POSITIVE SOLUTIONS TO A SUPERCRITICAL ...

POSITIVE SOLUTIONS TO A SUPERCRITICAL ELLIPTIC PROBLEM WHICH CONCENTRATE ALONG A THIN SPHERICAL HOLE MÓNICA CLAPP, JORGE FAYA, AND ANGELA PISTOIA

Abstract. We consider the supercritical problem v = jvjp

2

in

v

;

v=0

on @

;

where is a bounded smooth domain in RN ; N 3; p > 2 := N2N2 ; and is obtained by deleting the -neighborhood of some sphere which is embedded in . In some particular situations we show that, for > 0 small enough, this problem has a positive solution v and that these solutions concentrate and blow up along the sphere as ! 0. Our approach is to reduce this problem to a critical problem of the form u = Q(x) juj n

4 2

u

in

;

u=0

on @

;

in a punctured domain := fx 2 : jx 0 j > g of lower dimension, by means of some Hopf map. We show that, if is a bounded smooth domain in Rn ; n 3, 0 2 ; Q 2 C 2 ( ) is positive and rQ( 0 ) 6= 0 then, for > 0 small enough, this problem has a positive solution u ; and that these solutions concentrate and blow up at 0 as ! 0. Key words: Nonlinear elliptic problem; supercritical problem; nonautonomous critical problem; positive solutions; domains with a spherical perforation, blow-up along a sphere. MSC2010: 35J60, 35J20.

1. Introduction We are interested in the supercritical problem (1.1)

p 2

v = jvj

v

in D;

v=0

on @D;

where D is a bounded smooth domain in R ; N 3; and p > 2 ; with 2 := N2N2 the critical Sobolev exponent. Existence of a solution to this problem is a delicate issue. Pohozhaev’s identity [20] implies that (1.1) does not have a nontrivial solution if D is strictly starshaped and p 2 . On the other hand, Kazdan and Warner [10] showed that in…nitely many radial solutions exist for every p 2 (2; 1) if D is an annulus. For p = 2 Bahri and Coron [2] established the existence of at least one positive solution to problem (1.1) in every domain D having nontrivial reduced homology with Z=2-coe¢ cients. However, in the supercritical case this is not enough to guarantee existence. In fact, for each 1 k N 3; Passaseo [18, 19] exhibited domains having the homotopy N

Date : March 3, 2013. M. Clapp and J. Faya are supported by CONACYT grant 129847 and PAPIIT grant IN106612 (Mexico). A. Pistoia is supported by Università degli Studi di Roma "La Sapienza" Accordi Bilaterali "Esistenza e proprietà geometriche di soluzioni di equazioni ellittiche non lineari" (Italy). 1

2

M ÓNICA CLAPP, JORGE FAYA, AND ANGELA PISTOIA

type of a k-dimensional sphere in which problem (1.1) does not have a nontrivial k) solution for p 2N;k := 2(N N k 2 : Existence may fail even in domains with richer topology, as shown in [4]. The …rst nontrivial existence result for p > 2 was obtained by del Pino, Felmer and Musso [5] in the slightly supercritical case, i.e. for p > 2 but close enough to 2 : For p slightly below 2N;1 solutions in certain domains, concentrating at a boundary geodesic as p ! 2N;1 ; were constructed in [7]. A fruitful approach to produce solutions to the supercritical problem (1.1) is to reduce it to some critical or subcritical problem in a domain of lower dimension, either by considering rotational symmetries, or by means of maps which preserve the laplacian, or by a combination of both. This approach has been recently taken in [1, 4, 11, 12, 15, 21] to produce solutions of (1.1) in di¤erent types of domains. We shall also follow this approach to obtain a new type of solutions in domains with thin spherical perforations. We start with some notation. Let O(N ) be the group of linear isometries of RN . If is a closed subgroup of O(N ), we denote by x := fgx : g 2 g the -orbit of x 2 RN . A domain D in RN is called -invariant if x D for all x 2 D; and a function u : D ! R is called -invariant if u is constant on every x: We denote by D := fx 2 D : gx = x 8g 2 g the set of -…xed points in D. We consider the problem (}Q; ) in

8
0 : u=0 := fx 2

: jx

0j

in ; in ; on @ ;

