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Nonexistence of bounded energy solutions for a critical exponent problem
J. Differential Equations 215 (2005) 19 – 36 www.elsevier.com/locate/jde
Nonexistence of bounded energy solutions for a critical exponent problem Florin Catrina Department of Mathematics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, USA Received 19 May 2004; revised 1 December 2004 Available online 20 January 2005
Abstract In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in RN . The weights depend on a positive parameter , which is allowed to increase to infinity. While for small values of solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as increases to infinity. We also prove a multiplicity result for high energy solutions. © 2004 Elsevier Inc. All rights reserved. Keywords: Bounded energy; Critical exponent
1. Introduction In this paper we are concerned with the problem
∗ ∗ −div |x|−2a ∇u = |x|−2 a u2 −1 , u > 0 in RN , u ∈ Da (RN ).
E-mail address: [email protected]. 0022-0396/$ - see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2004.12.002
(1)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Throughout we shall assume 3 N,
a<
N −2 , 2
and
2∗ =
2N . N −2
(2)
For convenience, we use interchangeably either the parameter a or
=
N − 2 − 2a N −2 > 0, and we denote 0 = . 2 2
Here Da (RN ) denotes the Hilbert space obtained as the completion of smooth functions with compact support in RN under the norm induced by the inner product (u, v)a =
RN
|x|−2a ∇u · ∇v dx.
Problems of this type and more general ones, have been the subject of many recent studies. There are several motivations that spark interest in such equations. One is that the problems are variational. A second reason is the invariance under a non-compact group of symmetries. Non-compactness is due to dilations about the origin. This makes the variational treatment non-trivial. The third reason is the existence of the parameter a, which makes the structure of solutions change. One more interesting aspect of (1) is the critical Sobolev exponent in the nonlinearity. This allows a local “almost invariance" under dilations at any point. The un-weighted case (when a = 0) of (1) is known as the Yamabe problem in (See [1]) RN . Existence of a solution implies the existence of a metric on RN , conformal to the standard metric, but with positive constant scalar curvature. Yamabe [15] stated the problem of finding metrics with constant scalar curvature conformal with one given on a Riemannian manifold, as a first step in an attempt to solving the Poincaré conjecture. In fact, as will be explained in Section 3, problem (1) may be also viewed as a modification of the Yamabe problem on C = R × SN−1 ⊂ RN+1 . It turns out that after a suitable transformation, the equation on C reads −v + 2 v = v 2
∗ −1
(3)
(recall that = N−2−2a ). Exactly this type of equations have been heavily studied 2 on bounded domains in RN with Neumann boundary conditions. This work had been started by Lin et al. [11] in connection with the Gierer–Meinhardt model in mathematical biology. We also remark that the cylinder C has nontrivial topology. As is noted in [2], this fact contributes to existence of more solutions in problems involving critical exponents. Being at the intersection of several extremely rich areas of research, it is natural to inquire which of the existing results continue to hold for our problem.
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
21
As suggested before, we shall take a variational approach. Solutions of (1) are critical points of the energy functional
−2a |∇u|2 dx N |x| Ea (u) = R ∗. −2∗ a |u|2∗ dx 2/2 RN |x|
(4)
We summarize several known facts about problem (1) in the theorem below. For proofs, please see [13,5], and the references therein. Theorem 1. (i) The functions u,a (x) = and Ua (x) =
N −2−2a 2
Ua (x) are solutions of (1) when > 0
N(N−2−2a)2 N−2
1 + |x|
N −2 4
2(N −2−2a) N −2
N 2−2
with energy
Ea (u,a ) =
N (N − 2 − 2a) (N − 2)
2(N −1) N
N −2 N
N2 N−1 2 N2 . 2(N )
(ii) For 0 a < N−2 2 , i.e. ∈ (0, 0 ], the infimum of energy (4) is achieved by the functions u,a above. (iii) For a < 0, i.e. > 0 , the infimum of energy (4) is
N2 N−1 2 N2 S0 = N (N − 2) 2(N ) and it is not achieved. N −2−2a (iv) Possibly after a dilation u(x) = 2 u(x), any solution u(x) of (1) satisfies the modified inversion symmetry
u
x |x|2
= |x|N−2−2a u(x).
Moreover, for any ∈ SN−1 , and any s > 0, the function f (s) = s strictly decreasing on the interval (1, ∞). The main new theorem in this paper is
N −2−2a 2
u(s ) is
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Theorem 2 (Nonexistence). For any integer N 3 and any positive constant M, there exists M > 0 such that for any > M , problem (1) has no solution with energy E (u) M. As the existence of radial solutions shows, the nonexistence result above does not hold if one lifts the bounded energy condition. In fact we show that in most cases, nonradial solutions should be expected. Precisely, we prove Theorem 3 (Multiplicity). For N 4 there exists N > 0 such that for all > N problem (1) has a nonradial solution (different than u,a in Theorem 1). The method of proof for this result indicates that although solutions with low energy do not exist, there is a rich variety of solutions for large values of . It is instructive to compare the structure of solutions of (1) with the subcritical case. Similar problems which have a power nonlinearity but with a subcritical exponent were studied in [6]. It is very interesting that in this case, as tends to infinity, in any dimension N 2 the number of distinct solutions increases to infinity. The technique was to restrict the energy functional to subspaces of functions invariant under group actions with discrete orbits. We proved that sequences along which the energy tends to a (local) minimum in these subspaces are convergent to critical points. In the case of critical exponent however, by increasing the parameter , Palais–Smale sequences have the tendency to concentrate and blow-up at the orbits where we try to locate most of the mass. One way to overcome this loss of compactness is to work in subspaces of functions invariant under group actions with continuous orbits. In these situations, one regains compactness but the solutions obtained have energy increasing to infinity as various powers of the parameter. In the proof of Theorem 2 we follow closely the arguments in [7], where the authors use a localized version of Pohozaev’s identity to prove nonexistence of positive solutions with bounded energy for Eq. (3) set on compact Riemannian manifolds. For results in the same spirit, we also refer to [3,4,8,12]. The a priori symmetry of solutions given by Theorem 1 (iv), plays an essential role in overcoming the fact that the domain is not compact, and in adapting the method in [7]. In Theorem 3, we employ previous compactness under symmetry results such as the ones developed in [9,10,14] to argument the fact that the infimum of the energy in certain subspaces is achieved. We then estimate the rate of increase of the energy to show that the solutions obtained are not radial.
