Sharp Sobolev inequalities with interior norms

Meijun Zhu Department of Mathematics The University of British Columbia Vancouver, B.C. Canada V6T 1Z2

Abstract

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain Ω ∈ ℝ^{n}. Let S_{1} and S be the sharp constants corresponding to Sobolev embedding and trace inequalities respectively. If n ≥ 4, there exist constants A(Ω), A_{1}(Ω) > 0 such that ∀ u ∈ H^{1}(Ω)

||u||_{2n/(n−2),Ω}^{2} ≤ 2^{2/n}S_{1}||∇u||_{2,Ω}^{2} + A(Ω)||u||_{2n/(n−1),Ω}^{2}

and

||u||_{2(n−1)/(n−2),∂Ω}^{2} ≤ S||∇u||_{2,Ω}^{2} + A_{1}(Ω)||u||_{2n/(n−1),Ω}^{2};

If n = 3, for any k_{3} > 3, there exist constants A(Ω, k_{3}), A_{1}(Ω, k_{3}) > 0 such that ∀ u ∈ H^{1}(Ω)

||u||_{2n/(n−2),Ω}^{2} ≤ 2^{2/n}S_{1} · ||∇u||_{2,Ω}^{2} + A(Ω, k_{3})||u||_{k_{3},Ω}^{2}

and

||u||_{2(n−1)/(n−2),∂Ω}^{2} ≤ S||∇u||_{2,Ω}^{2} + A_{1}(Ω, k_{3})||u||_{k_{3},Ω}^{2}.

0 Introduction

Let Ω ∈ ℝ^{n} (n ≥ 3) be a smooth bounded domain. In our recent papers [9] and [10], we established the following inequalities. See also [3].

Theorem A. There exist constants C(Ω), C_{1}(Ω) > 0 such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ 2^{2/n}S_{1} ∫_{Ω} |∇u|^{2}dv + C(Ω) ∫_{∂Ω} u^{2}ds, ∀ u ∈ H^{1}(Ω) and

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ S ∫_{Ω} |∇u|^{2}dv + C_{1}(Ω) ∫_{∂Ω} u^{2}ds, ∀ u ∈ H^{1}(Ω),

where and throughout this paper we denote p + 1 = 2n/(n − 2), q + 1 = 2(n − 1)/(n − 2) as the critical exponents corresponding to Sobolev embedding and trace inequalities, 1/S_{1} = inf{||∇u||_{2,ℝ^{n}}^{2} : u ∈ C_{0}^{∞}(ℝ^{n}), ∫_{ℝ^{n}} |u|^{p+1} = 1} = πn(n−2)(Γ(n/2)/Γ(n))^{2/n} and 1/S = inf{||∇u||_{2,ℝ_{+}^{n}}^{2} : u ∈ C^{∞}(ℝ_{+}^{n}), ∫_{∂ℝ_{+}^{n}} |u|^{q+1} = 1} = (n − 2)/2 · (2π^{n/2}/(Γ(n/2)))^{1/(n−1)} as the corresponding sharp constants.

It can be shown by the work of X. Wang [14] (see also [2]) that one can not expect the existence of constants D(Ω) and D_{1}(Ω), such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ 2^{2/n}S_{1} ∫_{Ω} |∇u|^{2}dv + D(Ω) ∫_{Ω} u^{2}dv

or

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ S ∫_{Ω} |∇u|^{2}dv + D_{1}(Ω) ∫_{Ω} u^{2}dv

holds for all u ∈ H^{1}(Ω). Then a quite natural question arise: can one raise the interior L^{2} norm in the right hand sides of the above two inequalities to a high L^{k} norm (obviously we do not want this k to exceed the critical exponent p + 1 ) such that the above inequalities still hold for some constants? Testing standard bubble functions centered at any point on the boundary suggests that it is true for k = 2n/(n − 1). In fact, we will prove in this paper the following two theorems.

Theorem 0.1 If n ≥ 4 and k = 2n/(n − 1), there exists a constant A(Ω) > 0 such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ 2^{2/n}S_{1} ∫_{Ω} |∇u|^{2}dv + A(Ω)(∫_{Ω} |u|^{k}dv)^{\frac{2}{k}} , ∀ u ∈ H^{1}(Ω).

(0.1)

If n = 3, for any k_{3} > 3 = 2n/(n − 1), there exists a constant A(Ω, k_{3}) > 0 such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ 2^{2/n}S_{1} ∫_{Ω} |∇u|^{2}dv + A(Ω, k_{3})(∫_{Ω} |u|^{k_{3}}dv)^{\frac{2}{k_{3}}} , ∀ u ∈ H^{1}(Ω). (0.2)

Theorem 0.2 If n ≥ 4 and k = 2n/(n−1), there exists a constant A_{1}(Ω) > 0 such that

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ S ∫_{Ω} |∇u|^{2}dv + A_{1}(Ω)(∫_{Ω} |u|^{k}dv)^{\frac{2}{k}} , ∀ u ∈ H^{1}(Ω). (0.3)

If n = 3, for any k_{3} > 3 = 2n/(n − 1), there exists a constant A_{1}(Ω, k_{3}) > 0 such that

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ S ∫_{Ω} |∇u|^{2}dv + A_{1}(Ω, k_{3})(∫_{Ω} |u|^{k_{3}}dv)^{\frac{2}{k_{3}}} , ∀ u ∈ H^{1}(Ω). (0.4)

Later on, throughout this paper we always assume that k = 2n/(n − 1) and k_{3} > 3. By Hölder inequality, we only need to show that (0.2) and (0.4) hold for some k_{3} close to 3. Thus we also assume that k_{3} is close to 3.

Remark 0.1 By checking the standard bubble function concentrating at some point on the boundary with positive mean curvature, we see that one can not expect (0.1) or (0.3) to hold for any exponent less that 2n/(n − 1).

Remark 0.2 We do not know whether (0.2) or (0.4) still holds for k_{3} = 3.

The proofs of Theorem 0.1 and 0.2 are quite different from our early works [9] and [10]. We have not been able to get some L^{∞} estimates by using Moser iteration method as we did in [9] and [10]. Here, we borrow some ideas from [4] and [3] which were used earlier by O. Rey [12] to prove a conjecture of Brezis and Peletier. Suppose that Theorem 0.1 is not true. If we set

I_{α}(u) = \frac{∫_{Ω} |∇u|^{2} + α(∫_{Ω} u^{k})^{\frac{2}{k}}}{(∫_{Ω} |u|^{p+1})^{\frac{2}{p+1}}} ,

we know that

inf_{u∈H^{1}(Ω)\\{0\}} I_{α}(u) < \frac{1}{2^{2/n}S_{1}} ,  ∀α > 1.  (0.5)

Then the minimizer of I_{α}(u) is attained at some point u_{α}. We normalize it so that ||u_{α}||_{p+1} = 1. u_{α} will blow up as α → ∞. We will show that such u_{α} is close to a standard bubble function concentrating at some point on the boundary in some sense. As we point out early, any standard bubble function concentrating at some point on the boundary does not satisfy (0.5) when α is sufficiently large. We can conclude that u_{α} can not satisfy (0.5) as α is sufficiently large either, hence derive a contradiction.

