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 we study the existence problem for extremals of the Sobolev trace inequality W^{1,p}(Ω) → L^{p_{∗}}(∂Ω) where Ω is a bounded smooth domain in ℝ^{N}, p_{∗} = p(N − 1)/(N − p) is the critical Sobolev exponent and N > p > 1.

1. Introduction.

Let Ω ⊂ ℝ^{N} be a bounded smooth domain. Relevant for the study of boundary value problems for differential operators are the two following Sobolev inequalities. For each 1 ≤ q ≤ p(N −1)/(N −p) ≡ p_{∗}, we have a continuous inclusion W^{1,p}(Ω) ↪ L^{q}(∂Ω), and for each 1 ≤ r ≤ pN/(N − p) ≡ p^{∗}, W_{0}^{1,p}(Ω) ↪ L^{r}(Ω), hence the following inequalities hold:

S_{q}∥u∥_{L^{q}(∂Ω)}^{p} ≤ ∥u∥_{W^{1,p}(Ω)}^{p},  S̄_{r}∥u∥_{L^{r}(Ω)}^{2} ≤ ∥u∥_{W_{0}^{1,p}(Ω)}^{p}.

These inequalities are known as the Sobolev trace theorem and the Sobolev embedding theorem respectively. The best constants for these embeddings are the largest S and S̄ such that the above inequalities hold, that is,

(1.1)  S_{q} = inf_{v∈W^{1,p}(Ω)\W_{0}^{1,p}(Ω)} \frac{∫_{Ω} |∇v|^{p} + |v|^{p} dx}{(∫_{∂Ω} |v|^{q} dσ)^{p/q}}

and

(1.2)  S̄_{r} = inf_{v∈W_{0}^{1,p}(Ω)\\{0\}} \frac{∫_{Ω} |∇v|^{p} dx}{(∫_{Ω} |v|^{r} dx)^{p/r}} .

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}, taking v(x) = u(μx) in (1.2) and changing variables we get

S̄_{r}(μΩ) = μ^{(rN−pr−pN)/r}S̄_{r}(Ω).

Key words and phrases. nonlinear boundary conditions, Sobolev trace embedding.

2000 Mathematics Subject Classification. 35J65, 35B33.

Supported by Fundacion Antorchas, CONICET and ANPCyT PICT No. 05009.

On the other hand, S_{q} is not homogeneous under dilatations. In fact we have

(1.3)  S_{q}(μΩ) = μ^{β} inf_{v∈W^{1,p}(Ω)\W_{0}^{1,p}(Ω)} \frac{∫_{Ω} μ^{−p}|∇v|^{p} + |v|^{p} dx}{(∫_{∂Ω} |v|^{q} dσ)^{p/q}},

where β = (Nq − pN + p)/q.

For 1 ≤ q < p_{∗} and 1 ≤ r < p^{∗} the embeddings are compact, so we have existence of extremals, i.e. functions where the infimum is attained. These extremals are weak solutions of the following problems

(1.4)  \{ \begin{matrix}Δ_{p}u = |u|^{p−2}u & in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{q−2}u & on ∂Ω, \end{matrix}

where Δ_{p}u = div(|∇u|^{p−2}∇u) is the p−laplacian, \frac{∂}{∂ν} is the outer unit normal derivative, and

(1.5)  \{ \begin{matrix}−Δ_{p}u = λ|u|^{r−2}u & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}

The asymptotic behavior of S_{q}(μΩ) in expanding (μ → ∞) and contracting domains (μ → 0), was studied in [7] and [11]. In [7] it is proved that for expanding domains and q > p = 2, S_{q}(μΩ) → S_{q}(ℝ_{+}^{N}). In [11] it is shown that

(1.6)  lim_{μ→0+} \frac{S_{q}(μΩ)}{μ^{β}} = \frac{|Ω|}{|∂Ω|^{p/q}} .