2. Nonexistence By taking u(x) = |x|a v(x) in (1) we get the equivalent problem for v
−v + h(x)v = v 2 N
∗ −1
,
v > 0 in R , v ∈ D0 (RN ),
(5)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
23
where h(x) =
2 − 20 N − 2 − 2a . , and as before = 2 2 |x|
Theorem 1.4 in [5] (see part (iv) in Theorem 1 above) can be restated in our context as: Theorem 4. Eventually after a dilation v(x) → 0 v(x), any solution v of (5) has the property that for any ∈ SN−1 ⊂ RN , the function N −2 f (s) = s 2 v(s ) satisfies
1 = f (s) f s
and
f (s) is strictly decreasing f or s > 1.
Similarly to the localization of the Pohozaev identity in [7] (see also [4]), we establish the following: Lemma 5. Let ∈ SN−1 ⊂ RN , and for a fixed, let v be a solution of (5). For any 0 < < 21 , there exists a positive constant C depending only on , such that
(2 − 20 )
B ()
v 2 dx C
B2 ()\B ()
v 2 dx +
B2 ()\B ()
∗ v 2 dx .
Proof. For notational convenience, we shall make a change of variables x → x − , so that the origin becomes the centers of the balls B and B2 , and Eq. (5) reads the same, but now h(x) =
2 − 20 . |x − |2
Define a smooth cut-off function 0 1, such that ≡ 1 on B , ≡ 0 outside B2 . We shall assume that has radial symmetry and as a function of r = |x| is a decreasing function. Moreover, we assume that there exists a constant C > 0 independent of such that | |
C ,
| |
C
2
, and | |
C
3
.
(6)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
We apply the identity
|∇v|2 N −2 N −2 div (x · ∇v)∇v − x+ v∇v = x · ∇v + v v, 2 2 2 to the function v where v is a solution of (5). By integration we get
RN
N −2 x · ∇(v) + v (v) dx, 2
0= which we re-write as 0 =
RN
x · ∇(v) +
+2
RN
N −2 v v dx 2
(x · ∇v)(∇ · ∇v) dx + R,
where R =
RN
+
2v(x · ∇ )(∇ · ∇v) dx +
RN
(x · ∇(v))v dx
N −2 v(v + 2∇ · ∇v) dx. 2 RN
Since pointwise we have (x · ∇v)(∇ · ∇v) 0, it follows that −
RN
x · ∇(v) +
N −2 v v dx R. 2
Substituting for −v from (5) and performing some integration by parts, we obtain
RN
1 2 ∗ 2 2 h + x · ∇h v dx + (x · ∇ )v 2 dx R. 2 N RN
Since for any x ∈ B2 it holds that 1−x· 1 − 2 1 (2 − 20 ) h + x · ∇h = (2 − 20 ) 2 |x − |4 (1 + 2)4 and x · ∇ − 2C,
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
25
we get that (2 − 20 )
1 − 2 (1 + 2)4
B
v 2 dx R +
4C N
∗
B2 \B
v 2 dx.
(7)
On the other hand, after some integration by parts, we have
N v 2 ∇ · ∇(x · ∇ ) + N 2 R 1 N −2 2 + x · ∇() + |∇ | dx. 2 2
R =−
(8)
Therefore, from (6) and (8) we get that there exists a constant C > 0 such that R
C
2
B2 \B
v 2 dx.
Substituting in (7), we obtain (2 − 20 )
1 − 2 (1 + 2)4
B
v 2 dx
C
2
which concludes the proof of the lemma.
B2 \B
v 2 dx +
4C N
∗
B2 \B
v 2 dx,
(9)
Lemma 6. Let 0 1 be a smooth cut-off function with support ⊂ RN . For any v solution of (5) it holds that
RN
v 2 (h2 − |∇ |2 ) dx
2∗
RN
(v) dx
2 2∗
v
2∗
dx
2∗ −2 ∗ 2
− S0 .
Proof. Multiply (5) by 2 v to get ∗ −div 2 v∇v + ∇v · ∇(2 v) + h2 v 2 = 2 v 2 , which is 1 ∗ −div 2 v∇v + 2 |∇v|2 + ∇(v 2 ) · ∇(2 ) + h2 v 2 = 2 v 2 . 2
(10)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
By using the identity 1 2 |∇v|2 + ∇(v 2 ) · ∇(2 ) = |∇(v)|2 − v 2 |∇ |2 , 2 in (10) and integrating, we get |∇(v)|2 dx + N
R
N
R
v 2 (h2 − |∇ |2 ) dx =
(11)
∗
N
R
2 v 2 dx.
(12)
Using Sobolev inequality
S0
2∗
RN
(v) dx
2 2∗
RN
|∇(v)|2 dx,
on the left-hand side of (12), and Hölder inequality RN
2 2∗ −2
(v) v
dx
RN
(v) dx
on the right-hand side, we get that
2∗ 2 ∗ S0 (v)2 dx + N
R
RN
2∗
2∗
(v) dx
2 2∗
N
R
which is the inequality we want to prove.
v
2∗
dx
2∗ −2 ∗ 2
,
v 2 (h2 − |∇ |2 ) dx
2 2∗
v
2∗
dx
2∗ −2 ∗ 2
,
The proof of Theorem 2 will proceed by contradiction. From now on we shall assume that there exists a sequence n → ∞ and corresponding solutions {vn } of (5) satisfying the symmetry ensured by Theorem 4, with energy En (vn ) M. We convene to denote chosen subsequences of {vn } by the same symbols, rather than {vnk }. Multiplying (5) by vn and integrating by parts we obtain that ∗ |∇vn |2 + hn (x)vn2 dx = vn2 dx. (13) RN
RN
From (13) we have
S0 En (vn ) =
RN
=
|∇vn |
2
R
N
∗ vn2
dx
+ hn (x)vn2 dx
2∗ −2 ∗ 2
M.
2∗ −2 ∗ 2
(14)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
27
Following [7], we say x ∈ RN is a concentration point for the sequence {vn } if and only if for any > 0, lim sup n→∞
∗
B (x)
vn2 dx > 0.
We remark that 0 ∈ RN is a possible concentration point, and in this eventuality, due to the assumption that the functions {vn } have the symmetry in Theorem 4, infinity will also be a concentration point for {vn } in the sense that for any > 0,
∗
lim sup
RN \B 1 (x)
n→∞
vn2 dx > 0.
Associated to the sequence {vn }, denote K = {x ∈ RN | x = 0, x is a concentration point }. We shall prove that K is non-empty finite set contained in SN−1 ⊂ RN . Lemma 7. Denote
1 = {x ∈ RN |
1 1 < |x| < 2} and 2 = {x ∈ RN | < |x| < 4}. 2 4
For sufficiently large n we have 2
∗
vn2 dx S02
2∗ ∗ −2
.