The crucial step in our proof is to estimate the distance of the blow-up minimum energy solution sequence {u_{α}} to a subspace of H^{1}(Ω) spanned by some standard bubble functions. Previously, in order to make sure that these calculations can be carried out, one needed some restrictions on the dimension of the domain. Sometimes this kind of restriction is not essential, for instance, in the proof of Sharp Sobolev trace inequalities (compare [9] with [3]). Therefore, we need some more delicate analysis to deal with the lower dimensional cases (for n = 3, 4). Following the observation of Z. Wang [15], we can show that (0.1) and (0.3) still hold for n = 4. However, for n = 3, we need k_{3} > 2n/(n − 1). We tend to believe that (0.2) and (0.4) still hold for k_{3} = 3. As a byproduct result, we would like to point out that one can prove all results in [3] by using the same method as in this paper for n ≥ 4.

We prove Theorem 0.1 and 0.2 simultaneously by contradiction and organize this paper as follows. In Section 1, under the contrary assumption, we show that minimum energy solutions will blow up at some point on the boundary and give a rough asymptotic behavior for the sequence. In Section 2, we give some lower bound energy estimates for those minimizers and derive a contradiction to the assumption.

Acknowledgment. I would like to thank Professor H. Brezis for bringing this problem to my attention.

1 Asymptotic analysis

We first quote some well-known inequalities.

Lemma 1.1 For all δ > 0, there exists some constant \overline{C}(δ) depending on δ such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ (2^{2/n}S_{1} + δ) ∫_{Ω} |∇u|^{2}dv + \overline{C}(δ) ∫_{Ω} |u|^{2}dv, ∀ u ∈ H^{1}(Ω). (1.6)

Lemma 1.2 For all δ > 0, there exists some constant \overline{C}_{1}(δ) depending on δ such that

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ (S +δ) ∫_{Ω} |∇u|^{2}dv +\overline{C}_{1}(δ) ∫_{Ω} |u|^{2}dv, ∀ u ∈ H^{1}(Ω). (1.7)

The proofs of the above two inequalities can be found in Aubin’s book [5]. Due to Hölder inequality, we have some easy consequences of Lemma 1.1 and 1.2. For convenience, we set τ = k for n ≥ 4 and τ = k_{3} for n = 3 throughout this paper.

Corollary 1.1 For all δ > 0, there exists some constant C̃(δ) depending on δ such that

(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} ≤ (2^{2/n}S_{1} + δ) ∫_{Ω} |∇u|^{2}dv + C̃(δ)(∫_{Ω} |u|^{τ}dv)^{\frac{2}{τ}} , ∀ u ∈ H^{1}(Ω). (1.8)

Corollary 1.2 For all δ > 0, there exists some constant C̃_{1}(δ) depending on δ such that

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ (S + δ) ∫_{Ω} |∇u|^{2}dv + C̃_{1}(δ)(∫_{Ω} |u|^{τ}dv)^{\frac{2}{τ}} , ∀ u ∈ H^{1}(Ω). (1.9)

In order to prove Theorem 0.2, we also need the following inequality. One can find a proof in Adam’s book [1].

Lemma 1.3 For all δ > 0, there exists a constant Ĉ(δ) depending on δ such that

(∫_{∂Ω} |u|^{q+1}ds)^{\frac{2}{q+1}} ≤ δ ∫_{Ω} |∇u|^{2}dv + Ĉ(δ)(∫_{Ω} |u|^{p+1}dv)^{\frac{2}{p+1}} , ∀ u ∈ H^{1}(Ω). (1.10)

We now begin to prove Theorem 0.1 and 0.2 simultaneously by contradiction. The argument will go through this section as well as the next section. Suppose the contrary of Theorem 0.1 and 0.2. We have : for all α ≥ 1, that

inf_{u∈H^{1}(Ω)\\{0\}} I_{α}(u) := inf_{u∈H^{1}(Ω)\\{0\}} \frac{∫_{Ω} |∇u|^{2} + α(∫_{Ω} u^{τ})^{\frac{2}{τ}}}{(∫_{Ω} |u|^{p+1})^{\frac{2}{p+1}}} := ξ_{α} < \frac{1}{2^{2/n}S_{1}} (1.11)

and

inf_{u∈H^{1}(Ω)\\{0\}} II_{α}(u) := inf_{u∈H^{1}(Ω)\\{0\}} \frac{∫_{Ω} |∇u|^{2} + α(∫_{Ω} u^{τ})^{\frac{2}{τ}}}{(∫_{∂Ω} |u|^{q+1})^{\frac{2}{q+1}}} := ξ_{1,α} < \frac{1}{S} . (1.12)

Lemma 1.4 For any fixed α ≥ 1, under condition (1.11) (or (1.12)), inf I_{α}(u) (or inf II_{α}) is achieved by some \overline{u} ∈ H^{1}(Ω) (or \overline{u}_{1} ∈ H^{1}(Ω)) .

Proof. We only show that inf I_{α}(u) is attained under the condition (1.11). Using Corollary 1.2, one can prove that inf II_{α} is attained by a similar argument.

We use the standard method to prove this lemma, see for example [6] or [13]. Let {u^{(m)}} be a minimizing sequence with ∥u^{(m)}∥_{p+1} = (∫_{Ω} |u^{(m)}|^{p+1}dv)^{1/(p+1)} = 1, and u^{(m)} ≥ 0. Clearly, ∥u^{(m)}∥_{H^{1}(Ω)} ≤ C. After passing to a subsequence, u^{(m)} converges weakly to some \overline{u} ∈ H^{1}(Ω), \overline{u} ≥ 0. It is not difficult to see that

∫_{Ω} |u^{(m)}|^{p+1} − ∫_{Ω} |u^{(m)} − \overline{u}|^{p+1} − ∫_{Ω} \overline{u}^{p+1} → 0, as m → ∞,  (1.13)

thus

∫_{Ω} |u^{(m)} − \overline{u}|^{p+1} ≤ 1 + o(1),  ∫_{Ω} \overline{u}^{p+1} ≤ 1 + o(1),  (1.14)

where o(1) denotes some quantity tending to zero as m tends to ∞. By compact embedding theorems, Corollary 1.1, (1.11), (1.13) and (1.14), we have, for any fixed δ_{1} > 0, that

