k'k = 1; with t3 < t1 :

(t0 ) > I" (t1 ') ;

The right inequality is ful…lled in (2.1). According to Lemma 3, inequality (2.3) and inequality (1.2), one has I" (t3 ') = I (1) (t3 ') + I (2) (t3 ') Z a 1 A (x) > (t0 ) + ] dvg 2 2 M (" + (t3 ')2 )2[ Z C(x) 1 + dvg : p 1 M (" + (t ')2 ) p 2 1

(2.12)

(2.13)

3

Now, by assumption 0 12 Z 1 @ p A(x)dvg A 2]

k S a SmaxB(x) x2M 4n

2+2] 2 2]

M

and

1 p

1

Z

2

p

C(x)dvg

M

So since

2]

(2

a)

p+1 2]

2+2] 2 2]

SmaxB(x) x2M

we get

0 12 Z 1 @ p A(x)dvg A 2]

2

a a

M

and

1 p

1

Z

M

p C(x)dvg

(aS)

= t2+2 0

1 ] S:t20

2

2]

(2

a)

k 4n

p 1 2]

p+1 2]

p+1 2 2]

SmaxB(x) x2M

]

k 2 t (2 4n 0

(aS)

p 1 2]

a)

k p+1 t 4n o

:

16

M OHAM M ED BENALILI AND ALI ZOUAOUI

Thus if we let

?

(2.14)

as in (2.7) then it follows by Lemma 4 that, 8" 2 ]0; Z 1 1 a A(x) dvg (t0 ) 2] 2] 2 4 (" + (t ')2 ) 2

(2.15)

1 p

1

Z

M

[

3

M

and also

?

C(x) (" + (t3

:')2 )

p

1

1 2

dvg

2

a 4

(t0 ):

Finally, by combination of (2.12), (2.14) and (2.15), we get I" (t3 ')

(t0 ):

Hence our result. 2 nd step. The in…mum of the functional I" is reached. Denote by K = u 2 Hk2 (M ) : ku t1 'k t1 t3 the closed ball centred at the origin t1 ' of radius t1 t3 in Hk2 (M ). In this section we will show that c" = inf K I" ( c" < (t0 ) ) is achieved. By relations (2.11), we get inf I"

(t0 ) and inf I" <

(t0 ) .

K

@K

Applying the Ekeland’s lemma to the functional I" : Hk2 (M ) ! R, we obtain for any such that 0 < < inf I" inf I" K

@K

a sequence (u ) in K such that I" (u ) < inf I" + K

and (2.16)

I" (u ) < I" (u) + ku

u k for u 6= u :

Hence we deduce that I" (u ) < inf I" +

inf I" +

< inf I"

K

K

@K

and consequently Meanwhile, by (2.16) we obtain

u 2 K:

kDI" (u )kH 2 (M )0

:

k

So we infer the existence of a sequence (um )m K such that I" (um ) ! c" = inf K I" and DI" (um ) ! 0 strongly in the dual space of Hk2 (M ). That is to say (um )m is a Palais-Smale sequence which converges in Hk2 (M ) to a weak solution v" of equation (1.1) such that c" = I" (v" ) (t0 ): By the same arguments as in the proof of Theorem 1 and Theorem4, we get that equation (1.1) has a smooth positive solution v" 2 K. Since the "-approximating solutions are obtained as a weak limit of sequences of functions from K, it follows by the weak lower semi-continuity of the norm that these "-approximating solutions are in K: As in turn the second solution v of the equation (0.2) is obtained as a limit of a sequence of "-approximating solutions, so v 2 K and and again by the same reasons as in the proof Theorem 1

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

17

and Theorem4, we get that v is positive and smooth: 3 rd -step. The two solutions are distinct. To show that the two solutions u and v are di¤erent, we will verify that their respective energies are di¤erent. We had already shown in the formula (1.13) that I" (t0 ')

(t0 ) +

1 2]

Z

A (x) 2

M

1

dvg + 2]

p

" + (t0 ')

1

Z

C(x) p

M

2

" + (to ')

Now, as in the proof of Lemma 4, we have Z

A(x)dvg

A (x) 2]

2

M

2

R p

M

dvg

" + (t0 ')

2] 2

k"k 2] + t2o k'2 k 2] 2

2

2

R p

A(x)dvg

M

" [V (M )] Letting ?

