Uniqueness results through a priori estimates part II: Dirichlet problem ^{∗}

Meijun Zhu Department of Mathematics and Statistics McMaster University Hamilton, ON. Canada L8S 4K1 zhum@icarus.math.mcmaster.ca

Abstract

Global L^{∞} bound and uniqueness results about the Dirichlet problem: −Δu − αu = u^{(n+2)/(n−2)}, u ≥ 0 in Ω, u = 0 on ∂Ω are obtained, where Ω ⊂ ℝ^{n} (n ≥ 3) is a bounded smooth domain and α ∈ (0, λ_{1}) is close to λ_{1}, the first eigenvalue of −Δ with Dirichlet boundary condition.

0 Introduction

This is the second part of our study on the uniqueness property of some nonlinear elliptic problems when some parameters in front of linear terms tend to the first eigenvalues of the corresponding linear problems. In this part, we are interested in the following Dirichlet boundary problem:

\{ \begin{matrix}−Δu − αu = u^{\frac{n+2}{n−2}} , u > 0, & in Ω \\ u = 0 & on ∂Ω. \end{matrix}  (0.1)

where Ω ⊂ ℝ^{n} (n ≥ 3) is a smooth bounded domain and α is a constant. Denote λ_{1} as the first eigenvalue of −Δ with Dirichlet boundary condition. It is well known that (0.1) has no solution for α ≥ λ_{1} or α < 0 when Ω is starshaped . Throughout this paper we always assume α ∈ (0, λ_{1}).

∗1991 Mathematical Subject Classification: 35J65, 35B40

Equation (0.1) arises naturally from some geometric and physical problems and has been studied extensively by many authors, see for example the celebrating paper of Brezis and Nirenberg [2]. In this paper we shall address the asymptotic behaviors and the uniqueness of solutions to equation (0.1) as α → λ_{1}. The local uniqueness property about (0.1) may follow from the standard bifurcation result (see Section 1 for a different argument). But the global uniqueness is unknown. If u^{(n+2)/(n−2)} is replaced by u^{p} in (0.1) and p is a subcritical power (that is 1 < p < (n + 2)/(n − 2)), through a simple blowup argument (by using a Liouville type theorem due to Gidas and Spruck [4]), one can show that L^{∞} norms of solutions to (0.1) are uniformly bounded as α → λ_{1}. However, such argument breaks down in critical cases, since −Δu = u^{(n+2)/(n−2)} does have positive solutions in ℝ^{n}.

When Ω is the unit disk B_{1}(0), Atkinson and Peletier [1] considered the asymptotic behaviors of solutions to (0.1) as α → λ_{1}. In this case, in view of a standard result of Gidas, Ni and Nirenberg [3], one knows that all solutions to (0.1) are radial. Therefore (0.1) becomes an ordinary differential equation. By using shooting method, Atkinson and Peletier were able to show that L^{∞} norms of solutions tend to 0 as α → λ_{1}. Obviously, such argument will break down once Ω is not a disk.

The purpose of this paper is, by using energy independent a priori estimates, to show the following:

Theorem 0.1 Let Ω be a smooth bounded domain in ℝ^{n} and u_{α} be a solution to (0.1). Then ||u_{α}||_{L^{∞}} → 0 as α → λ_{1}. Furthermore, there exists a constant α_{0} > 0 such that, for α ∈ (λ_{1} − α_{0}, λ_{1}), (0.1) has a unique solution.

For energy independent a priori estimates, we refer readers to [10], [11], [7]-[9] and references therein.

The main step in the proof of the above theorem is to show that L^{∞} norms of solutions to (0.1) are uniformly bounded as α → λ_{1}. We outline it as follows. Let u_{α} be a solution of (0.1). First of all, by using the method of moving planes, one can show that there exist constants δ_{0} > 0 and C > 0 such that for α ∈ (λ_{1}/2, λ_{1})

0 < u_{α}(x) ≤ C, ∀x ∈ {x ∈ Ω : dist(x, ∂Ω) ≤ δ_{0}}.

The main difficulty is to show that u_{α} is uniformly bounded in {x ∈ Ω | dist(x, ∂Ω) ≥ δ_{0}}. We use a contradiction argument. Suppose not, up to some subsequence, ||u_{α}||_{L^{∞}} → ∞ as α → λ_{1}. We can show, through some tedious arguments, that after passing to some further subsequence of {u_{α}}, ||u_{α}||_{L^{∞}} · u_{α}(x) → G(x, {q_{j}^{∗}}_{j=1}^{k}) in C^{1}(K) for any compact set K in Ω \ {q_{j}^{∗}}_{j=1}^{k}, where G(x, {q_{j}^{∗}}_{j=1}^{k}) is the Green’s function in Ω with Dirichlet boundary condition and some discrete poles at {q_{j}^{∗}}_{j=1}^{k} ⊂ {x ∈ Ω dist(x, ∂Ω) ≥ δ_{0}}. However, it is not difficult to show that there is no such Green’s function, we thus derive a contradiction!

We organize this paper as follows. In Section 1, under the assumptions of the a priori L^{∞} bound, we prove the uniqueness part in Theorem 0.1. In Section 2, we focus on obtaining those a priori estimates. Later on, throughout this paper, we always assume that Ω ⊂ ℝ^{n} (n ≥ 3) is a bounded domain and use C, C_{0}, C_{1}, ⋯, to represent various positive constants.