Proof. Let
+ = {x ∈ RN | 1 < |x| < 2} and − = {x ∈ RN |
1 < |x| < 1}. 2
Since we assume that {vn } are “normalized” as in Theorem 4, we have that x (|x|) > f x (2|x|) for all x ∈ + f |x| |x|
and
x (|x|) > f x f |x| |x|
1 |x| 2
for all x ∈ − .
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
That means that for all n, vn (x) > 2
N −2 2
vn (2x) on +
(15)
and vn (x) > 2−
N −2 2
vn
1 x 2
on − .
(16)
Inequalities (15) and (16) imply that for all n,
vn2 vn2 dx < dx. 2 2 1 |x| 2 \1 |x|
(17)
Define a smooth cut-off function 0 1, such that ≡ 1 on 1 , and ≡ 0 outside 2 . If the lemma does not hold, then by using the function above in Lemma 6 we obtain that there exists a subsequence n → ∞ such that RN
vn2 (hn 2 − |∇ |2 ) dx 0.
That is (2n − 20 )
2
2 2 v dx |x|2 n
2 \1
vn2 |∇ |2 dx.
From this inequality and inequality (17) it follows that vn2 vn2 vn2 2 2 dx 16||∇ || dx 16||∇ || dx. ∞ ∞ 2 2 2 1 |x| 2 \1 |x| 1 |x|
(2n − 20 )
But this implies that n is bounded, which contradicts our assumption that the Lemma does not hold. The same argument used to prove (17) shows 2 \1
∗ vn2
dx <
1
∗
vn2 dx.
(18)
Together with Lemma 7, (18) implies that for n large ∗ 1 2∗2−2 ∗ < vn2 dx. S 2 0 1
(19)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
29
Assuming that K is empty, we have that for any x ∈ 1 , there exists a = (x) < such that ∗ lim vn2 dx = 0.
1 4
n→∞ B (x)
Since 1 is compact, one can therefore find x1 , . . . , xk ∈ 1 , and 0 < 1 , . . . , k < 41 , such that
1 ⊂ ∪ki=1 Bi (xi ) and
∗ lim v2 n→∞ B (x ) n i i
dx = 0.
Therefore, for ε > 0 there exists nε such that for all n nε and for all i = 1, . . . , k, we have ∗ vn2 dx < ε. Bi (xi )
Taking ε = non-empty.
2∗ ∗
S02 −2 2k
, we obtain a contradiction with (19), which proves that K ∩ 1 is
Lemma 8. Let x ∈ K and consider a positive such that 2 < |x|. Then lim sup n→∞
∗
B2 (x)
vn2 dx S02
2∗ ∗ −2
.
Proof. Note that (14) implies that
2∗
vn2 M 2∗ −2 dx < . 2 2n − 20 RN |x|
(20)
Consider a smooth cut-off function 0 1, such that ≡ 1 on B (x), and ≡ 0 outside B2 (x). In Lemma 6 we disregard the positive term on the left-hand side, and making use of (20), we obtain
o(1) = −
N
R
vn2 |∇ |2 dx
∗
N
R
(vn )2 dx
2 2∗
∗
B2 (x)
vn2 dx
2∗ −2 ∗ 2
− S0 .
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Since lim sup n→∞
∗
B (x)
vn2 dx > 0,
we obtain the claim of the lemma.
2∗ 2∗ −2
Therefore K is finite, for otherwise, there exist k > , x1 , . . . , xk ∈ K, and > 0 such that B2 (xi ) ∩ B2 (xj ) = ∅ when i = j . But then for n large RN
k
∗
vn2 dx >
i=1
∗
B2 (xi )
2M S0
2∗
vn2 dx > M 2∗ −2
which contradicts (14). Because of the symmetry and the monotonicity property in Theorem 4, if x ∈ K then |x|x 2 ∈ K, and the whole line segment joining x and |x|x 2 will be included in K. If |x| = 1 we obtain a contradiction with K being finite. We can now follow the arguments in [4] to prove Lemma 9. Let be a compact domain in RN which does not contain concentration points for {vn }. Then vn → 0 uniformly on , as n → ∞. Proof. First we claim that
∗ lim v2 n→∞ n
dx = 0.
(21)
Let 0 be the distance between and the set of concentration points. Since any x ∈ is not a concentration point, there exists 0 < = (x) < 0 such that lim
n→∞
∗
B (x)
vn2 dx = 0.
Since is compact, one can therefore find x1 , . . . , xk ∈ , and 0 < 1 , . . . , k < 0 , such that
⊂ ∪ki=1 Bi (xi ) and
∗ v2 lim n→∞ B (x ) n i i
from which (21) follows.
dx = 0
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
31
Let 0 < 2 < 0 , and consider
= ∪x∈ B2 (x). Fix r 1. According to (21) there exists n,r such that for all n n,r we have 1 S0
∗ vn2
dx
2∗ −2 ∗ 2
1 . r +1
(22)
For x ∈ , define a smooth cut-off function 0 1, such that ≡ 1 on B (x), ≡ 0 outside B2 (x), and |∇ | 2 . Multiply (5) by 2 vnr and integrate by parts. For convenience we shall disregard the indices. We obtain RN
∇v · ∇(2 v r ) + h2 v r+1 dx =
RN
2 v 2
∗ +r−1
dx.
Disregarding the positive term on the left-hand side, and by Hölder and then by Sobolev inequality we obtain RN
2 r
∇v · ∇( v ) dx
B2 (x)
1 S0
v
2∗
dx
B2 (x)
v
2∗
2∗ −2 ∗ 2
RN
dx
2∗ r+1 2∗ 2 v 2 dx
2∗ −2 ∗ 2
RN
r+1 2 ∇ v 2 dx.
By taking the difference of the identities below r+1 2 2 r + 1 2 r−1 v |∇v|2 + 2v r ∇v · ∇ ∇ v 2 dx = N 2 r + 1 RN R 2 r+1 + v |∇ |2 dx r +1 and RN
2 r
∇v · ∇( v ) dx =
RN
r 2 v r−1 |∇v|2 + 2v r ∇v · ∇ dx,
one obtains r+1 2 2 r − 1 2 r−1 2 ∇v · ∇(2 v r ) − v |∇v|2 ∇ v dx = N r + 1 RN 2 R 2 v r+1 |∇ |2 dx. + r +1
(23)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Therefore r+1 2 2 ∇ v 2 dx r + 1 RN ∇v · ∇(2 v r ) dx + RN
2 v r+1 |∇ |2 dx. r + 1 RN
(24)
For n n,r , using (22) and (23) in (24), we obtain 1 r + 1 RN
r+1 2 ∇ vn 2 dx 2 v r+1 |∇ |2 dx. r + 1 RN n
By Sobolev inequality it follows that S0
RN
r+1 2
2∗
vn
2 2∗
2
dx
RN
vnr+1 |∇ |2 dx
and since ||∇ ||2∞ 4/2 , we obtain
S0 2N+3
1
N
B (x)
2∗ (r+1) 2
vn
dx
2 2∗
1 (2)N
B2 (x)
vnr+1 dx.