> g;

where n 3; is a bounded smooth domain in Rn which is invariant under the action of some closed subgroup of O(n); 0 2 ; and the function Q 2 C 2 ( ) ; is also is -invariant and satis…es minx2 Q(x) > 0: Note that, since 0 2 -invariant. We will prove the following result. Theorem 1.1. Assume that rQ( 0 ) 6= 0: Then there exists 0 > 0 such that, for each 2 (0; 0 ); problem (}Q; ) has a -invariant solution u which concentrates and blows up at the point 0 as ! 0: Now we describe two situations where one can apply this result to obtain solutions of supercritical problems which concentrate and blow up at a sphere. For N = 2; 4; 8; 16 we write RN = K K, where K is either the real numbers R, the complex numbers C, the quaternions H or the Cayley numbers O: The set of units SK := f# 2 K : j#j = 1g; which is a group if K = R, C or H and a quasigroup with unit if K = O, acts on RN by multiplication on each coordinate, i.e. #(z1 ; z2 ) := (#z1 ; #z2 ): The orbit space of RN with respect to this action turns out to be Rdim K+1 and the projection onto the orbit space is the Hopf map hK : RN = K K ! R K = Rdim K+1 given by 2

hK (z1 ; z2 ) := (jz1 j

2

jz2 j ; 2z1 z2 ).

A SUPERCRITICAL ELLIPTIC PROBLEM

3

What makes this map special is that it preserves the laplacian. Maps with this property are called harmonic morphisms [3, 8, 23]. More precisely, the following statement holds true. It can be derived by straightforward computation (cf. Proposition 4.1) or from the general theory of harmonic morphisms as in [4]. Proposition 1.2. Let N = 2; 4; 8; 16 and let D be an SK -invariant bounded smooth domain in RN = K2 such that 0 2 = D: Set U := hK (D). If u is a solution to problem ( p 2 1 juj u in U; u = 2jxj (1.2) u=0 on @U; then v := u hK is an SK -invariant solution of problem (1.1). Conversely, if v is an SK -invariant solution of problem (1.1) and v = u hK ; then u solves (1.2). We apply this result as follows: Let N = 4; 8; 16 and let be an SK -invariant bounded smooth domain in RN = K2 such that 0 2 = : Fix a point z0 2 and for each > 0 small enough let := fz 2

: dist(z; SK z0 ) > g

where SK z0 := f#z : # 2 SK g: This is again an SK -invariant bounded smooth domain in K2 : We consider the supercritical problem 8 dim K+3 < v = v dim K 1 in ; 1 (} ) v>0 in ; : u=0 on @ :

Then, Theorem 1.1 with n := dim K + 1; = f1g, := hK ( ); 0 := hK (z0 ) and 1 Q(x) := 2jxj ; together with Proposition 1.2, immediately yields the following result.

Theorem 1.3. There exists 0 > 0 such that, for each 2 (0; 0 ); the supercritical problem (}1 ) has an SK -invariant solution v which concentrates and blows up along the sphere SK z0 as ! 0.

Now let O(m) O(m) act on R2m Rm Rm in the obvious way and O(m) m+1 act on the last m coordinates of R R Rm : We write the elements of R2m as m (y1 ; y2 ) with yi 2 R and the elements of Rm+1 as x = (t; ) with t 2 R; 2 Rm : Recently Pacella and Srikanth showed that the real Hopf map provides a one-toone correspondence between [O(m) O(m)]-invariant solutions of a supercritical problem in a domain in R2m and O(m)-invariant solutions of a critical problem in some domain in Rm+1 . In [16] they proved the following result.

Proposition 1.4. Let N = 2m; m 2; and D be an [O(m) bounded smooth domain in R2m such that 0 2 = D: Set U := f(t; ) 2 R

O(m)]-invariant

Rm : hR (jy1 j ; jy2 j) = (t; j j) for some (y1 ; y2 ) 2 Dg:

If u(t; ) = u(t; j j) is an O(m)-invariant solution of problem ( p 2 1 u = 2jxj juj u in U; (1.3) u=0 on @U;

then v(y1 ; y2 ) := u(hR (jy1 j ; jy2 j)) is an [O(m) O(m)]-invariant solution of problem (1.1). Conversely, if v(y1 ; y2 ) = v(jy1 j ; jy2 j) is an [O(m) O(m)]-invariant solution of problem (1.1) and v = u hR ; then u(t; ) = u(t; j j) is an O(m)-invariant solution of problem (1.2).

