e 2005 London Mathematical Society
Bull. London Math. Soc. 37 (2005) 119–125
C
DOI: 10.1112/S0024609304003819
ON THE EXISTENCE OF EXTREMALS FOR THE SOBOLEV
TRACE EMBEDDING THEOREM WITH CRITICAL EXPONENT
JULIÁN FERNÁNDEZ BONDER and JULIO D. ROSSI
Abstract
In this paper, the existence problem is studied for extremals of the Sobolev trace inequality
W 1,p (Ω) → Lp ∗ (∂Ω), where Ω is a bounded smooth domain in RN , p∗ = p(N − 1)/(N − p) is the
critical Sobolev exponent, and 1 < p < N .
1. Introduction.
Let Ω ⊂ R be a bounded smooth domain. The following two Sobolev inequalities
are relevant to the study of boundary-value problems for differential operators.
For each 1 q p(N −1)/(N −p) ≡ p∗ , we have a continuous inclusion W 1,p (Ω) →
Lq (∂Ω), and for each 1 r pN/(N − p) ≡ p∗ , we have W01,p (Ω) → Lr (Ω); hence
the following inequalities hold:
N
Sq upLq (∂Ω) upW 1, p (Ω) ;
S̄ r u2Lr (Ω) upW 1, p (Ω) .
0
These inequalities are known, respectively, as the Sobolev trace theorem and the
Sobolev embedding theorem. The best constants for these embeddings are the
largest S and S̄ such that the above inequalities hold; that is,
|∇v|p + |v|p dx
Ω
Sq =
inf 1, p
(1.1)
p/q
v∈W 1, p (Ω)\W0
(Ω)
|v|q dσ
∂Ω
and
|∇v|p dx
S̄ r =
inf
v∈W01, p (Ω)\{0}
Ω
p/r .
(1.2)
|v| dx
r
Ω
One big difference between these two quantities is the fact that S̄ r is homogeneous
under dilatations of the domain; that is, if we define µΩ = {µx| x ∈ Ω}, by taking
v(x) = u(µx) in (1.2) and changing variables, we obtain
S̄ r (µΩ) = µ(rN −pr−pN )/r S̄ r (Ω).
Received 17 October 2003; revised 20 April 2004.
2000 Mathematics Subject Classification 35J65 (primary), 35B33 (secondary).
Supported by Fundacion Antorchas, CONICET, ANPCyT PICT Nos 05009 and 10608, and
UBA X066.
120
julián fernández bonder and julio d. rossi
On the other hand, Sq is not homogeneous under dilatations. In fact, we have
µ−p |∇v|p + |v|p dx
β
Ω
Sq (µΩ) = µ
inf 1, p
(1.3)
p/q ,
v∈W 1, p (Ω)\W0
(Ω)
|v|q dσ
∂Ω
where β = (N q − pN + p)/q.
For 1 q < p∗ and 1 r < p∗ , the embeddings are compact, so we see the
existence of extremals (that is, functions where the infimum is attained). These
extremals are weak solutions of the following problems:

in Ω,
∆p u = |u|p−2 u
(1.4)
|∇u|p−2 ∂u = λ|u|q−2 u
on ∂Ω,
∂ν
p−2
where ∆p u = div(|∇u| ∇u) is the p−laplacian, ∂/∂ν is the outer unit normal
derivative, and
in Ω,
−∆p u = λ|u|r−2 u
(1.5)
u=0
on ∂Ω.
The asymptotic behavior of Sq (µΩ) in expanding and contracting domains
(µ → ∞ and µ → 0, respectively) is studied in [13] and [8]. In [13] it is proved for
expanding domains and p = 2, q > p, that Sq (µΩ) → Sq (RN
+ ). In [8] it is shown
that
Sq (µΩ)
|Ω|
=
.
(1.6)
lim
β
µ→0+
µ
|∂Ω|p/q
The behavior of the extremals for (1.1) in expanding and contracting domains is
also studied in [13] and [8]. For expanding domains, it is proved in [13] (again in
the case p = 2, q > p) that the extremals develop a peak near a point where the
mean curvature of the boundary maximizes. For contracting domains, we find that
the extremals, when rescaled to the original domain as v(x) = u(µx), x ∈ Ω, and
normalized with vLq (∂Ω) = 1, are nearly constant in the sense that
lim v =
µ→0
1
|∂Ω|1/q
in W 1,p (Ω).
Another big difference between the Sobolev trace theorem and the Sobolev
embedding theorem arises in the behavior of extremals. We see that if Ω is a ball,
then Ω = B(0, µ), as the extremals do not change sign; from results of [11] the
extremals for (1.2) are radial, while (at least for large µ) extremals for (1.1) are
not, since they develop peaking concentration phenomena, as is described in [13].
As for the symmetry properties of the extremals of the Sobolev trace constant, it
is proved in [10] that if Ω is a ball of sufficiently small radius, then the extremals are
radial functions. Also in [10], the authors use this result to prove that there exists
a radial extremal for the immersion H 1 (B(0, µ)) → L2∗ (∂B(0, µ)) if the radius µ is
small enough. See also [1] for other geometric conditions that lead to the existence
of extremals in the case p = 2.