ξ_{α} = ∫_{Ω} |∇u^{(m)}|^{2} + α(∫_{Ω} |u^{(m)}|^{τ})^{\frac{2}{τ}} + o(1)
= ∫_{Ω} |∇(u^{(m)} − \overline{u})|^{2} + ∫_{Ω} |∇\overline{u}|^{2} + α∥\overline{u}∥_{τ,Ω}^{2} + o(1)
= ∫_{Ω} |∇(u^{(m)} − \overline{u})|^{2} + \frac{C̃(δ_{1})}{2^{2/n}S_{1}+δ_{1}} (∫_{Ω} |u^{(m)} − \overline{u}|^{τ})^{2/τ} + ∫_{Ω} |∇\overline{u}|^{2}
+α∥\overline{u}∥_{τ,Ω}^{2} + o(1)
≥ \frac{1}{2^{2/n}S_{1}+δ_{1}} (∫_{Ω} |u^{(m)} − \overline{u}|^{p+1})^{\frac{2}{p+1}} + ξ_{α}(∫_{Ω} \overline{u}^{p+1})^{\frac{2}{p+1}} + o(1)
≥ \frac{1}{2^{2/n}S_{1}+δ_{1}} ∫_{Ω} |u^{(m)} − \overline{u}|^{p+1} + ξ_{α} ∫_{Ω} \overline{u}^{p+1} + o(1)
= (\frac{1}{2^{2/n}S_{1}+δ_{1}} − ξ_{α}) ∫_{Ω} |u^{(m)} − \overline{u}|^{p+1} + ξ_{α} + o(1).

Due to (1.11), by choosing δ_{1} > 0 small so that \frac{1}{2^{2/n}S_{1}+δ_{1}} − ξ_{α} ≥ δ_{1}/2, we have from the above that ∫_{Ω} |u^{(m)} − \overline{u}|^{p+1} = o(1). It follows easily that \overline{u} is a minimum point of I_{α}.

From Lemma 1.4, we know that there exist u_{α}, u_{1,α} ∈ H^{1}(Ω) satisfying

ξ_{α} = ∫_{Ω} |∇u_{α}|^{2} + α(∫_{Ω} |u_{α}|^{τ})^{\frac{2}{τ}} ,  ∫_{Ω} u_{α}^{p+1} = 1, u_{α} ≥ 0,  (1.15)
ξ_{1,α} = ∫_{Ω} |∇u_{1,α}|^{2} + α(∫_{Ω} |u_{1,α}|^{τ})^{\frac{2}{τ}} ,  ∫_{∂Ω} u_{1,α}^{q+1} = 1, u_{1,α} ≥ 0, (1.16)

and the following equations respectively

\{ \begin{matrix}−Δu_{α} = ξ_{α}u_{α}^{p} − α(∫_{Ω} u_{α}^{τ})^{\frac{2}{τ} −1}u_{α}^{τ−1} & in Ω, \\ \frac{∂u_{α}}{∂ν} = 0 & on ∂Ω, \end{matrix}

\{ \begin{matrix}−Δu_{1,α} = −α(∫_{Ω} u_{1,α}^{τ})^{\frac{2}{τ} −1}u_{1,α}^{τ−1} & in Ω, \\ \frac{∂u_{1,α}}{∂ν} = ξ_{1,α}u_{1,α}^{q} & on ∂Ω. \end{matrix}

As in [10] or [16], by using (1.15), (1.16), Corollary 1.1 and 1.2 we can get

Lemma 1.5

lim_{α→∞} ξ_{α} = \frac{1}{2^{2/n}S_{1}} , lim_{α→∞} ξ_{1,α} = \frac{1}{S}  (1.17)

and

lim_{α→∞} α∥u_{α}∥_{τ,Ω}^{2} = 0, lim_{α→∞} α∥u_{1,α}∥_{τ,Ω}^{2} = 0.  (1.18)

Set

\overline{u}_{α} = ξα^{\frac{1}{p−1}} u_{α}, \overline{u}_{1,α} = ξ1,α^{\frac{1}{q−1}} u_{1,α}.

Then ∫_{Ω} \overline{u}_{α}^{p+1} = ξ_{α}^{n/2}, ∫_{∂Ω} \overline{u}_{1,α}^{q+1} = ξ_{1,α}^{n−1} and \overline{u}_{α}, \overline{u}_{1,α} satisfy the following equations

\{ \begin{matrix}−Δ\overline{u}_{α} = \overline{u}_{α}^{p} − α(∫_{Ω} \overline{u}_{α}^{τ})^{\frac{2}{τ} −1}\overline{u}_{α}^{τ−1} & in Ω, \\ \frac{∂\overline{u}_{α}}{∂ν} = 0 & on ∂Ω, \end{matrix}  (1.19)

\{ \begin{matrix}−Δ\overline{u}_{1,α} = −α(∫_{Ω} \overline{u}_{1,α}^{τ})^{\frac{2}{τ} −1}\overline{u}_{1,α}^{τ−1} & in Ω, \\ \frac{∂\overline{u}_{1,α}}{∂ν} = \overline{u}_{1,α}^{q} & on ∂Ω. \end{matrix}  (1.20)

It follows from Cherrier [8] that \overline{u}_{α} and \overline{u}_{1,α} are smooth up to the boundary. Let x_{α}, x_{1,α} be some maximum points of \overline{u}_{α} and \overline{u}_{1,α} respectively. By the maximum principle, we know that x_{1,α} ∈ ∂Ω. Set

\{ \begin{matrix}μ_{α} = \overline{u}_{α}^{− \frac{2}{n−2}} (x_{α}), Ω_{α} = \frac{Ω−x_{α}}{μ_{α}} , \\ v_{α}(x) = μα^{\frac{n−2}{2}} \overline{u}_{α}(μ_{α}x + x_{α}) for x ∈ Ω_{α}, \end{matrix}

\{ \begin{matrix}μ_{1,α} = \overline{u}_{1,α}^{− \frac{2}{n−2}} (x_{1,α}), Ω_{1,α} = \frac{Ω−x_{1,α}}{μ_{1,α}} , \\ v_{1,α}(x) = μ1,α^{\frac{n−2}{2}} \overline{u}_{1,α}(μ_{1,α}x + x_{1,α}) for x ∈ Ω_{1,α}. \end{matrix}

Then

(∫_{Ω} \overline{u}_{α}^{τ})^{\frac{2}{τ}} = μ_{α}^{σ}(∫_{Ω_{α}} v_{α}^{τ})^{\frac{2}{τ}} , (∫_{Ω} \overline{u}_{1,α}^{τ})^{\frac{2}{τ}} = μ_{1,α}^{σ}(∫_{Ω_{1,α}} v_{1,α}^{τ})^{\frac{2}{τ}}  (1.21)

where and throughout this paper, we denote

σ = 2n/τ − (n − 2).