4


We apply this result as follows: Let be an [O(m) O(m)]-invariant bounded smooth domain in R2m such that 0 2 = and (y0 ; 0) 2 : We write S0m 1 := f(y; 0) : jyj = jy0 jg for the [O(m) O(m)]-orbit of (y0 ; 0); and for each > 0 small enough we set := fx 2 : dist(x; S0m 1 ) > g:

This is again an [O(m) O(m)]-invariant bounded smooth domain in R2m : We consider the supercritical problem 8 m+3 < v = v m 1 in ; (}2 ) v>0 in ; : u=0 on @ : Then, Theorem 1.1 with n = m + 1; := f(t; ) 2 R

R

m

= O(m);

: hR (jy1 j ; jy2 j) = (t; j j) for some (y1 ; y2 ) 2 g;

:= (jy0 j ; 0; : : : ; 0) and Q(x) = yields the following result. 0

1 2jxj ;

together with Proposition 1.4, immediately

Theorem 1.5. There exists 0 > 0 such that, for each 2 (0; 0 ); problem (} ) has an [O(m) O(m)]-invariant solution v which concentrates and blows up along the (m 1)-dimensional sphere S0m 1 as ! 0. The proof of Theorem 1.1 uses the well-known Ljapunov-Schmidt reduction, adapted to the symmetric case. In the following section we sketch this reduction, highlighting the places where the symmetries play a role. In section 3 we give an expansion of the reduced energy functional and use it to prove Theorem 1.1. We conclude with some remarks concerning Proposition 1.4. 2. The finite dimensional reduction For every bounded domain U in Rn we take Z (u; v) := ru rv; kuk := U

Z

U

1=2 2

jruj

;

as the inner product and its corresponding norm in H01 (U): If we replace U by Rn these are the inner product and the norm in D1;2 (Rn ): We write Z r kukr := ( juj )1=r U

r

for the norm in L (U), r 2 [1; 1): If U is -invariant for some closed subgroup H01 (U)

:= fu 2

H01 (U)

of O(n) we set

: u is -invariantg

and, similarly, for D1;2 (Rn ) and Lr (U) : It is well known that the standard bubbles n

U

;

(x) = [n(n

2)]

n

2 2

2 4

(

2

+ jx

j2 )

n

2 2

are the only positive solutions of the equation U = Up

in Rn ;

2 (0; 1);

2 Rn ;


where p := nn+22 : Thus, the function W the equation

;

:=

0U ;

W = Q( 0 )W p

(2.1)

; with

5

0

:= [Q( 0 )] p

1 1

, solves

in Rn :

Let 2 jx j2 ; 2 2 ( + jx j )n=2 n 2 xj j j 2) 2 ; j = 1; : : : ; n: n (n ; 2 ( + jx j2 )n=2 The space generated by these n+1 functions is the space of solutions to the problem

(2.2)

n

@U ; = @ @U ; := = @ j

0 ;

:=

2

n

= pU p;

(2.3)

4

2

n

2

1

2 D1;2 (Rn ):

;

Note that U ; 2 D1;2 (Rn ) and, similarly, for every j = 0; 1; : : : ; n; j ;

2 D1;2 (Rn )

i¤

2 (Rn )

i¤

2 (Rn ) :

Let be a -invariant bounded smooth domain in Rn ; Q 2 C 2 ( ) be positive and -invariant, and 0 2 . For > 0 small enough set := fx 2

: jx

0j

(P W ) =

W

in

n

> g:

Consider the orthogonal projection P : D (R ) ! H01 ( then P W is the unique solution to the problem (2.4)

1;2

;

), i.e. if W 2 D1;2 (Rn )

P W = 0 on @

:

H01 (

1;2

A consequence of the uniqueness is that P W 2 ) if W 2 D (Rn ) : We denote by G(x; y) the Green function of the Laplace operator in with zero Dirichlet boundary condition and by H(x; y) its regular part, i.e. ! 1 G(x; y) = n H(x; y) ; n 2 jx yj

where n is a positive constant depending only on n: The following estimates will play a crucial role in the proof of Theorem 1.1. Lemma 2.1. Assume that := 0 + ; and de…ne

! 0 as

! 0 and n

2

! 0. Fix

2 Rn ; set n 2

n

: n 2 0j (1 + j j2 ) 2 jx Then there exists a positive constant c such that the following estimates hold true for every x 2 r B( 0 ; ): R(x) := P U

;

(x)

U

;

(x) +

n

2

H(x; ) +

= o( ) as

n

2

n

2

2

jR(x)j

c

j@ R(x)j

c

j@ i R(x)j

c

n

2 2

n

4 2

n 2

Proof. See Lemma 3.1 in [9].