=

"

2

#

2 2]

2

2 2]

1

a

+

2] 2

2

t2o S 2]

:

2

V (M ) 2]

where V (M ) denotes the volume of M and 2

= S 2] t20 we get for 0 < " < Z

A (x)

M

2

2]

dvg

" + (t0 ')

0 12 Z p 2 a @ A(x)dvg A ] : 2St20 M

Taking into account of the condition (0.9), we infer that Z A (x) 1 a a 1 dvg 1 (t0 ): 2] 2] M 4 2 2] 2 " + (t0 ') Similarly we have Z

M

C (x) 2

" + (t0 ')

p 1 dvg

R p C(x)dvg

2

M

p

2

1 2

k"k 2] + t2o k'2 k 2] 2

R p C(x)dvg

2

M

" [V (M )]

2 2]

+ t2o S

p 2 2]

1 2

:

1 2

dvg :

18

M OHAM M ED BENALILI AND ALI ZOUAOUI

Letting in this case

2 2]

2

=

1

2 a

V (M )

2

= S 2] t20 .

, where

2 2]

, we obtain

For 0 < " < Z

C (x) 2

M

0 12 Z p @ C(x)dvg A

p 1 dvg

" + (t0 ')

2

a

p 1 2]

2S

t1o

p

:

M

and taking into mind the assumption (0.10), we deduce that Z p 1 1 C (x) 1 a 2] dv (2 a) g p 1 p 1 M 4 (p 1) 2 2 " + (t0 ')

(t0 ):

Consequently I" (t0 ') Hence, putting

=

I(u)

=

1 (2 4

1+ 1 4

(2

a)

a)

1 p 1

a 2

1 p p 1 2]

1

a 2

+

a 2:2]

p 1 2]

+

a 2:2]

(t0 ):

>0

lim I" (u" ) = lim C" = lim inf max I" ( (t)) "!0+

"!0+

(1 + ) (t0 ) > (t0 )

"!0+ 2 t2[0;1]

lim I" (v" ) = I(v):

"!0+

where denotes the set of smooth paths joining u1 = t1 ' and u2 = t2 ': Observe that the path (t) = t' with t 2 [t1 ; t2 ] belongs to and t0 2 [t1 ; t2 ] : So u and v are distinct solutions of our equation. 4 th -step. The conditions of Theorem 2 intersect. Indeed, let us rewrite the condition (0.9) of Theorem 2 12 0 Z 1 @ p k 2+2] (2.17) A(x)dvg A t aS: 2] 2n 0 M

By Hölder inequality, we get Z p A(x)dvg

M

(2.18)

=

Z

M

s

A(x) 2] ' 2 dvg '2]

Z

1 Z 2 ] A(x) dv '2 dvg g ] 2 M ' M 1 Z 2 2] A(x) 2 k'k2] dv : g ] 2 M '

From inequality (0.4) and the fact that k'k = 1, it comes that 0 12 Z Z 1 @ p S A(x) A A(x)dv dvg g ] ] 2 2 M '2 ] M

1 2

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

with the condition (0.7) of Theorem 2, we obtain 0 12 Z p 1 @ A(x)dvg A S:C(n; p; k) SmaxB(x) x2M 2]

19

2+2] 2 2]

.

M

where C(n; p; k) is given by (1.7). Thus 12 0 Z 1 @ p (2.19) A(x)dvg A 2]

S

2k 1 4n

2] 2+2] t0 :

M

By combining the two conditions (2.17) and (2.19), we …nd the following double inequality 0 12 Z 2k 1 2] 2+2] 1 @ p k 2+2] A(x)dvg A S t0 : t0 aS < ] 4n 2 4n M

Which in turn is equivalent to

2 2]

4nt0 2] S

ka

0 12 Z p @ A(x)dvg A

(2k

1)

2]

.