So we arrive at the purpose of this paper: to show the existence of extremals for
the Sobolev trace theorem with the critical exponent in a general smooth bounded
domain Ω. Our main result is as follows.
trace embedding
Theorem A.
Let Ω be a bounded smooth domain in RN such that
|Ω|
1
,
<
K(N, p)
|∂Ω|p/p∗
121
(1.7)
where K(N, p) is given by (2.2). Then there exists an extremal for the immersion
W 1,p (Ω) → Lp∗ (∂Ω).
For the proof of Theorem A, we use the same approach as that employed in [2]
(see also [16]), appropriately adapted to our new context.
Existence results for elliptic problems with critical Sobolev exponents have rightly
attracted a great deal of attention since the pioneer work [5], and this is an intensive
area of research nowadays. The best Sobolev inequalities have been sought by many
authors, and now constitute a classical subject, going back at least as far as [2, 3].
The other key ingredient in the proof is a result given in [4], where the author
computes the optimal constant in the Sobolev trace inequality (see Theorem 2.1
below). See also [12] for a similar result in the case p = 2. For more references on
Sobolev inequalities, see [6].
Remark 1.8. Let Ω be any smooth bounded domain in RN , and let
Ωµ = µΩ = {µx | x ∈ Ω},
where µ > 0. We observe that when µ is small enough, Ωµ verifies the hypotheses
of Theorem A, and hence there is an extremal for the immersion W 1,p (Ωµ ) →
Lp∗ (∂Ωµ ). More precisely, this applies when
µ<
|∂Ω|1/p∗
1
.
1/p
K(N, p)
|Ω|1/p
Remark 1.9. We observe that with the same ideas and computations, we can
consider a problem of the form

in Ω,
∆p u = |u|p−2 u
(1.10)
|∇u|p−2 ∂u = a(x)up∗ −1 on ∂Ω,
∂ν
∞
with a ∈ L (∂Ω) bounded away from zero. This corresponds to a Sobolev trace
immersion with a weight a on the boundary. In this case, the condition on Ω and
the weight function a is
p/p∗
p/p∗
1
<
a dσ
.
|Ω| sup a
K(N, p) ∂Ω
∂Ω
Remark 1.11. Observe that from the proof of Theorem A, we obtain the
existence of extremals for every domain Ω that satisfies
1
.
(1.12)
Sp∗ = Sp∗ (Ω) <
K(N, p)
Condition (1.7) is the simplest geometric condition that ensures that (1.12) holds.
2. Proof of Theorem A
For the proof of Theorem A, we use the following theorem, due to Biezuner [4].
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julián fernández bonder and julio d. rossi
Theorem 2.1. For every ε > 0, there exists a constant B = B(ε) > 0 such that
p/p∗
p∗
p
v dσ
(K(N, p) + ε) |∇v| dx + B v p dx
∂Ω
Ω
for every v ∈ W 1,p (Ω), where
1
=
K(N, p)p
Ω
inf
∇w∈Lp (RN
+)
RN
+
|∇w|p dx
|w| dx
p∗
w∈Lp ∗ (∂RN
+)
∂RN
+
p/p∗ .
(2.2)
Remark 2.3. The constant K(N, p) in Theorem 2.1 is sharp.
Now, to prove our result, we will use the same approach as that used by [2] (see
also [16]). Let us consider, for each 1 < q p∗ , the quotient
|∇u|p + up dx
Ω
Qq (u) = (2.4)
p/q
uq dσ
∂Ω
and
Sq =
inf
u∈W 1, p (Ω)\W01, p (Ω)
Qq (u).
To simplify the notation, we denote
Q(u) = Qp∗ (u)
and
S = Sp∗ .
(2.5)
We have the following lemma.
Lemma 2.6. If q < p∗ , the constant Sq is attained.
Proof.
This follows from the compactness of the immersion W 1,p (Ω) → Lq (∂Ω).
From now on, we will denote by uq an extremal related to Sq , normalized such that
uq Lq (∂Ω) = 1. For the proof of the theorem, we need the following proposition,
which is a straightforward modification of [16].
Proposition 2.7. Let qi → r p∗ and assume that
sup uqi (x) A.
x∈Ω
Then there exists a subsequence uqi j such that uqi j → u in the sense of distributions
and u is a solution of

in Ω,
∆p u = |u|p−2 u
∂u
|∇u|p−2
= λ|u|r−2 u on ∂Ω,
∂ν
where λ = lim Sqi j . Moreover, uLr (∂Ω) = 1.
The next proposition, we believe, is of independent interest.
123
trace embedding
Proposition 2.8. The constant Sq is continuous with respect to q in 1 q p∗ .
Proof. Let v ∈ W 1,p (Ω) \ W01,p (Ω) be fixed. As v ∈ Lp∗ (∂Ω), it follows from
Lebesgue’s dominated convergence theorem that Qq (v) is a continuous function
with respect to q.