Acknowledgments: I would like to thank H.Brezis, Y.Y. Li, R. Nussbaum and P. Rabinowitz for some useful conversations during the preparation of this paper. This work was carried out when I was a postdoctoral fellow at the University of British Columbia and Pacific Institute for mathematical sciences. I would like to thank N. Ghoussoub, C. Gui and Pacific Institute for mathematical sciences for their generous supports.

1 Small L^{∞} norm implies uniqueness

The existence of one solution to (0.1) for α close to λ_{1} was obtained in [2]. In order to derive a global uniqueness result, we need the following a priori estimates.

Theorem 1.1 There exist constants δ_{1} > 0 and C = C(δ_{1}) > 0 such that, if u_{α} is a solution to (0.1) for α ∈ (λ_{1} − δ_{1}, λ_{1}), then

||u_{α}||_{L^{∞}(Ω)} ≤ C.

The proof of Theorem 1.1 is the main part of this paper and will be given in next section.

Notice that the only solution to

\{ \begin{matrix}−Δu − λ_{1}u = u^{\frac{n+2}{n−2}} , u ≥ 0, & in Ω \\ u = 0 & on ∂Ω \end{matrix}

is the trivial one. A quick consequence of Theorem 1.1 is the following corollary, which yields the first part of Theorem 0.1.

Corollary 1.1 Let u_{α} be a solution of (0.1), then ||u_{α}||_{L^{∞}} → 0 as α → λ_{1}.

We are now ready to give the proof of the uniqueness part in Theorem 0.1 by using the above corollary.

Proof of uniqueness. We prove it through a contradiction argument. Suppose not, then there are sequences α_{i} → λ_{1} as i → ∞, {u_{α_{i}}} and {v_{α_{i}}} such that for all i ≥ 1, u_{α_{i}} > 0 and v_{α_{i}} > 0 solve (0.1) for α = α_{i} and ||u_{α_{i}} − v_{α_{i}}||_{L^{∞}} ≠ 0.

Without loss of generality, we may assume that ||u_{α_{i}} −v_{α_{i}}||_{L^{∞}} = max\overline{Ω}(u_{α_{i}} −v_{α_{i}}). Denote A_{α_{i}} = max\overline{Ω}(u_{α_{i}} − v_{α_{i}}) and z_{α_{i}} = (u_{α_{i}} − v_{α_{i}})/A_{α_{i}}. Then z_{α_{i}} satisfies

\{ \begin{matrix}−Δz_{α_{i}} − α_{i}z_{α_{i}} = c_{α_{i}}z_{α_{i}} & in Ω \\ z_{α_{i}} = 0 & on ∂Ω, \end{matrix}

where c_{α_{i}}(x) = \frac{n+2}{n−2}ξα_{i}^{\frac{4}{n−2}} and ξ_{α_{i}} is a positive function between u_{α_{i}} and v_{α_{i}}. It follows from Corollary 1.1 that ||c_{α_{i}}||_{L^{∞}} → 0 as i → ∞. Noticing ||z_{α_{i}}||_{L^{∞}} = 1, using standard elliptic estimates, we know that z_{α_{i}} → z_{0} in C^{2}(\overline{Ω}) and z_{0} satisfies

−Δz_{0} − λ_{1}z_{0} = 0 in Ω
z_{0} = 0  on ∂Ω
max\overline{Ω}z_{0} = 1.

It follows that z_{0} is a first eigenfunction of −Δ, therefore

z_{0} > 0 in Ω, ∂z_{0}/∂ν < 0 on ∂Ω  (1.1)

where ν is the exterior unit normal on ∂Ω.

On the other hand, since u_{α_{i}} and v_{α_{i}} satisfy (0.1), we have

∫_{Ω} −Δu_{α_{i}}v_{α_{i}} − ∫_{Ω} α_{i}u_{α_{i}}v_{α_{i}} = ∫_{Ω} uα_{i}^{\frac{n+2}{n−2}} v_{α_{i}}

and

∫_{Ω} −Δv_{α_{i}}u_{α_{i}} − ∫_{Ω} α_{i}v_{α_{i}}u_{α_{i}} = ∫_{Ω} vα_{i}^{\frac{n+2}{n−2}} u_{α_{i}}.

Combining with the boundary condition, we have

∫_{Ω} u_{α_{i}}v_{α_{i}}(uα_{i}^{\frac{4}{n−2}} − vα_{i}^{\frac{4}{n−2}} ) = 0.

Therefore, for all α_{i}, there exists x_{α_{i}} ∈ Ω such that z_{α_{i}}(x_{α_{i}}) = 0. Since z_{0}(x) > 0 in Ω, we know that x_{α_{i}} → x_{0} ∈ ∂Ω. Let x̃_{α_{i}} be one of the closest points to x_{α_{i}} on ∂Ω. Then the direction of \overline{x_{α_{i}} x̃_{α_{i}}} is close to the exterior unit normal ν_{0} at x_{0} as i → ∞. Since z_{α_{i}}(x̃_{α_{i}}) = 0, it then follows from the mean value theorem and z_{α_{i}} → z_{0} in C^{2}(\overline{Ω}) that

\frac{∂z_{0}}{∂ν} |_{x=x_{0}} = 0.

This contradicts to (1.1).

