Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents

Advances in Mathematics 298 (2016) 484–533

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Advances in Mathematics www.elsevier.com/locate/aim

Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents Woocheol Choi a,∗,1 , Seunghyeok Kim b , Ki-Ahm Lee a,c a

Department of Mathematical Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-747, Republic of Korea b Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile c Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, 130-722, Republic of Korea

a r t i c l e

i n f o

Article history: Received 5 December 2014 Accepted 7 March 2016 Available online xxxx Communicated by Ovidiu Savin MSC: primary 35B40 secondary 35B33, 35B40, 35J15 Keywords: Asymptotic behavior of solutions Critical exponents Linearized problem Multi-bubble solutions

a b s t r a c t The objective of this paper is to obtain qualitative characteristics of multi-bubble solutions to the Lane–Emden–Fowler equations with slightly subcritical exponents given any dimension n ≥ 3. By examining the linearized problem at each m-bubble solution, we provide a number of estimates on the first (n + 2)m-eigenvalues and their corresponding eigenfunctions. Specifically, we present a new and unified proof of the classical theorems due to Bahri–Li–Rey (1995) [2] and Rey (1999) [24] which state that if n ≥ 4 or n = 3, respectively, then the Morse index of a multi-bubble solution is governed by a certain symmetric matrix whose component consists of a combination of Green’s function, the Robin function, and their first and second derivatives. © 2016 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (W. Choi), [email protected] (S. Kim), [email protected] (K.-A. Lee). 1 Current address: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemungu, Seoul 02455, Republic of Korea 130-722. http://dx.doi.org/10.1016/j.aim.2016.03.043 0001-8708/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we perform a qualitative analysis on the problem ⎧ p− ⎪ in Ω, ⎨ −Δu = u u >0 in Ω, ⎪ ⎩ u =0 on ∂Ω,

(1.1 )

where Ω is a bounded domain contained in Rn (n ≥ 3), p = (n + 2)/(n − 2), and  > 0 is a small parameter. When  > 0, the compactness of the Sobolev embedding H01 (Ω) → Lp+1− (Ω) allows one to find its extremal function, and hence, a positive least energy solution u ¯ for (1.1 ). However, if  = 0, this does not hold true anymore and existence of solutions strongly depends on topological or geometric properties of the domain (see for instance [1] and [10]). For example, if Ω is star-shaped and  = 0, then an application of the Pohožaev identity [22] gives nonexistence of a nontrivial solution for (1.1 ). As a particular consequence of this, the supremum of u ¯ diverges to ∞ as  → 0 for such domains. In the work of Brezis and Peletier [5], they deduced the precise asymptotic behavior of u ¯ when the domain Ω is the unit ball, and this result was extended to general domains by Han [15] and Rey [23], in which they independently proved that u ¯ blows-up at the unique point x0 that is a critical point of the Robin function of the domain. Later, Grossi and Pacella [14] investigated the related eigenvalue problem, obtaining estimates for its first (n + 2)-eigenvalues, asymptotic behavior of the corresponding eigenvectors and the Morse index of u ¯ . Since our result is closely related to their conclusion, we describe it in a detailed fashion. Let us denote by G = G(x, y) (x, y ∈ Ω) Green’s function of −Δ with Dirichlet boundary condition satisfying −ΔG(·, y) = δy in Ω

and G(·, y) = 0 on ∂Ω,

and by H(x, y) its regular part, i.e., H(x, y) =

γn 1 . − G(x, y) where γn = |x − y|n−2 (n − 2) |S n−1 |

(1.2)

We also define the Robin function τ (x) = H(x, x), and the bubble Uλ,ξ with the concentration rate λ > 0 and the center ξ = (ξ1 , · · · , ξn ) ∈ Rn ,  Uλ,ξ (x) = βn

λ λ2 + |x − ξ|2

 n−2 2

for x ∈ Rn

where βn = (n(n − 2))

n−2 4

(1.3)

which are solutions of the equation  −ΔU = U

p

in R , n

u > 0 in R

n

|∇U |2 < ∞.

and Rn

(1.4)

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In addition, given a solution u to (1.1 ), let μ be the -th eigenvalue of

−Δv = μ(p − )up−1− v in Ω, v =0 on ∂Ω,

(1.5)

provided that the sequence of eigenvalues is arranged in nondecreasing order permitting duplication, and v the corresponding L∞ (Ω)-normalized eigenfunction (namely, v L∞ (Ω) = 1). Theorem A (Grossi and Pacella [14]). Given n ≥ 3, we consider the eigenvalue problem (1.5) at a positive least energy solution u = u ¯ to (1.1 ). For the point x ∈ Ω such that u (x ) = u L∞ (Ω) (x → x0 as  → 0 by [15] and [23]), we set ⎛

⎞

vˇ (x) = v ⎝x +

x u 

p−1− 2 L∞ (Ω)

⎠

p−1−

ˇ  = u  ∞2 (Ω − x ). for arbitrary x ∈ Ω L (Ω)

(1) If 2 ≤  ≤ n +1, then there exist nonzero vectors (d,1 , · · · , d,n ) ∈ Rn and a constant 1 > 0 such that C vˇ →

n 

d,k

∂U1,0 1 in Cloc (Rn ), ∂ξk

d,k

∂G (·, x0 ) in C 1 (Ω \ {x0 }) . ∂yk

k=1 n−1 1 − n−2 v → C

n  k=1

Moreover, if ρ2 ≤ ρ3 ≤ · · · ≤ ρn+1 are the eigenvalues of the Hessian D2 τ (x0 ) of the Robin function at x0 , then  n  n μ = 1 − c˜0 ρ  n−2 + o  n−2 for some suitable c˜0 > 0 as  → 0. (2) Assume  = n + 2. Then vˇ(n+2) → dn+2

∂U1,0 1 in Cloc (Rn ) ∂λ

and

μ(n+2) = 1 + c˜1  + o()

as  → 0

for some c˜1 > 0. Consequently, if x0 is a nondegenerate critical point of the Robin function τ , then the Morse index of u ¯ is equal to 1 + (the Morse index of x0 as a critical point of τ ). As the next step to understand equation (1.1 ), one can imagine more general type of solutions so called multi-bubbles. Let {k }∞ k=1 be a sequence of small positive numbers

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1 such that k → 0 as k → ∞ and {uk }∞ k=1 a bounded sequence in H0 (Ω) of solutions m for (1.1 ) with  = k which blow-up at m ∈ N points {x10 , · · · , xm0 } ⊂ Ω . Struwe [27] obtained a concentration-compactness type result which implies that the sequence of solutions {uk }∞ k=1 can be written in the form

uk = u0 +

m 

P Uζik ,xik + Rk

(1.6)

i=1

after extracting a subsequence if necessary. Here u0 is a solution of (1.1 ) with  = 0, the function P Uλ,ξ is a projected bubble in H01 (Ω), namely, a function which solves ΔP Uλ,ξ = ΔUλ,ξ

in Ω,

P Uλ,ξ = 0 on ∂Ω,

(1.7)

and Rk is a remainder term whose H01 (Ω)-norm converges to 0 as k → ∞. Also, {ζik }k∈N and {xik }k∈N are a sequence of positive numbers and that of elements in Ω, respectively, such that ζik → ∞ and xik → xi0 ∈ Ω as k → ∞ for each fixed i = 1, · · · , m. As it is mentioned in [2], a further description of (1.6) was obtained in Schoen [25] as an application of the moving sphere method. Namely, if there exists at least one blow-up region so that m = 0 in the expression (1.6), then u0 ≡ 0 and any two concentration points should not coincide. On the other hand, Han [15] found a uniform bound of solutions near the boundary which shows that xi0 ∈ Ω. Besides, according to Bahri–Li–Rey [2] and Rey [24], there is a certain matrix determined by the concentration points and Green’s function such that if it is non-degenerate, then the solutions blowing-up at some point should have the form uk =

m 

P Uλik αk 0 ,xik + Rk

(1.8)

i=1

where α0 = 1/(n − 2). Moreover the limit (λ10 , · · · , λm0 , x10 , · · · , xm0 ) ∈ (0, ∞)m × Ωm of the sequence {(λ1k , · · · , λmk , x1k , · · · , xmk )}∞ k=1 as k → ∞ is a critical point of the function Υm (λ1 , · · · , λm , x1 , · · · xm )  m  m   n−2 n−2 τ (xi )λi − G(xi , xj )(λi λj ) 2 − c2 log(λ1 · · · λm ) = c1 i=1

(1.9)

i,j=1 i=j

provided that n ≥ 3. The positive numbers c1 and c2 in the above function are ⎛ c1 = ⎝

 Rn

⎞2 p ⎠ U1,0

(n − 2)2 and c2 = 4n

 p+1 U1,0 . Rn

(1.10)

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Conversely, by applying the Lyapunov–Schmidt reduction method, Musso and Pistoia [19] proved that if n ≥ 3 and (λ10 , · · · , λm0 , x10 , · · · , xm0 ) ∈ (0, ∞)m × Ωm is a C 1 -stable critical point of H in the sense of Y. Li [16], then there is a multi-bubbling solution of (1.1 ) having the form (1.8) which blows-up at each point xi0 with the rate of the concentration λi0 (i = 1, · · · , m). This generalizes the existence result also achieved in paper [2], where the authors used the gradient flow of critical points at infinity to get solutions. Our interest lies on the derivation of certain asymptotic behaviors of multiple bubbling solutions {u } to (1.1 ) satisfying (1.8) when  converges to 0. (Precisely speaking, sequences of parameters k , αik , λik and xik in (1.8) should be substituted by , αi , λi and xi , respectively, such that αi → 1, λi → λi0 and xi → xi0 as  → 0. Hereafter, such a substitution is always assumed.) In particular, we shall examine the behavior of eigenpairs (μ , v ) to the linearized problem (1.5) at u for 1 ≤  ≤ (n + 2)m as Grossi and Pacella did for single bubbles in [14]. Firstly, we concentrate on behavior of the first m-eigenvalues and eigenvectors. Given i,  ∈ N, 1 ≤ i ≤ m, let v˜i be a dilation of v defined as for each x ∈ Ωi := (Ω − xi ) /(λi α0 )

v˜i (x) = v (xi + λi α0 x)

(1.11)

where α0 = 1/(n − 2) again. Theorem 1.1. Suppose that n ≥ 3. Let  > 0 be a small parameter, {u } a family of solutions for (1.1 ) of the form (1.8) and μ the -th eigenvalue of problem (1.5) for some 1 ≤  ≤ m. Denote also as ρ1 the -th eigenvalue of the symmetric matrix   A1 = A1ij 1≤i,j≤m given by

A1ij

=

n−2 − (λi0 λj0 ) 2 G (xi0 , xj0 ) −C0 +

λn−2 i0 τ (xi0 )

if i = j, if i = j,

where C0 = c2 /(c1 (n−2)) > 0.

(1.12)

Then we have μ =

n−2 + b1  + o() n+2

 where b1 =

n−2 n+2

2 +

(n − 2)3 c1 1 ρ 4n(n + 2)c2 

(1.13)

as  → 0. Moreover, there exists a nonzero column vector  n−2 T n−2 c = λ102 c1 , · · · , λm02 cm ∈ Rm such that for each i ∈ {1, · · · , m} the function v˜i converges to ci U1,0 weakly in H 1 (Rn ). This c becomes an eigenvector corresponding to the eigenvalue ρ1 of A1 , and it holds that cT1 · cT2 = 0 for 1 ≤ 1 = 2 ≤ m.

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Next, we study the next mn-eigenvalues and corresponding eigenvectors. The first theorem for these eigenpairs concerns with asymptotic behaviors of the eigenvectors. Let   us define a symmetric m × m matrix M1 = m1ij 1≤i,j≤m by

m1ij =

−G (xi0 , xj0 )

if i = j,

−(n−2) C0 λi0

if i = j.

+ τ (xi0 )

(1.14)

By Lemma 2.1 below, it can be checked  that M1 is positive definite and in particular ij invertible. We denote its inverse by m1 . 1≤i,j≤m

Theorem 1.2. Assume that n ≥ 3 and m + 1 ≤  ≤ (n + 1)m. Then, for each i ∈ {1, · · · , m}, there exists a vector (d,i,1 , · · · , d,i,n ) ∈ Rn , which is nonzero for some i, such that v˜i → −

n  k=1

d,i,k

∂U1,0 ∂xk

1 in Cloc (Rn )

(1.15)

and ⎡ − n−2 v (x) → C1 ⎣ n−1

m  n m  

mij 1

i=1 j=1 k=1

⎛ 1 ∂τ × ⎝− λn−1 d,j,k (xj0 ) + 2 j0 ∂xk +

m  n 



⎞ λn−1 l0 d,l,k

l=j

∂G (xj0 , xl0 )⎠ G(x, xi0 ) ∂yk



λn−1 i0 d,i,k

i=1 k=1

∂G (x, xi0 ) ∂yk

in C 1 (Ω \ {x10 , · · · , xm0 }) as  → 0. Here C1 = βnp

(1.16)  n+2   n

|x|2 dx Rn (1+|x|2 )(n+4)/2

> 0.

If d ∈ Rmn denotes a nonzero vector defined by  n−2 n−2 n−2 d = λ102 d,1,1 , · · · , λ102 d,1,n , λ202 d,2,1 , · · · , n−2

n−2

n−2

2 d,m−1,n , λm02 d,m,1 , · · · , λm02 d,m,n λ(m−1)0

T ,

(1.17)

then we can give a further description on it. Our next theorem is devoted to this fact as well as a quite precise estimate of the eigenvalues. Set an m × mn matrix P = (Pit )1≤i≤m,1≤t≤mn and a symmetric mn × mn matrix Q = (Qst )1≤s,t≤mn as follows.