M

or 2nt0 2

a 2 So, if we choose

2]

2]

] 2] :kS 2

such that

0 12 Z p @ A(x)dvg A

1

1 2k

:

M

2a < 3

2]


we obtain 1 a 3 1 < 2] < 1 2 4 2k 2 provided that k 2. In other words smooth functions A(x) that ful…ll the assumptions (0.7) and (0.9) of Theorem 2 are those that satisfy the following double inequality 1 a1 A(x) 1 a2 2k (2.20)

]

2] akSt2+2

2] k

2]

St2+2

]

where a1 = 4nV (M0 )2 , a2 = 2nV (M0)2 M. In the same manner as above we get 1 p

1

0

12 Z p @ C(x)dvg A

with k

1 p

1

p

1 1

p

1

S

p 1 2]

S

p 1 2]

Z

C (x) dvg p 1 M ' Z C (x) p 1 k'k dvg p 1 M ' Z C (x) dvg : p 1 M '

p 1 k'kp 1

M

1

2 and V (M ) is the volume of

20

M OHAM M ED BENALILI AND ALI ZOUAOUI

Thus by (0.8), (0.10) and (1.10), we obtain 0 12 Z p 2] p+1 p 1 k 1 @ (2 a) 2] (aS) 2] C(x)dvg A tp+1 4n o p 1

S

(2k 1) 4n

p 1 2]

p 1 p+1 to

M

or

2]

(2

a)

2

2] p 1

! p ]1 2

a

p+1 2]

2]

2n (p

1) kS

p 1 2]

p 1 p+1 to

]

0 12 Z p @ C(x)dvg A

1

M

Taking in mind (2.20), we get for p < 1 + 2 su¢ ciently close to 1 + 2] ! p ]1 2 2] p+1 a 1 1 3 < (2 a) 2] 1 < ] 2] 2 2 4 2k 2p 1 ]

2 < a and k 2. So functions C(x) candidates to conditions provided that 2a 3 < (0.8) and (0.10) of Theoerem 2 are those that ful…ll

c1 (p 1)(2 a)

p 1 2]

(aS)

where c1 = 4nV (M )2 V (M ) is the volume of M:

C(x)

p 1 2]

ktp+1 o

(1

1 )c2 2k p

and c2 =

1

(p 1)k p 1 S 2] tp+1 o 2nV (M )2

with k

2 and

3. Nonexistence of solution In this section we will be placed in a closed ball B(0; R) of Hk2 (M ) centered at the origin 0 and of radius R > 0, we prove that under some condition (inequality 0.11 of Theorem 3), equation (0.2) has no solution i.e. we will give the proof of Theorem 3. Proof. ( Proof of Theorem 3) Suppose that there exists a smooth positive solution u 2 Hk2 (M ) such that kukHk2 (M ) R. By multiplying both sides of equation (1.1) by u and integrating over M , we get Z Z ] A (x) C (x) uPg (u)dvg = B (x) u2 + 2] + p 1 dvg : u u M M rR uPg (u)dvg is a norm equivalent to kukHk2 (M ) , there exists a And since kuk = M

constant S1 > 0 such that Then it follows that Z (3.1)

M

kuk ]

B (x) u2 +

S1 kukHk2 (M ) .

A (x) C (x) + p 1 u u 2]

dvg

Moreover Hölder’s inequality allows us to write 1 21 0 0 Z p Z Z C(x) A @ B(x)up (3.2) B (x) C (x)dvg @ dv g up 1 M M

M

2

(S1 R) .

1

1 21

dvg A .

1 2k

:

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

By applying again the Hölder’s inequality, one …nds Z Z p 1 1 p 1 B (x) up 1 dvg = B (x) 2] B (x) 2] u M

M

Z

2]

p+1 2]

B (x) dvg

Z

(3.3)

2]

p+1 2]

M

Z

M

it comes that

Z

1 2]

2] p 1

p 1

u p 1 2]

2]

B (x) u dvg

2]

p+1 2]

B (x) dvg

:

Z

p 1 2]

2]

B (x) u dvg

M

p B (x) C (x)dvg

Since A is of positive values, then Z

2

Z

p

2] 2]

1

B (x) dvg

M

1 p 2]

.