(25)
Starting with r = 2∗ − 1, and by iterating (25), we get that for any x ∈ and any n n,r , we have
S0 2N+3
N 2
∗ vn2 (x)
1 (2)N
B2 (x)
∗ vn2
1 ∗ dx v 2 dx. (2)N n
From (21)
∗
vn2 dx → 0,
hence the pointwise convergence on is uniform.
Proof of Theorem 2. Let 0 < < 1/4 be sufficiently small so that balls centered at points in K with radius 4 are disjoint. We define a smooth cut-off function 0 1, such that ≡ 1 on 1 \ ∪∈K B2 (), and ≡ 0 on ∪∈K B () and outside 2 . Multiplying (5) by 2 vn and integrating we obtain
N
R
2 |∇vn |2 + 2vn (∇ · ∇vn ) + hn 2 vn2 dx =
∗
N
R
2 vn2 dx.
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
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Therefore
2 |∇vn |2 dx − 2
RN
+
RN
hn 2 vn2 dx
1
1
2
2 |∇vn |2 dx
RN
RN
|∇ |2 vn2 dx
2
∗
RN
2 vn2 dx.
∗ By adding RN |∇ |2 vn2 dx − RN 2 vn2 dx on both sides above, we obtain
∗ (hn − vn2 −2 )2 vn2 dx N
R
|∇ |2 vn2 dx.
N
R
Therefore
(2n − 20 )
1 \∪∈K B2 ()
∗
1 vn2 −2 − |x|2 2n − 20
||∇ ||2∞
∈K
vn2 dx
B2 ()\B ()
vn2 dx +
2 \1
vn2 dx .
From Lemma 9 we have that for sufficiently large n, ∗
1 1 vn (x)2 −2 2− 2 on 1 \ ∪∈K B2 () 8 |x| n − 20 and using (17) we obtain (2n − 20 )
1 \∪∈K B2 ()
vn2 dx C
1
vn2 dx.
(26)
We obtain a complementary inequality from Lemma 5. We take = 2 , therefore the balls with radius 2 centered at points in K are disjoint. For ∈ K, and for sufficiently large n, using that vn → 0 uniformly on B2 () \ B (), we get (2n
− 20 )
B2 ()
vn2 dx C
1
vn2 dx.
(27)
Summing (26) and (27) for all ∈ K, we obtain that n is bounded, which contradicts our assumption.
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
3. Multiplicity Proof of Theorem 3. As in [5], consider the cylinder C = R × SN−1 ⊂ RN+1 . Define the conformal diffeomorphism
N
ϑ : R \ {0} → C
x given by ϑ(x) = − ln |x|, |x|
= (t, ).
The transformation Υa : H (C) → Da (RN )
Υa v = u
with
u(x) = |x|− v(ϑ(x))
is a Hilbert space isomorphism. Here H (C) is obtained by completion of smooth functions with compact support in C under the norm ||v||2 =
C
|∇v|2 + 2 v 2 d .
We remark that as a function space, H (C) is independent of . We keep the index however, to indicate which inner product is used. If u is solution of (1) then v = Υa−1 u satisfies the equation
− v + 2 v = v 2
∗ −1
,
(28)
v > 0 in C, v ∈ H (C). One can check that for any v ∈ H (C) and Υa (v) = u ∈ Da (RN ), we have
|∇v|2 + 2 v 2 d ∗ , 2∗ d 2/2 v C
C
E (u) = F (v) := therefore S =
inf
u∈Da (RN )\{0}
E (u) =
inf
v∈H (C )\{0}
F (v).
Let C be the cylinder of radius and v(t, ) = w(t, ). Then C
F (v) = I (w) :=
|∇w|2 + w 2 d 2/2∗ . ∗ 2 C w d
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
35
Consider N = N1 +N2 , with N1 , N2 2. Let the group G = Z2 ×O(N1 )×O(N2 ) act by isometries on C , where the Z2 part takes (t, ) to (t , ) and O is the orthogonal group acting on . Denote by HG (C ) the subspace in H (C ), of functions left unchanged by the action of G. Let S,G =
inf
w∈HG (C )\{0}
I (w).
It is by now standard (see [14,9]) that S,G is achieved by a function w . We claim that 2 , and let (x , . . . , x ) = for sufficiently large, w is not radial. Let r 2 = x12 +· · ·+xN 1 N1 1 (r) be a smooth non-negative function, with support in the unit ball in RN1 . Consider on C a nonzero test function w = f (t)(r) ∈ HG (C ) where f (−t) = f (t) and f (t) ≡ 0 for |t| 1. Let
= {(t, ) ∈ C | t ∈ (−1, 1) and 21 + · · · + 2N1 < 1} and note that it has measure increasing with as ( ) = O N2 −1 . We then estimate 2(N2 −1) 2∗ −2 S,G I (w) = O ( ) 2∗ = O N . 2(N −1) N
Since the energy of a radial function increases as large, S,G cannot be attained by a radial solution.
, we have that for sufficiently
In closing, we remark that a more detailed analysis in dimensions N 4 may employ the method of [10] to obtain that the number of solutions of (1) tends to infinity as → ∞. It would be interesting to classify all solutions of (1) as increases to infinity. Another very interesting question is the following. Is it true that in dimension N = 3 any solution of (1) has spherical symmetry? Acknowledgments The author would like to thank the referee for suggestions toward improving the organization of the paper, and for bringing to our attention some of the references below. References [1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, New York, 1998.