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Pi,(j−1)n+k =

⎧ n n ∂G ∂G 2 2 ⎪ ⎪ ⎨λj0 ∂y (xi0 , xj0 ) = λj0 ∂x (xj0 , xi0 ) if i = j, k

k

n 1 ∂τ ⎪ ⎪ 2 ⎩−λi0 (xi0 ) 2 ∂xk

(1.18)

if i = j,

for i, j ∈ {1, · · · , m} and k ∈ {1, · · · , n}, and

Q(i−1)n+k,(j−1)n+q

⎧ n ∂2G ⎪ ⎪ (xi0 , xj0 ) if i = j, (λi0 λj0 ) 2 ⎪ ⎪ ⎪ ∂xk ∂yq ⎪ ⎪ ⎨ n λ ∂2τ = − i0 (xi0 ) ⎪ 2 ∂xk ∂xq ⎪ ⎪ ⎪ n+2  n−2 ⎪ ∂2G ⎪ 2 ⎪ λl02 (xi0 , xl0 ) if i = j, ⎩ + λi0 ∂x ∂x l=i

k

(1.19)

q

for i, j ∈ {1, · · · , m} and k, q ∈ {1, · · · , n}. Theorem 1.3. Assuming that n ≥ 3, let A2 be an mn × mn symmetric matrix A2 = P T M−1 1 P + Q. Then as  → 0 we have  n  n μ = 1 − b2 ρ2  n−2 + o  n−2

(1.20)

for some b2 > 0 (whose value is computed in (6.1)) where ρ2 is the ( − m)-th eigenvalue of the matrix A2 . Furthermore the vector d ∈ Rmn is an eigenvector corresponding to the eigenvalue ρ2 of A2 , which satisfies dT1 · dT2 = 0 for m + 1 ≤ 1 = 2 ≤ (n + 1)m. Remark 1.4. If the number of blow-up points is m = 1, then P = 0 and so the matrix A2 is reduced to 12 λn10 D2 τ (x10 ) which is consistent with Theorem A. See also Remark 5.6. Lastly, the -th eigenpair for (n + 1)m + 1 ≤  ≤ (n + 2)m can be examined. Let   A3 = A3ij 1≤i,j≤m be a symmetric matrix whose components are given by A3ij =

n−2 − (λi0 λj0 ) 2 G (xi0 , xj0 ) if i = j, C0 + λn−2 i0 τ (xi0 )

if i = j.

(1.21)

Theorem 1.5. Suppose that n ≥ 3. Moreover, for each (n + 1)m + 1 ≤  ≤ (n + 2)m, let ρ3 be the ( − m(n + 1))-th eigenvalue of A3ij , which will be shown to be positive. Then there exist a nonzero vector  n−2 T n−2 ˆ  = λ 2 d,1 , · · · , λ 2 d,m ∈ Rm d 10 m0 and a positive number b3 such that

(1.22)

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 v˜i  d,i

∂U1,0 ∂λ

491

 weakly in H 1 (Rn )

and μ = 1 + b3 ρ3  + o()

as  → 0.

ˆ T = 0 for ˆ  is a corresponding eigenvector to ρ3 , and it holds that d ˆT · d Furthermore, d  1 2 (n + 1)(m + 1) ≤ 1 = 2 ≤ (n + 2)m. As a result, we obtain the following corollary. Corollary 1.6. Let ind(u ) and ind0 (u ) be the Morse index and the augmented Morse index of the solution u to (1.1 ), respectively. Also for the matrix A2 in Theorem 1.3, ind(−A2 ) and ind0 (−A2 ) are similarly understood. Then m ≤ m + ind(−A2 ) ≤ ind(u ) ≤ ind0 (u ) ≤ m + ind0 (−A2 ) ≤ (n + 1)m for sufficiently small  > 0 provided that n ≥ 3. Therefore if A2 is nondegenerate, then so is u and ind(u ) = m + ind(−A2 ) ∈ [m, (n + 1)m]. Remark 1.7. By the discussion before, our results hold for solutions found by Musso and Pistoia in [19]. Moreover, if k → 0 as k → ∞, any H01 (Ω)-bounded sequence {uk }∞ k=1 of solutions for (1.1 ) with  = k has a subsequence to which our work can be applied. This recovers the work of Bahri–Li–Rey [2] and Rey [24] in a different way. Besides Theorems 1.1, 1.2, 1.3 and 1.5 provide sharp asymptotic behaviors of the eigenpairs (μ , v ) as  → 0 which were not dealt with in [2,24]. In this article we compute each component of the matrix A2 explicitly, which turns out to be complicated. It would be worthwhile to compare our description with one given in [2,24]. As aforementioned, Grossi and Pacella [14] studied qualitative behaviors of single blow-up solutions of (1.1 ) and our results are the generalization of Theorem A towards multi-bubble solutions. The main difficulty in our proof is originated from the delicate interaction between different bubbles. In particular, we have to inspect the decay of solutions u and eigenfunctions v near each blow-up point in a careful way. In order to get the sharp decay of u , we will utilize the method of moving spheres which has been used on equations from conformal geometry and related areas. (See for example [7,9,18,21,25].) Furthermore we shall make use of the Moser–Harnack type estimate and an iterative comparison argument to find an almost sharp decay of v (see Lemma 2.8). Before starting the proof of our main theorems, we would like to mention about related results obtained for the Gelfand problem

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−Δu = λeu in Ω, u =0 on ∂Ω,

where Ω is a bounded smooth domain in R2 and λ > 0 is a small parameter. In [11],  given uλ an one-bubble solution satisfying λ Ω euλ → 8π as λ → 0, Gladiali and Grossi obtained the asymptotic behavior of the eigenvalues μ for the problem

−Δv = λμeuλ v in Ω, v =0 on ∂Ω,

and the Morse index of uλ as a by-product. Recently, such a type of results has been extended to solutions with multiple blow-up points in [13] – while the decay estimation of each bubble is less significant than ours – and further qualitative properties of the first m eigenfunctions have been described in [12] when m designates the number of blow-up points. Also, we believe that there should be analogue to our main results for solutions of the Brezis–Nirenberg problem [4] ⎧ p ⎪ ⎨ −Δu = u + u in Ω, u >0 in Ω, ⎪ ⎩ u =0 on ∂Ω, where Ω is a bounded smooth domain of Rn (n ≥ 5), if asymptotic forms of the solutions are written as u =

m 

P Uλi 1/(n−4) ,xi + R

i=1

for λi → λi0 > 0, xi → xi0 and R → 0 in H01 (Ω) as  → 0. This type of solutions was obtained by Musso and Pistoia [19], and Takahashi [28] analyzed the linear problem of one-bubble solutions. Strategy for the proof. For readers’ convenience, we give a summary of our proofs of the theorems on the eigenpairs (μ , v ). We shall first deal with the case 1 ≤  ≤ m, and then m + 1 ≤  ≤ m(n + 1) and m(n + 1) + 1 ≤  ≤ m(n + 2) in order. Studying each case following the order is not only natural, but also turns out to be mandatory, because the information of previous pairs is necessary to investigate the next pairs, especially, in utilizing the orthogonality property of eigenfunctions. The procedure to study the eigenpairs in respective cases consists of the following three steps: (1) Estimate of the eigenvalue μ . (2) Weak limit characterization of the rescaled eigenfunction v˜i . (3) Application of Green’s function and bilinear identities to u and v .

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First of all, we obtain estimates of eigenvalues μ using the min–max principle with properly selected approximate eigenfunctions (see (3.1)). Then we can find the limit of μ as  → 0. By employing it and a nondegeneracy result (Lemma 2.10), we get a weak limit characterization of v˜i . Consequently, we express u and v in terms of Green’s function and plug them into certain bilinear identities, which provides us refined informations of their weak limits and eigenvalues in terms of the eigenpairs of the matrices A1 , A2 , and A3 . The following table presents the place to which each step corresponds. Let us denote U = U1,0 . Cases

1≤≤m

m + 1 ≤  ≤ m(n + 1)

Estimate of μ

p−1 + o(1)

less than 1 + O( n−2 ) (Proposition  4.1),

n

n(m + 1) + 1 ≤  ≤ m(n + 2) 1+o(1) (Lemma 7.1)

n−1

1 + O  n−2 Weak limit characterization of v˜i Expression of u and v with Green’s function Bilinear identities

Span{U }

(Corollary 4.6)   ∂U ∂U Span ∂x1 , · · · , ∂x n (Proposition 4.4)

  Span ∂U ∂λ (Lemma 7.2)

(3.3), (3.4)

Lemmas 4.3, 5.3

Lemma 4.3

(3.2)

(4.15), (4.24), (5.8)

(4.15)

As this table indicates, a more careful analysis is required to handle the case m + 1 ≤  ≤ m(n + 1). Hence we give an account of this case. The first issue is to get rid of the possibility that the weak limit contains the factor ∂U ∂λ . For this aim, we shall assume the contrary. Then we combine (4.8) of Lemma 4.3 and a bilinear identity (4.15) to deduce asymptotic formula (4.20) of μ , which leads a contradiction to the a priori estimate n μ = 1 +O( n−2 ). The next issue is to get the formula (1.16), and it again needs a refined analysis as it is difficult to improve directly the estimate O() of ki1 in (5.3) using just the weak limit characterization. Nevertheless we can make use of the bilinear Pohožaev n−1 identity to show ki1 = O( n−2 ) and express ki1 in terms of kj2 . This will enable us to apply (6.4), which completes the study for the case m + 1 ≤  ≤ (n + 1)m. Structure of the paper. In Section 2, we gather all preliminary results necessary to deduce our main theorems. This section in particular includes estimates of the decay of the solutions u or the eigenfunctions v outside of the concentration points {x10 , · · · , xm0 }. In Section 3, we prove Theorem 1.1 which deals with the first m-eigenvalues and eigenfunctions of problem (1.5). A priori bounds for the first (n + 1)m-eigenvalues and the limit behavior (1.15) of expanded eigenfunction v˜i are found in Section 4. Based on these results, we compute an asymptotic expansion (1.16) of the -th eigenvectors ( = m + 1, · · · , (n + 1)m) and that of its corresponding eigenvalues (1.20) in Sections 5

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and 6 respectively. The description of the vector d is also obtained as a byproduct during the derivation of (1.20). Section 7 is devoted to study the next m-eigenpairs, i.e., the -th eigenvalues and eigenfunctions ( = (n + 1)m + 1, · · · , (n + 2)m). Finally, we present the proof of Proposition 2.3 in Appendix A, which is conducted with the moving sphere method. Notations. – Big-O notation and little-o notation are used to describe the limit behavior of a certain quantity as  → 0. – B n (x, r) is the n-dimensional open ball whose center is located  at x and radius is r. n−1 n−1   is its surface area. Also, S is the (n − 1)-dimensional unit sphere and S – C > 0 is a generic constant which may vary from line to line, while numbers with subscripts such as c1 or C1 have positive fixed values. – For any number c ∈ R, c = c+ − c− where c+ , c− ≥ 0 are the positive or negative part of c, respectively. – For any vector v, its transpose is denoted as vT . – Throughout the paper, the symbol α0 always denotes 1/(n − 2). 2. Preliminaries In this section, we collect some results necessary for our analysis. For the rest of the paper, we write x1 , · · · , xm to denote the concentration points, dropping out the subscript 0. The same omission also applies to the concentrate rates λ1 , · · · , λm .   Lemma 2.1. If we set a matrix M2 = m2ij 1≤i,j≤m by m2ij =

−G(xi , xj ) τ (xi )

if i = j, if i = j,

(2.1)

then it is a non-negative definite matrix. Proof. See Appendix A of Bahri, Li and Rey [2].

2

Fix any i ∈ {1, · · · , m} and decompose u in the following way. u = Uλi α0 ,xi + (P Uλi α0 ,xi − Uλi α0 ,xi ) +



P Uλi α0 ,xi + R .

(2.2)

n−2 2 = . p−1− 2 − (n − 2)/2

(2.3)

j=i

Then we rescale it to define u ˜i (x) = (λi α0 )σ u (xi + λi α0 x)

where σ =

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It immediately follows that {˜ ui } is a family of positive C 2 -functions defined in B n (0, −α0 r0 ) for some r0 > 0 small enough (determined in the next lemma), which are solutions of −Δu = up− . Moreover it has the following property. ui } satisfies ˜ ui L∞ (B n (0,−α0 r0 )) ≤ c for some small r0 > 0 Lemma 2.2. The sequence {˜ 1 n and converges to U1,0 weakly in H (R ) as  → 0. Proof. For fixed i, let us denote f˜(x) = (λi α0 )σ f (xi + λi α0 x) for x ∈ Ωi = (Ω − xi ) /(λi α0 ). Set also Uj = Uλj α0 ,xj for all j ∈ {1, · · · , m}. Then f˜H 1 (Ωi ) = (1 + o(1))f H 1 (Ω) and u ˜i − U1,0 =



  i + R  PUj + PUi − U

in Ωi

(2.4)

j=i

by (2.2). Observe with the maximum principle that 0 ≤ P Ui ≤ Ui in Ω and P Uλ,ξ (x) = Uλ,ξ (x) − C2 λ

n−2 2

 n−2  H(x, ξ) + o λ 2

  in C Ω , 0

 p U1,0 >0

C2 := Rn

holds for any small λ > 0 and ξ ∈ Ω away from the boundary. Thus we get from (1.4) and (1.7) that    P Ui − Ui 2H 1 (Ω) = |∇P Ui |2 − 2 ∇P Ui · ∇Ui + |∇Ui |2 Ω

Ω

 Uip P Ui +

=−

Ω



Ω

|∇Ui |2 Ω





Uip (P Ui − Ui ) −

=− Ω

 Uip+1 +

Ω

|∇Ui |2 = o(1) Ω

and  P Ui 2H 1 (Ω)

 Uip P Ui

= Ω

p+1 U1,0

≤ Rn

so that the last three terms in the right-hand side of (2.4) go to 0 strongly in H01 (Ωi ) ⊂ H 1 (Rn ). On the other hand, we have            p− → 0 U ∇P U j · ∇ϕ ≤ ϕL∞ (Ω)  j   supp(ϕ)  supp(ϕ) as  → 0 for any test function ϕ ∈ Cc∞ (Rn ). Therefore u ˜i  U1,0 weakly in H 1 (Rn ).