Also v_{α} satisfies

\{ \begin{matrix}−Δv_{α} = v_{α}^{p} − αμ_{α}^{σ}(∫_{Ω_{α}} v_{α}^{τ})^{\frac{2}{τ} −1}v_{α}^{τ−1}, 0 ≤ v_{α} ≤ 1, & in Ω_{α}, \\ \frac{∂v_{α}}{∂ν} = 0 & on ∂Ω_{α}, \end{matrix} (1.22)

and v_{1,α} satisfies

\{ \begin{matrix}−Δv_{1,α} = −αμ_{1,α}^{σ}(∫_{Ω_{1,α}} v_{1,α}^{τ})^{\frac{2}{τ} −1}v_{1,α}^{τ−1}, 0 ≤ v_{1,α} ≤ 1, & in Ω_{1,α}, \\ \frac{∂v_{1,α}}{∂ν} = v_{1,α}^{q} & on ∂Ω_{1,α}. \end{matrix}

(1.23)

Since τ < p + 1, v_{α} ≤ 1, we know that as α → ∞,

∫_{Ω_{α}} v_{α}^{τ} ≥ ∫_{Ω_{α}} v_{α}^{p+1} = ∫_{Ω} \overline{u}_{α}^{p+1} = ξ_{α}^{n/2} ≥ C > 0.

It follows from (1.18), (1.21) and the above that

αμ_{α}^{σ} → 0, as α → ∞.  (1.24)

Since ∫_{Ω} |∇u_{1,α}|^{2} ≤ C, we know from Lemma 1.3 and (1.16) that ∫_{Ω} u_{1,α}^{p+1} ≥ C > 0. Combining this with (1.21), we have

αμ_{1,α}^{σ} → 0, as α → ∞.  (1.25)

Therefore, by standard elliptic estimates we know that v_{α} → v in C^{3}(\overline{Ω_{α}} ∩ B_{R}(0)) for all fixed R > 1, where v satisfies

\{ \begin{matrix}−Δv = v^{p} & in ℝ_{+,T}^{n} , \\ \frac{∂v}{∂ν} = 0 & on ∂ℝ_{+,T}^{n} , \\ v(0) = 1,  0 ≤ v(x) ≤ 1, \end{matrix}  (1.26)

T = \overline{lim}_{α→∞}dist(x_{α}, ∂Ω)/μ_{α} and ℝ_{+,T}^{n} = \{x = (x′, x_{n}) : x_{n} > −T \}. If T = ∞, (1.26) has no boundary condition. And, for v_{1,α}, we know that v_{1,α} → v_{1} in C^{3}(\overline{Ω_{1,α}} ∩ B_{R}(0)) for all fixed R > 1, where v_{1} satisfies

\{ \begin{matrix}−Δv_{1} = 0 & in ℝ_{+}^{n}, \\ \frac{∂v_{1}}{∂ν} = v_{1}^{q} & on ∂ℝ_{+}^{n}, \\ v_{1}(0) = 1,  0 ≤ v_{1}(x) ≤ 1. \end{matrix}

Due to [11],

v_{1}(x) = (\frac{(n − 2)^{2}}{|x′|^{2} + (x_{n} + n − 2)^{2}} )^{\frac{n−2}{2}} .

From [7], we know that T = 0 or +∞.

Lemma 1.6 Let T and v be given as the above, then T = 0 and

v(x) = (\frac{n(n − 2)}{n(n − 2) + |x|^{2}} )^{\frac{n−2}{2}} .  (1.27)

Furthermore, as α large enough, x_{α} ∈ ∂Ω.

Proof. If T = +∞, we know the explicit form of v, for example, see [7]. Multiplying (1.26) by v and integrating by parts, we obtain

∫_{ℝ^{n}} |∇v|^{2} = ∫_{ℝ^{n}} v^{p+1}.

From the definition of S_{1} we have

∫_{ℝ^{n}} |∇v|^{2} ≥ \frac{1}{S_{1}} (∫_{ℝ^{n}} v^{p+1})^{\frac{2}{p+1}} .

Therefore, we know that ∫_{ℝ^{n}} v^{p+1} ≥ S_{1}^{−(p+1)/(p−1)} = S_{1}^{−n/2}.

On the other hand, for any fixed R > 1,

∫_{B_{R}(0)} v_{α}^{p+1} = ∫_{B_{μαR}(x_{α})} \overline{u}_{α}^{p+1} ≤ ξ_{α}^{n/2}.

Sending α → ∞, we obtain ∫_{B_{R}(0)} v^{p+1} ≤ ξ_{α}^{n/2}, thus ∫_{ℝ^{n}} v^{p+1} ≤ ξ_{α}^{n/2} ≤ \frac{1}{2}S_{1}^{−n/2} (due to the form of v). Contradiction!

It follows that T = 0 and v(x) is given by (1.27), see for example [7] or [11].

As α large enough, we claim that x_{α} ∈ ∂Ω. If not, since 0 is the maximum point of v_{α} we know that ∇v_{α}(0) = 0. Also due to ∂v_{α}/∂ν = 0, T = 0, v_{α} → v in C^{3}(\overline{Ω_{α}}∩B_{R}(0)) and the mean value theorem, we know that ∂^{2}v/∂x_{n}^{2}(0) = 0, which contradicts to (1.27).

An easy consequence of the above discussion and lemma is the following.

Lemma 1.7

lim_{α→∞} ∫_{Ω_{α}} |∇v_{α} − ∇v|^{2} = lim_{α→∞} ∫_{Ω_{α}} |v_{α} − v|^{p+1} = 0,

and

lim_{α→∞} ∫_{Ω_{1,α}} |∇v_{1,α} − ∇v_{1}|^{2} = lim_{α→∞} ∫_{∂Ω_{1,α}} |v_{1,α} − v_{1}|^{q+1} = 0.

Proof. See [10] or [16].

2 lower bound estimates of the energy

Let

U_{α} = (\frac{n(n − 2)μ_{α}}{n(n − 2)μ_{α}^{2} + |x − x_{α}|^{2}} )^{\frac{n−2}{2}} ,

and

V_{α} = (\frac{(n − 2)^{2}μ_{1,α}}{|x′ − x′_{1,α}|^{2} + (x_{n} − x_{1,α,n} + (n − 2)μ_{1,α})^{2}} )^{\frac{n−2}{2}}

where we denote x_{1,α} = (x′_{1,α}, x_{1,α,n}). The following is a consequence of Lemma 1.7.

Corollary 2.1

lim_{α→∞} ∫_{Ω} |∇\overline{u}_{α} − ∇U_{α}|^{2} = lim_{α→∞} ∫_{Ω} |\overline{u}_{α} − U_{α}|^{p+1} = 0  (2.28)

and

lim_{α→∞} ∫_{Ω} |∇\overline{u}_{1,α} − ∇V_{α}|^{2} = lim_{α→∞} ∫_{∂Ω} |\overline{u}_{1,α} − V_{α}|^{q+1} = 0.  (2.29)

Proof. By rescaling, we have: ∫_{Ω} |∇\overline{u}_{α} − ∇U_{α}|^{2} = ∫_{Ω_{α}} |∇v_{α} − ∇v|^{2} and ∫_{Ω} |\overline{u}_{α} − U_{α}|^{p+1} = ∫_{Ω_{α}} |v_{α} − v|^{p+1}. (2.29) can be proved in a similar way.