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
[2] H. Brezis, Elliptic equations with limiting Sobolev exponents—the impact of topology, Comm. Pure Appl. Math. 39 (1986) S17–S39. [3] M. Ben Ayed, K. El Mehdi, M. Hammami, A nonexistence result for Yamabe type problems on thin annuli, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (5) (2002) 715–744. [4] D. Castorina, G. Mancini, Non existence of bounded-energy solutions for some semilinear elliptic equations with a large parameter, Rend. Sem. Mat. Univ. Padova 110 (2003) 147–160. [5] F. Catrina, Z.-Q. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001) 229– 258. [6] F. Catrina, Z.-Q. Wang, Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN , Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2) (2001) 157–178. [7] O. Druet, E. Hebey, M. Vaugon, Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case, Nonlinear Anal. Ser. A: Theory Methods 51 (1) (2002) 79–94. [8] N. Ghoussoub, C. Gui, M. Zhu, On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations 26 (11–12) (2001) 1929–1946. [9] E. Hebey, M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl. (9) 76 (10) (1997) 859–881. [10] Y.-Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations 83 (2) (1990) 348–367. [11] C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1) (1988) 1–27. [12] O. Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl. 81 (2002) 655–696. [13] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (2) (1996) 241–264. [14] D. Weiyue, On a conformally invariant elliptic equation on RN , Comm. Math. Phys. 107 (1986) 331–335. [15] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21–37.
Nonexistence of bounded energy solutions for a critical exponent problem Florin Catrina Department of Mathematics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, USA Received 19 May 2004; revised 1 December 2004 Available online 20 January 2005
Abstract In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in RN . The weights depend on a positive parameter , which is allowed to increase to infinity. While for small values of solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as increases to infinity. We also prove a multiplicity result for high energy solutions. © 2004 Elsevier Inc. All rights reserved. Keywords: Bounded energy; Critical exponent
1. Introduction In this paper we are concerned with the problem
∗ ∗ −div |x|−2a ∇u = |x|−2 a u2 −1 , u > 0 in RN , u ∈ Da (RN ).
E-mail address: [email protected]. 0022-0396/$ - see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2004.12.002
(1)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Throughout we shall assume 3 N,
a<
N −2 , 2
and
2∗ =
2N . N −2
(2)
For convenience, we use interchangeably either the parameter a or
=
N − 2 − 2a N −2 > 0, and we denote 0 = . 2 2
Here Da (RN ) denotes the Hilbert space obtained as the completion of smooth functions with compact support in RN under the norm induced by the inner product (u, v)a =
RN
|x|−2a ∇u · ∇v dx.
Problems of this type and more general ones, have been the subject of many recent studies. There are several motivations that spark interest in such equations. One is that the problems are variational. A second reason is the invariance under a non-compact group of symmetries. Non-compactness is due to dilations about the origin. This makes the variational treatment non-trivial. The third reason is the existence of the parameter a, which makes the structure of solutions change. One more interesting aspect of (1) is the critical Sobolev exponent in the nonlinearity. This allows a local “almost invariance" under dilations at any point. The un-weighted case (when a = 0) of (1) is known as the Yamabe problem in (See [1]) RN . Existence of a solution implies the existence of a metric on RN , conformal to the standard metric, but with positive constant scalar curvature. Yamabe [15] stated the problem of finding metrics with constant scalar curvature conformal with one given on a Riemannian manifold, as a first step in an attempt to solving the Poincaré conjecture. In fact, as will be explained in Section 3, problem (1) may be also viewed as a modification of the Yamabe problem on C = R × SN−1 ⊂ RN+1 . It turns out that after a suitable transformation, the equation on C reads −v + 2 v = v 2
∗ −1
(3)
(recall that = N−2−2a ). Exactly this type of equations have been heavily studied 2 on bounded domains in RN with Neumann boundary conditions. This work had been started by Lin et al. [11] in connection with the Gierer–Meinhardt model in mathematical biology. We also remark that the cylinder C has nontrivial topology. As is noted in [2], this fact contributes to existence of more solutions in problems involving critical exponents. Being at the intersection of several extremely rich areas of research, it is natural to inquire which of the existing results continue to hold for our problem.
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
21
As suggested before, we shall take a variational approach. Solutions of (1) are critical points of the energy functional
−2a |∇u|2 dx N |x| Ea (u) = R ∗. −2∗ a |u|2∗ dx 2/2 RN |x|
(4)
We summarize several known facts about problem (1) in the theorem below. For proofs, please see [13,5], and the references therein. Theorem 1. (i) The functions u,a (x) = and Ua (x) =
N −2−2a 2
Ua (x) are solutions of (1) when > 0
N(N−2−2a)2 N−2
1 + |x|
N −2 4
2(N −2−2a) N −2
N 2−2
with energy
Ea (u,a ) =
N (N − 2 − 2a) (N − 2)
2(N −1) N
N −2 N
N2 N−1 2 N2 . 2(N )
(ii) For 0 a < N−2 2 , i.e. ∈ (0, 0 ], the infimum of energy (4) is achieved by the functions u,a above. (iii) For a < 0, i.e. > 0 , the infimum of energy (4) is
N2 N−1 2 N2 S0 = N (N − 2) 2(N ) and it is not achieved. N −2−2a (iv) Possibly after a dilation u(x) = 2 u(x), any solution u(x) of (1) satisfies the modified inversion symmetry
u
x |x|2
= |x|N−2−2a u(x).
Moreover, for any ∈ SN−1 , and any s > 0, the function f (s) = s strictly decreasing on the interval (1, ∞). The main new theorem in this paper is
N −2−2a 2
u(s ) is
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Theorem 2 (Nonexistence). For any integer N 3 and any positive constant M, there exists M > 0 such that for any > M , problem (1) has no solution with energy E (u) M. As the existence of radial solutions shows, the nonexistence result above does not hold if one lifts the bounded energy condition. In fact we show that in most cases, nonradial solutions should be expected. Precisely, we prove Theorem 3 (Multiplicity). For N 4 there exists N > 0 such that for all > N problem (1) has a nonradial solution (different than u,a in Theorem 1). The method of proof for this result indicates that although solutions with low energy do not exist, there is a rich variety of solutions for large values of . It is instructive to compare the structure of solutions of (1) with the subcritical case. Similar problems which have a power nonlinearity but with a subcritical exponent were studied in [6]. It is very interesting that in this case, as tends to infinity, in any dimension N 2 the number of distinct solutions increases to infinity. The technique was to restrict the energy functional to subspaces of functions invariant under group actions with discrete orbits. We proved that sequences along which the energy tends to a (local) minimum in these subspaces are convergent to critical points. In the case of critical exponent however, by increasing the parameter , Palais–Smale sequences have the tendency to concentrate and blow-up at the orbits where we try to locate most of the mass. One way to overcome this loss of compactness is to work in subspaces of functions invariant under group actions with continuous orbits. In these situations, one regains compactness but the solutions obtained have energy increasing to infinity as various powers of the parameter. In the proof of Theorem 2 we follow closely the arguments in [7], where the authors use a localized version of Pohozaev’s identity to prove nonexistence of positive solutions with bounded energy for Eq. (3) set on compact Riemannian manifolds. For results in the same spirit, we also refer to [3,4,8,12]. The a priori symmetry of solutions given by Theorem 1 (iv), plays an essential role in overcoming the fact that the domain is not compact, and in adapting the method in [7]. In Theorem 3, we employ previous compactness under symmetry results such as the ones developed in [9,10,14] to argument the fact that the infimum of the energy in certain subspaces is achieved. We then estimate the rate of increase of the energy to show that the solutions obtained are not radial.