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We now attempt to attain a priori L∞ -estimate for {˜ ui } . Firstly we fix a suffi1 ciently small r0 . In fact, the choice r0 = 2 min {|xi − xj | : i, j = 1, · · · , m and i = j} > 0 would suffice. Then for any number η > 0, one can find r > 0 small such that ! ! ! p−1− ! ui ≤ η is valid for any |x| ≤ −α0 r0 provided  > 0 sufficiently small. !˜ ! n n L 2 (B (x,r))

Hence the Moser iteration technique applies as in [15, Lemma 6], deducing n ˜ ui L(p+1) n−2 ≤ (B n (x,r/2))

C C ˜ ui Lp+1 (B n (x,r)) ≤ ˜ ui H 1 (Ωi ) r r

where the rightmost value is uniformly bounded in  > 0. Also it is notable that C > 0 is independent of x, r or u ˜i . As a result, we observe from the elliptic regularity [15, Lemma 7] that |u(x)| ≤ uL∞ (B n (x,r/4)) ≤ CuLp+1 (B n (x,r/2)) where C > 0 depends only on r and the supreme of

  n . This ˜ ui L(p+1) n−2 n (B (x,r/2)) 

completes the proof. 2

This lemma will be used in a crucial way to deduce a local uniform estimate near each blow-up point x1 , · · · , xm of u . Proposition 2.3. There exist numbers C > 0 and small δ0 ∈ (0, r0 ) independent of  > 0 such that u ˜i (x) ≤ CU1,0 (x)

  for all x ∈ B n 0, −α0 δ0

(2.5)

for all sufficiently small  > 0. A closely related result to Proposition 2.3 appeared in [17] as an intermediate step to deduce the compactness property of the Yamabe equation, the problem proposed by Schoen who also gave the positive answer for locally conformally flat manifolds (see [26]). Even though the proof of this proposition, based on the moving sphere method, can be achieved by adapting the argument presented in [17] with a minor modification, we provide it in Appendix A to promote clear understanding of the reader. From the next lemma to Lemma 2.6, we study the behavior of solutions u of (1.1 ) outside the blow-up points {x1 , · · · , xm }. For the sake of notational convenience, we set n Ar = Ω \ ∪m i=1 B (xi , r)

for any r > 0.

(2.6)

Lemma 2.4. Suppose that {u } is a family of solutions for (1.1 ) satisfying the asymptotic behavior (1.8). Then for any r > 0, we have u (x) = o(1) uniformly for x ∈ Ar as  → 0. Proof. Let a = up−1− so that −Δu = a u in Ω. Then we see from (1.8) that

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a L n2 (A

r/4 )

m ! ! ! ! p−1− ≤C !P Uλi α0 ,xi ! i=1

m ! ! ! p−1− ! ≤C !Uλi α0 ,xi ! i=1

497

 n

L 2 (Ar/4 )

+

R p−1− n Lp+1− 2 (Ar/4 ) 

n

L 2 (Rn \B n (xi ,r/4))

+

p−1− R H 1 (Ω)

  = O 2α0 + o(1) = o(1). Therefore we can proceed the Moser iteration argument as in the proof of [15, Lemma 6] to get a Lq (Ar/2 ) = o(1) for some q > n/2, and then the standard elliptic regularity result (see [15, Lemma 7]) implies u L∞ (Ar ) = o(1). 2 We can improve this result by combining the kernel expression of u and Proposition 2.3. Lemma 2.5. Fix r > 0 small. Then there holds u (x) = O

√  

(2.7)

uniformly for x ∈ Ar . Proof. Without any loss of generality, we may assume that r ∈ (0, δ0 ) where δ0 > 0 is the number picked up in Proposition 2.3 so that (2.5) holds. Thus if we fix i ∈ {1, · · · , m}, then we have the bound    n−2  −1 u (x) = (λi α0 )−σ u ˜i (λi α0 ) (x − xi ) ≤ CUλi α0 ,xi (x) ≤ C 2 α0 √ valid for each x such that r/2 ≤ |x−xi | ≤ r. It says that u (x) ≤ C  for all x ∈ Ar/2 \Ar . By Green’s representation formula, one may write  G(x, y)up− (y)dy + 

u (x) =

m  i=1

Ar/2

 G(x, y)up− (y)dy. 

(2.8)

B n (xi ,r/2)

Let us estimate each of the term in the right-hand side. If we set b = max{u (x) : x ∈ Ar }, then we find 

 G(x, y)up− (y)dy ≤ C  Ar/2

  √ p−  √ p−  G(x, y) bp− +  +C  dy ≤ C bp− (2.9)  

Ar/2

for any x ∈ Ar . Besides, (2.5) gives us that

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 G(x, y)up− (y)dy ≤ C(r)  B n (xi ,r/2)

up− (y)dy 

B n (xi ,r/2)



Uλp− (y)dy α i  0 ,xi

≤ C · C(r)

(2.10)

B n (xi ,r/2)  n−2  α0 2

≤ C · C(r)

√ = C · C(r) 

for each i and x ∈ Ar , where C(r) = max{G(x, y) : x, y ∈ Ω, |x − y| ≥ r/2}. Hence, by combining (2.9) and (2.10), we get  √  b ≤ C bp− +  .  √ Since it is guaranteed by Lemma 2.4 that b = o(1), this shows that b ≤ C . The lemma is proved. 2 The following result will be used to obtain the asymptotic formulas of the eigenvalues. Lemma 2.6. Suppose that u satisfies equation (1.1 ) and the asymptotic behavior (1.8). Then we have m 

− 2 · u (x) = C2 1

n−2 2

λi

G(x, xi ) + o(1)

(2.11)

i=1

in C 2 (Ω \ {x1 , · · · , xm }). Here C2 =

 Rn

p U1,0 > 0.

Proof. Take any r > 0 small for which Lemma 2.5 holds and decompose u (x) as in (2.8) for x ∈ Ar . Then we have   ⎞ ⎛     1   p−1−  −2  G(x, y)up− (y)dy  ≤ C 2 ⎝ G(x, y)dy ⎠ = o(1).       Ar/2 Ω

(2.12)

Also, if we write 

 G(x, y)up− (y)dy = G(x, xi )  B n (xi ,r/2)



up− (y)dy 

B n (xi ,r/2)

+

(G(x, y) − G(x, xi ))up− (y)dy 

B n (xi ,r/2)

for i ∈ {1, · · · , m}, it follows from Lemma 2.2 and the dominated convergence theorem that

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− 2 1

n−2 2



n−2 2

p U1,0 (y)dy = λi

up− (y)dy → λi 

C2

and from the mean value theorem that       1   −2  p− (G(x, y) − G(x, xi ))u (y)dy       B n (xi ,r/2)  1 ≤ − 2 |G(x, y) − G(x, xi )|up− (y)dy  ≤ − 2

(2.13)

Rn

B n (xi ,r/2)

B n (x

499

(2.14)

i ,r/2)



1

sup ∇y G(x, ty + (1 − t)xi ) · |y − xi |up− (y)dy ≤ Cr. 

x∈Ar , B n (xi ,r/2) t∈(0,1)

Therefore, combining (2.8), (2.12), (2.13) and (2.14), we confirm that C2

m 

n−2 2

λi

G(x, xi ) − Cr ≤ lim inf − 2 u (x) ≤ lim sup − 2 u (x)

i=1

1

1

→0

≤ C2

m 

→0

n−2 2

λi

G(x, xi ) + Cr.

i=1

Since r > 0 is arbitrary, (2.11) holds in C 0 (Ω \ {x1 , · · · , xm }). Also, the C 2 -convergence comes from the elliptic regularity. This proves the lemma. 2 In Lemma 2.7 and Lemma 2.8, we conduct a decay estimate for solutions of the eigenvalue problem (1.5). Lemma 2.7. For a fixed  ∈ N, let {μ } be the family of -th eigenvalues for problem (1.5), and v an L∞ (Ω)-normalized eigenfunction corresponding to μ . Then for any r > 0 the function v converges to zero uniformly in Ar as  → 0. Proof. For x ∈ Ar , we write v (x) = μ (p − )

 G(x, y)up−1− v (y)dy + Ar/2

m  i=1

 G(x, y)up−1− v (y)dy. (2.15)

B n (xi ,r/2)

From Lemma 2.5, we have   ⎞ ⎛        2    p−1−   G(x, y) up−1− v (y)dy  ≤ C ·  2 ⎝ G(x, y)dy ⎠ = O  n−2 .     Ar/2 Ω

(2.16)

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Also, we utilize (2.5) to obtain that          p−1−    G(x, y) u v (y)dy  ≤ C(r) up−1− (y)dy    B n (xi ,r/2)  B n (xi ,r/2)  Uλp−1− (y)dy ≤ C · C(r) α i  0 ,0 B n (0,r)

(2.17)

⎧  2  n−2 ⎪ if n ≥ 5, ⎪ ⎨O  = O(| log |) if n = 4, ⎪ ⎪ ⎩ O() if n = 3 for any 1 ≤ i ≤ m where the definition of C(r) can be found in the sentence after (2.10). Putting estimates (2.16) and (2.17) into (2.15) validates that v = o(1) uniformly in Ar . 2 Lemma 2.8. Assume that 0 ∈ Ω, fix  ∈ N and set    v˜ = v (α0 x) and d (x) = dist x, −α0 x1 , · · · , −α0 xm for x ∈ Ω := −α0 Ω. Then for any ζ > 0 small, we can pick a constant C = C(ζ) > 0 independent of  > 0 such that |˜ v (x)| ≤

C 1 + d (x)n−2−ζ

for all x ∈ Ω .

(2.18)

In particular, if i ∈ {1, · · · , m} is given and {˜ vi } is a family of dilated eigenfunctions for (1.1 ) defined as in (1.11), then |˜ vi (x)| ≤

C 1 + |x|n−2−ζ

for all |x| ≤ −α0 r

(2.19)

and v = O() in Ar for some r > 0 small. Proof. We will obtain our assertion by suitably modifying the proof of Lemmas A.5, B.3 and Proposition B.1 of Cao, Peng and Yan [6], in which the authors investigated the p-Laplacian version of the Brezis–Nirenberg problem. Let u ˜ = u (α0 ·) and x ˜i = −α0  xi . Notice that v˜ solves −Δ˜ v = a v˜

in Ω

where a = μ (p − )2α0 u ˜p−1− ≥ 0.

From Proposition 2.3 and Lemma 2.5, we realize that a ≤ C|x|−4+(n−2) holds in each annulus B n (˜ xi , δ0 −α0 ) \ B n (˜ xi , R) provided i ∈ {1, · · · , m} and R > 1 large, and

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501

n a ≤ C4α0 in Ω \ ∪m xi , δ0 −α0 ). Hence, given any η > 0, there exists a large i=1 B (˜ R(η) > 1 such that



R := Ω \ where A

n

|a | 2 dx < η

m "

B n (˜ xi , R).

(2.20)

i=1

R(η) A

Suppose that ζ > 0 is selected to be small enough. Then one can apply the Moser iteration technique (see e.g. [15]) to get a small number η > 0 and large q > p + 1 such that if (2.20) holds, there is a constant C > 0 independent of R, η or v˜ satisfying ˜ v Lq A

R



≤

C (R − 2R(η))

n−2 2 −ζ

· ˜ v Lp+1 A



2R(η)

for any R > 2R(η). On the other hand, it is possible to get that ˜ v Lp+1 A



2R(η)

≤

CR−2ζ by taking a smaller ζ if necessary. Thus standard elliptic regularity theory gives |˜ v (x)| ≤ ˜ v L∞ (B n (x,1)) ≤ C˜ v Lq A



R−1

≤

C (R − 2R(η) − 1)

n−2 2 −ζ

· ˜ v Lp+1 A

2R(η)



≤

C R

n−2 2 +ζ

(2.21)

R , R ≥ 3R(η). for all x ∈ A Having (2.21) in mind, we now prove (2.18) by employing the comparison principle iteratively. Assume that it holds |˜ v (x)| ≤ Dj

m  i=1

1 |x − x ˜i |qj

R , for all x ∈ A

(2.22)

some Dj > 0 and 0 < qj < n − 2 to be determined soon (j ∈ N). Since we have (n − 2)(p − 1 − ) > 3 for small  > 0, Proposition 2.3, Lemma 2.5 and (2.22) tell us that  j > 0 whose choice is affected by only Dj , n and  such that there exists some D −Δ(˜ v )± (x) = μ (p −

)˜ up−1− (˜ v )± (x)

j ≤D

m  i=1

1 |x − x ˜i |qj +3

R . for any x ∈ A

Select any number 0 < η˜ < min(1, n − 2 − qj ) and set a function χj (x) = Dj+1

m  i=1

1 |x − x ˜i |qj +˜η

for x ∈ Rn

n where Dj+1 > 0 is a number so large that χj ≥ |˜ v | on ∪m xi , R). Then one can i=1 ∂B (˜ compute

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−Δχj (x) = Dj+1 (qj + η˜) ((n − 2) − (qj + η˜))

m  i=1

j ≥D

m 

1 |x − x ˜i |qj +˜η+2 (2.23)