n 2

n+1 (1 + ) + n 2 jx 0j n 2 n+1 (1 + ) + n 2 jx 0j n 2 n (1 + ) + 2+ n 2 jx 0j

n 2

2

+

2

+

; n 2

;

n 2 n 1

:

6


For each

> 0 and (d; ) 2

Vd; := P W

=

;

0P

(Rn ) set (see (2.1))

:= (0; 1) U

with

;

:= d

The map (d; ) 7! Vd; is a C 2 -embedding of whose tangent space at Vd; is j ;

Kd; := spanfP

n n

2 1

;

:=

0

+

:

as a submanifold of H01 (

: j = 0; 1; : : : ; ng:

Note that, since 0 ; 2 (Rn ) ; also 2 (Rn ) and, therefore, Kd; H01 ( write Kd;;? := f 2 H01 ( ) : ( ; P j; ) = 0 for j = 0; 1; : : : ; ng for the orthogonal complement of Kd; in H01 ( ) , and d; : H01 ( and d;;? : H01 ( ) ! Kd;;? for the orthogonal projections, i.e. d;

(u) :=

n X

j ;

(u; P

j ;

)P

) ,

;? d; (u)

;

:= u

d;

) : We

) ! Kd;

(u):

j=0

2n

Let i : L n+2 ( ) ! H01 ( ) be the adjoint operator to the embedding i : 2n H01 ( ) ,! L n 2 ( ), i.e. v = i (u) if and only if Z (v; ') = u' 8' 2 Cc1 ( ) if and only if (2.5)

v=u

in

;

v=0

on @

:

Sobolev’s inequality yields a constant c > 0; independent of , such that (2.6)

ki (u)k

c kuk

2n

8u 2 L n+2 (

2n n+2

); 8 > 0:

Note again that i (u) 2 H01 (

2n 2

)

if u 2 L n

(

) :

We rewrite problem (}Q; ) in the following equivalent way: u = i [Q(x)f (u)] ; u 2 H01 ( );

(2.7)

where f (s) := (s+ )p and p := nn+22 . We shall look for a solution to problem (2.7) of the form (2.8)

u = Vd; +

with (d; ) 2

As usual, our goal will be to …nd (d; ) 2 enough, (2.9)

;? d; [Vd;

and

and

2 Kd;;? :

2 Kd;;? such that, for

+

i (Qf (Vd; + ))] = 0

[Vd; +

i (Qf (Vd; + ))] = 0:

and (2.10)

d;

small


7

First we will show that, for every (d; ) 2 and small enough, there exists an unique 2 Kd;;? which satis…es (2.9). To this aim we consider the linear operator

Ld; : Kd;;? ! Kd;;? de…ned by

;? d;

Ld; ( ) :=

i [Qf 0 (Vd; ) ]:

It has the following properties. Proposition 2.2. For every compact subset D of such that, for each 2 (0; 0 ) and each (d; ) 2 D; (2.11)

Ld; ( )

ck k

for all

there exist

0

> 0 and c > 0

2 Kd;;? ;

and the operator Ld; is invertible. Proof. The argument given in [9] to prove Lemma 5.1 carries over with minor changes to our situation. The following estimates may be found in [13]. Lemma 2.3. For each a; b; q 2 R with a 0 and q constant c such that the following inequalities hold

1 there exists a positive

q

q

c minfjbj ; aq 1 jbjg if 0 < q < 1; q 1 q c(jaj jbj + jbj ) if q 1:

aq j

jja + bj

Again, the argument given to prove similar results in the literature carries over with minor changes to prove the following result. We include it this time to illustrate this fact and also because some of the estimates will be used later on. Proposition 2.4. For every compact subset D of there exist 0 > 0 and c > 0 such that, for each 2 (0; 0 ) and for each (d; ) 2 D; there exists a unique d; 2 Kd;;? H01 ( ) which solves equation (2.9) and satis…es (2.12) Moreover, the function (d; ) 7! Proof. Note that

c

d; d;

n n

2 1

:

is a C 1 -map.