2 A priori estimates

We devote to prove Theorem 1.1 in this section by using energy independent a priori estimates. Throughout this section, we always assume that u_{α} is a solution to (0.1).

Our first lemma says that u_{α} has a uniformly upper L^{∞} bound near the boundary of Ω.

Lemma 2.1 There exist some positive constants δ_{0} and C = C(δ_{0}) such that for all α ∈ (λ_{1}/2, λ_{1}),

u_{α}(y) ≤ C, ∀y ∈ {y ∈ Ω | dist(y, ∂Ω) ≤ δ_{0}}.

This lemma can be proved by the standard moving plane method combining with some interior integral estimates. We refer readers to [5] (Page 163-164) and shall omit details here.

From now on, we begin to prove Theorem 1.1 by contradiction. Suppose that Theorem 1.1 is false. We have some sequences α and corresponding solutions u_{α} to (0.1) such that ||u_{α}||_{L^{∞}} → ∞ as α → λ_{1}.

2.1 Definition and basic properties on blowup points

We classify blowup points of u_{α} as in [7].

Definition 2.1 Let δ_{0} be the constant given in Lemma 2.1. A point ȳ ∈ Ω is called an isolated blowup point of {u_{α}} if there exist 0 < r̄ ≤ δ_{0}/2, C̃ > 0, and a sequence y_{α} ∈ Ω tending to ȳ as α → λ_{1}, such that, y_{α} is a local maximum of u_{α}, u_{α}(y_{α}) → +∞ and

u_{α}(y) ≤ C̃|y − y_{α}|^{− \frac{n−2}{2}} for all y ∈ Br̄(y_{α}).  (2.1)

Due to Lemma 2.1, we easily see that an isolated blowup point has a distance to ∂Ω larger than δ_{0}.

Let y_{α} → ȳ be an isolated blowup point of {u_{α}}. For any 0 < r ≤ δ_{0}/2, set

ū_{α}(r) = \frac{1}{|∂B_{r}(y_{α})|} ∫_{∂B_{r}(y_{α})} u_{α}.  (2.2)

Definition 2.2 ȳ ∈ Ω is called an isolated simple blowup point, if ȳ is an isolated blowup point of {u_{α}}, such that, there exists a 0 < ρ ≤ r̄/3 (independent of α), r^{\frac{n−2}{2}} ū_{α}(r) has precisely one critical point in (0, ρ) for α close to λ_{1}.

We present some basic properties on isolated and isolated simple blowup points.

Lemma 2.2 Suppose that y_{α} → ȳ ∈ Ω is an isolated blowup point. Then for any 0 < r < \frac{1}{3} r̄, we have the following Harnack inequality:

max_{y∈B_{2r}(y_{α})\B_{\frac{r}{2}} (y_{α})} u_{α}(y) ≤ C_{1} min_{y∈B_{2r}(y_{α})\B_{\frac{r}{2}} (y_{α})} u_{α}(y),  (2.3)

where C_{1} is a positive constant independent of r and α.

Lemma 2.3 Suppose that y_{α} → ȳ ∈ Ω is an isolated blowup point. Then for any R >> 1 we have, after passing to a subsequence {u_{α}} that

||u_{α}(y_{α})^{−1}u_{α}(u_{α}^{− \frac{2}{n−2}} (y_{α})x + y_{α}) − (1 + k(n)|x|^{2})^{− \frac{n−2}{2}} ||_{C^{2}(B_{2R}(0))} → 0  (2.4)

as α → λ_{1}, where k(n) = (n(n − 2))^{−1} .

Lemma 2.4 Suppose that R >> 1 and {u_{α}} are those given in Lemma 2.3, y_{α} → ȳ is an isolated simple blowup point. Then there exist 0 < δ, ϵ_{1} << 1 and ρ_{0} = ρ_{0}(δ) > 0, such that for α ∈ (λ_{1} − ϵ_{1}, λ_{1}),

u_{α}(y) ≤ C_{2}u_{α}(y_{α})^{\frac{2δ}{n−2} −1}|y − y_{α}|^{2−n+δ} for Ru_{α}^{− \frac{2}{n−2}} (y_{α}) ≤ |y − y_{α}| ≤ ρ_{0}, (2.5)

where C_{2} is some positive constant independent of α.

We refer readers to [7] or [9] for the proofs of Lemma 2.2-2.4.

We then can show the following proposition.

Proposition 2.1 Suppose that y_{α} → ȳ is an isolated simple blowup point and {u_{α}} is the same subsequence as given in Lemma 2.3. Then there exist constant C > 0 and 0 < ϵ_{2} < 1, such that for α ∈ (λ_{1} − ϵ_{2}, λ_{1}),

u_{α}(y)u_{α}(y_{α}) ≤ C|y − y_{α}|^{2−n}, ∀|y − y_{α}| ≤ ρ  (2.6)

where ρ is given in Definition 2.2.