Direct calculation shows that there exist C_{0}, C_{1,0} > 0, such that

||U_{α}||_{τ}^{2} = (C_{0} + o_{μ_{α}}(1))μ_{α}^{σ}  (2.30)
||V_{α}||_{τ}^{2} = (C_{1,0} + o_{μ_{1,α}}(1))μ_{1,α}^{σ}  (2.31)

where o_{μ_{α}}(1), o_{μ_{1,α}}(1) → 0 as μ_{α}, μ_{1,α} → 0. If we write J_{α}(u) = ||∇u||_{2}^{2}/||u||_{p+1}^{2}, J_{1,α}(u) = ||∇u||_{2}^{2}/||u||_{q+1,∂Ω}^{2}, it follows from [3] that there exist positive constants A_{n}, A_{1,n}, which depend only on n, such that as α large enough

J_{α}(U_{α}) = \frac{1}{2^{2/n}S_{1}} − \frac{n − 2}{2}A_{n}H(x_{α})β_{1}(μ_{α}) + o(1)β_{1}(μ_{α}),  (2.32)

J_{1,α}(V_{α}) = \frac{1}{S} − \frac{n − 2}{2}A_{1,n}H(x_{1,α})β_{1}(μ_{1,α}) + o(1)β_{1}(μ_{1,α}),  (2.33)

where H(x_{α}), H(x_{1,α}) are the mean curvatures of ∂Ω at point x_{α} and x_{1,α}, and

β_{1}(t) = \{ \begin{matrix}t log \frac{1}{t} & n = 3, \\ t & n ≥ 4. \end{matrix}

Let

U_{ϵ,y} = (\frac{n(n − 2)ϵ}{n(n − 2)ϵ^{2} + |x − y|^{2}} )^{\frac{n−2}{2}} ,  (2.34)

and

V_{ϵ,y} = (\frac{(n − 2)^{2}ϵ}{|x′ − y′|^{2} + (x_{n} − y_{n} + (n − 2)ϵ)^{2}} )^{\frac{n−2}{2}} .  (2.35)

Define

\{ \begin{matrix}M = \{CU_{ϵ,y} : C ∈ ℝ, y ∈ ∂Ω, ϵ > 0\}, \\ d(φ, M) = inf\{||∇φ − ∇ψ||_{2}^{2}, : ψ ∈ M\}, \end{matrix}

\{ \begin{matrix}M_{1} = \{CV_{ϵ,y} : C ∈ ℝ, y ∈ ∂Ω, ϵ > 0\}, \\ d_{1}(φ, M_{1}) = inf\{||∇φ − ∇ψ||_{2}^{2}, : ψ ∈ M_{1}\}. \end{matrix}

We quote the following lemma from [4].

Lemma 2.1 Let δ > 0 and {ϕ_{α}} ⊂ H^{1}(Ω) be a sequence such that ϕ_{α} ⇀ 0 in H^{1}(Ω) and d(ϕ_{α}, M)^{2} ≤ ||∇ϕ_{α}||_{2}^{2} − 2δ, then there exists a α_{0} > 0 such that for all α > α_{0}, d(ϕ_{α}, M) is achieved by some C_{α}U_{ϵ_{α},y_{α}} ∈ M.

A similar proof yields the following lemma concerning M_{1} and d_{1}.

Lemma 2.2 Let δ > 0 and {ϕ_{α}} ⊂ H^{1}(Ω) be a sequence such that ϕ_{α} ⇀ 0 in H^{1}(Ω) and d_{1}(ϕ_{α}, M_{1})^{2} ≤ ||∇ϕ_{α}||_{2}^{2} − 2δ, then there exists a α_{1,0} > 0 such that for all α > α_{1,0}, d_{1}(ϕ_{α}, M_{1}) is achieved by some C_{1,α}V_{ϵ_{1,α},y_{1,α}} ∈ M_{1}.

Due to Corollary 2.1, applying the above two lemmas to \overline{u}_{α}, \overline{u}_{1,α}, we know that as α large enough there exist C_{α}U_{ϵ_{α},y_{α}} ∈ M, C_{1,α}V_{ϵ_{1,α},y_{1,α}} ∈ M_{1}, such that

d(\overline{u}_{α}, M) = ||∇\overline{u}_{α} − C_{α}∇U_{ϵ_{α},y_{α}}||_{2}^{2}, d_{1}(\overline{u}_{1,α}, M_{1}) = ||∇\overline{u}_{1,α} − C_{1,α}∇V_{ϵ_{1,α},y_{1,α}}||_{2}^{2}.

Set

\overline{u}_{α} = C_{α}U_{ϵ_{α},y_{α}} + w_{ϵ_{α},y_{α}}, \overline{u}_{1,α} = C_{1,α}V_{ϵ_{1,α},y_{1,α}} + h_{ϵ_{1,α},y_{1,α}}.  (2.36)

It follows from Corollary 2.1 that as α → ∞

||∇w_{ϵ_{α},y_{α}}||_{2}^{2} = ||∇\overline{u}_{α} − C_{α}∇U_{ϵ_{α},y_{α}}||_{2}^{2} ≤ ||∇\overline{u}_{α} − ∇U_{α}||_{2}^{2} → 0,
||∇h_{ϵ_{1,α},y_{1,α}}||_{2}^{2} = ||∇\overline{u}_{1,α} − C_{1,α}∇V_{ϵ_{1,α},y_{1,α}}||_{2}^{2} ≤ ||∇\overline{u}_{1,α} − ∇V_{α}||_{2}^{2} → 0.

Also, it is easy to see that ||w_{ϵ_{α},y_{α}}||_{τ} → 0, ||h_{ϵ_{1,α},y_{1,α}}||_{τ} → 0. Therefore as α → ∞,

||w_{ϵ,y_{α}}|| → 0, ||h_{ϵ_{1,α},y_{1,α}}|| → 0,  (2.37)

where ||f||^{2} = ||∇f||_{2}^{2} + ||f||_{2}^{2}. Similarly, due to ||C_{α}∇U_{ϵ_{α},y_{α}} − ∇U_{α}||_{2}^{2} → 0, ||C_{1,α}∇V_{ϵ_{1,α},y_{1,α}} − ∇V_{α}||_{2}^{2} → 0, we can see that as α → ∞,

\begin{matrix}C_{α} → 1, y_{α} → x_{α}, ϵ_{α}/μ_{α} → 1, \\ C_{1,α} → 1, y_{1,α} → x_{1,α}, ϵ_{1,α}/μ_{1,α} → 1. \end{matrix}  (2.38)

Later on, for convenience, we use U_{ϵ}, V_{ϵ_{1}}, w_{ϵ}, h_{ϵ_{1}}, ϵ and ϵ_{1} to represent U_{ϵ_{α},y_{α}}, V_{ϵ_{1,α},y_{1,α}}, w_{ϵ_{α},y_{α}}, h_{ϵ_{1,α},y_{1,α}}, ϵ_{α} and ϵ_{1,α} respectively.