2 Kd;;? solves equation (2.9) if and only if

is a …xed point of

the operator Td; : Kd;;? ! Kd;;? de…ned by Td; ( ) = (Ld; )

1

;? d; i

[Qf (Vd; + )

Qf 0 (Vd; )

Q( 0 )( 0 U

;

)p ] :

We will prove that Td; is a contraction on a suitable ball. To this aim, we …rst show that there exist 0 > 0 and c > 0 such that for, each 2 (0; 0 ); (2.13)

k k

c

n n

2 1

)

Td; ( )

c

n n

From Proposition 2.2 we have that, for some c > 0 and (Ld; )

1

c

8(d; ) 2 D:

2 1

:

small enough,

8


Using (2.6) we obtain Td; ( )

c kQ [f (Vd; + )

f 0 (Vd; ) ]

c kQ [f (Vd; + )

f (Vd; )

+ c kQf (Vd; )

Q( 0 U

Q( 0 )( 0 U f 0 (Vd; ) ]k

)p k

;

2n n+2

2n n+2

2n n+2

p 0

+c

)p k

;

Q( 0 )] U p;

[Q

:

2n n+2

Using the mean value theorem, Lemma 2.3 and the Hölder inequality we have that, for some t 2 (0; 1), kQ [f (Vd; + )

f (Vd; )

f 0 (Vd; ) ]k

c k[f 0 (Vd; + t )

2n n+2

c kf 0 (Vd; + t ) c(k

+k

p k2

2n n+2

f 0 (Vd; )kn=2 k k2

4

c(k k2 + k k2n 2 k2

f 0 (Vd; )] k

2

) k k2

):

Moreover, using Lemma 2.1 one can show that kQf (Vd; ) (2.14)

c c ;

Z

Q( 0 U U p; 1 (P

;

U

)p k

c (P U

2n n+2

U

;

)

;

2n n+2

+c

)p

;

Z

U p;

jP U

;

2n n+2

U

p+1

n+2 2n

j

;

see inequality (6.4) in [9]. Finally, setting y = x = x 0 and e := fy 2 Rn : y + 2 g; and using the mean value theorem, for some t 2 (0; 1) we obtain [Q

Q( 0 )] U p;

2n n+2

=

Z

=

(2.15)

c :

e

Z

jQ( y + e

c

0

n n

2 1

2n

Q( 0 )j n+2 U p+1 (y)dy

0)

jhrQ(t y + t

This proves statement (2.13). Next we show that we may choose operator Td; : f 2 Kd;;? : k k

+

n+2 2n

2n

n+2 U p+1 (y)dy 0 ); y + ij

+

> 0 such that, for each g ! f 2 Kd;;? : k k

c

2 (0; n n

2 1

n+2 2n

0 );

the

g

is a contraction and, therefore, has a unique …xed point, as claimed. n 2 If 1 ; 2 2 f 2 Kd;;? : k k c n 1 g, using again the mean value theorem we obtain Td; (

1)

Td; (

2)

c kf (Vd; +

1)

= c k[f 0 (Vd; + (1 c kf 0 (Vd; + (1

f (Vd; + t) t)

1 1

+

+

2) 2)

2)

f 0 (Vd; )( f 0 (Vd; )](

2 ))k 2 )k

1

f 0 (Vd; )k n k 2

1

1

2n n+2

2n n+2

2 k2


9

for some t 2 [0; 1]; and arguing as before we conclude that kf 0 (Vd; + (1

t)

1

+

2)

4

f 0 (Vd; )k n

c k(1

2

t)

1

+

2 k2

+ k(1

t)

1

+

n

2 k2 4

4

c k Hence, if

1 k2

+k

2 k2

+k

n

1 k2

2

+k

n

2 k2

is su¢ ciently small, it follows that Td; (

1)

Td; (

2)

k

2k

1

with 2 (0; 1). Finally, a standard argument shows that (d; ) 7! cludes the proof.

d;

is a C 1 -map. This con-

Consider the functional J : H01 ( ) ! R de…ned by Z Z 1 1 J (u) := jruj2 Qjujp+1 : 2 p+1 It is well known that the critical points of J are the solutions of problem (2.7). We ! R by de…ne the reduced energy functional Je :

(2.16)

Je (d; ) := J (Vd; +

d;

):

= f1g is the trivial group, we simply write Je instead of Je and instead of : Next we show that the critical points of Je are -invariant solutions of problem (2.7).

If

Proposition 2.5. If (d; ) 2 is a critical point of the function Je ; then Vd; + 1 ) is a critical point of the functional J and, therefore, a -invariant d; 2 H0 ( solution of problem (2.7). Proof. Assume …rst that is the trivial group. Then = (0; 1) Rn and the statement is proved using similar arguments to those given to prove Lemma 6.1 in [6] or Proposition 2.2 in [9]. If is an arbitrary closed subgroup of O(n); then is the set of -…xed points in of the action of on the space R Rn which is given by g(t; x) := (t; gx) for g 2 ; t 2 R, x 2 Rn : By the principle of symmetric criticality [17, 22], if (d; ) 2 is a critical point of the function Je ; then (d; ) is a critical point of e J : (0; 1) Rn ! R, and the result follows from the previous case. 3. The asymptotic expansion of the reduced energy functional

In order to …nd a critical point of Je we will use the following asymptotic expansion of the functional Je : (0; 1) Rn ! R.