Proof. We first show that (2.6) holds for all y = y_{α} + μe, where e ∈ ℝ^{n} is a unit vector, μ < σ_{1} and σ_{1} is some small constant which will be chosen in the following: Let φ be the first eigenfunction of −Δ in B_{2σ_{1}}(ȳ) with respect to Dirichlet boundary condition. It is well known that as σ_{1} becomes smaller, the corresponding first eigenvalue will become larger. Since α ≤ 2λ_{1}, we are able to choose σ_{1} ≤ ρ_{0} (ρ_{0} is given in Lemma 2.4) sufficiently small so that

−Δφ − αφ > 0 in B_{σ_{1}}(ȳ).  (2.7)

Let g = φ^{\frac{4}{n−2}} dy_{i}dy_{i}, L_{0} and L_{g} denote the conformal Laplacian operators of Euclidean metric and g respectively. It is well known that

L_{g}ϕ = φ^{− \frac{n+2}{n−2}} L_{0}(ϕφ), ∀ϕ ∈ C^{∞}(B_{σ_{1}}(ȳ)).  (2.8)

Denote ũ_{α} = u_{α}/φ. Easy to see that ũ_{α} satisfies

−Δ_{g}ũ_{α} + R_{g}ũ_{α} = αφ^{− \frac{4}{n−2}} · ũ_{α} + ũα^{\frac{n+2}{n−2}}

where R_{g} is the scalar curvature of B_{σ_{1}}(ȳ) with respect to metric g. Taking ϕ = 1 in (2.8), we have

R_{g} = −Δφ/φ^{\frac{n+2}{n−2}} .

Therefore, ũ_{α} satisfies:

−Δ_{g}ũ_{α} + k_{α}ũ_{α} = ũα^{\frac{n+2}{n−2}} in B_{σ_{1}}(ȳ)  (2.9)

where k_{α} = (−Δφ − αφ) · φ^{− \frac{n+2}{n−2}} . Due to (2.7), we know that k_{α} > 0 in B_{σ_{1}}(ȳ). Let w_{α}(y) = ũ_{α}^{−1}(y_{α} + μe) · ũ_{α}(y), then w_{α} satisfies

−Δ_{g}w_{α} + k_{α}w_{α} = ũ_{α}(y_{α} + μe)^{\frac{4}{n−2}} wα^{\frac{n+2}{n−2}} in B_{σ_{1}}(ȳ).  (2.10)

Notice u_{α}/C ≤ ũ_{α} ≤ Cu_{α} in B_{σ_{1}}(ȳ). It follows from Lemma 2.2 that for all compact set K ⊂ B_{σ_{1}}(ȳ) \ {ȳ}, there exists some constant C(K) such that

C(K)^{−1} ≤ w_{α} ≤ C(K) on K.

Due to Lemma 2.4, we know that ũ_{α}(y_{α} +μe)^{\frac{4}{n−2}} → 0. It then follows from standard elliptic theories that after passing to some subsequence, as α → λ_{1},

w_{α} → w in C_{loc}^{2}(B_{σ_{1}}(ȳ) \ {ȳ}),

where w(y) satisfies

−Δ_{g}w + kw = 0, w > 0 in B_{σ_{1}}(ȳ) \ {ȳ}

with k(y) = (−Δφ − λ_{1}φ) · φ^{− \frac{n+2}{n−2}} ≥ 0.

Notice that for any fixed e and σ_{1}

lim_{α→λ_{1}} ũ_{α}^{−1}(y_{α} + σ_{1}e)r^{\frac{n−2}{2}} \overline{ũ_{α}(r)} = r^{\frac{n−2}{2}} w̄(r)

where

\overline{ũ_{α}(r)} = \frac{1}{|∂B_{r}(ȳ)|} ∫_{∂B_{r}(ȳ)} ũ_{α},  w̄(r) = \frac{1}{|∂B_{r}(ȳ)|} ∫_{∂B_{r}(ȳ)} w.

Since ȳ is an isolated simple blowup point, we know that r^{\frac{n−2}{2}} w̄(r) is nonincresing for 0 < r < ρ. This implies that w(y) has a singular point at ȳ. It follows from Corollary 9.1 in [9] that there exists a constant m > 0 such that

lim_{γ→0} ∫_{∂B_{γ}(\overline{y})} ∇_{g}w · ν = −m.

Therefore as γ sufficiently small,

− ∫_{B_{γ}(y_{α})} Δw_{α} = − ∫_{∂B_{γ}(y_{α})} ∇w_{α} · ν = − ∫_{∂B_{γ}(\overline{y})} ∇_{g}w · ν + o_{α}(1) > m/2 (2.11)

where o_{α}(1) → 0 as α → λ_{1}.

On the other hand, since k_{α}(y) ≥ 0, we have from (2.10) that for γ < σ_{1},

\begin{matrix}− ∫_{B_{γ}(y_{α})} Δ_{g}w_{α}(y) = ∫_{B_{γ}(y_{α})}[ũ_{α}(y_{α} + σ_{1}e)^{−1}ũ_{α}(y)^{\frac{n+2}{n−2}} − k_{α}(y)w_{α}(y)]dy \\ ≤ ũ_{α}(y_{α} + σ_{1}e)^{−1} ∫_{B_{γ}(y_{α})} ũ_{α}(y)^{\frac{n+2}{n−2}} dy \\ ≤ Cu_{α}(y_{α} + σ_{1}e)^{−1} ∫_{B_{γ}(y_{α})} u_{α}(y)^{\frac{n+2}{n−2}} dy. \end{matrix} (2.12)

Set x = u_{α}(y_{α})^{\frac{2}{n−2}} · (y − y_{α}) and r_{α} = Ru_{α}(y_{α})^{− \frac{2}{n−2}} . Using Lemma 2.3, we have

∫_{|y−y_{α}|≤r_{α}} uα^{\frac{n+2}{n−2}} ≤ Cu_{α}(y_{α})^{\frac{n+2}{n−2}} ∫_{|y−y_{α}|≤r_{α}}(1 + k(n)|x|^{2})^{− \frac{n+2}{2}} dy
= Cu_{α}(y_{α})^{\frac{n+2}{n−2}} ∫_{|x|≤R}(1 + k(n)|x|^{2})^{− \frac{n+2}{2}} · u_{α}(y_{α})^{− \frac{2n}{n−2}} dx
≤ Cu_{α}(y_{α})^{−1}.