1 ≥ −Δ(˜ v )± (x), |x − x ˜i |qj +3

i=1

R x∈A

by taking a larger Dj+1 if necessary. However χj > 0 and v˜ = 0 on ∂Ω , hence χj ≥ |˜ v |  on ∂ AR . Consequently, by (2.23) and the maximum principle, it follows that |˜ v (x)| ≤ χj (x) = Dj+1

m  i=1

1 , |x − x ˜i |qj +˜η

R . x∈A

Letting q1 = n−2 2 + ζ in (2.21), choosing an appropriate D1 > 0 and repeating this comparison procedure, we can deduce |˜ v (x)| ≤ C

m  i=1

1 , |x − x ˜i |q

R x∈A

given any 1 < q < n − 2. This proves (2.18). Finally, (2.19) and the claim that v = O() in Ar is a straightforward consequence of (2.18). The proof is completed. 2 By utilizing (2.5), (2.19), (2.7), the fact that v = O() in Ar and regularity theory, we immediately establish a decay estimate for the derivatives of u ˜i and v˜i . Lemma 2.9. For any k ∈ {1, · · · , n}, there exists a universal constant C > 0 such that    ∂u  C  ˜i (x)  ≤  ∂xk  1 + |x|n−2

and

   ∂˜  C  vi (x)  ≤  ∂xk  1 + |x|n−2−ζ

for all |x| ≤ −α0 r

for ζ, r > 0 small. Moreover we have |∂k u | , |∂k ∂l u | = O

√  

and

|∂k v | = O()

for all k, l = 1, · · · , n

as  → 0 in any compact subset of Ar . Finally, we recall two well-known results. The first lemma states the nondegeneracy property of the standard bubble U1,0 . We refer to [3] for its proof. Lemma 2.10. The space of solutions to the linear problem  −Δv =

p−1 pU1,0 v

in R

n

|∇v|2 < ∞

and Rn

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503

is spanned by xn x1 n , ··· , n (1 + |x|2 ) 2 (1 + |x|2 ) 2

and

1 − |x|2 n . (1 + |x|2 ) 2

The next lemma lists some formulas regarding the derivatives of Green’s function. The proof can be found in [14,15]. Lemma 2.11. For ξ ∈ Ω, it holds that 

 (x − ξ, ν) ∂Ω

 

∂G (x, ξ) ∂ν

∂G (x, ξ) ∂ν

2 dS = (n − 2)τ (ξ),

2 νk (x)dS =

∂τ (ξ), ∂xk

k = 1, · · · , n

∂Ω

and 

∂ ∂G (x, ξ) ∂xk ∂yl



 1 ∂2τ ∂G (x, ξ) dS = (ξ), ∂ν 2 ∂xk ∂xl

k, l = 1, · · · , n.

∂Ω

Here ν is the outward normal unit vector to ∂Ω and dS is the surface measure ∂Ω. 3. Proof of Theorem 1.1 In this section, we present estimates for the first m eigenvalues and eigenfunctions of (1.5), which are the contents of Theorem 1.1. For the set of the concentration points {x1 , · · · , xm } ⊂ Ωm , let us fix a small number r > 0 such that for any 1 ≤ i = j ≤ m and any  > 0 small the following holds: B n (xi , 4r) ⊂ Ω

and B n (xi , 4r) ∩ B n (xj , 4r) = ∅.

For each 1 ≤ i ≤ m, we set φi (x) = φ(x −xi ) where a cut-off function φ ∈ Cc∞ (B n (0, 3r)) satisfies φ ≡ 1 in B n (0, 2r) and 0 ≤ φ ≤ 1 in B n (0, 3r). Define also u,i = φi u ,

ψ,i,k = φi

∂u (1 ≤ k ≤ n) ∂xk

 ψ,i,n+1 = φi · (x − xi ) · ∇u + in Ω.

and

2u p−1−

 (3.1)

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The following lemma serves as a main ingredient for the proof of Theorem 1.1. Lemma 3.1. Fix  ∈ N. Suppose that {v } is a family of normalized eigenfunctions of (1.5) corresponding to the -th eigenvalue μ . Then there exists at least one i0 ∈ {1, · · · , m} such that v˜i0  (see (1.11) for its definition) converges to a nonzero function in the weak H 1 (Rn )-sense. Proof. Lemma 2.8 ensures that there exist a large R > 0 and a small r > 0 such that |˜ vi | ≤ 1/2 for R ≤ |x| ≤ −α0 r. Suppose that v˜i  0 weakly in H 1 (Rn ) as  → 0 for all 1 ≤ i ≤ m. Then each v˜i tends to 0 uniformly in B n (0, R) by elliptic regularity. Since we already know that v → 0 uniformly on Ar from Lemma 2.7, it follows that v L∞ (Ω) ≤ 1/2. However v L∞ (Ω) = 1 by its own definition, hence a contradiction arises. 2 Given Lemma 3.1, we are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Let V be a vector space whose basis consists of {u,i : 1 ≤ i ≤ m}. By the Courant–Fischer–Weyl min–max principle, we have 

|∇f (x)|2 dx Ω   W⊂H0 (Ω), f ∈W\{0} (p − ) f 2 up−1− (x)dx Ω dimW=m  |∇f (x)|2 dx Ω  ≤ max .  f ∈V\{0} (p − ) f 2 up−1− (x)dx Ω

μm =

min 1

max

m If we denote a nonzero element f ∈ V by f = i=1 ai u,i for some (a1 , · · · , am ) = 0, then the fact that u,i1 and u,i2 have disjoint supports for any 1 ≤ i1 =

i2 ≤ m implies  

Ω

(p − )

|∇f |2

Ω

f 2 up−1−

m 

=

(p −

2 i=1 Ω |∇ (ai u,i )| m  2 ) i=1 Ω (ai u,i ) up−1−



|∇(ai u,i )|2  1≤i≤m (p − ) (a u )2 up−1− Ω i ,i   |∇(φi u )|2 1 Rn |∇U1,0 |2 1 Ω = max =  2 p+1− → ·  p+1 1≤i≤m (p − ) p p φ u U Ω i  Rn 1,0 ≤ max

Ω

as  → 0.

Thus we know that μm ≤ p−1 + o(1), and particularly if we let μ = lim μ , then →0

μ ≤ p−1 for any 1 ≤  ≤ m. Fix  ∈ {1, · · · , m}. By Lemma 3.1 there is an index i0 ∈ {1, · · · , m} such that v˜i0  converges H 1 (Rn )-weakly to a nonzero function V . A direct computation shows

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p−1− −Δ˜ vi0  = μ (p − )˜ ui v˜i0 

505

in Ωi0 

where the function u ˜i and the set Ωi0  are defined in (2.3) and (1.11), respectively. Thus it follows from Lemma 2.2 that V ∈ H 1 (Rn ) \ {0} is a solution of p−1 −ΔV = μ pU1,0 V

in Rn .

Note that U1,0 can be characterized as a mountain pass solution to (1.4) and so has the Morse index 1. Consequently, in light of the estimate for μ in the previous paragraph, the only possibility is μ = p−1 . On the other hand, for any i, we also see that v˜i converges to a function W weakly p−1 in H 1 (Rn ) so that W solves −ΔW = U1,0 W in Rn . Thus there is a nonzero vector  n−2  n−2 c = λ1 2 c1 , · · · , λm2 cm ∈ Rm such that v˜i  ci U1,0 weakly in H 1 (Rn ) for each i ∈ {1, · · · , m}. Let us prove (1.13) now. Fixing i, we multiply (1.1 ) (or (1.5) with v = v ) by v   (or u ) to get the identity, say, I (or II respectively). Also we denote by I and II the identities which can be obtained after integrating I and II over B n (xi , r). Subtracting   I from II and utilizing Green’s identity (4.12) below, we see then 

 ∂B n (x

 ∂v ∂u v − u dS = (μ (p − ) − 1) ∂ν ∂ν



B n (x

i ,r)



 v (x)dx up− 

(3.2)

i ,r)

for each i ∈ {1, · · · , m} and any r > 0 sufficiently small. Moreover, if we set the functions n−2 2

C2−1 g˜i (x) = −λi

H(x, xi ) +



n−2

λj 2 G(x, xj ),

j=i

˜ i (x) = −λn−2 ci H(x, xi ) + C2−1 h i



λn−2 cj G(x, xj ) j

j=i

which are harmonic near xi , then (the proof of) Lemma 2.6 permits us to obtain that n−2 2

− 2 u (x) = C2 λi 1

γn + g˜i (x) + o(1) |x − xi |n−2

(3.3)

and −1

v (x) γn ˜ i (x) + o(1) = C2 λn−2 ci +h i μ (p − ) |x − xi |n−2

(3.4)

for x ∈ B n (xi , 2r). Therefore, by inserting (3.3) and (3.4) into (3.2), and then using the mean value formula for harmonic functions and ∇λ Υ(λ1 , · · · , λm , x1 , · · · , xm ) = 0, one discovers

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506

⎡  1  ⎤  −1    ∂ − 2 u   ∂  v 1  ⎣ −1 v − − 2 u ⎦ dS ∂ν ∂ν

 ∂B n (xi ,r)

%

 = −(n − 2)C2 γn ∂B n (xi ,r)

n−2 1 ˜ i (x) λi 2 h n−1 |x − xi |

& 1 n−2 λ c g ˜ (x) dS + o(1) i i |x − xi |n−1 i  n−2   ˜ i (xi ) → (n − 2)C2 γn S n−1  λn−2 ci g˜i (xi ) − λi 2 h i −

⎡

⎞

⎛



⎝ = c1 ⎣λn−2 i

n−2 2

λj

G(xi , xj )⎠ ci −

j=i

 =

3(n−2) 2

c1 λi



⎤ n−2 2

λi

λn−2 G(xi , xj )cj ⎦ j

j=i

τ (xi ) −

n−2 c2 λi 2 n−2

 ci − c1



n−2 2

λi

λn−2 G(xi , xj )cj j

j=i

as  → 0. Also, an application of the dominated convergence theorem with Lemmas 2.2 and 2.8, Proposition 2.3 and the observation that v˜i → ci U1,0 pointwise give us that

− 2 1





n−σ (p−) (n−σ (p−))α0 − 12

up− v = λi 



  B 0,r(λi α0 )−1

B n (xi ,r) n−2 2

→ ci λi

u ˜p− ˜i i v

4nc2 (n − 2)2

(refer to (1.10)). From these estimates, we deduce  λn−2 τ (xi ) − i

c2 (n − 2)c1  =



n−2 2

λi

 n−2    n−2 ci − (λi λj ) 2 G(xi , xj ) λj 2 cj

4nc2 (n − 2)2 c1

j=i



 · lim

→0

μ (p − ) − 1 



n−2 2

λi

  n−2  ci := ρ1 λi 2 ci ,

or equivalently, A1 c = ρ1 c . This justifies (1.13). We also showed that cT is an eigenvector corresponding to the eigenvalue ρ1 at the same time. Finally, to verify the last assertion of the theorem, we assume that 1 = 2 . Since v1  and v2  are orthogonal each other, we have

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

−1

0 = lim −1 (μ1  (p − ))

 ∇v1  · ∇v2 

→0

Ω

⎛ m  −1 ⎜ = lim  ⎝ →0

= lim

→0

i=1 m  i=1

⎟ up−1− v1  v2  ⎠

up−1− v1  v2  + n Ω\∪m i=1 B (xi ,r)

i ,r)

 λn−2 i Bn

Thus cT1 · cT2 = 0.

⎞





B n (x



0,(λi

507

α0 )−1 r



p−1− u ˜i v˜1 i v˜2 i

=

m  

n−2 2

λi

(3.5)

  n−2  p+1 2 c1 i λi c2 i U1,0 .

i=1

Rn

2

4. Upper bounds for the -th eigenvalues and asymptotic behavior of the -th eigenfunctions, m + 1 ≤  ≤ (n + 1)m The objective of this section is to provide estimates of the -th eigenvalues and their corresponding eigenfunctions when m + 1 ≤  ≤ (n + 1)m. Their refinement will be accomplished in the subsequent sections based on the results deduced in this section. In the first half of this section, our interest will lie on achieving upper bounds of the eigenvalues μ for m + 1 ≤  ≤ (n + 1)m, as the following proposition depicts. Proposition 4.1. Suppose that m + 1 ≤  ≤ (n + 1)m. Then  n  μ ≤ 1 + O  n−2 . Proof. We define a linear space V spanned by {u,i : 1 ≤ i ≤ m} ∪ {ψ,i,k : 1 ≤ i ≤ m, 1 ≤ k ≤ n} (refer to (3.1)) so that any nonzero function f ∈ V \ {0} can be written as f=

m 

fi

with

fi = ai0 u,i +

i=1

n 

aik ψ,i,k

k=1

where at least one number aik (1 ≤ i ≤ m and 0 ≤ k ≤ n) is nonzero. By the variational characterization of the eigenvalue μ , we have  μ((n+1)m) =

min 1



Ω

max

f ∈W\{0} W⊂H0 (Ω), dimW=(n+1)m



(p − )

|∇f |2

Ω

f 2 up−1−

 ≤ max

f ∈V\{0}

(p − )

|∇fi |2  2 p−1− := max max ai . f ∈V\{0} 1≤i≤m (p − ) f ∈V\{0} 1≤i≤m f u Ω i 

≤ max

max

Ω



Ω

|∇f |2

Ω

f 2 up−1−

508

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

 n  Hence it suffices to show that each ai is bounded by 1 + O  n−2 . As a matter of fact, this can be achieved along the line of the proof of [14, Proposition 3.2], but we provide a brief sketch here since our argument slightly simplifies the known proof. Fix i ∈ {1, · · · , m}. For the sake of notational simplicity, we write a = ai , φ = φi n ∂u and ak = aik for 0 ≤ k ≤ n. Denote also z = k=1 ak ∂x so that fi = a0 φu + φz . k 2 2 After multiplying (1.1 ) by φ u or φ z , and integrating the both sides over Ω, one can deduce 



 |∇φ|2 u2

|∇(φu )| = 2

Ω

Ω

φ2 up+1− 

+

(4.1)

Ω

and 

 ∇(φu ) · ∇(φz ) = Ω

 |∇φ|2 u z +

Ω

 φ∇φ · (u ∇z − z ∇u ) +

Ω

φ2 up− z . 