From [4] and [3], we have

∫_{Ω} U_{ϵ}^{p}w_{ϵ} = O(1)β_{2}(ϵ)||w_{ϵ}||,  (2.39)
∫_{∂Ω} V_{ϵ_{1}}^{q}h_{ϵ_{1}} = O(1)β_{2}(ϵ_{1})||h_{ϵ_{1}}||  (2.40)

where

β_{2}(t) = \{ \begin{matrix}t^{1/2} & n = 3 \\ t(log 1/t)^{2/3} & n = 4 \\ t & n ≥ 5. \end{matrix}

Lemma 2.3 For any C > 0, there exist γ_{1} = γ_{1}(C), γ_{1,1} = γ_{1,1}(C) > 0, such that

\begin{aligned}∫_{Ω} |∇w_{ϵ}|^{2} & +αC ∫_{Ω} w_{ϵ}^{2} \\ & ≥ (p + γ_{1}) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2} + O(1)β_{2}(ϵ)^{2}||w_{ϵ}||^{2}, \end{aligned}  (2.41)
\begin{aligned}∫_{Ω} |∇h_{ϵ_{1}}|^{2} & +αC ∫_{Ω} h_{ϵ_{1}}^{2} \\ & ≥ (q + γ_{1,1}) ∫_{∂Ω} V_{ϵ_{1}}^{q−1}h_{ϵ_{1}}^{2} + O(1)β_{2}(ϵ_{1})^{2}||h_{ϵ_{1}}||^{2}. \end{aligned}  (2.42)

Inequality (2.41) was proved in [4]. (2.42) can be proved in the same way. We refer readers to the proofs of Lemma 3.3-3.4 in [4] for more details. Note: in our case, due to (1.24), (1.25), (2.38) and σ ≤ 1, we know that αϵ_{α} → 0, αϵ_{1,α} → 0.

Using Hölder inequality we have the following corollary.

Corollary 2.2 There exist γ_{2}, γ_{1,2} ∈ (0, 1), such that

\begin{aligned}γ_{2}( ∫_{Ω} |∇w_{ϵ}|^{2}+ & α(∫_{Ω} |w_{ϵ}|^{τ})^{2/τ}) \\ & ≥ p ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2} + O(1)β_{2}(ϵ)^{2}||w_{ϵ}||^{2}, \end{aligned}

\begin{aligned}γ_{1,2}( ∫_{Ω} |∇h_{ϵ_{1}}|^{2} & +α(∫_{Ω} h_{ϵ_{1}}^{τ})^{2/τ}) \\ & ≥ q ∫_{∂Ω} V_{ϵ_{1}}^{q−1}h_{ϵ_{1}}^{2} + O(1)β_{2}(ϵ_{1})^{2}||h_{ϵ_{1}}||^{2}. \end{aligned}

When τ = 2 the following lemma was originally used by Z. Wang [15] to extend some results in [4] to n ≥ 5. In our case, we can extend it to lower dimensional domains including n = 3, 4.

Lemma 2.4 For any γ > 0,

∫_{Ω} U_{ϵ}^{τ−1}|w_{ϵ}| ≤ o_{ϵ}(1)ϵ^{\frac{τσ}{2}} + γ||w_{ϵ}||_{τ}^{τ},
∫_{Ω} V_{ϵ_{1}}^{τ−1}|h_{ϵ_{1}}| ≤ o_{ϵ_{1}}(1)ϵ_{1}^{\frac{τσ}{2}} + γ||h_{ϵ_{1}}||_{τ}^{τ}.

where o_{ϵ}(1), o_{ϵ_{1}}(1) → 0 as ϵ, ϵ_{1} → 0.

Proof. We only give a proof for the first inequality. The second one can be proved similarly.

Let us first prove it for n ≥ 4. In this case, τ = k = 2n/(n − 1), σ = 2n/τ − (n − 2) = 1.

Choosing r ∈ (\frac{2n}{n+2}, \frac{2n}{n+1}), we have

\begin{aligned}∫_{Ω} U_{ϵ}^{k−1}|w_{ϵ}| & ≤ (∫_{Ω} U_{ϵ}^{(k−1)r})^{1/r}(∫_{Ω} |w_{ϵ}|^{r′})^{1/r′} \\ & = ϵ^{− \frac{(k−1)(n−2)}{2} + \frac{n}{r}} (∫_{Ω_{ϵ}}(\frac{n(n−2)}{n(n−2)+|x|^{2}} )^{\frac{n−2}{2} ·(k−1)r}dx)^{1/r}||w_{ϵ}||_{r′}, \end{aligned}

where 1/r + 1/r′ = 1.

Direct calculation shows that \frac{n}{(n−2)(k−1)} < \frac{2n}{n+2} if n ≥ 4, thus (n − 2)(k − 1)r − n > 0. It follows that

∫_{Ω} U_{ϵ}^{k−1}|w_{ϵ}| ≤ Cϵ^{− \frac{(k−1)(n−2)}{2} + \frac{n}{r}} ||w_{ϵ}||_{r′}.

Since r ∈ (\frac{2n}{n+2}, \frac{2n}{n+1}), we know that r′ ∈ (k, p + 1). By Hölder inequality, we have

||w_{ϵ}||_{r′} ≤ ||w_{ϵ}||_{k}^{a} · ||w_{ϵ}||_{p+1}^{1−a}

where

a = \frac{\frac{1}{r′} − \frac{1}{p+1}}{\frac{1}{k} − \frac{1}{p+1}} ∈ (0, 1).  (2.43)

Therefore, for any γ > 0, there exists a C(γ) > 0 such that

\begin{aligned}∫_{Ω} U_{ϵ}^{k−1}|w_{ϵ}| & ≤ Cϵ^{− \frac{(k−1)(n−2)}{2} + \frac{n}{r}} ||w_{ϵ}||_{p+1}^{1−a}||w_{ϵ}||_{k}^{a} \\ & ≤ γ||w_{ϵ}||_{k}^{k} + C(γ)ϵ^{(− \frac{(k−1)(n−2)}{2} + \frac{n}{r})· \frac{k}{k−a}} ||w_{ϵ}||p+1^{\frac{(1−a)k}{k−a}} . \end{aligned}

Claim: for k = \frac{2n}{n−1},

(−\frac{(k − 1)(n − 2)}{2} + \frac{n}{r}) · \frac{k}{k − a} = \frac{k}{2}.

From (2.43), we know

\frac{a}{2n} = \frac{1}{r′} − \frac{1}{p + 1},

that is

\frac{1}{r} − \frac{1}{n} = \frac{1}{2} − \frac{a}{2n}.

Therefore

\frac{1}{r} − \frac{(n − 2)(k − 1)}{2n} = \frac{k}{2n} − \frac{a}{2n}.

The claim follows from the above easily.

Note ||w_{ϵ}||_{p+1} ≤ C||w_{ϵ}|| → 0 and \frac{(1−a)k}{k−a} > 0. We have that for n ≥ 4

∫_{Ω} U_{ϵ}^{k−1}|w_{ϵ}| ≤ o_{ϵ}(1)ϵ^{\frac{k}{2}} + γ||w_{ϵ}||_{k}^{k}.