Using Lemma 2.4, we have

∫_{r_{α}≤|y−y_{α}|≤γ} uα^{\frac{n+2}{n−2}} ≤ C ∫_{r_{α}≤|y−y_{α}|≤γ}(u_{α}(y_{α})^{\frac{2δ}{n−2} −1}|y − y_{α}|^{2−n+δ})^{\frac{n+2}{n−2}} dy
≤ Cr_{α}^{−2+ \frac{n+2}{n−2} ·δ}u_{α}(y_{α})^{\frac{2δ}{n−2} · \frac{n+2}{n−2} − \frac{n+2}{n−2}}
= CR^{−2+ \frac{n+2}{n−2} ·δ}u_{α}(y_{α})^{−1}.

Therefore

∫_{|y−y_{α}|<γ} uα^{\frac{n+2}{n−2}} ≤ Cu_{α}(y_{α})^{−1}.  (2.13)

It follows from (2.11), (2.12) and (2.13) that

u_{α}(y_{α} + μe) · u_{α}(y_{α}) ≤ C(μ).

We have established Proposition 2.1 for all |y − y_{α}| = μ with μ ≤ σ_{1}.

For |y − y_{α}| < r_{α}, Proposition 2.1 follows from Lemma 2.3 directly.

We now use rescaling to establish Proposition 2.1 for r_{α} ≤ |y−y_{α}| ≤ σ_{1}. We ague by contradiction. Suppose the contrary, then there exist a sequence ŷ_{α} satisfying r_{α} ≤ |ŷ_{α} − y_{α}| ≤ σ_{1} and |ŷ_{α} − y_{α}| → 0 as α → λ_{1}, such that

lim_{α→λ_{1}} u_{α}(ŷ_{α})u_{α}(y_{α})|ŷ_{α} − y_{α}|^{n−2} = ∞.

Set r̂_{α} = |ŷ_{α} − y_{α}|/σ_{1}, and û_{α}(z) = r̂α^{\frac{n−2}{2}} u_{α}(y_{α} + r̂_{α}z). Then \{û_{α}\} satisfies

−Δû_{α} − αr̂_{α}^{2}û_{α} = ûα^{\frac{n+2}{n−2}} in B_{σ_{1}/r̂_{α}}(0).

It is easy to see that {0} is an isolated simple blowup point of û_{α} . We repeat the above procedure (notice that αr̂_{α} → 0), and can show that

û_{α}(0)û_{α}(z) ≤ C ∀|z| = σ_{1}.

Choosing z = (ŷ_{α} − y_{α})/r̂_{α}, we have from the above that

û_{α}(0)û_{α}(\frac{ŷ_{α} − y_{α}}{r̂_{α}}) ≤ C.

That is

lim_{α→λ_{1}} u_{α}(ŷ_{α})u_{α}(y_{α})|ŷ_{α} − y_{α}|^{n−2} < ∞.

Contradiction!

When σ_{1} ≤ |y − y_{α}| ≤ ρ, (2.6) follows from Harnack inequality. We have hereby established Proposition 2.1.

2.2 Isolated blowup points must be isolated simple blowup points

In this subsection, we focus on showing the following proposition.

Proposition 2.2 Let y_{α} → ȳ be an isolated blowup point of u_{α}. Then ȳ must be an isolated simple blowup point.

In our current case there is no difference between n = 3 and n ≥ 4. Notice that the coefficient in front of linear term in (0.1) has a different sign to that in the equation studied in [12].

Proof of Proposition 2.2. Without loss of generality, we assume that u_{α} is the subsequence satisfying (2.4) for some R >> 1. It follows from Lemma 2.3 that r^{\frac{n−2}{2}} ū_{α}(r) has precisely one critical point in the interval 0 < r < r_{α} = Ru_{α}(y_{α})^{− \frac{2}{n−2}} . If ȳ is not an isolated simple blowup point and μ_{α} is the second critical point of r^{\frac{n−2}{2}} ū_{α}(r), we have μ_{α} ≥ r_{α} and μ_{α} → 0 as α → λ_{1}.

Set

ξ_{α}(y) = μα^{\frac{n−2}{2}} u_{α}(μ_{α}y + y_{α}), for |y| < \frac{δ_{0}}{2μ_{α}}

where δ_{0} is given in Lemma 2.1. Then ξ_{α} satisfies

−Δξ_{α}(y) − μ_{α}^{2}αξ_{α}(y) = ξ_{α}(y)^{\frac{n+2}{n−2}}  for |y| < \frac{δ_{0}}{2μ_{α}} ,
|y|^{\frac{n−2}{2}} ξ_{α}(y) ≤ C  for |y| < \frac{δ_{0}}{2μ_{α}} ,

lim_{α→λ_{1}}ξ_{α}(0) = ∞,

r^{\frac{n−2}{2}} ξ̄_{α}(r) has precisely one critical point in 0 < r < 1,

\frac{d}{dr}\{r^{\frac{n−2}{2}} ξ̄_{α}(r)\}|_{r=1} = 0.