(4.2)

Ω

Similarly, testing −Δz = (p − )up−1− z with φ2 z , one finds that 



 |∇φ|2 z2

|∇(φz )| = 2

Ω

+ (p − )

Ω

φ2 up−1− z2 .

(4.3)

Ω

Then (4.1)–(4.3) yields a = 1 + b/c where ⎛ b = −(p − 1 − ) ⎝a20 



 φ2 up+1− 

Ω

 +

|∇φ|2 u2

a20 Ω



φ∇φ · (u ∇z − z ∇u ) + 2a0

+ 2a0

Ω

φ2 up− z ⎠ 

Ω



|∇φ|2 z2

+

+ 2a0

⎞

Ω

|∇φ|2 u z

(4.4)

Ω

and ⎛ c = (p − ) ⎝a20





 φ2 up+1− + 2a0 

Ω

φ2 up− z +  Ω

⎞ φ2 up−1− z2 ⎠ .

(4.5)

Ω

Our aim is to find an upper bound of b and a lower bound of c. Let us estimate b first. We see at once that  −(p − 1 −

φ2 up+1− < −Ca20 . 

)a20 Ω

Also, if we let a ¯ = (a1 , · · · , an ), then (2.7) guarantees

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509

          n     ∂u   2 p−  a0 φ2 up−  z  = a0 ak φ u    ∂xk     j=1 Ω Ω      n 2   p+1− a0 ∂φ p+1−   ≤ Ca0 |¯ = ak u a| 2 .  p + 1 −  ∂x k   k=1

Ω

Moreover we have that  |∇φ|2 u2 ≤ Ca20 .

a20 Ω

On the other hand, for D1 = B n (xi , 3r) \ B n (xi , 2r) and D2 = B n (xi , 4r) \ B n (xi , r), we easily discover 

 |∇φ|2 z2

≤C

 z2

≤ C|¯ a|

2

D1

Ω





|∇u | ≤ C|¯ a| 2

2

D1

 up+1− + u2 ≤ C|¯ a|2  

D2

and 

 |∇z |2 ≤ C D1



 z2 + up−1− z2 ≤ C

D2

 z2 ≤ C|¯ a|2  D2

(cf. (4.1) and (4.3)), which implies         2a0 φ∇φ · (u ∇z − z ∇u ) + 2a0 |∇φ|2 u z  ≤ Ca0 |¯ a|.     Ω

Ω

Utilizing these estimates and the Cauchy–Schwarz inequality we deduce b ≤ C|¯ a|2 .

(4.6)

To obtain a lower bound of c, we note that          2     p+1− 1 ∂u ∂φ  p+1−    φ2 up− = ≤ C 2 u     ∂xk   p + 1 −  ∂xk   Ω

Ω

and that Lemma 2.9 ensures ⎛

 φ2 up−1−

∂u ∂u = λ−2 i  ∂xk ∂xl

2 − n−2

Ω

for 1 ≤ k, l ≤ n. Hence we conclude that

⎝ δkl n

 Rn

⎞ p−1 U1,0 |∇U1,0 |2 + o(1)⎠

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W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

c ≥ Ca20 − Ca0 |¯ a|

p+1− 2

+ C|¯ a|2 − n−2 ≥ 2

2 C 2 − n−2 |¯ a|  . 2

(4.7)

 n  Consequently, a combination of (4.6) and (4.7) asserts that a ≤ 1 + O  n−2 . This completes the proof of the lemma. 2 Corollary 4.2. For m + 1 ≤  ≤ (n + 1)m, we have the following limit lim μ = 1.

→0

Proof. By Lemma 3.1 we can find i1 ∈ {1, · · · , m} such that v˜i1  converges weakly to a nonzero function V . Then, as in the proof of Theorem 1.1, we observe that V solves p−1 −ΔV = μ pU1,0 V

in Rn

where μ = lim→0 μ . Also, owing to Proposition 4.1, we have μ ≤ 1. Since the Morse index of U1,0 is 1, it should hold that μ = p−1 or 1. Assume that μ = p−1 . Then the proof of Theorem 1.1 again gives us that there is  n−2

n−2

a vector b = λ1 2 b1 , · · · , λm2 bm = 0 such that v˜i  bi U1,0 weakly in H 1 (Rn ). Furthermore b · c1 = 0 for any 1 ≤ 1 ≤ m, but this is impossible since {c1 , · · · , cm } already spans Rm . Hence μ = 1, which finishes the proof. 2

Next, we provide a general convergence result of the -th L∞ (Ω)-normalized eigenfunction v . We recall its dilation v˜i defined in (1.11). Lemma 4.3. Suppose that m + 1 ≤  ≤ (n + 1)m. (1) For any i ∈ {1, · · · , m} there exists a vector (d,i,1 , · · · , d,i,n+1 ) ∈ Rn+1 such that the function v˜i converges to n 

 d,i,k

k=1

∂U1,0 ∂ξk



 + d,i,n+1

∂U1,0 ∂λ



weakly in H 1 (Rn ) (see (1.3) for the definition of Uλ,ξ ). In addition, there is at least one i1 ∈ {1, · · · , m} such that (d,i1 ,1 , · · · , d,i1 ,n+1 ) = 0. (2) As  → 0 we have −1 v → C3

m 

d,i,n+1 λn−2 G(·, xi ) i

i=1

where C3 = p

 Rn

p−1 U1,0



∂U1,0 ∂λ

 > 0.

in C 1 (Ω \ {x1 , · · · , xm })

(4.8)

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511

Proof. It is not hard to show the first statement with Lemmas 3.1 and 2.10, and Corollary 4.2. Hence let us consider the second statement. For r > 0 fixed small, assume that a point x ∈ Ω belongs to Ar where Ar is the set in (2.6). According to Green’s representation formula and Lemmas 2.5 and 2.7, −1 v (x) = −1 μ (p − )



m  i=1

G(x, y)up−1− (y)v (y)dy + o(1).

B n (xi ,r/2)

Besides, Proposition 2.3 with Lemmas 2.8 and 4.3 (1) allow us to obtain lim 

−1

 G(x, y)up−1− (y)v (y)dy

→0

B n (xi ,r/2)



= λn−2 lim i

→0   B n 0,(λi α0 )−1 r/2

  p−1− G (x, xi + λi α0 y) u ˜i v˜i (y)dy 



p−1 U1,0 (y)

G(x, xi ) = d,i,n+1 λn−2 i Rn

∂U1,0 ∂λ

(4.9)

 (y)dy.

Thus the lemma is proved. 2 In fact, we can refine the first statement of the above lemma to arrive at (1.15), which is the main result of the latter part of this section. Proposition 4.4. Let m + 1 ≤  ≤ (n + 1)m. For each i ∈ {1, · · · , m} and (d,i,1 , · · · , d,i,n ) ∈ Rn , the function v˜i converges to n  k=1

 d,i,k

∂U1,0 ∂ξk

 =−

n  k=1

 d,i,k

∂U1,0 ∂xk



weakly in H 1 (Rn ). As a preparation for its proof, we first consider the following auxiliary lemma. Lemma 4.5. Fix 1 ≤ i ≤ m. For a small r > 0 (any choice of r < min{dist(xj , xl ) : 1 ≤ j = l ≤ m}/2 is available) and 1 ≤ j, l ≤ m, we define  r Ijl;i

= ∂B n (xi ,r)



%   & ∂ n−2 (x − xi ) · ∇G(x, xj ) + G(x, xj ) G(x, xl ) ∂ν 2   &  % n−2 ∂ G(x, xl ) dS. (4.10) G(x, xj ) − (x − xi ) · ∇G(x, xj ) + 2 ∂ν

512

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

r Then Ijl;i is independent of r > 0 and its value is computed as

⎧ ⎪ 0 ⎪ ⎪   ⎪ ⎪ ⎪ n−2 ⎪ ⎪ G(xi , xj ) ⎨ 2   = ⎪ n−2 ⎪ ⎪ G(xi , xl ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩−(n − 2)τ (x ) i

r Ijl;i

if j = i and l = i, if j = i and l = i, (4.11) if j = i and l = i, if j = l = i.

Proof. Assuming 0 < r2 < r1 are small enough and putting f (x) = (x − xi ) · ∇G(x, xj ) + G(x, xj ), g(x) = G(x, xl ) and D = B n (xi , r1 ) \ B n (xi , r2 ) into Green’s identity  

∂g ∂f g− f ∂ν ∂ν



 (Δf · g − Δg · f ) dx,

dS =

(4.12)

D

∂D r we see that Ijl;i is constant because

 %  & n−2 G(x, xj ) = 0 and Δ (x − xi ) · ∇G(x, xj ) + 2

ΔG(x, xl ) = 0

(4.13)

r for all x = xj , xl . Thus it suffices to find the value Ijl;i = limr→0 Ijl;i . (1) If j, l = i, then Ijl;i = 0. This follows simply by applying (4.12) for D = B n (xi , r) since (4.13) holds for any x ∈ B n (xi , r). (2) If j = i and l = i, then we have    n−2 ∂ G(x, xj ) G(x, xi )dS Ijl;i = Iji;i = lim − r→0 2 ∂ν ∂B n (xi ,r)



= lim

r→0 ∂B n (xi ,r)



=

n−2 2



n−2 2

 G(x, xj ) ·

n−2 dS (n − 2) |S n−1 | |x − xi |n−1



G(xi , xj ).

(3) Suppose that j = i and l = i. In this case, we deduce %   &  n−2 ∂ (x − xi ) · ∇G(x, xi ) + G(x, xi ) G(x, xl )dS Ijl;i = Iil;i = lim r→0 ∂ν 2 ∂B n (xi ,r)



= lim

r→0 ∂B n (xi ,r)

n−2 · G(x, xl )dS = 2 |S n−1 | |x − xi |n−1



n−2 2

 G(xi , xl ).

(4) If k = l = j, then Green’s identity, the fact that G(x, xi ) = 0 on ∂Ω and Lemma 2.11 lead

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

  Ijl;i = Iii;i =

513

%   & ∂ n−2 (x − xi ) · ∇G(x, xi ) + G(x, xi ) G(x, xi ) ∂ν 2

∂Ω

 %  &  n−2 ∂ − (x − xi ) · ∇G(x, xi ) + G(x, xi ) G(x, xi ) dS 2 ∂ν

 =−

[(x − xi ) · ∇G(x, xi )]

∂ G(x, xi )dS = −(n − 2)τ (xi ). ∂ν

∂Ω

All the computations made in (1)–(4) show the validity of (4.11).

2

Proof of Proposition 4.4. Fix i ∈ {1, · · · , m} and let wi (x) = (x − xi ) · ∇u +

2u p−1−

for x ∈ Ω,

(4.14)

be a solution of −Δwi = (p − )up−−1 wi 

in Ω.

Then by (4.12) it satisfies 

 ∂B n (x

 ∂v ∂wi v − wi dS = (μ − 1)(p − ) ∂ν ∂ν

 up−1− wi v

B n (x

i ,r)

(4.15)

i ,r)

for r > 0 small, where ν is the outward normal unit vector to the sphere ∂B n(xi , r). n ∂U1,0 In light of Lemma 4.3 (1), we already know that v˜i  + k=1 d,i,k ∂ξk   ∂U1,0 1 n d,i,n+1 ∂λ weakly in H (R ) as  → 0. Thus we only need to verify that d,i,n+1 = 0 for all i ∈ {1, · · · , m} in order to establish Proposition 4.4. Assume to the contrary that d,i,n+1 = 0 for some i. We will achieve a contradiction by showing that an estimate of μl − 1 obtained through (4.15) does not match to one found in Proposition 4.1. To reduce the notational complexity, we use di or d,i to denote d,i,n+1 in this proof. Let us observe from Lemma 2.6 and (4.14) that 

− 12

wi (x) → C2

m  j=1

n−2 2

λj

%   & n−2 (x − xi ) · ∇G(x, xj ) + G(x, xj ) 2

in C 1 (Ω \ {x1 , · · · , xm })

(4.16)

as  → 0. Combining this with (4.8) we get lim − 2 3



→0

∂B n (xi ,r)



 m  n−2 ∂wi ∂v r v − wi dS = C2 C3 λj 2 λn−2 dl Ijl;i l ∂ν ∂ν j,l=1

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

514

r where Ijl;i is the value defined in (4.10). By inserting (4.11) into the above identity, we further find that    ∂v ∂wi − 32 v − wi dS lim  →0 ∂ν ∂ν ∂B n (xi ,r)

⎡   n−2  n−2 ⎣ λi 2 = C2 C3 λn−2 dl G(xi , xl ) l 2 l=i

⎛

 n − 2

di ⎝ + λn−2 i ) = C2 C3

2

j=i

n−2 2 

− λn−2 di i



n−2 2

λi



⎞⎤ n−2 2

λj

n−2 2

G(xi , xj ) − (n − 2)λi

λn−2 dj G(xi , xj ) j

j=i

n−2 2



n−2 2

λi

τ (xi )⎠⎦

τ (xi ) +

− n−2 C0 λi 2



 .

Here C0 = c2 /((n − 2)c1 ) > 0 as in (1.12), and we employed the fact that (λ1 , · · · , λm , x1 , · · · , xm ) is a critical point of the functional Υm (see (1.9)) so as to obtain the second equality. Borrowing the notation of the matrix A3 in (1.21), the left-hand side of (4.15) can be described in a legible way. lim 





− 32

→0

 ∂wi ∂v v − wi dS ∂ν ∂ν

∂B n (xi ,r)



= −C2 C3

n−2 2

 m

 n−2  A3ij λj 2 dj .