A (x) dvg > 0 u2]

M

(S1 R)

2]

B (x) u dvg + D

M

and if we set

Z

1 p 2]

2]

B (x) u dvg

M

t=

Z

]

B (x) u2 dvg

M

we obtain

(RS1 )2

f (t)

where f (t) = t + Dt

1 p 2]

:

f takes a minimum at t0 =

p

1 2]

2] 2] +p

1

D

and consequently 8t > 0, f (t)

Z

M

so it comes that

Z

2

dvg A

:

1 p Z 2] C(x) 2] dv D B (x) u dv : g g p 1 M u M and therefore (3.1) becomes Z Z Z ] ] A (x) dv + D B (x) u2 dvg (S1 R)2 B (x) u2 dvg + g 2] u M M M

2

1 p ]1

M

M

Letting

D=

B (x)

Z

B (x) dvg

Z

2

p B (x) C (x)dvg

dvg

0 Z @

M

So Z

1

M

M

21

2] + p 1 min f (t) = f (t0 ) = t>0 p 1

p

1 2]

2] 2] +p

D

1

.

C(x) dvg : up 1

22

M OHAM M ED BENALILI AND ALI ZOUAOUI

Finally, replacing D by its value, we obtain 2

(RS1 )

2] + p 1 p 1

p

1

2] 2] +p

2]

1

Z

M

and if we set

2] 2] +p

2:

p

B (x) C (x)dvg

2] + p 1 C(n; p; k) = p 1

1

Z

p 1 2] 2] +p 1

B (x) dvg

M

p

1

2] 2] +p

1

2]

then it comes 2

(RS1 )

C(n; p; k)

R

M

!2: ] 2] p 2 +p B (x) C (x)dvg R B (x) dvg M

1

Z

B (x) dvg :

M

Acknowledgement 1. The authors are thankful to the referee for the valuable suggestions towards the improvement of this paper. References [1] T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Di¤erential Equations18, (2003), 253–268. [2] M. Benalili, On the singular $Q$-curvature type equation. J. Di¤erential Equations 254 (2013) no.2, 547-598. [3] C. Birahim Ndiaye; Constant Q-curvature metrics in arbitrary dimension. J. Funct. Anal. 251 no. 1(2007), 1-58. [4] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant $Q$-curvature. Ann. of Math. (2) 168 (2008) no. 3, 813-856. [5] C.R. Graham, R. Jenne, L.J. Masson, G.A.J. Sparling, Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. 46 (1992), 557-565. [6] E. Hebey, F. Pacard, D. Pollack, A variational Analysis of Einstein-scalar Field Lichnerowicz Equations on compact Riemannian Manifolds, Commun. Math. Phys. 278, (2008), 117-132. [7] E. Hebey, F. Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coe¢ cients, Calc.Var. 13, (2001) p.p. 491-517. [8] E. Hebey, G. Veronelli, The Lichnerowicz equation in the closed case of the Einstein-Maxwell theory. Trans. Amer. Math. Soc., 366, (2014) no. 3, 1179-1193. [9] A. Maalaoui, On a fourth order Lichnerowicz type equation involving the Paneitz-Branson operator. Ann. Global An. Geom. 42, no 3, (2012) 391-4012. [10] S. Mazumdar, GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutions, J. Di¤erential Equations 261(2016), no. 9, 4997–5034. [11] I. Meghea, Ekeland Variational Principles with Generalizations and Variants, Old City Publishing, 2009. [12] Q.A. Ngô, X. Xu, Existence results for the Einstein-scalar …eld Lichnerowicz equation on compact Riemannian manifolds. Adv. Math. 230 (2012) no. 4-6, 2379-2415. [13] S.M. Paneitz, A quartic conformally di¤erential operator for arbitrary pseu-Riemannian manifolds. Symmetry Integrability Geom. Methods and Appl.,4, (2008), Paper 036. [14] B. Premoselli, E¤ective multiplicity for the Einstein-scalar …eld Lichnerowicz equation. Calc. Var. Partial Di¤erential Equations 53 (2015) no. 1-2, 29-64. [15] B. Premoselli, The Einstein-scalar …eld equation constraint in the positive case. Comm. Math. Phys. 386 (2014) no. 2, 543-557. [16] P. Rabinowitz, Minimax methods in critical point theory with applications to di¤erential equations, CBMS Regional Conference in Mathematics, 65. (1986) American Mathematical Society, Providence, RI. [17] F. Robert, Admissible Q curvatures under isometries for the conformal GJMS operators, Contemporary Mathematics, Volume in the honor of Jean-Pierre Gossez.

.

SINGULAR ELLIPTIC EQUATION INVOLVING THE GJM S OPERATOR...

Dept. Maths, Faculty of Sciences, University UABB, Tlemcen, Algeria E-mail address : [email protected] E-mail address : [email protected]

23