(2.14)

It follows that {0} is an isolated simple blowup point of ξ_{α}.

From Proposition 2.1 and Harnack inequality, we know that for all compact set K ⊂ ℝ^{n} \ {0}, there exists a constant C(K) > 0 such that

C(K)^{−1} ≤ ξ_{α}(0)ξ_{α}(y) ≤ C(K) in K.  (2.15)

It follows from standard elliptic theories that

ξ_{α}(0)ξ_{α}(y) → H(y) in C_{loc}^{2}(ℝ^{n} \ {0}),  (2.16)

where H(y) satisfies

ΔH(y) = 0 in ℝ^{n} \ {0}.

Thus H(y) = a|y|^{2−n} + b(y) with a ≥ 0 and b(y) being a regular harmonic function in ℝ^{n}. Since H(y) ≥ 0, we know from the maximum principle that b(y) ≥ 0, thus b(y) ≡ b ≥ 0. From (2.14) and (2.16), we have

\frac{d}{dr}\{r^{\frac{n−2}{2}} H(r)\}|_{r=1} = 0.

It follows that b = a ≥ 0. Due to (2.15), we know a = b > 0.

We need to use the following Pohozaev identity. We refer readers to [6] or [12] for the proof.

Lemma 2.5 Assume that v satisfies

−Δv − αv = v^{\frac{n+2}{n−2}} , v ≥ 0, in Ω.

If B_{σ}(x_{0}) ⊂ Ω, then

∫_{B_{σ}(x_{0})} αv^{2} = ∫_{∂B_{σ}(x_{0})} B(y, σ, v, ∇v) + σ ∫_{∂B_{σ}(x_{0})}(\frac{αv^{2}}{2} + \frac{n − 2}{2n}v^{\frac{2n}{n−2}} )

where

B(y, σ, v, ∇v) = \frac{n − 2}{2}v \frac{∂v}{∂ν} − \frac{σ}{2} |∇v|^{2} + σ(\frac{∂v}{∂ν})^{2},

ν denotes the exterior unit normal on ∂B_{σ}(x_{0}).

We now continue our proof of Proposition 2.2. Denote B_{σ} = B_{σ}(0). Applying Lemma 2.5 to (2.14) for some small σ, we have

μ_{α}^{2} ∫_{B_{σ}} αξ_{α}^{2} = ∫_{∂B_{σ}} B(y, σ, ξ_{α}, ∇ξ_{α}) + σ ∫_{∂B_{σ}}(\frac{μ_{α}^{2}αξ_{α}^{2}}{2} + \frac{n − 2}{2n}ξα^{\frac{2n}{n−2}} ),  (2.17)

where

B(y, σ, ξ_{α}, ∇ξ_{α}) = \frac{n − 2}{2}ξ_{α} \frac{∂ξ_{α}}{∂ν} − \frac{σ}{2} |∇ξ_{α}|^{2} + σ(\frac{∂ξ_{α}}{∂ν})^{2}.

From (2.16) we have

\begin{matrix}lim_{α→λ_{1}}ξ_{α}(0)^{2} ∫_{∂B_{σ}} B(y, σ, ξ_{α}, ∇ξ_{α}) = ∫_{∂B_{σ}} B(y, σ, H, ∇H) \\ = −\frac{(n−2)^{2}a^{2}}{2} |∂B_{1}| < 0. \end{matrix}  (2.18)

Also from (2.16) and μ_{α} → 0 as α tends to λ_{1}, we know

lim_{α→λ_{1}}ξ_{α}(0)^{2} · σ ∫_{∂B_{σ}}(\frac{μ_{α}^{2}αξ_{α}(y)^{2}}{2} + \frac{n − 2}{2n}ξ_{α}(y)^{\frac{2n}{n−2}} ) = 0.

Multiply both sides of (2.17) by ξ_{α}(0)^{2} and send σ to 0, α to λ_{1}. The right hand side of (2.17) becomes a negative number. However the left hand side of (2.17) is greater than or equals to zero. Contradiction! We have thus established Proposition 2.2.

2.3 Finite isolated simple blowup points

In this subsection, we show that u_{α} has only isolated blowup points and these points must stay away from each other uniformly. Combing with previous subsection we know that u_{α} has only finite isolated simple blowup points.

By a slight modification of the proof of Lemma 3.1 in [11] (or see [9]) and using Lemma 2.1, we have the following proposition.

Proposition 2.3 Assume that λ_{1}/2 < α < λ_{1} and u is a solution to (0.1). For any ϵ > 0 and R > 0, there exist C^{∗}(ϵ, R), C_{∗}(ϵ, R) > 0, such that: if

max_{Ω} u(y) > C^{∗}(ϵ, R),

then there exist finite points in Ω denoted as

BL(u) = {q_{1}, q_{2}, ⋯, q_{k}}

such that

i) Each q_{l} is a local maximum point of u. Let r_{l} = Ru^{− \frac{2}{n−2}} (q_{l}), then B_{r_{l}}(q_{l}) ∩ B_{r_{m}}(q_{m}) = ∅ for l ≠ m and

||u^{−1}(q_{l})u(u^{− \frac{2}{n−2}} (q_{l})x + q_{l}) − v(x)||_{C^{2}(B_{2R}(0))} < ϵ,

where v(x) = (1 + \frac{|x|^{2}}{n(n−2)})^{− \frac{n−2}{2}} .

ii)

u(y) ≤ C_{∗}\{dist(y, BL(u))\}^{− \frac{n−2}{2}} , ∀y ∈ Ω.