Now we consider the case of n = 3. In this case τ = k_{3} > 3.

We choose r_{1} satisfying (here we use the assumption that k_{3} is close to 3)

\frac{6}{5} < \frac{n}{k_{3} − 1} < r_{1} < \frac{k_{3}}{k_{3} − 1}.

Thus r_{1}′ = r_{1}/(r_{1} − 1) ∈ (k_{3}, p + 1).

Using Hölder inequality, we have

\begin{aligned}∫_{Ω} U_{ϵ}^{k_{3}−1}|w_{ϵ}| & ≤ (∫_{Ω} U_{ϵ}^{(k_{3}−1)r_{1}})^{1/r_{1}}(∫_{Ω} |w_{ϵ}|^{r_{1}′})^{1/r_{1}′} \\ & = ϵ^{− \frac{(k_{3}−1)(n−2)}{2} + \frac{n}{r_{1}}} (∫_{Ω_{ϵ}}(\frac{n(n−2)}{n(n−2)+|x|^{2}} )^{\frac{n−2}{2} ·(k_{3}−1)r_{1}}dx)^{1/r_{1}}||w_{ϵ}||_{r_{1}}′. \end{aligned}

Since r_{1} > n/(k_{3} − 1), we know

∫_{Ω_{ϵ}}(\frac{n(n − 2)}{n(n − 2) + |x|^{2}} )^{\frac{n−2}{2} ·(k_{3}−1)r_{1}}dx ≤ C.

Therefore, using Hölder inequality, we know that for any γ > 0, there exists a C(γ) > 0, such that

\begin{aligned}∫_{Ω} U_{ϵ}^{k_{3}−1}|w_{ϵ}| & ≤ Cϵ^{− \frac{(k_{3}−1)(n−2)}{2} + \frac{n}{r_{1}}} ||w_{ϵ}||_{p+1}^{1−a_{1}}||w_{ϵ}||_{k_{3}}^{a_{1}} \\ & ≤ γ||w_{ϵ}||_{k_{3}}^{k_{3}} + C(γ)ϵ^{(− \frac{(k_{3}−1)(n−2)}{2} + \frac{n}{r_{1}} )· \frac{k_{3}}{k_{3}−a_{1}}} ||w_{ϵ}||p+1^{\frac{(1−a_{1})k_{3}}{k_{3}−a_{1}}} , \end{aligned}

where

a_{1} = \frac{\frac{1}{r_{1}′} − \frac{1}{p+1}}{\frac{1}{k_{3}} − \frac{1}{p+1}} ∈ (0, 1).  (2.44)

Claim:

(−\frac{(k_{3} − 1)(n − 2)}{2} + \frac{n}{r_{1}} ) · \frac{k_{3}}{k_{3} − a_{1}} = \frac{k_{3}σ}{2} .

From (2.44), we have

a_{1} = \frac{(5r_{1} − 6)k_{3}}{(6 − k_{3})r_{1}} ,

thus

k_{3} − a_{1} = k_{3} · \frac{r_{1} − r_{1}k_{3} + 6}{6r_{1} − r_{1}k_{3}}.

It follows that (note n = 3 and σ = \frac{6}{k_{3}} − 1)

(−\frac{(k_{3}−1)(n−2)}{2} + \frac{n}{r_{1}} ) · \frac{k_{3}}{k_{3}−a_{1}}
= \frac{6−r_{1}k_{3}+r_{1}}{2r_{1}} · \frac{k_{3}}{k_{3}−a_{1}}
= \frac{6−r_{1}k_{3}+r_{1}}{2r_{1}} · \frac{k_{3}(6r_{1}−r_{1}k_{3})}{k_{3}(r_{1}−r_{1}k_{3}+6)}
= \frac{k_{3}}{2} · (\frac{6}{k_{3}} − 1)
= \frac{k_{3}σ}{2}.

Notice ||w_{ϵ}||_{p+1} → 0. We have

∫_{Ω} U_{ϵ}^{k_{3}−1}|w_{ϵ}| ≤ o_{ϵ}(1)ϵ^{\frac{k_{3}σ}{2}} + γ||w_{ϵ}||_{k_{3}}^{k_{3}}.

We hereby complete the proof of Lemma 2.4.

We still need several elementary inequalities. Some of them are obvious and we list them here without providing proofs.

Lemma 2.5 There exist C_{1} > 0, C_{2} > 0 such that for all x ≥ 0 and x+y ≥ 0,

(x + y)^{τ} ≥ |y|^{τ} + C_{1}x^{τ} − C_{2}x^{τ−1}|y|.  (2.45)

Proof. The following elementary inequality is easy to check: there exist C_{1} > 0, C_{2} > 0, such that: ∀a with 1 + a ≥ 0

(1 + a)^{τ} ≥ |a|^{τ} + C_{1} − C_{2} · |a|.

If x ≠ 0, (2.45) follows from the above directly. If x = 0, (2.45) is obvious.

Lemma 2.6 For any 0 < γ < 1, there exists a C_{3}(γ) > 0, such that

(x + y)^{2/τ} ≥ γx^{2/τ} + C_{3}(γ)y^{2/τ}, ∀x, y ≥ 0.

Lemma 2.7 For any γ > 0, there exist C_{4}(γ), C_{1,4}(γ) > 1 such that for any x ≥ 0 and x + y ≥ 0

(x + y)^{p+1} ≤ x^{p+1} + (p + 1)x^{p}y + \frac{(p + γ)(p + 1)}{2}x^{p−1}y^{2} + C_{4}(γ)|y|^{p+1}

and

(x + y)^{q+1} ≤ x^{q+1} + (q + 1)x^{q}y + \frac{(q + γ)(q + 1)}{2}x^{q−1}y^{2} + C_{1,4}(γ)|y|^{q+1}.

Proof. We give a proof for the first inequality and skip the second one. We only need to show that for any γ > 0, there exists a C_{4}(γ) > 1 such that for any 1 + a ≥ 0

(1 + a)^{p+1} ≤ 1 + (p + 1)a + \frac{(p + γ)(p + 1)}{2}a^{2} + C_{4}(γ)|a|^{p+1}.

Let f(a) = 1 + (p + 1)a + \frac{(p+γ)(p+1)}{2}a^{2} + C|a|^{p+1} − (1 + a)^{p+1}. Easy to check

f′(a) = (p + 1)[1 + (p + γ)a + C|a|^{p−1}a − (1 + a)^{p}],

and

f′′(a) = (p + 1)p[\frac{p + γ}{p} + C|a|^{p−1} − (1 + a)^{p−1}].

If −1 ≤ a ≤ (1 + γ/p)^{1/(p−1)} − 1, we know that f′′(a) ≥ 0 for any C > 0. If a > (1 + γ/p)^{1/(p−1)} − 1, choosing C large enough, we still have f′′(a) ≥ 0. Since f′(0) = 0 , we know that as C large enough, f(a) ≥ f(0) = 0 for a ≥ −1.