The behavior of the extremals for (1.1) in expanding and contracting domains is also studied in [7] and [11]. For expanding domains, it is proved in [7] (again in the case q > p = 2) that the extremals develop a peak near a point where the mean curvature of the boundary maximizes. For contracting domains, we have that the extremals, when rescaled to the original domain as v(x) = u(μx), x ∈ Ω, and normalized with ∥v∥_{L^{q}(∂Ω)} = 1, are nearly constant in the sense that

lim_{μ→0} v = \frac{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. Namely, if Ω is a ball, Ω = B(0, μ), as the extremals do not change sign, from results of [12] 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 [7].

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)) → L^{2_{∗}}(∂B(0, μ)) if the radius μ is small enough. See also [1] for other geometric conditions that leads to existence of extremals in the case p = 2.

So we arrive at the purpose of this article: to find existence of extremals for the Sobolev trace theorem with the critical exponent in a general smooth bounded domain Ω. Our main result is the following:

Theorem A. Let Ω be a bounded smooth domain in ℝ^{N} such that

(1.7)  \frac{|Ω|}{|∂Ω|^{p/p_{∗}}} < \frac{1}{K(N, p)},

where K(N, p) is given by (2.2). Then there exists an extremal for the immersion W^{1,p}(Ω) → L^{p_{∗}}(∂Ω).

For the proof of Theorem A we use the same approach as in [2] (see also [16]), properly adapted to our new context.

Existence result for elliptic problems with critical Sobolev exponents have deserved a great deal of attention since the pioneer work [5] and is a intensive area of research nowadays. Best Sobolev inequalities have been studied by many authors and is by now a classical subject. It at least goes back to [2, 3].

The other key ingredient in the proof is the result of [4] where the author compute the optimal constant in the Sobolev trace inequality (see Theorem 2.1 below). See also [13] 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 ℝ^{N} and let

Ω_{μ} = μΩ = {μx | x ∈ Ω},

where μ > 0. We observe that when μ is small enough, precisely

μ < \frac{1}{K(N, p)^{1/p}} \frac{|∂Ω|^{1/p_{∗}}}{|Ω|^{1/p}} ,

then Ω_{μ} verifies the hypotheses of Theorem A and hence there is an extremal for the immersion W^{1,p}(Ω_{μ}) → L^{p_{∗}}(∂Ω_{μ}).

Remark 1.9. We observe that with the same ideas and computations, we can consider a problem of the form

(1.10)  \{ \begin{matrix}Δ_{p}u = |u|^{p−2}u & in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = a(x)u^{p_{∗}−1} & on ∂Ω, \end{matrix}

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

|Ω| (sup_{∂Ω} a)^{p/p_{∗}} < \frac{1}{K(N, p)} (∫_{∂Ω} a dσ)^{p/p_{∗}} .

Remark 1.11. Observe that from the proof of Theorem A, we obtain the existence of extremals for every domain Ω that satisfies

(1.12)  S_{p_{∗}} = S_{p_{∗}}(Ω) < \frac{1}{K(N, p)}.

Condition (1.7) is the simplest geometric condition that ensures (1.12).

4 J. FERNÁNDEZ BONDER AND J.D. ROSSI

2. Proof of Theorem A

For the proof of Theorem A we use the following Theorem due to [4] Theorem 2.1. There exists a constant B > 0 such that,

(∫_{∂Ω} v^{p_{∗}} dσ)^{p/p_{∗}} ≤ K(N, p) ∫_{Ω} |∇v|^{p} dx + B ∫_{Ω} v^{p} dx

for every v ∈ W^{1,p}(Ω), where

(2.2)  \frac{1}{K(N, p)^{p}} = inf_{∇w∈L^{p}(ℝ_{+}^{N}), w∈L^{p∗} (∂ℝ_{+}^{N})} \frac{∫_{ℝ_{+}^{N}} |∇w|^{p} dx}{(∫_{∂ℝ_{+}^{N}} |w|^{p_{∗}} dx′)^{p/p_{∗}}} .

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 used by [2] (see also [16]). Let us consider, for each 1 < q ≤ p_{∗} the quotient

(2.4)  Q_{q}(u) = \frac{∫_{Ω} |∇u|^{p} + u^{p} dx}{(∫_{∂Ω} u^{q} dσ)^{p/q}}

and

S_{q} = inf_{u∈W^{1,p}(Ω)\W_{0}^{1,p}(Ω)} Q_{q}(u).