(4.17)

j=1

On the other hand, counting on Proposition 2.3 and Lemmas 2.2 and 4.3, we can compute its right-hand side as follows. & %  2u (x) − 12 p−1− v (x)dx lim  u (x) (x − xi ) · ∇u (x) + →0 p−1− B n (xi ,r)



n−2 2

= lim λi →0

n−2 2

= λi



  B n 0,(λi α0 )−1 r

di Rn

where C4 = deduce that

 Rn

& % 2˜ ui (y) p−1− v˜i (y)dy u ˜i (y) y · ∇˜ ui (y) + p−1−

(4.18)

&  % n−2 ∂U1,0 2U1,0 (y) p−1 (y)dy = −λi 2 di C4 U1,0 (y) y · ∇U1,0 (y) + p−1 ∂λ

p−1 U1,0



∂U1,0 ∂λ

2 > 0. Consequently, (4.17), (4.18) and (4.15) enable us to

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

ˆ1 = A3 d 

2pC4 lim (n − 2)C2 C3 →0



μ − 1 

⎞ n−2 λ1 2 d,1 ⎟ ˆ1 = ⎜

0. where d ··· ⎠= ⎝ 

515

⎛

 ˆ1 d 

(4.19)

n−2 2

λm d,m

 T ˆ1 Multiplying a row vector d in the both sides yields   lim

→0

μ − 1 



⎞ T ˆ2 ˆ 2 M2 d d  (n − 2) C2 C3 ⎜  ⎟ = ·⎝ + C0 ⎠  2 2(n + 2)C4  ˆ 1 d  ⎛

2

(4.20)



  ˆ 2 = λn−2 d,1 , · · · , λn−2 d,m T and M2 is the matrix introduced in Lemma 2.1. where d m 1  However the right-hand side of (4.20) is positive due to Lemma 2.1, and this contradicts the bound of μ provided in Proposition 4.1. Hence it should hold that d,i = 0 for all i. The proof is finished. 2 This result improves our knowledge on the limit behavior of the -th eigenvalues (see Corollary 4.2) for m + 1 ≤  ≤ (n + 1)m, which is essential in the next section. Corollary 4.6. For m + 1 ≤  ≤ (n + 1)m, one has  n−1  |μ − 1| = O  n−2

as  → 0.

(4.21)

Proof. By Proposition 4.4 and Lemma 4.3 (1), there is i1 ∈ {1, · · · , m} such that v˜i1  

n 

 d,i1 ,k

k=1

∂U1,0 ∂ξk

 weakly in H 1 (Rn )

where (d,i1 ,1 , · · · , d,i1 ,n ) = 0. Without any loss of generality, we may assume that d,i1 ,1 = 0. By differentiating the both sides of (1.1 ), we get −Δ

∂u ∂u = (p − )up−1− . ∂x1 ∂x1

(4.22)

∂u Let us multiply (4.22) by v and (1.5) by ∂x , respectively, integrate both of them over 1 n B (xi1  , r) for a small fixed r > 0 and subtract the first equation from the second to derive *   +  ∂u ∂v ∂ ∂u v − dS ∂ν ∂x1 ∂x1 ∂ν ∂B n (xi1  ,r)



= (p − ) (μ − 1) B n (x

up−1− i1  ,r)

∂u v . ∂x1

(4.23)

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516

  By Lemma 2.9, its left-hand side is O 3/2 while the right-hand side is computed as  up−1−



∂u n−(σ +1)−2 v = (λi1 α0 ) ∂x1

B n (xi1  ,r)

B n (0,(λi α0 )−1 r)

⎛ n−4 2 i1

= −λ



n−4 2(n−2)

u ˜ip−1− 1

⎝d,i1 ,1

Rn

Therefore, if we denote C5 =



U p−1 Rn 1,0



∂U1,0 ∂x1



 p−1 U1,0

∂U1,0 ∂x1

2

∂u ˜i1  v˜i1  ∂x1 ⎞

(4.24)

+ o(1)⎠ .

2 > 0, we deduce that

 3 , n−4 n−4 O  2 = −λi12  2(n−2) (p + o(1)) lim (μ − 1) (d,i,1 C5 + o(1)) , →0

which leads the desired estimate (4.21).

2

5. A further analysis on asymptotic behavior of the -th eigenfunctions, m + 1 ≤  ≤ (n + 1)m In view of Lemma 4.3 and the proof of Proposition 4.4, we know that −1 v → 0 as  → 0 uniformly in Ω outside of the blow-up points {x1 , · · · , xm }. Motivated by the argument in [13], we prove its improvement (1.16) here, which is stated once more in the following proposition. Proposition 5.1. Let M1 and P be the matrices defined in (1.14) and (1.18), respectively. Also we remind a column vector d ∈ Rmn in (1.17) and set two row vectors G(x) and  G(x) by G(x) = (G(x, x1 ), · · · , G(x, xm )) ∈ Rm ,  n  n 2  G(x) = λ12 ∇y G(x, x1 ), · · · , λm ∇y G(x, xm ) ∈ Rmn

(5.1)

for any x ∈ Ω. If m + 1 ≤  ≤ (n + 1)m, then   n−1  − n−2 v (x) → C1 G(x)M−1 1 P + G(x) d ,

(5.2)

in C 1 (Ω \ {x1 , · · · , xm }) as  → 0 where C1 > 0 is a constant in Theorem 1.2.  Remark 5.2. If we write (5.2) in terms of the components of the vectors G(x) and G(x), −1 and matrices M1 and P, we get (1.16). We will present the proof by dividing it into several lemmas. The first lemma is a variant of Lemmas 2.6 and 4.3 (2).

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

517

Lemma 5.3. Given a small fixed number r > 0, it holds that u (x) =

m 

 n  κi0 G(x, xi ) + o  2(n−2)

i=1

and  n−1   v (x) = (κi1 G(x, xi ) + κi2 · ∇y G(x, xi )) + o  n−2 μ (p − ) i=1 m

(5.3)

in C 1 (Ω \ {x1 , · · · , xm }) as  → 0 where 

√  =O  ,

up− 

κi0 =

 up−1− v = O()

κi1 =

B n (xi ,r)

B n (xi ,r)

and κi2 = (κi21 , · · · , κi2n ) ∈ Rn is a row vector such that 

 n−1    (y − xi ) up−1− v (y)dy = O  n−2

κi2 = B n (x

(5.4)

i ,r)

(note that κi0 , κi1 and κi2 depend also on  or ). Proof. The proof is similar to Lemmas 2.6 and 4.3 (2), so we just briefly sketch why (5.3) holds in C 0 (K) for any compact subset K of Ω \ {x1 , · · · , xm }. For x ∈ Ar (see (2.6)), a combination of Green’s representation formula and the Taylor expansion of G(x, y) in the y-variable shows that 

 v (x) = μ (p − ) i=1 m

(G(x, xi ) + (y − xi ) · ∇y G(x, xi )

B n (xi ,r/2)

  p−1−   n   u v (y)dy + O  n−2 . + O |y − xi |2 Also, by means of Proposition 2.3 and Lemma 2.8, we have     |y − xi |2 ·  up−1− v (y) dy B n (xi ,r/2)



 p−1−   ˜ |x|2 ·  u v˜ (x) dx

n

= (λi α0 )

  B n 0,(λi α0 )−1 r/2 −

≤ C

n n−2

C

1 n−2

tn+1 1+

0

t(n+2)−(n−2)

 n  dt = O  n−2

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for each i, from which the desired result follows. The order of ki0 , ki1 and κi2 can be computed as in (2.13) or (4.9). 2 Let us write u and v in the following way. For each i = 1, · · · , m,  n  κi0 γn + g (x) + o  2(n−2) where i |x − xi |n−2  κj0 G(x, xj ), gi (x) = −κi0 H(x, xi ) +

u (x) =

(5.5)

j=i

and  n−1  κi1 γn x − xi v (x) = + (n − 2)γ κ · + h (x) + o  n−2 n i2 i μ (p − ) |x − xi |n−2 |x − xi |n

(5.6)

where hi (x) = − (κi1 H(x, xi ) + κi2 · ∇y H(x, xi ))  + (κj1 G(x, xj ) + κj2 · ∇y G(x, xj )) .

(5.7)

j=i

Note that gi and hi are harmonic in a neighborhood  n−1  of xi . With these decompositions we now compute κi1 , will be shown to be O  n−2 , by applying the bilinear version of the Pohožaev identity which the next lemma describes. Lemma 5.4.  For any point ξ ∈ Rn , a positive number r > 0 and functions f, g ∈  C 2 B n (ξ, r) , it holds that  [((x − ξ) · ∇f ) Δg + ((x − ξ) · ∇g) Δf ] B n (ξ,r)

 =r



 ∂f ∂g 2 − ∇f · ∇g + (n − 2) ∂ν ∂ν

∂B n (ξ,r)

 ∇f · ∇g

(5.8)

B n (ξ,r)

where ν is the outward unit normal vector on ∂B n (ξ, r). Proof. This follows from an elementary computation. See the proof of [20, Proposition 5.5] in which the author considered it when n = 2. 2   ij Lemma 5.5. Recall the definition of M1 in (1.14) and its inverse M−1 1 = m1 Then it holds for m + 1 ≤  ≤ (n + 1)m that

1≤i,j≤m

.

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

− n−2 κi1

519

n−1

⎛

m 

n−1 ⎝− 1 − n−2 κj2 · ∇τ (xj ) + mij 1 2 j=1

=



⎞ − n−2 κl2 · ∇y G(xj , xl )⎠ + o(1). (5.9) n−1

l=j

Remark 5.6. If m = 1, one has that Υ1 (λ1 , x1 ) = c1 τ1 (x1 )λn−2 −c2 log λ1 (refer to (1.9)). 1 n−1 n−2 Therefore (5.9) and 0 = ∂x1 Υ1 (λ1 , x1 ) = c1 (∂x1 τ ) (x1 )λ1 imply − n−2 κi1 = o(1). Proof. Fixing a sufficiently small number r > 0, we take ξ = xi , f = u and g = v for (5.8). Then from (1.1 ), (1.5) and the estimate  (1 − μ )

[(x − xi ) · ∇u ] up−1− v

B n (xi ,r)

 =O 

n−1 n−2



1 2

⎛ n−2 2

·  λi

⎝−

n 

 d,i,k

k=1

Rn

⎞  n−1 1  ∂U 1,0 p−1 (x · ∇U1,0 ) U1,0 + o(1)⎠ = o  n−2 + 2 ∂xk

where Proposition 4.4 and Corollary 4.6 are made use of, one finds that the left-hand side of (5.8) is equal to 



  (x − xi ) · ∇ up− v + (1 − μ )(p − ) 

− B n (xi ,r)

[(x − xi ) · ∇u ] up−1− v

B n (xi ,r)





 1

n−1

up− v + o  n−2 + 2 . 

=n B n (xi ,r)

As a result, (5.8) reads as   ∂u ∂v − ∇u · ∇v + (n − 2) 2 ∂ν ∂ν

 r ∂B n (xi ,r)



=2



n−1

 1



∂u v ∂ν

∂B n (xi ,r)

up− v + o  n−2 + 2 

B n (xi ,r) −1



= 2 [μ (p − ) − 1]



(5.10)   n−1 1  ∂u ∂v v − u dS + o  n−2 + 2 ∂ν ∂ν

∂B n (xi ,r)

where the latter equality is due to Green’s identity (4.12). We compute the rightmost side of (5.10) first. Since gi , hi and (x − xi ) · ∇gi are harmonic near xi (see (5.5) and (5.7) to remind their definitions), a direct computation with (5.5)–(5.7), the mean value formula and Green’s identity (4.12) shows that

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520





 ∂u ∂v v − u dS ∂ν ∂ν

∂B n (xi ,r)

⎡

  ⎢ = μ (p − ) ⎣(n − 2)γn S n−1  (κi1 gi (xi ) − κi0 hi (xi )) 

(n − 2)γn + κi2 · rn

(5.11)

∂gi dS (x − xi ) ∂ν

∂B n (xi ,r)



(n − 2)(n − 1)γn κi2 · rn+1

+



 1

⎤

n−1 ⎥ (x − xi )gi dS + o  n−2 + 2 ⎦ .

∂B n (xi ,r)

x−xi n Moreover, both gi and |x−x n are harmonic in B (xi , r) \ {xi }, so Green’s identity i | again implies that the value



 I1r := κi2 · ∂B n (xi ,r)



%

= κi2 · ∂B n (xi ,r)

 x − xi ∂gi x − xi + (n − 1) gi dS |x − xi |n ∂ν |x − xi |n+1

x − xi ∂gi ∂ − |x − xi |n ∂ν ∂ν



x − xi |x − xi |n



& gi dS

(5.12)

is independent of r > 0. Thus, taking the limit r → 0 and applying the Taylor expansion of gi , we find that it is equal to

I10 := lim I1r r→0

n  κi2k = lim r→0 rn+1 k,l=1

 xk xl [(∂l gi ) (xi ) + O(|x|)] dS ∂B n (0,r)

n  κi2k + (n − 1) lim r→0 rn+1 k=1

=n

n  k,l=1



) xk gi (xi ) +

∂B n (0,r)

 κi2k (∂l gi ) (xi )

n 

  2 xl (∂l gi ) (xi ) + O |x| dS

l=1

  xk xl dS = S n−1  κi2 · ∇gi (xi ).