2. Nonexistence By taking u(x) = |x|a v(x) in (1) we get the equivalent problem for v
−v + h(x)v = v 2 N
∗ −1
,
v > 0 in R , v ∈ D0 (RN ),
(5)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
23
where h(x) =
2 − 20 N − 2 − 2a . , and as before = 2 2 |x|
Theorem 1.4 in [5] (see part (iv) in Theorem 1 above) can be restated in our context as: Theorem 4. Eventually after a dilation v(x) → 0 v(x), any solution v of (5) has the property that for any ∈ SN−1 ⊂ RN , the function N −2 f (s) = s 2 v(s ) satisfies
1 = f (s) f s
and
f (s) is strictly decreasing f or s > 1.
Similarly to the localization of the Pohozaev identity in [7] (see also [4]), we establish the following: Lemma 5. Let ∈ SN−1 ⊂ RN , and for a fixed, let v be a solution of (5). For any 0 < < 21 , there exists a positive constant C depending only on , such that
(2 − 20 )
B ()
v 2 dx C
B2 ()\B ()
v 2 dx +
B2 ()\B ()
∗ v 2 dx .
Proof. For notational convenience, we shall make a change of variables x → x − , so that the origin becomes the centers of the balls B and B2 , and Eq. (5) reads the same, but now h(x) =
2 − 20 . |x − |2
Define a smooth cut-off function 0 1, such that ≡ 1 on B , ≡ 0 outside B2 . We shall assume that has radial symmetry and as a function of r = |x| is a decreasing function. Moreover, we assume that there exists a constant C > 0 independent of such that | |
C ,
| |
C
2
, and | |
C
3
.
(6)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
We apply the identity
|∇v|2 N −2 N −2 div (x · ∇v)∇v − x+ v∇v = x · ∇v + v v, 2 2 2 to the function v where v is a solution of (5). By integration we get
RN
N −2 x · ∇(v) + v (v) dx, 2
0= which we re-write as 0 =
RN
x · ∇(v) +
+2
RN
N −2 v v dx 2
(x · ∇v)(∇ · ∇v) dx + R,
where R =
RN
+
2v(x · ∇ )(∇ · ∇v) dx +
RN
(x · ∇(v))v dx
N −2 v(v + 2∇ · ∇v) dx. 2 RN
Since pointwise we have (x · ∇v)(∇ · ∇v) 0, it follows that −
RN
x · ∇(v) +
N −2 v v dx R. 2
Substituting for −v from (5) and performing some integration by parts, we obtain
RN
1 2 ∗ 2 2 h + x · ∇h v dx + (x · ∇ )v 2 dx R. 2 N RN
Since for any x ∈ B2 it holds that 1−x· 1 − 2 1 (2 − 20 ) h + x · ∇h = (2 − 20 ) 2 |x − |4 (1 + 2)4 and x · ∇ − 2C,
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
25
we get that (2 − 20 )
1 − 2 (1 + 2)4
B
v 2 dx R +
4C N
∗
B2 \B
v 2 dx.
(7)
On the other hand, after some integration by parts, we have
N v 2 ∇ · ∇(x · ∇ ) + N 2 R 1 N −2 2 + x · ∇() + |∇ | dx. 2 2
R =−
(8)
Therefore, from (6) and (8) we get that there exists a constant C > 0 such that R
C
2
B2 \B
v 2 dx.
Substituting in (7), we obtain (2 − 20 )
1 − 2 (1 + 2)4
B
v 2 dx
C
2
which concludes the proof of the lemma.
B2 \B
v 2 dx +
4C N
∗
B2 \B
v 2 dx,
(9)
Lemma 6. Let 0 1 be a smooth cut-off function with support ⊂ RN . For any v solution of (5) it holds that
RN
v 2 (h2 − |∇ |2 ) dx
2∗
RN
(v) dx
2 2∗
v
2∗
dx
2∗ −2 ∗ 2
− S0 .
Proof. Multiply (5) by 2 v to get ∗ −div 2 v∇v + ∇v · ∇(2 v) + h2 v 2 = 2 v 2 , which is 1 ∗ −div 2 v∇v + 2 |∇v|2 + ∇(v 2 ) · ∇(2 ) + h2 v 2 = 2 v 2 . 2
(10)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
By using the identity 1 2 |∇v|2 + ∇(v 2 ) · ∇(2 ) = |∇(v)|2 − v 2 |∇ |2 , 2 in (10) and integrating, we get |∇(v)|2 dx + N
R
N
R
v 2 (h2 − |∇ |2 ) dx =
(11)
∗
N
R
2 v 2 dx.
(12)
Using Sobolev inequality
S0
2∗
RN
(v) dx
2 2∗
RN
|∇(v)|2 dx,
on the left-hand side of (12), and Hölder inequality RN
2 2∗ −2
(v) v
dx
RN
(v) dx
on the right-hand side, we get that
2∗ 2 ∗ S0 (v)2 dx + N
R
RN
2∗
2∗
(v) dx
2 2∗
N
R
which is the inequality we want to prove.
v
2∗
dx
2∗ −2 ∗ 2
,
v 2 (h2 − |∇ |2 ) dx
2 2∗
v
2∗
dx
2∗ −2 ∗ 2
,
The proof of Theorem 2 will proceed by contradiction. From now on we shall assume that there exists a sequence n → ∞ and corresponding solutions {vn } of (5) satisfying the symmetry ensured by Theorem 4, with energy En (vn ) M. We convene to denote chosen subsequences of {vn } by the same symbols, rather than {vnk }. Multiplying (5) by vn and integrating by parts we obtain that ∗ |∇vn |2 + hn (x)vn2 dx = vn2 dx. (13) RN
RN
From (13) we have
S0 En (vn ) =
RN
=
|∇vn |
2
R
N
∗ vn2
dx
+ hn (x)vn2 dx
2∗ −2 ∗ 2
M.
2∗ −2 ∗ 2
(14)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
27
Following [7], we say x ∈ RN is a concentration point for the sequence {vn } if and only if for any > 0, lim sup n→∞
∗
B (x)
vn2 dx > 0.
We remark that 0 ∈ RN is a possible concentration point, and in this eventuality, due to the assumption that the functions {vn } have the symmetry in Theorem 4, infinity will also be a concentration point for {vn } in the sense that for any > 0,
∗
lim sup
RN \B 1 (x)
n→∞
vn2 dx > 0.