Proposition 3.1. The asymptotic expansion Je (d; ) = c0 + Q( 0 )

2 p

1

F (d; )

n n

2 1

+ o(

n n

2 1

)

2

2

10


holds true C 1 -uniformly on compact subsets of Rn ! R is given by (3.1)

F (d; ) :=

8 >
:

D

1 (1+j j2 )d

1 (1+j j2 )d

for some positive constants c0 ; ;

rQ( 0 ) Q( 0 ) ;

D

n 2

, where the function F : (0; 1)

rQ( 0 ) Q( 0 ) ;

E

d E

if n = 3; d

if n

4:

and .

Proof. We write J (Vd; +

d;

)=

1 Vd; + 2

2 d;

1 p+1

Z

p+1

Q Vd; +

d;

Z p = J (Vd; ) + 0 (U p; (P U ; ) ) d; Z 1 p p [Q Q( 0 )] (P U ; ) d; + 0 2 d; Z 1 p+1 p+1 jVd; j Q Vd; + d; p+1

2

(p + 1)Vd;p

d;

:

Then, using Hölder’s inequality and inequalities (2.12), (2.14) and (2.15) we obtain J (Vd; + (3.2)

2(n

d;

2)

n 1 ) = J (Vd; ) + O Z Z 1 p+1 p 2 1 = 0 U ; (P U ; ) jP U ; j 2 p+1 Z 2(n 1 p+1 p+1 n [Q Q( 0 )] jP U ; j +O p+1 0

2) 1

:

Next, we compute the …rst summand on the right-hand side of equality (3.2). From Lemma 2.1 we have that Z 1 p+1 U p; (P U ; ) jP U ; j p+1 Z Z Z 1 1 p 1 p+1 p U; U ; (P U ; U; ) = 2(p + 1) 2 p+1 Z Z n 2 p 1 1 p+1 = U1;0 U p; (P U ; U ; ) + o( n 1 ) 2(p + 1) 2 Z Z n 2 p 1 1 p+1 n 1 ); = U + Up ; + o( 2(p + 1) Rn 1;0 2 Rn ;

1 2

Z

jP U

p+1

;

j

where (3.3)

;

(x) :=

n

n

2 2

H(x; ) +

n 2

1 n

n

2 2

(1 + j j2 )

n

2 2

jx

n 2 0j

:

U p+1 ;


Setting x = + y we have Z U p; n ; Rn

=

n

Z

Rn

U p;

(x)(

n

2 2

H(x; ))dx +

n

Z

Rn

Z

U p;

n

n 2

n 2

1

(x)

n

2 2

p (y)H( y + + 0 ; + 0 )dy U1;0 Z n 2 1 p + n U1;0 (y) dy n 2 n 2 jy jn 2 (1 + j j2 ) 2 Rn Z 1 p = n U1;0 H( 0 ; 0 ) n 2 (1 + o(1)) + n g( ) n

=

11

(1 + j j2 )

n

2 2

jx

n 2 0j

Rn

Rn

where the function g : Rn ! R is de…ned by Z 1 g( ) := n 2 (1 + j j2 ) 2 Rn jy

1 jn

2

n 2 2

(1 + o(1));

p U1;0 (y)dy:

U = U p in Rn ; an easy computation shows that 1 1 g( ) = n n 2 U1;0 ( ) = 2 (1 + j j2 )n (1 + j j ) 2

Since

2

:

To compute the second summand on the right-hand side of equality (3.2) we use the Taylor expansion Q( y +

0

+

) = Q( 0 ) + hrQ( 0 ); y + i + O( 2 (1 + jyj2 ))

to obtain Z

Z n [Q Q( 0 )] jP U ; jp+1 = [Q Q( 0 )] U p+1 + o( n ; Z n 2 p+1 = (Q( y + 0 + ) Q( 0 ))U1;0 (y)dy + o( n 1 ) e Z Z hrQ( 0 ); yi p+1 dy + O = hrQ( 0 ); iU1;0 (y)dy + (1 + jyj2 )n n n R R Z p+1 = hrQ( 0 ); i U1;0 (1 + o(1));

because

R

2 1

)

2(n 2) n 1

Rn

Rn

hrQ( 0 );yi (1+jyj2 )n dy

= 0. Collecting all the previous information we obtain

Je (d; ) = J (Vd; + d; ) 8 D E p p 0) > < c0 + 02 c1 H( 0 ; 0 )d + c2 g( ) d1 c3 rQ( ; d + o( ) if n = 3; Q( 0 ) = D E n 2 n 2 > : c0 + 02 c2 g( ) n1 2 c3 rQ( 0 ) ; d n 1 + o( n 1 ) if n 4; d Q( 0 )

as claimed.