(5.13)

∂B n (0,1)

n−1

1

n−2 + 2 , However the quantity  n−1κ i2 · ∇gi (xi ) is negligible in the sense that its order is  because κi2 = O  n−2 and that ∇x Υm (λ1 , · · · , λm , x1 , · · · , xm ) = 0 means

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521

 1  1 lim − 2 ∇gi (xi ) = − lim − 2 κi0 (∇x H) (xi , xi ) →0 →0  1   lim − 2 κj0 (∇x G) (xi , xj ) + ⎛

j=i

→0

⎞

(5.14)

 n−2 1 n−2 = ⎝− λi 2 (∇x τ ) (xi ) + λj 2 (∇x G) (xi , xj )⎠ C2 = 0 2 j=i

where C2 =

 Rn

p U1,0 as before. Hence we can conclude that

 n−1 1  I10 = o  n−2 + 2 .

(5.15)

Regarding the leftmost side of (5.10), one gets in a similar fashion to the derivation of (5.11) that 

∂u ∂v dS ∂ν ∂ν

∂B n (xi ,r)

⎡

 n−1  2 2  κi0 κi1 ⎢ (n − 2) γn S = μ (p − ) ⎣ rn−1 (n − 2)(n − 1)γn κi2 · rn+1

−

 (x − xi )

∂gi dS ∂ν

(5.16)

∂B n (xi ,r)

⎤   n−1 1 ∂gi ∂hi ⎥ dS + o  n−2 + 2 ⎦ . ∂ν ∂ν

 + ∂B n (xi ,r)

Furthermore, we have  ∇u · ∇v dS ∂B n (xi ,r)

⎡

 n−1  2 2  κi0 κi1 n(n − 2)γn ⎢ (n − 2) γn S = μ (p − ) ⎣ − κi2 · n−1 r rn+1 +

(n − 2)γn κi2 · rn 

∂B n (x

i ,r)

 1

⎤

n−1 ⎥ ∇gi · ∇hi dS + o  n−2 + 2 ⎦ i ,r)

∂gi dS ∂ν

∇gi dS 

+

(x − xi )

∂B n (xi ,r)



∂B n (x



(5.17)

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522

and 

∂u v dS ∂ν

∂B n (xi ,r)

)

    (n − 2)γn2 S n−1  κi0 κi1 = μ (p − ) − − (n − 2)γn S n−1  κi0 hi (xi ) n−2 r 

(n − 2)γn κi2 · + rn

∂gi dS + (x − xi ) ∂ν

∂B n (xi ,r)

(5.18) ⎤   n−1 1 ∂gi ⎥ hi dS + o  n−2 + 2 ⎦ . ∂ν



∂B n (xi ,r)

Therefore putting (5.11) and (5.15)–(5.18) into (5.10) gives that ⎡



⎢ (μ (p − ) − 1) ⎣2r

∂B n (x



(n − 2)γn ∂gi ∂hi dS − κi2 · ∂ν ∂ν rn−1

 ∇gi dS

∂B n (x

i ,r)

i ,r)

  ∇gi · ∇hi dS − (n − 2)2 γn S n−1  κi0 hi (xi )

−r ∂B n (xi ,r)

(5.19)

⎤



∂gi ⎥ hi dS ⎦ ∂ν

+ (n − 2) ∂B n (xi ,r)

,

 n−1 1   = 2 (n − 2)γn S n−1  (κi1 gi (xi ) − κi0 hi (xi )) + o  n−2 + 2 . Noticing that each component of ∇gi is harmonic, we obtain 

1

r

κ · n−1 i2

 n−1 1    ∇gi dS = S n−1  κi2 · ∇gi (xi ) = o  n−2 + 2 ,

∂B n (xi ,r)

where the second equality was deduced in (5.14). Also, by setting f = gi , g = hi and ξ = xi in the bilinear Pohožaev identity (5.8), one can verify that ⎛ ⎜ r⎝

⎞

 2

∂gi ∂hi ⎟ − ∇gi · ∇hi ⎠ dS + (n − 2) ∂ν ∂ν

∂B n (xi ,r)



∂gi hi dS = 0. ∂ν

∂B n (xi ,r)

Subsequently, (5.19) is reduced to   1   n−1   1 2κi1 − 2 gi (xi ) = [2 − (μ (p − ) − 1) (n − 2)] − 2 κi0 hi (xi ) + o  n−2 .

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523

Now we employ ∇λ Υm (λ1 , · · · , λm , x1 , · · · , xm ) = 0 to see that ⎡ 

− 12

n−2 2

gi (xi ) = C2 ⎣−τ (xi )λi

+



⎤ n−2 2

G(xi , xj )λj

⎦ + o(1) = −

j=i

C2 c2 n−2 2

c1 (n − 2)λi

+ o(1)

n−2

and that − 2 κi0 = λi 2 C2 + o(1), where C2 > 0 is the constant that appeared in (5.14) and c1 , c2 > 0 are the numbers in (1.10). Consequently, we have 1



−(n−2)

C0 λi

  n−1  + o(1) κi1 = hi (xi ) + o  n−2 % & 1 = − κi1 τ (xi ) + κi2 · ∇τ (xi ) 2  n−1   (κj1 G(xi , xj ) + κj2 · ∇y G(xi , xj )) + o  n−2 , + j=i

which can be rewritten as ⎞  1 − κ12 · ∇τ (x1 ) + κj2 · ∇y G(x1 , xj ) ⎟ ⎜ 2 κ11 j=1 ⎟ ⎜  n−1  ⎟ ⎜ .. ⎟ ⎜ . .. (M1 + o(1)) ⎝ . ⎠ = ⎜ ⎟ + o  n−2 . ⎟ ⎜  ⎠ ⎝ 1 κm1 κj2 · ∇y G(xm , xj ) − κm2 · ∇τ (xm ) + 2 j=m ⎞

⎛

This is nothing but (5.9).

⎛

2

Proof of Proposition 5.1. According to (5.4) and Proposition 4.4, we have 

− n−1 n−2

κi2k = 

− n−1 n−2



  (y − xi )k up−1− v (y)dy

B n (xi ,r)

⎛

d,i,k ⎝− = λn−1 i



Rn

⎞ ∂U 1,0 ⎠ p−1 x1 U1,0 + o(1) ∂x1

= λn−1 d,i,k p−1 C1 + o(1) i for any i ∈ {1, · · · , m} and k ∈ {1, · · · , n}. Hence the proposition follows from (5.3), Corollary 4.2 (or Corollary 4.6) and Lemma 5.5. 2 6. Characterization of the -th eigenvalues, m + 1 ≤  ≤ (n + 1)m Our goal in this section is to perform the proof of Theorem 1.3. For the convenience, we restate it in the following proposition.

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Proposition 6.1. Let A2 be the matrix which was introduced in the statement of Theorem 1.3 and ρ2 the ( − m)-th eigenvalue of A2 . For m + 1 ≤  ≤ (n + 1)m, the -th eigenvalue μ for linear problem (1.5) satisfies that  n  n μ = 1 − b2 ρ2  n−2 + o  n−2

where

b2 = (C1 C2 )/(pC5 ) > 0.

(6.1)

In addition, the nonzero vector d ∈ Rmn defined via (1.17) is an eigenfunction of A2 corresponding to ρ2 and satisfies dT1 · dT2 = 0 if m + 1 ≤ 1 = 2 ≤ (n + 1)m. The next lemma contains a key computation for the proof of Proposition 6.1. Lemma 6.2. Define %

 r Jjl;ik =

∂ ∂νx



 & ∂G ∂G ∂G (x, xj ) G(x, xl ) − (x, xj ) (x, xl ) ∂xk ∂xk ∂νx

(6.2)

∂B n (xi ,r)

and %

 r Kjl;ikq =

∂ ∂νx



∂G (x, xj ) ∂xk



∂G ∂G ∂ (x, xl ) − (x, xj ) ∂yq ∂xk ∂νx



&

∂G (x, xl ) ∂yq

∂B n (xi ,r)

(6.3) for each i, j, l ∈ {1, · · · , m} and k, q ∈ {1, · · · , n}, where the outward unit normal derivative ∂ν∂x acts over the x-variable of Green’s function G = G(x, y). Then they are the value independent of r > 0 and calculated as

r Jjl;ik

⎧ ⎪ 0 if j ⎪ ⎪ ⎪ ∂G ⎪ ⎪ ⎪ (xi , xl ) if j ⎪ ⎪ ⎨ ∂xk = ∂G ⎪ (xi , xj ) if j ⎪ ⎪ ⎪ ∂xk ⎪ ⎪ ⎪ ⎪ ∂τ ⎪ ⎩− (xi ) if j ∂xk ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∂2G ⎪ ⎪ (xi , xl ) ⎪ ⎪ ⎪ ⎨ ∂xk ∂yq

r = Kjl;ikq

∂2G ⎪ (xi , xj ) ⎪ ⎪ ⎪ ⎪ ∂xk ∂xq ⎪ ⎪ ⎪ ⎪ 1 ∂2τ ⎪ ⎪ (xi ) ⎩− 2 ∂xk ∂xq

= i and l = i, = i and l = i,

= i and l = i,

and

= l = i, if j = i and l = i, if j = i and l = i, if j = i and l = i, if j = l = i.

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525

r Proof. As explained in the proof of Lemma 4.5, the integral Jjl;ik in (6.2) is independent r of r > 0, so one may take r → 0 to find its value. We compute each Jjl;ik by considering four mutually exclusive cases categorized according to the relation of indices j, l and i. r (1) If j, l = i, then Jjl;ik vanishes. (2) Suppose that j = i and l = i. Since



∂ ∂νx

∂G (x, xi ) ∂xk



(x − xi )k (x − xi ) · ∇x = (n − 2)(n − 1)γn − n+1 r r



∂H(x, xi ) ∂xk



on ∂B n (xi , r) and   G(x, xl ) = G(xi , xl ) + (x − xi ) · ∇x G(xi , xl ) + O |x − xi |2 near the point xi , we discover %   &  ∂ ∂G ∂G ∂G ∂G r Jil;ik = (x, xi ) G(x, xl ) − (x, xi ) (x, xl ) = (xi , xl ). ∂νx ∂xk ∂xk ∂νx ∂xk ∂B n (xi ,r)

(3) In the case that j = i and l = i, a similar argument in (2) applies, yielding r Jji;ik =

∂G (xi , xj ). ∂xk

(4) Assume that j = l = i. Then Green’s identity (4.12) and Lemma 2.11 show that %

 r Jii;ik =

∂ ∂νx



 & ∂G ∂G ∂G (x, xi ) G(x, xi ) − (x, xi ) (x, xi ) dS ∂xk ∂xk ∂νx

∂B n (xi ,r)

 =−

∂G ∂G (x, xi ) (x, xi )dS = − ∂xk ∂νx

∂Ω

 

2

∂G (x, xi ) ∂νx

νk (x)dS = −

∂τ (xi ). ∂xk

∂Ω

We can deal with (6.3) in a similar manner, which we left to the reader. 2 Proof of Proposition 6.1. We reconsider (4.23), but in this time we allow to put any i ∈ {1, · · · , m} and xk (k ∈ {1, · · · , n}) in the place of i0 and x1 , respectively. By n−1 1 multiplying − 2 − n−2 on both sides, we obtain 

 n−1  ⎤  1  ⎧  1 ⎫ −2 −2 ⎨ ⎬   ∂ − n−2 v ∂  u u ∂    n−1 ⎦ dS ⎣∂ · − n−2 v − · ⎭ ∂ν ⎩ ∂xk ∂xk ∂ν ⎡

∂B n (xi ,r)

 = (p − )

μ − 1 n  n−2



⎡ ⎢ · ⎣

(n−4) − 2(n−2)

⎤

 up−1− B n (xi ,r)

∂u ⎥ v ⎦ . (6.4) ∂xk

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526

The right-hand side of (6.4) can be computed as in (4.24), which turns out to be 

μ − 1 n  n−2

,

n−4 2

−λi

d,i,k pC5 + o(1) .

Meanwhile, if we let λ ∈ Rm be a nonzero column vector  n−2 n−2 T λ = λ102 , · · · , λm02 , then (2.11) in Lemma 2.6 can be written in a vectorial form as −1/2 u (x) → C2 G(x)λ (see (5.1)). Hence, with the aid of Proposition 5.1 and Lemma 6.2, it is possible to take  → 0 in the left-hand side of (6.4) to derive ⎡



⎢ C1 C2 λ ⎣



T

∂ ∂G (x) ∂ν ∂xk

T

 G(x) −

T 

∂G (x) ∂xk

∂B n (xi ,r)



 + 4

∂ ∂G (x) ∂ν ∂xk

T

 G(x) −



3

∂G (x) ∂ν

dx · M−1 1 P

⎤ T   3 ∂G ∂G ⎥ (x) dx⎦ d (x) ∂xk ∂ν

∂B n (xi ,r)

5 = C1 C2 λT Jik M−1 1 P + Kik d r where Jik is an m × m matrix having Jjl;ik defined in (6.2) as its components, namely,     r Jik = Jjl;ik for each fixed i, k ∈ {1, · · · , m}, and Kik = Kjb;ik 1≤j≤m,1≤b≤mn 1≤j,l≤m

is an m × mn matrix whose components are

n 2

r Kj,(l−1)n+q;ik = λl Kjl;ikq =

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ n ⎪ ∂2G ⎪ ⎪λl2 (xi , xl ) ⎪ ⎪ ⎪ ⎨ ∂xk ∂yq n ∂2G 2 ⎪ λ (xi , xj ) ⎪ i ⎪ ∂xk ∂xq ⎪ ⎪ ⎪ ⎪ ⎪ n 1 ⎪ ∂2τ ⎪ ⎪ (xi ) ⎩−λi2 2 ∂xk ∂xq

if j = i and l = i, if j = i and l = i, if j = i and l = i, if j = l = i,

for j, l, i ∈ {1, · · · , m} and q, k ∈ {1, · · · , n}. From direct computations especially using that ⎧ n ∂G ⎪ 2 ⎪ ⎪ ⎨λi ∂xk (xi , xj )   T λi λ Jik = n ∂τ n 1 ∂τ  n−2 ∂G ⎪ j ⎪ ⎪ λl 2 (xi , xj ) − λi2 (xi ) = −λi2 (xi ) ⎩λi ∂x ∂x 2 ∂xk k k l=i

if i = j, if i = j,

W. Choi et al. / Advances in Mathematics 298 (2016) 484–533

for λT Jik =



λT Jik 4

A 2 d = P

T

 1

527

   , · · · , λT Jik ∈ Rm , we conclude m

M−1 1 P

    μ − 1 pC5 lim d = ρ2 d + Q d = − n C1 C2 →0  n−2 5

with matrices M1 , P and Q given in (1.14), (1.18) and (1.19). The claim that dT1 ·dT2 = 0 can be proved as in the proof of Theorem 1.1, or particularly, (3.5). The proof is done. 2 7. Estimates for the -th eigenvalues and eigenfunctions, (n + 1)(m + 1) ≤  ≤ (n + 2)m We now establish Theorem 1.5 by obtaining a series of lemmas. In the first lemma we will compute the limit of the -th eigenvalues as  → 0 when (n+1)(m+1) ≤  ≤ (n+2)m. Lemma 7.1. If (n + 1)(m + 1) ≤  ≤ (n + 2)m, we have lim μ = 1.