To simplify the notation, we denote

(2.5) Q(u) = Q_{p_{∗}}(u) and S = S_{p_{∗}}.

We have the following Lemma 2.6. If q < p_{∗}, the constant S_{q} is attained. Proof. It follows from the compactness of the immersion W^{1,p}(Ω) ↪ L^{q}(∂Ω). □

From now on, we will denote by u_{q} an extremal related to S_{q}, normalized such that ∥u_{q}∥_{L^{q}(∂Ω)} = 1.

For the proof of the Theorem, we need the following proposition, which is a straightforward modification of [16]. Proposition 2.7. Let q_{i} → r ≤ p_{∗} and assume that

sup_{x∈\overline{Ω}} u_{q_{i}}(x) ≤ A.

Then there exists a subsequence u_{q_{i_{j}}} such that u_{q_{i_{j}}} → u in the sense of distributions and u is a solution of

\{ \begin{matrix}Δ_{p}u = |u|^{p−2}u & in Ω \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{r−2}u & on ∂Ω, \end{matrix}

where λ = lim S_{q_{i_{j}}}. Moreover ∥u∥_{L^{r}(∂Ω)} = 1.

The next proposition we believe that has independent interest.

Proposition 2.8. The constant S_{q} is continuous with respect to q in 1 ≤ q ≤ p_{∗}.

Proof. Let v ∈ W^{1,p}(Ω) \ W_{0}^{1,p}(Ω) be fixed. As v ∈ L^{p_{∗}}(∂Ω), it follows from the Lebesgue’s dominated convergence Theorem that Q_{q}(v) is a continuous function with respect to q.

From this fact it follows that S_{q} is an upper semicontinuous function for q ∈ [1, p_{∗}]. In fact, by definition, for any ε > 0 there exists v ∈ W^{1,p}(Ω) \ W_{0}^{1,p}(Ω) such that Q_{q}(v) < S_{q} + ε. On the other hand, S_{r} ≤ Q_{r}(v) and, as Q_{r}(v) → Q_{q}(v) as r → q, it follows that

(2.9)  lim sup_{r→q} S_{r} < S_{q} + ε,

as we wanted to show.

Let us now show the left continuity on [1, p_{∗}]. For 1 ≤ r < q ≤ p_{∗} we have

(∫_{∂Ω} v^{r} dσ)^{1/p} ≤ (∫_{∂Ω} v^{q} dσ)^{1/q} |∂Ω|^{\frac{1}{r} − \frac{1}{q}} ,

hence

Q_{r}(v) ≥ Q_{q}(v)|∂Ω|^{\frac{1}{r} − \frac{1}{q}}

and then

(2.10)  S_{r} ≥ S_{q}|∂Ω|^{\frac{1}{r} − \frac{1}{q}} .

Now the claim follows by taking limit in (2.10) together with (2.9).

It remains to prove the continuity of S_{q} in [1, p_{∗}). To this end observe that the functions u_{q} are uniformly bounded for 1 ≤ q ≤ q_{0} < p_{∗}, so we can apply Proposition 2.7.

Assume that there exists r ∈ [1, p_{∗}) such that S_{q} is not continuous in r. Then there exists a sequence q_{i} → r such that lim_{i→∞} S_{q_{i}} = λ ≠ S_{r}. By (2.9) we must have λ < S_{r}, but, by Proposition 2.7, there exists a function u satisfying (2.11) and ∥u∥_{L^{r}(∂Ω)} = 1 and hence Q_{r}(u) = λ from where it follows that λ ≥ S_{r}, a contradiction.

This finishes the proof. □

Now we are ready to prove the main theorem.