Proof of theorem 1.1. We will show that the function F de…ned in (3.1) has a critical point (d0 ; 0 ) 2 = (0; 1) (Rn ) which is stable under C 1 -perturbations. Then, we deduce from Proposition 3.1 that the functional Je has a critical point in for small enough, so the result follows from Proposition 2.5.

!

dx

12

M ÓNICA CLAPP, JORGE FAYA, AND ANGELA PISTOIA 0) Let n = 3: Set 0 := rQ( 2 R3 : Q( 0 ) and consider the half space H := f h 0 ; i > 0g. For each 2 H there exists a unique d = d( ); given by s

d( ) =

(1 + j j2 )(

h 0 ; i)

2 (0; 1);

such that Fd (d; ) = 0. Moreover, Fdd (d( ); ) > 0 for any 2 H. Consider the function Fe : H ! R de…ned by s h 0; i 2 : Fe( ) := F (d( ); ) = 2 1 + j j2 The point

0

0

:= @

q

2

2

+ 2

2

j 0j

j 0j

1 A

0

is a strict maximum point of Fe: Setting d0 := d( 0 ) we deduce from Lemma 5.7 in [14] that (d0 ; 0 ) is a C 1 -stable critical point of the function F . Note that, since and Q is -invariant, rQ( 0 ) 2 (Rn ) : Hence, (d0 ; 0 ) 2 : 0 2 If n 4 arguing as in the previous case we easily conclude that, if 0

:=

rQ( 0 ) ; jrQ( 0 )j

d0 :=

(n 2) Q( 0 ) n 2 2 jrQ( 0 )j

1 n

1

then (d0 ; 0 ) is a C 1 -stable critical point of the function F and (d0 ; concludes the proof.

; 0)

2

: This

4. Final remarks One may wonder whether Proposition 1.4 is also true in other dimensions. We show that this is not so. If N = k1 + k2 we write the elements of RN as (y1 ; y2 ) with yi 2 Rki ; and the elements of Rm+1 as (t; ) with t 2 R; 2 Rm : Proposition 4.1. Let N = k1 + k2 ; D be an [O(k1 ) O(k2 )]-invariant bounded smooth domain in RN such that 0 2 = D; and f 2 C 0 (R): Set U := f(t; ) 2 R

Rm : hR (jy1 j ; jy2 j) = (t; j j) for some (y1 ; y2 ) 2 Dg

and let u 2 C 2 (U), u(t; ) = u(t; j j); be an O(m)-invariant solution of equation 1 (4.1) u= f (u) 2 jxj in U. Then v(y1 ; y2 ) := u(hR (jy1 j ; jy2 j)) is an [O(k1 ) equation (4.2)

O(k2 )]-invariant solution of

v = f (v)

in D if and only if k1 = k2 = m: Proof. A straighforward computation shows that a function v(y1 ; y2 ) = v(jy1 j ; jy2 j) solves equation (4.2) in f(y1 ; y2 ) 2 D : y1 6= 0; y2 6= 0g if and only if v solves (4.3)

v

k1 1 @v z1 @z1

k2 1 @v = f (v) z2 @z2


in D0 := fz = (z1 ; z2 ) 2 R2 : z1 ; z2 > 0; z1 = jy1 j ; z2 = jy2 j, (y1 ; y2 ) 2 Dg: Similarly, a function u(t; ) = u(t; j j) solves equation (4.1) in f(t; ) 2 U : if and only if u solves m 1 @u 1 (4.4) u = f (u) x2 @x2 2 jxj

13

6= 0g

in U0 := fx = (x1 ; x2 ) 2 R2 : x2 > 0; x2 = j j ; (x1 ; ) 2 U g: Assuming that u solves equation (4.4) in U0 , we show next that v := u hR solves equation (4.3) in D0 if and only if k1 = k2 = m: A straightforward computation yields k1 1 @v k2 1 @v v z1 @z1 z2 @z2 # " # ! " k1 1 z2 k2 1 z1 @u k1 1 k2 1 @u 2 + : = 2 jzj u 2 2 2 2 @x1 jzj jzj jzj z1 jzj z2 @x2 2