Lemma 2.8 For any r > 0

(1 + x)^{−r} ≥ 1 − rx, ∀x ≥ 0.

Now we can give the lower bound energy estimates for I_{α}(\overline{u}_{α}) and II_{α}(\overline{u}_{1,α}). Let us first look at I_{α}(\overline{u}_{α}). It follows from Lemma 2.7 that

\begin{aligned}||\overline{u}_{α}||_{p+1}^{p+1} & ≤ C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1} + (p + 1)C_{α}^{p}| ∫_{Ω} U_{ϵ}^{p}w_{ϵ}| \\ & +\frac{(p+γ)(p+1)}{2}C_{α}^{p−1} ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2} + C_{4}(γ)||w_{ϵ}||^{p+1}. \end{aligned}

From Lemma 2.8 and (2.39), we know that

||\overline{u}_{α}||_{p+1}^{−2}

≥ C_{α}^{−2}||U_{ϵ}||_{p+1}^{−2}(1 + \frac{(p+γ)(p+1)C_{α}^{p−1} ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{2C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} + \frac{(p+1)C_{α}^{p}| ∫_{Ω} U_{ϵ}^{p}w_{ϵ}|+C_{4}(γ)||w_{ϵ}||^{p+1}}{C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} )^{− \frac{2}{p+1}}
≥ C_{α}^{−2}||U_{ϵ}||_{p+1}^{−2}(1 − \frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} − C| ∫_{Ω} U_{ϵ}^{p}w_{ϵ}| − C||w_{ϵ}||^{p+1})
≥ C_{α}^{−2}||U_{ϵ}||_{p+1}^{−2}(1 − \frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} − Cβ_{2}(ϵ)||w_{ϵ}|| − C||w_{ϵ}||^{p+1}).

Using (2.36) and lemma 2.5 we have that

\begin{aligned}∫_{Ω} \overline{u}_{α}^{τ} & = ∫_{Ω}(C_{α}U_{ϵ} + w_{ϵ})^{τ} \\ & ≥ ∫_{Ω} |w_{ϵ}|^{τ} + C_{1} ∫_{Ω} C_{α}^{τ}U_{ϵ}^{τ} − O(1)(∫_{Ω} U_{ϵ}^{τ−1}|w_{ϵ}|). \end{aligned}

Due to (2.30), (2.38), Lemma 2.4 and Lemma 2.6, for any 0 < γ_{3} < 1, there exists a C(γ_{3}) > 0 such that

(∫_{Ω} \overline{u}_{α}^{τ})^{2/τ} ≥ γ_{3}||w_{ϵ}||_{τ}^{2} + C(γ_{3})ϵ^{σ}.

Since ∫_{Ω} ∇U_{ϵ} · ∇w_{ϵ} = 0, by using (2.32), we have

I_{α}(\overline{u}_{α}) ≥ \{C_{α}^{2}||∇U_{ϵ}||_{2}^{2} + ||∇w_{ϵ}||_{2}^{2} + αγ_{3}||w_{ϵ}||_{τ}^{2} + αC(γ_{3})ϵ^{σ}\}
·C_{α}^{−2}||U_{ϵ}||_{p+1}^{−2}(1 − \frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} − Cβ_{2}(ϵ)||w_{ϵ}|| − C||w_{ϵ}||^{p+1})
≥ \{J_{α}(U_{ϵ}) + αC · C(γ_{3})ϵ^{σ} + \frac{||∇w_{ϵ}||_{2}^{2}+αγ_{3}||w_{ϵ}||_{τ}^{2}}{C_{α}^{2}||U_{ϵ}||_{p+1}^{2}} \}
·(1 − \frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} − C(γ_{4})β_{2}(ϵ)^{2} − γ_{4}||w_{ϵ}||^{2} − C||w_{ϵ}||^{p+1})
≥ \frac{1}{2^{2/n}S_{1}} − \frac{n−2}{2}A_{n}H(x_{α})β_{1}(ϵ) + αC · C(γ_{3})ϵ^{σ} − C_{5} · γ_{4}||w_{ϵ}||^{2}
−C_{5} · C(γ_{4})β_{2}(ϵ)^{2} − C_{5} · C||w_{ϵ}||^{p+1} + \frac{||∇w_{ϵ}||_{2}^{2}+αγ_{3}||w_{ϵ}||_{τ}^{2}}{C_{α}^{2}||U_{ϵ}||_{p+1}^{2}}
−\frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{2^{2/n}S_{1}C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} + high order terms.

Note that ||U_{ϵ}||_{p+1}^{p−1} → 1/(2^{2/n}S_{1}) and C_{α} → 1. If we choose γ small and γ_{3} close to 1 such that γ_{3}p/(p + γ) > γ_{2}, by Corollary 2.2, we have

\begin{matrix}\frac{||∇w_{ϵ}||_{2}^{2}+αγ_{3}||w_{ϵ}||_{τ}^{2}}{C_{α}^{2}||U_{ϵ}||_{p+1}^{2}} − \frac{(p+γ) ∫_{Ω} U_{ϵ}^{p−1}w_{ϵ}^{2}}{2^{2/n}S_{1}C_{α}^{p+1}||U_{ϵ}||_{p+1}^{p+1}} \\ ≥ \overline{C}(γ_{3})(||∇w_{ϵ}||_{2}^{2} + α||w_{ϵ}||_{τ}^{2} + o(1)β_{2}(ϵ)^{2}) \end{matrix}

for some \overline{C}(γ_{3}) > 0. Then we choose γ_{4} small enough such that C_{5}·γ_{4} < \overline{C}(γ_{3}). Note ||w_{ϵ}||^{p+1} = o(1)||w_{ϵ}||^{2}. We have for some C, C(γ_{3}) > 0 independent of α, that

\begin{aligned}I_{α}(\overline{u}_{α}) & ≥ \frac{1}{2^{2/n}S_{1}} − \frac{n−2}{2}H(x_{α})A_{n}β_{1}(ϵ) + αCC(γ_{3}))ϵ^{σ} − O(1)β_{2}(ϵ)^{2} \\ & > \frac{1}{2^{2/n}S_{1}} , as α → ∞. \end{aligned}

Therefore

I_{α}(u_{α}) > \frac{1}{2^{2/n}S_{1}} as α → ∞,

This contradicts to (1.11). Thus we complete the proof of Theorem 0.1. Similar calculation shows that for some C̃, C̃(γ_{3}) > 0 independent of α,

\begin{aligned}II_{α}(\overline{u}_{1,α}) & ≥ \frac{1}{S} − \frac{n−2}{2}H(x_{1,α})A_{1,n}β_{1}(ϵ) + αC̃C̃(γ_{3}))ϵ^{σ} − O(1)β_{2}(ϵ)^{2} \\ & > \frac{1}{S} , as α → ∞. \end{aligned}

Thus

II_{α}(u_{1,α}) > \frac{1}{S} as α → ∞,

This contradicts to (1.12), we hereby complete the proof of Theorem 0.2.

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