We now state our main proposition in this subsection.

Proposition 2.4 Assume that λ_{1}/2 < α < λ_{1} and u_{α} is a solution to (0.1). Then for ϵ > 0 sufficiently small and R > 1 sufficiently large, there exists a constant δ^{∗} = δ^{∗}(ϵ, R, C^{∗}), such that : if max_{Ω}u_{α}(x) > C^{∗}, we have

|q_{l} − q_{m}| ≥ δ^{∗}, for all 1 ≤ l ≠ m ≤ k,

where C^{∗}, q_{l} = q_{l}(u_{α}), q_{m} = q_{m}(u_{α}), k = k(u_{α}) are the ones defined in Proposition 2.3.

Proof. We prove it by contradiction. Suppose the contrary, then for some small ϵ > 0, large R, α_{i} and {u_{i}(y)} satisfying

\{ \begin{matrix}−Δu_{i} − α_{i}u_{i} = ui^{\frac{n+2}{n−2}} , u_{i} > 0, & in Ω \\ u_{i} = 0 & on ∂Ω, \end{matrix}  (2.19)

with max_{Ω} u_{i} ≥ C^{∗}, we have

lim_{i→∞} min_{l≠m}|q_{l}(u_{i}) − q_{m}(u_{i})| = 0.

Without loss of generality, we can assume that

δ_{i} := |q_{1}(u_{i}) − q_{2}(u_{i})| = min_{l≠m}|q_{l}(u_{i}) − q_{m}(u_{i})| → 0.

In view of Proposition 2.3 (i), we have that B_{Ru_{i}(q_{1})^{− \frac{2}{n−2}}} (q_{1}) ∩ B_{Ru_{i}(q_{2})^{− \frac{2}{n−2}}} (q_{2}) = ∅. It follows that u_{i}(q_{1}), u_{i}(q_{2}) → ∞.

Let w_{i}(x) = δi^{\frac{n−2}{2}} u_{i}(δ_{i}x + q_{1}), Ω_{i} = (Ω − q_{1})/δ_{i}. Without loss of generality, we assume σ_{0} = 2 in Lemma 2.1. Therefore B_{\frac{1}{δ_{i}}} (0) ⊂ Ω_{i}. It follows that w_{i}(x) satisfies

\{ \begin{matrix}−Δw_{i}(x) − δ_{i}^{2}α_{i}w_{i}(x) = w_{i}(x)^{\frac{n+2}{n−2}} & in B_{\frac{1}{δ_{i}}} (0) \\ w_{i}(x) > 0 & in B_{\frac{1}{δ_{i}}} (0). \end{matrix}  (2.20)

From Proposition 2.3 i) we know

δ_{i} > max\{Ru_{i}(q_{1})^{− \frac{2}{n−2}} , Ru_{i}(q_{2})^{− \frac{2}{n−2}} \}.  (2.21)

and

u_{i}(y) ≤ C_{∗}dist(y, \{q_{1}, q_{2}, ...q_{k}\})^{− \frac{n−2}{2}} for all y ∈ Ω.  (2.22)

Denote x_{l} = (q_{l} − q_{1})/δ_{i} for 2 ≤ l ≤ k, we have (using (2.21) and (2.22)) that

w_{i}(0), w_{i}(x_{2}) ≥ R;

Each x_{l} is a local maximum point of w_{i}; w_{i}(x)dist(x, \{x_{1}, x_{2}, ⋯, x_{k}\})^{\frac{n−2}{2}} ≤ C_{∗} ∀x ∈ Ω_{i}.

As in [9], one can show that

w_{i}(0) → ∞, and w_{i}(x_{2}) → ∞.

According to our definition, we know that {0} and x_{2} → x̄ are two isolated blowup points for w_{i}(x). Therefore, from Proposition 2.2 we know that these two points are exactly two isolated simple blowup points for w_{i}(x).

After passing to a subsequence, we denote S̃ as the set of blowup points of w_{i}(x), and have

min{|x̄_{m} − x̄_{l}| : x̄_{m}, x̄_{l} ∈ S̃} ≥ 1.

Without loss of generality, we can assume that w_{i} satisfies (2.4) for some R >> 1 and assume (passing to some further subsequence) that

w_{i}(0) = min_{x∈S̃} w_{i}(x).