Proof of Theorem A. First, we claim that there exists a sequence q_{j} ↗ p_{∗} such that u_{q_{j}} → u weakly in W^{1,p}(Ω) and strongly in L^{p}(∂Ω). Moreover, the limit u satisfies

(2.11)  \{ \begin{matrix}Δ_{p}u = |u|^{p−2}u & in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = Su^{p_{∗}−1} & on ∂Ω. \end{matrix}

In order to see this, we observe that

∥u_{q}∥_{W^{1,p}(Ω)} ≤ S_{q} ≤ C

by Proposition 2.8 (see also [9]). Hence, as W^{1,p}(Ω) → L^{p_{∗}}(∂Ω) continuously,

∥u_{q}∥_{L^{p∗} (∂Ω)} ≤ C.

Moreover, as p_{∗}/(p_{∗} − 1) < p_{∗}/(q − 1), we have u_{q}^{q−1} is uniformly bounded in L^{p_{∗}/(p_{∗}−1)}(∂Ω). As a consequence of these uniform bounds, there exists a sequence q_{j} ↗ p_{∗} and a function u ∈ H^{1}(Ω) such that

(2.12)

u_{q_{j}} ⇀ u  weakly in W^{1,p}(Ω),
u_{q_{j}} → u  strongly in L^{p}(Ω) and in L^{p}(∂Ω),
u_{q_{j}} → u  a.e. Ω and a.e. ∂Ω,
u_{q_{j}}^{q_{j}−1} ⇀ u^{p_{∗}−1} weakly in L^{p_{∗}/(p_{∗}−1)}(∂Ω).

Now, let ψ ∈ W^{1,p}(Ω). We have, by (2.12),

∫_{∂Ω} u_{q_{j}}^{q_{j}−1}ψ dσ → ∫_{∂Ω} u^{p_{∗}−1}ψ dσ,
∫_{Ω} ∇u_{q_{j}}∇ψ + u_{q_{j}}ψ dx → ∫_{Ω} ∇u∇ψ + uψ dx.

By the continuity of S_{q} (Proposition 2.8) we have

lim_{q_{j}↗p_{∗}} S_{q_{j}} = S = S_{p_{∗}},

therefore u ∈ H^{1}(Ω) is a weak solution of (2.11) and the claim follows.

Now, as u_{q} ≥ 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 ∥u∥_{L^{p}(Ω)} ≠ 0. In fact, by Theorem 2.1, there exists a constant B such that

(∫_{∂Ω} v^{p_{∗}} dσ)^{p/p_{∗}} ≤ K(N, p) ∫_{Ω} |∇v|^{p} dx + B ∫_{Ω} |v|^{p} dx

for every v ∈ W^{1,p}(Ω). Recall that u_{q} are normalized such that ∥u_{q}∥_{L^{q}(∂Ω)} = 1, so, by Hölder’s inequality,

\begin{aligned}|∂Ω|^{\frac{p}{p∗} − \frac{p}{q}} (∫_{∂Ω} u_{q}^{q} dσ)^{p/q} & ≤ (∫_{∂Ω} u_{q}^{p_{∗}} dσ)^{p/p_{∗}} \\ & ≤ K(N, p) ∫_{Ω} |∇u_{q}|^{p} dx + B ∫_{Ω} u_{q}^{p} dx \end{aligned}

and hence

(2.13)  |∂Ω|^{\frac{p}{p∗} − \frac{p}{q}} ≤ K(N, p)S_{q} + (B − K(N, p)) ∫_{Ω} u_{q}^{p} dx.

Passing to the limit q_{j} ↗ p_{∗} in (2.13) we arrive at

1 ≤ K(N, p)S + (B − K(N, p)) ∫_{Ω} u^{p} dx

therefore, if S < K(N, p)^{−1} we are done. By taking v ≡ 1 in (1.1), we get

S ≤ \frac{|Ω|}{|∂Ω|^{p/p_{∗}}} .

Hence, if

\frac{|Ω|}{|∂Ω|^{p/p_{∗}}} < K(N, p)^{−1},

we have the existence of an extremal. □

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Julián Fernández Bonder Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jfbonder@dm.uba.ar
Web page: http://mate.dm.uba.ar/∼jfbonder

Julio D. Rossi Departamento de Matemática, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile.

E-mail address: jrossi@riemann.mat.puc.cl