Associated to the sequence {vn }, denote K = {x ∈ RN | x = 0, x is a concentration point }. We shall prove that K is non-empty finite set contained in SN−1 ⊂ RN . Lemma 7. Denote
1 = {x ∈ RN |
1 1 < |x| < 2} and 2 = {x ∈ RN | < |x| < 4}. 2 4
For sufficiently large n we have 2
∗
vn2 dx S02
2∗ ∗ −2
.
Proof. Let
+ = {x ∈ RN | 1 < |x| < 2} and − = {x ∈ RN |
1 < |x| < 1}. 2
Since we assume that {vn } are “normalized” as in Theorem 4, we have that x (|x|) > f x (2|x|) for all x ∈ + f |x| |x|
and
x (|x|) > f x f |x| |x|
1 |x| 2
for all x ∈ − .
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
That means that for all n, vn (x) > 2
N −2 2
vn (2x) on +
(15)
and vn (x) > 2−
N −2 2
vn
1 x 2
on − .
(16)
Inequalities (15) and (16) imply that for all n,
vn2 vn2 dx < dx. 2 2 1 |x| 2 \1 |x|
(17)
Define a smooth cut-off function 0 1, such that ≡ 1 on 1 , and ≡ 0 outside 2 . If the lemma does not hold, then by using the function above in Lemma 6 we obtain that there exists a subsequence n → ∞ such that RN
vn2 (hn 2 − |∇ |2 ) dx 0.
That is (2n − 20 )
2
2 2 v dx |x|2 n
2 \1
vn2 |∇ |2 dx.
From this inequality and inequality (17) it follows that vn2 vn2 vn2 2 2 dx 16||∇ || dx 16||∇ || dx. ∞ ∞ 2 2 2 1 |x| 2 \1 |x| 1 |x|
(2n − 20 )
But this implies that n is bounded, which contradicts our assumption that the Lemma does not hold. The same argument used to prove (17) shows 2 \1
∗ vn2
dx <
1
∗
vn2 dx.
(18)
Together with Lemma 7, (18) implies that for n large ∗ 1 2∗2−2 ∗ < vn2 dx. S 2 0 1
(19)
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
29
Assuming that K is empty, we have that for any x ∈ 1 , there exists a = (x) < such that ∗ lim vn2 dx = 0.
1 4
n→∞ B (x)
Since 1 is compact, one can therefore find x1 , . . . , xk ∈ 1 , and 0 < 1 , . . . , k < 41 , such that
1 ⊂ ∪ki=1 Bi (xi ) and
∗ lim v2 n→∞ B (x ) n i i
dx = 0.
Therefore, for ε > 0 there exists nε such that for all n nε and for all i = 1, . . . , k, we have ∗ vn2 dx < ε. Bi (xi )
Taking ε = non-empty.
2∗ ∗
S02 −2 2k
, we obtain a contradiction with (19), which proves that K ∩ 1 is
Lemma 8. Let x ∈ K and consider a positive such that 2 < |x|. Then lim sup n→∞
∗
B2 (x)
vn2 dx S02
2∗ ∗ −2
.
Proof. Note that (14) implies that
2∗
vn2 M 2∗ −2 dx < . 2 2n − 20 RN |x|
(20)
Consider a smooth cut-off function 0 1, such that ≡ 1 on B (x), and ≡ 0 outside B2 (x). In Lemma 6 we disregard the positive term on the left-hand side, and making use of (20), we obtain
o(1) = −
N
R
vn2 |∇ |2 dx
∗
N
R
(vn )2 dx
2 2∗
∗
B2 (x)
vn2 dx
2∗ −2 ∗ 2
− S0 .
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Since lim sup n→∞
∗
B (x)
vn2 dx > 0,
we obtain the claim of the lemma.
2∗ 2∗ −2
Therefore K is finite, for otherwise, there exist k > , x1 , . . . , xk ∈ K, and > 0 such that B2 (xi ) ∩ B2 (xj ) = ∅ when i = j . But then for n large RN
k
∗
vn2 dx >
i=1
∗
B2 (xi )
2M S0
2∗
vn2 dx > M 2∗ −2
which contradicts (14). Because of the symmetry and the monotonicity property in Theorem 4, if x ∈ K then |x|x 2 ∈ K, and the whole line segment joining x and |x|x 2 will be included in K. If |x| = 1 we obtain a contradiction with K being finite. We can now follow the arguments in [4] to prove Lemma 9. Let be a compact domain in RN which does not contain concentration points for {vn }. Then vn → 0 uniformly on , as n → ∞. Proof. First we claim that
∗ lim v2 n→∞ n
dx = 0.
(21)
Let 0 be the distance between and the set of concentration points. Since any x ∈ is not a concentration point, there exists 0 < = (x) < 0 such that lim
n→∞
∗
B (x)
vn2 dx = 0.
Since is compact, one can therefore find x1 , . . . , xk ∈ , and 0 < 1 , . . . , k < 0 , such that
⊂ ∪ki=1 Bi (xi ) and
∗ v2 lim n→∞ B (x ) n i i
from which (21) follows.
dx = 0
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
31
Let 0 < 2 < 0 , and consider
= ∪x∈ B2 (x). Fix r 1. According to (21) there exists n,r such that for all n n,r we have 1 S0
∗ vn2
dx
2∗ −2 ∗ 2
1 . r +1
(22)
For x ∈ , define a smooth cut-off function 0 1, such that ≡ 1 on B (x), ≡ 0 outside B2 (x), and |∇ | 2 . Multiply (5) by 2 vnr and integrate by parts. For convenience we shall disregard the indices. We obtain RN
∇v · ∇(2 v r ) + h2 v r+1 dx =
RN
2 v 2
∗ +r−1
dx.
Disregarding the positive term on the left-hand side, and by Hölder and then by Sobolev inequality we obtain RN
2 r
∇v · ∇( v ) dx
B2 (x)
1 S0
v
2∗
dx
B2 (x)
v
2∗
2∗ −2 ∗ 2
RN
dx
2∗ r+1 2∗ 2 v 2 dx
2∗ −2 ∗ 2
RN
r+1 2 ∇ v 2 dx.