From this fact it follows that Sq is an upper semicontinuous function for q ∈
[1, p∗ ]. In fact, by definition, for any ε > 0 there exists v ∈ W 1,p (Ω) \ W01,p (Ω) such
that Qq (v) < Sq + ε. On the other hand, Sr Qr (v) and, as Qr (v) → Qq (v) as
r → q, it follows that
lim sup Sr < Sq + ε,
(2.9)
r→q
as we wanted to show.
Let us now show the left continuity on [1, p∗ ]. For 1 r < q p∗ , we have
1/p 1/q
r
q
v dσ
v dσ
|∂Ω|1/r−1/q ;
∂Ω
∂Ω
hence
Qr (v) Qq (v)|∂Ω|1/r−1/q
and then
Sr Sq |∂Ω|1/r−1/q .
(2.10)
Now the claim follows by taking the limit in (2.10) together with (2.9).
It remains to prove the continuity of Sq in [1, p∗ ). To this end, observe that
the functions uq are uniformly bounded for 1 q q0 < p∗ , so we can apply
Proposition 2.7.
Assume that there exists r ∈ [1, p∗ ) such that Sq is not continuous in r. Then
there exists a sequence qi → r such that limi→∞ Sqi = λ = Sr . By (2.9) we must
have λ < Sr , but, by Proposition 2.7, there exists a function u satisfying (2.11) and
uLr (∂Ω) = 1; hence Qr (u) = λ, from which it follows that λ Sr , a contradiction.
This completes the proof.
Now we are ready to prove the main theorem.
Proof of Theorem A. First, we claim that there exists a sequence qj p∗ such
that uqj → u weakly in W 1,p (Ω) and strongly in Lp (∂Ω). Moreover, the limit u
satisfies

in Ω,
∆p u = |u|p−2 u
(2.11)
∂u
|∇u|p−2
= Sup∗ −1 on ∂Ω.
∂ν
In order to see this, we observe that
uq W 1, p (Ω) Sq C,
by Proposition 2.8 (see also [9]). Hence, as W 1,p (Ω) → Lp∗ (∂Ω) continuously,
uq Lp ∗ (∂Ω) C.
Moreover, as p∗ /(p∗ − 1) < p∗ /(q − 1), we see that uqq−1 is uniformly bounded in
Lp∗ /(p∗ −1) (∂Ω).
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julián fernández bonder and julio d. rossi
As a consequence of these uniform bounds, there exist a sequence qj p∗ and a
function u ∈ H 1 (Ω) such that:
uqj u
uqj → u
uqj → u
uqqjj −1 up∗ −1
weakly in W 1,p (Ω);
strongly in Lp (Ω) and in Lp (∂Ω);
for almost every Ω and almost every ∂Ω;
weakly in Lp∗ /(p∗ −1) (∂Ω).
(2.12)
Now, let ψ ∈ W 1,p (Ω). We have, by (2.12),
qj −1
uqj ψ dσ →
up∗ −1 ψ dσ;
∂Ω
∂Ω
∇uqj ∇ψ + uqj ψ dx →
∇u∇ψ + uψ dx.
Ω
Ω
By the continuity of Sq (Proposition 2.8), we have
lim Sqj = S = Sp∗ ;
qj p∗
therefore, u ∈ W 1,p (Ω) is a weak solution of (2.11), and the claim follows.
Now, as uq 0, it follows that u 0 and, by classical regularity theory (see
[14]), u is smooth (C 1,α ) up to the boundary. By the strong maximum principle
and Hopf’s lemma (see [15]), it follows that either u > 0 or u ≡ 0. So, in order to
complete the proof of the theorem, we have to rule out the possibility of u ≡ 0. To
do this, we adapt the argument given in [2] to show that uLp (Ω) = 0. In fact, by
Theorem 2.1, given ε > 0, there exists a constant B = B(ε) such that
p/p∗
v p∗ dσ
(K(N, p) + ε) |∇v|p dx + B |v|p dx
∂Ω
1,p
Ω
Ω
for every v ∈ W (Ω). Recall that the uq are normalized such that uq Lq (∂Ω) = 1;
so, by Hölder’s inequality,
p/q p/p∗
|∂Ω|p/p∗ −p/q
uqq dσ
upq ∗ dσ
∂Ω
∂Ω
p
(K(N, p) + ε) |∇uq | dx + B upq dx
Ω
Ω
and hence
|∂Ω|p/p∗ −p/q (K(N, p) + ε)Sq + (B − K(N, p) − ε)
upq dx.
(2.13)
Ω
Passing to the limit qj p∗ in (2.13), we arrive at
1 (K(N, p) + ε)S + (B − K(N, p) − ε)
up dx.
Ω
Therefore, if S < K(N, p)−1 we complete the proof by choosing ε small enough. By
taking v ≡ 1 in (1.1), we obtain
S
Hence, if
|Ω|
.
|∂Ω|p/p∗
|Ω|
< K(N, p)−1 ,
|∂Ω|p/p∗
we have the existence of an extremal.
trace embedding
125
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Julián Fernández Bonder
Departemento de Matemática
FCEyN
UBA (1428) Buenos Aires
Argentina
[email protected]
Julio D. Rossi
Departamento de Matemática
Universidad Católica de Chile
Casilla 306, Correo 22
Santiago
Chile
[email protected]