From Proposition 2.1, we have

w_{i}(0)w_{i}(x) ≤ C(K) K ⊂⊂ ℝ^{n} \ S̃.  (2.23)

Thus

lim_{i→∞}w_{i}(0)w_{i}(x) = h^{∗}(x) in C_{loc}^{2}(ℝ^{n} \ S̃)

where h^{∗}(x) is a regular harmonic function in ℝ^{n} \ S̃. We know from Böcher’s theorem that

h^{∗}(x) = a_{1}|x|^{−1} + a_{2}|x − x̄_{1}|^{−1} + b^{∗}(x) x ∈ ℝ^{n} \ S̃

where a_{1}, a_{2} ≥ 0 and b^{∗} is some regular harmonic function in ℝ^{n}\{S̃\{0, x̄_{1}}}. Since \{0\} is an isolated simple blowup point, we know that r^{(n−2)/2}\overline{h^{∗}}(r) is nonincreasing near r = 0. It follows that h^{∗} has a singular point at {0}, that is a_{1} > 0. Similarly, we know a_{2} > 0. Also noticing h^{∗} ≥ 0 and h^{∗} → ∞ as x → S̃, we know that b^{∗} ≥ 0 as |x| → ∞ or x → S̃. It follows from the maximum principle that b^{∗}(x) ≥ 0 for all x ∈ ℝ^{n} \ {S̃ \ {0, x̄_{1}}}. Therefore, for some A > 0

h^{∗}(x) = a_{1}|x|^{−1} + A + O(|x|) as |x| → 0.  (2.24)

Choosing σ < 1 such that B_{σ}(0) ⊂ Ω_{i}, we apply Lemma 2.5 to (2.20) on B_{σ}(0) and get

\begin{matrix}δ_{i}^{2} ∫_{B_{σ}(0)} α_{i}w_{i}^{2}dx \\ = ∫_{∂B_{σ}(0)} B(x, σ, w_{i}, ∇w_{i}) + σ ∫_{∂B_{σ}(0)}(\frac{δ_{i}^{2}α_{i}w_{i}^{2}}{2} + \frac{n−2}{2n}wi^{\frac{2n}{n−2}} ). \end{matrix}  (2.25)

From (2.24), one can check that as σ small enough,

lim_{i→∞}w_{i}(0)^{2} ∫_{∂B_{σ}} B(σ, y, w_{i}, ∇w_{i}) = ∫_{∂B_{σ}} B(σ, y, h^{∗}, ∇h^{∗}) < 0.  (2.26)

From (2.23), we have

lim_{i→∞} w_{i}(0)^{2} · σ ∫_{∂B_{σ}(0)}(\frac{δ_{i}^{2}α_{i}w_{i}^{2}}{2} + \frac{n − 2}{2n}wi^{\frac{2n}{n−2}} ) = 0.

Multiply (2.25) by w_{i}(0)^{2} and send σ to zero and i to ∞. The left hand side of (2.25) is greater than or equals to zero, but the right hand side becomes a negative number. Contradiction! We have hereby established Proposition 2.4

2.4 Proof of Theorem 1.1

After completing the blowup analysis, we return to the proof of Theorem 1.1. Since ||u_{α}||_{L^{∞}} → ∞ as α → λ_{1}, according to Proposition 2.2 and 2.4, we know that after passing to a further subsequence of α, we can find k points in Ω: BL(u_{α}) = {q_{1}, ⋯, q_{k}}, q_{l} → q_{l}^{∗} as α → λ_{1} for l = 1, ⋯, k, which are isolated and simple blowup points. Without loss of generality, we assume that u_{α}(q_{1}) = min_{q∈BL(u_{α})} u_{α}(q). Let w_{α}(x) = u_{α}(q_{1}) · u_{α}(x). From Proposition 2.1, we know that for any K ⊂⊂ Ω \ BL(u_{α})

w_{α}(x) ≤ C in K.

Therefore,

w_{α}(x) → w_{0} in C^{2}(K)

and w_{0} > 0 satisfies

−Δw_{0} − λ_{1}w_{0} = 0 in Ω \ {q_{1}^{∗}, q_{2}^{∗}, ⋯, q_{k}^{∗}}
w_{0} = 0 on ∂Ω
dist(q_{i}, ∂Ω) ≥ σ_{0}, ∀i = 1, ⋯, k.

Since q_{1}^{∗}, q_{2}^{∗}, ⋯, q_{k}^{∗} are isolated simple blowup points, we know that w_{0}(x) → ∞ as x → {q_{1}^{∗}, q_{2}^{∗}, ⋯, q_{k}^{∗}}. It follows (see for example the proof of Proposition 9.1 in [9]) that

\{ \begin{matrix}−Δw_{0} − λ_{1}w_{0} = ∑_{l=1}^{k} a_{l}^{∗}δ_{q_{l}^{∗}} in Ω \\ w_{0} = 0 on ∂Ω \end{matrix}  (2.27)

where a_{l}^{∗} > 0.

However, we claim that there does not exist such a Green’s function. Therefore we derive a contradiction by assuming ||u_{α}||_{L^{∞}} → ∞ as α → λ_{1}, and complete the proof of Theorem 1.1.

To prove the above claim we use a contradiction argument. If there is a w_{0} satisfying (2.27), we multiply both sides of (2.27) by φ_{1}, the first eigenvalue of −Δ in Ω with Dirichlet boundary condition, and have:

0 = − ∫_{Ω} w_{0}Δφ_{1} − λ_{1} ∫_{Ω} w_{0}φ_{1} = ∑_{l=1}^{k} a_{l}^{∗} ∫_{Ω} δ_{q_{l}^{∗}}φ_{1} = ∑_{l=1}^{k} a_{l}^{∗}φ_{1}(q_{i}^{∗}) > 0.

Contradiction!

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