By taking the difference of the identities below r+1 2 2 r + 1 2 r−1 v |∇v|2 + 2v r ∇v · ∇ ∇ v 2 dx = N 2 r + 1 RN R 2 r+1 + v |∇ |2 dx r +1 and RN
2 r
∇v · ∇( v ) dx =
RN
r 2 v r−1 |∇v|2 + 2v r ∇v · ∇ dx,
one obtains r+1 2 2 r − 1 2 r−1 2 ∇v · ∇(2 v r ) − v |∇v|2 ∇ v dx = N r + 1 RN 2 R 2 v r+1 |∇ |2 dx. + r +1
(23)
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
Therefore r+1 2 2 ∇ v 2 dx r + 1 RN ∇v · ∇(2 v r ) dx + RN
2 v r+1 |∇ |2 dx. r + 1 RN
(24)
For n n,r , using (22) and (23) in (24), we obtain 1 r + 1 RN
r+1 2 ∇ vn 2 dx 2 v r+1 |∇ |2 dx. r + 1 RN n
By Sobolev inequality it follows that S0
RN
r+1 2
2∗
vn
2 2∗
2
dx
RN
vnr+1 |∇ |2 dx
and since ||∇ ||2∞ 4/2 , we obtain
S0 2N+3
1
N
B (x)
2∗ (r+1) 2
vn
dx
2 2∗
1 (2)N
B2 (x)
vnr+1 dx.
(25)
Starting with r = 2∗ − 1, and by iterating (25), we get that for any x ∈ and any n n,r , we have
S0 2N+3
N 2
∗ vn2 (x)
1 (2)N
B2 (x)
∗ vn2
1 ∗ dx v 2 dx. (2)N n
From (21)
∗
vn2 dx → 0,
hence the pointwise convergence on is uniform.
Proof of Theorem 2. Let 0 < < 1/4 be sufficiently small so that balls centered at points in K with radius 4 are disjoint. We define a smooth cut-off function 0 1, such that ≡ 1 on 1 \ ∪∈K B2 (), and ≡ 0 on ∪∈K B () and outside 2 . Multiplying (5) by 2 vn and integrating we obtain
N
R
2 |∇vn |2 + 2vn (∇ · ∇vn ) + hn 2 vn2 dx =
∗
N
R
2 vn2 dx.
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
33
Therefore
2 |∇vn |2 dx − 2
RN
+
RN
hn 2 vn2 dx
1
1
2
2 |∇vn |2 dx
RN
RN
|∇ |2 vn2 dx
2
∗
RN
2 vn2 dx.
∗ By adding RN |∇ |2 vn2 dx − RN 2 vn2 dx on both sides above, we obtain
∗ (hn − vn2 −2 )2 vn2 dx N
R
|∇ |2 vn2 dx.
N
R
Therefore
(2n − 20 )
1 \∪∈K B2 ()
∗
1 vn2 −2 − |x|2 2n − 20
||∇ ||2∞
∈K
vn2 dx
B2 ()\B ()
vn2 dx +
2 \1
vn2 dx .
From Lemma 9 we have that for sufficiently large n, ∗
1 1 vn (x)2 −2 2− 2 on 1 \ ∪∈K B2 () 8 |x| n − 20 and using (17) we obtain (2n − 20 )
1 \∪∈K B2 ()
vn2 dx C
1
vn2 dx.
(26)
We obtain a complementary inequality from Lemma 5. We take = 2 , therefore the balls with radius 2 centered at points in K are disjoint. For ∈ K, and for sufficiently large n, using that vn → 0 uniformly on B2 () \ B (), we get (2n
− 20 )
B2 ()
vn2 dx C
1
vn2 dx.
(27)
Summing (26) and (27) for all ∈ K, we obtain that n is bounded, which contradicts our assumption.
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
3. Multiplicity Proof of Theorem 3. As in [5], consider the cylinder C = R × SN−1 ⊂ RN+1 . Define the conformal diffeomorphism
N
ϑ : R \ {0} → C
x given by ϑ(x) = − ln |x|, |x|
= (t, ).
The transformation Υa : H (C) → Da (RN )
Υa v = u
with
u(x) = |x|− v(ϑ(x))
is a Hilbert space isomorphism. Here H (C) is obtained by completion of smooth functions with compact support in C under the norm ||v||2 =
C
|∇v|2 + 2 v 2 d .
We remark that as a function space, H (C) is independent of . We keep the index however, to indicate which inner product is used. If u is solution of (1) then v = Υa−1 u satisfies the equation
− v + 2 v = v 2
∗ −1
,
(28)
v > 0 in C, v ∈ H (C). One can check that for any v ∈ H (C) and Υa (v) = u ∈ Da (RN ), we have
|∇v|2 + 2 v 2 d ∗ , 2∗ d 2/2 v C
C
E (u) = F (v) := therefore S =
inf
u∈Da (RN )\{0}
E (u) =
inf
v∈H (C )\{0}
F (v).
Let C be the cylinder of radius and v(t, ) = w(t, ). Then C
F (v) = I (w) :=
|∇w|2 + w 2 d 2/2∗ . ∗ 2 C w d
F. Catrina / J. Differential Equations 215 (2005) 19 – 36
35
Consider N = N1 +N2 , with N1 , N2 2. Let the group G = Z2 ×O(N1 )×O(N2 ) act by isometries on C , where the Z2 part takes (t, ) to (t , ) and O is the orthogonal group acting on . Denote by HG (C ) the subspace in H (C ), of functions left unchanged by the action of G. Let S,G =
inf
w∈HG (C )\{0}
I (w).
It is by now standard (see [14,9]) that S,G is achieved by a function w . We claim that 2 , and let (x , . . . , x ) = for sufficiently large, w is not radial. Let r 2 = x12 +· · ·+xN 1 N1 1 (r) be a smooth non-negative function, with support in the unit ball in RN1 . Consider on C a nonzero test function w = f (t)(r) ∈ HG (C ) where f (−t) = f (t) and f (t) ≡ 0 for |t| 1. Let
= {(t, ) ∈ C | t ∈ (−1, 1) and 21 + · · · + 2N1 < 1} and note that it has measure increasing with as ( ) = O N2 −1 . We then estimate 2(N2 −1) 2∗ −2 S,G I (w) = O ( ) 2∗ = O N . 2(N −1) N
Since the energy of a radial function increases as large, S,G cannot be attained by a radial solution.
, we have that for sufficiently
In closing, we remark that a more detailed analysis in dimensions N 4 may employ the method of [10] to obtain that the number of solutions of (1) tends to infinity as → ∞. It would be interesting to classify all solutions of (1) as increases to infinity. Another very interesting question is the following. Is it true that in dimension N = 3 any solution of (1) has spherical symmetry? Acknowledgments The author would like to thank the referee for suggestions toward improving the organization of the paper, and for bringing to our attention some of the references below. References [1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, New York, 1998.
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F. Catrina / J. Differential Equations 215 (2005) 19 – 36
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