Note that jzj = jhR (z)j : So if u solves equation (4.4), we have that k1 1 @v k2 1 @v v = jvjp 2 v z1 @z1 z2 @z2 if and only if k2 1 k1 1 z2 k2 1 z1 m 1 k1 1 and + = 2 = 2 2 2 z1 z2 jzj jzj jzj z1 jzj z2 if and only if k1 = k2 = m: The argument given in [16] to prove Proposition 1.4 uses polar coordinates. Note that if we write z1 = r cos ; z2 = r sin ; x1 = cos '; x2 = sin '; then the Hopf map x = 12 hR (z) becomes 1 '=2 ; = r2 ; 2 which is the map considered in [16]. References [1] N. Ackermann, M. Clapp, A. Pistoia, Boundary clustered layers for some supercritical problems. J. Di¤ erential Equations, to appear. [2] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The e¤ect of the topology of the domain. Comm. Pure Appl. Math. 41 (1988), 253-294. [3] P. Baird, J.C. Wood, Harmonic morphisms between Riemannian manifolds. London Mathematical Society Monographs. New Series, 29. The Clarendon Press, Oxford University Press, Oxford, 2003. [4] M. Clapp, J. Faya, and A. Pistoia, Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents. Calc. Var. Partial Di¤ erential Equations, DOI: 10.1007/s00526-012-0564-6. [5] M. del Pino, P. Felmer, M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries. Bull. London Math. Soc. 35 (2003), 513–521. [6] M. del Pino, P. Felmer, M. Musso, Two-bubble solutions in the super-critical BahriCoron’s problem. Calc. Var. Partial Di¤ erential Equations 16 (2003), 113–145. [7] M. del Pino, M. Musso, F. Pacard, Bubbling along boundary geodesics near the second critical exponent. J. Eur. Math. Soc. (JEMS) 12 (2010), 1553–1605. [8] J. Eells, A. Ratto, Harmonic maps and minimal immersions with symmetries. Methods of ordinary di¤ erential equations applied to elliptic variational problems. Annals of Mathematics Studies, 130. Princeton University Press, Princeton, NJ, 1993.

14


[9] Y. Ge, M. Musso, A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains. Comm. Partial Di¤ erential Equations 35 (2010), 1419-1457. [10] J. Kazdan, F. Warner, Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 38 (1975), 557-569. [11] S. Kim, A. Pistoia, Clustered boundary layer sign changing solutions for a supercritical problem. J. London Math. Soc., to appear. [12] S. Kim, A. Pistoia, Boundary towers of layers for some supercritical problems. Preprint arXiv:1302.1217. [13] Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition. Comm. Partial Di¤ erential Equations 23 (1998), 487–545. [14] R. Molle, A. Pistoia, Concentration phenomena in elliptic problems with critical and supercritical growth. Adv. Di¤ erential Equations 8 (2003), 547–570. [15] F. Pacella, A. Pistoia, Bubble concentration on spheres for supercritical elliptic problems. Preprint. [16] F. Pacella, P.N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres. Preprint. arXiv:1210.0782 [17] R.S. Palais, The principle of symmetric criticality. Comm. Math. Phys. 69 (1979), 19–30. [18] D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains. J. Funct. Anal. 114 (1993), 97–105. [19] D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity. Di¤ erential Integral Equations 8 (1995), 577–586. [20] S.I. Pohozhaev, Eigenfunctions of the equation u+ f (u) = 0: Soviet Math. Dokl. 6 (1965), 1408-1411. [21] J. Wei, S. Yan, In…nitely many positive solutions for an elliptic problem with critical or supercritical growth. J. Math. Pures Appl. 96 (2011), 307–333. [22] M. Willem, Minimax theorems. Progress in Nonlinear Di¤erential Equations and their Applications 24, Birkhäuser, Boston, 1996. [23] J.C. Wood, Harmonic morphisms between Riemannian manifolds. Modern trends in geometry and topology, 397–414, Cluj Univ. Press, Cluj-Napoca, 2006. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F., Mexico E-mail address : [email protected] Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F., Mexico E-mail address : [email protected] Dipartimento di Metodi e Modelli Matematici, Universitá di Roma "La Sapienza", via Antonio Scarpa 16, 00161 Roma, Italy E-mail address : [email protected]