→0

Proof. By virtue of Corollary 4.2 or Corollary 4.6, it is enough to show that lim sup→0 μ ≤ 1. Referring to (3.1), we let V be a vector space whose basis is {u,i : 1 ≤ i ≤ m} ∪ {ψ,i,k : 1 ≤ i ≤ m, 1 ≤ k ≤ n + 1}. If we write f ∈ V \ {0} as f=

m 

fi

with

fi = ai0 u,i +

i=1

n+1 

aik ψ,i,k

k=1

for some (a10 , · · · , a1(n+1) , · · · , am0 , · · · , am(n+1) ) ∈ Rm(n+1) \ {0}, then we have  μ((n+2)m) =

min 1



Ω

max

f ∈W\{0} W⊂H0 (Ω), dimW=(n+2)m

(p − )

|∇f |2

Ω

f 2 up−1−

 ≤ max

f ∈V\{0}



Ω

(p − )

|∇f |2

Ω

f 2 up−1−



|∇fi |2  2 p−1− := max max ai , f ∈V\{0} 1≤i≤m (p − ) f ∈V\{0} 1≤i≤m f u Ω i 

≤ max

max

Ω

so it is sufficient to check that ai ≤ 1 + o(1). If we denote a = ai for a fixed i and modify n ∂u the definition of z in the proof of Proposition 4.1 into z = k=1 ak ∂x + an+1 wi , then k we again have a = 1 + b/c. (The definition of b, c and wi can be found in (4.4), (4.5) and (4.14).) Moreover computing each of the term of b and c as we did in the proof of Proposition 4.1, we find  2  b ≤ C |¯ a| + a2n+1 

 2  2 and c ≥ C− n−2 |¯ a|2 + Ca2n+1 ≥ C |¯ a| + a2n+1 ,

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from which one can conclude that μ((n+2)m) ≤ 1 +O(). For more detailed computations, we ask for the reader to check the proof of Theorem 1.4 in [14]. 2 The following lemma is the counterpart of Proposition 4.4 for (n + 1)(m + 1) ≤  ≤ (n + 2)m. Lemma 7.2. Let (n +1)(m +1) ≤  ≤ (n +2)m. For each i ∈ {1, · · · , m} and d,i,n+1 ∈ R, it converges to  v˜i  d,i,n+1

∂U1,0 ∂λ

 weakly in H 1 (Rn ).

Proof. Lemma 4.3 (1) holds in this case also by Lemma 7.1. Therefore it is enough to show that the vector d in (1.17) is zero. As in (3.5), the orthogonality of v and v1  for m + 1 ≤ 1 ≤ (n + 1)m implies T d · dT1 = 0. However, we also know from Proposition 6.1 that {dm+1 , · · · , d(n+1)m } serves a basis for Rmn . Hence d = 0, concluding the proof. 2 As a consequence, we reach at Proposition 7.3. Let A3 be the matrix (1.21). For (n + 1)(m + 1) ≤  ≤ (n + 2)m, if ρ3 is the ( − (m + 1)n)-th eigenvalue of A3 , then it is positive and the -th eigenvalue μ to problem (1.5) is estimated as μ = 1 + b3 ρ3  + o()

where b3 =

(n − 2)2 C2 C3 . 2(n + 2)C4

(7.1)

ˆ  in (1.22) is a corresponding eigenvector to ρ3 and Furthermore, the nonzero vector d  T T ˆ ˆ d1 · d2 = 0 if (n + 1)(m + 1) ≤ 1 = 2 ≤ (n + 2)m. Proof. Denote d,i = d,i,n+1 in the previous lemma. Then we can recover (4.8) from Lemma 7.1. Hence the arguments in the proof of Proposition 4.4 work, giving (4.20) ˆ  is an eigenvector and (4.19) to us again. From them, we conclude that ρ3 is positive, d 3 corresponding to ρ and (7.1) is valid. The last orthogonality assertion is deduced in the same way as one in Theorem 1.1. See (3.5). 2 Acknowledgments W. Choi was supported by the Global PhD Fellowship of the Government of South Korea 300-20130026. S. Kim was supported by FONDECYT Grant 3140530. K. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2014R1A2A2A01004618). K. Lee also holds a joint appointment with the Research Institute of Mathematics of Seoul National University.

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Appendix A. A moving sphere argument In this appendix, we show the following proposition by employing the moving sphere argument given in [17] (refer also to [8]). Note that it implies Proposition 2.3 at once. Proposition A.1. Let r0 > 0 be fixed and p = (n + 2)/(n − 2) as above. Suppose that a family {u } of positive C 2 -functions which satisfy −Δu = up− 

  in B n 0, −α0 r0 ,

u L∞ (B n (0,−α0 r0 )) ≤ c

for some c > 0, and lim u (x) = U1,0 (x)

→0

weakly in H 1 (Rn ).

(A.1)

Then there are constants C > 0 and 0 < δ0 < r0 independent of  > 0 such that u (x) ≤ CU1,0 (x)

  for all x ∈ B n 0, −α0 δ0 .

Before conducting its proof, we introduce Green’s function GR of −Δ in B n (0, R) for each R > 0 with zero Dirichlet boundary condition. By the scaling invariance, we have GR (x, y) = G1

x y 1 , R R Rn−2

for x, y ∈ B n (0, R).

Thus we can decompose Green’s function in B n (0, R) into its singular part and regular part as follows: GR (x, y) =

x y γn 1 , − n−2 H1 n−2 |x − y| R R R

for x, y ∈ B n (0, R).

(A.2)

See (1.2) for the definition of the normalizing constant γn . Now we begin to prove Proposition A.1. By (A.1) and elliptic regularity, for arbitrarily given ζ1 > 0 and any compact set K ⊂ Rn , there is 1 > 0 such that it holds u − U1,0 C 2 (K) ≤ ζ1

for  ∈ (0, 1 ).

(A.3)

Let us define the Kelvin transform of u :  uλ (x) =

λ |x|

n−2

  u xλ ,

xλ =

λ2 x |x|2

for |xλ | < −α0 r0

(A.4)

and the difference wλ = u − uλ between u and it. Then we have  −Δwλ = up− − 

λ |x|

(n−2)

 λ p−  p− u ≥ up− − uλ = ξ (x)wλ 

for |x| ≥ λ

(A.5)

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where ⎧ p−  λ p− ⎪ ⎨ u − u (x) if u (x) = uλ (x), u − uλ ξ (x) = ⎪ ⎩ if u (x) = uλ (x). (p − )up−1− (x) Lemma A.2. For any ζ2 > 0, there exist small constants δ1 > 0 and 2 > 0 such that min u (y) ≤ (1 + ζ2 )U1,0 (r)

|y|=r

for 0 < r := |x| ≤ −α0 δ1 and any  ∈ (0, 2 ).

(A.6)

Proof. We first choose a candidate δ1 ∈ (0, r0 ) for which (A.6) will have the validity. Fix a sufficiently small value η1 > 0 and a number R0 > 0 such that  uλ (x)

≤

ζ2 1+ 4

 βn |x|2−n

for any 0 < λ ≤ 1 + η1 and |x| ≥ R0

(A.7)

provided  > 0 small. (Here βn = (n(n − 2))p−1 is the constant appeared in (1.3).) Take λ λ1 = 1 − η1 and λ2 = 1 + η1 . Because U1,0 = Uλ2 ,0 for any λ > 0 and u → U1,0 in 1 n C -uniformly over compact subsets of R as  → 0, by enlarging R0 > 0 if necessary, we can find a number η2 > 0 so small that wλ1 (x) > 0 for λ1 < |x| ≤ R0 ,

uλ 1 (x) ≤ (1 − 2η2 )βn |x|2−n

for |x| ≥ R0

(A.8)

and  up− (x)dx 

  η2  p ≥ 1− U1,0 (x)dx 2

(A.9)

Rn

B n (0,R0 )

for sufficiently small  > 0. On the other hand, provided δ1 > 0 small enough, we have the following inequality u (x) ≥ (1 − η2 )βn |x|2−n

for R0 ≤ |x| ≤ −α0 δ1 .

(A.10)

Indeed, if we choose a function u ˆ which solves −Δˆ u = up− 

  in B n 0, −α0

and u ˆ = 0 on



 |x| = −α0 ,

then the comparison principle tells us that u ≥ u ˆ . Since Green’s function is always positive, we can make  η2 γn −α0 (n−2)  · H1 −α0 x, −α0 y ≤ 4 |x − y|n−2

  for x, y ∈ B n 0, −α0 δ1

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by taking δ1 small. Also, the relation |x − y| ≤ (1 − 1/l)|x| holds for |x| ≥ lR0 and |y| ≤ R0 given any l ∈ (1, ∞), and so we see from (A.2) and (A.9) that  up− (y)G−α0 (x, y)dy 

u ˆ (x) = B n (0,−α0 )



⎛

η2  ⎜ ≥ 1− ⎝ 2



βn |x|n−2

η2  ≥ 1− 4

 up− (y) 

B n (0,−α0 δ1 )

⎞

B n (0,R0 )

= (1 − η2 )



γn dy |x − y|n−2

⎛ ⎞  γ γn ⎟ n p up− (y)dy ⎠ n−2 ≥ (1 − η2 ) ⎝ U1,0 (y)dy ⎠ n−2  |x| |x| Rn

for lR0 ≤ |x| ≤ −α0 δ1

by choosing l large enough. If |x| ≤ lR0 , the uniform convergence of u to U1,0 implies u (x) ≥ (1 − η2 )βn |x|2−n for  > 0 sufficiently small. This shows the validity of (A.10). Fixing δ1 > 0 for which (A.10) is valid, suppose that (A.6) does not hold on the con −α0  ∞ trary. Then there are sequences {k }∞ δ1 k=1 and {rk }k=1 such that k → 0, rk ∈ 0, k and min uk (x) > (1 + ζ2 )U1,0 (rk ).

|x|=rk

Set uk = uk for brevity. Since uk → U1,0 uniformly on any compact set, it should hold that rk → ∞. Therefore  min uk (x) ≥

|x|=rk

1+

ζ2 2

 βn rk2−n .

(A.11)

To deduce a contradiction, let us apply the moving sphere method to wkλ = uk − uλk ¯ k by for the parameters λ1 ≤ λ ≤ λ2 . Define λ ¯ k = sup {λ ∈ [λ1 , λ2 ] : wμ ≥ 0 in Σμ for all λ1 ≤ μ ≤ λ} λ k where Σμ = {x ∈ Rn : μ < |x| < rk }. ¯ k = λ2 for sufficiently large k ∈ N. First of all, putting together with We claim that λ ¯ k ≥ λ1 . Recall from (A.5) that (A.8) and (A.10), we discover that wkλ1 > 0 in Σλ1 , so λ ¯

¯

¯

−Δwkλk + (ξk )− wkλk ≥ (ξk )+ wkλk ≥ 0

in Σλ¯ k .

¯

Moreover, from (A.11) and (A.7) we have wkλk > 0 on ∂B n (0, rk ). Thus by the maximum principle and Hopf’s lemma we have ¯

¯

wkλk > 0 in Σλ¯ k

and

  ∂wkλk ¯k < 0 on ∂B n 0, λ ∂ν

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where ν is the unit outward normal vector. However this means wkμ ≥ 0 in Σμ even after taking a slightly larger value of μ than ¯ k . Hence our claim is justified. Consequently, the maximality of λ λ2 to wk ≥ 0 in Σλ2 allows one to get

¯ k < λ2 , then that if λ ¯ λk , which contradicts taking a limit k → ∞

λ2 U1,0 (x) ≥ U1,0 (x) in |x| ≥ λ2 ,

but it cannot be possible since λ2 > 1. Thus (A.6) should be true. 2 The following lemma completes our proof of Proposition A.1. Lemma A.3. For some constant C > 0 and parameter δ0 ∈ (0, δ1 ), we have u (x) ≤ CU1,0 (x)

for |x| ≤ −α0 δ0

provided that  > 0 is sufficiently small. Here δ1 > 0 is the number chosen in the proof of the previous Lemma. Proof. Argue as in the proof of Lemma 2.4 in [17] employing Lemma A.2 above. In that paper, the statement of the lemma as well as its proof is written for a sequence {uk }∞ k=1 of solutions, but they apply to a family {u } as well. To proceed our proof, we substitute Gk , Rk and vk in [17] with Dirichlet Green’s function G−α0 δ1 of −Δ in B n (0, −α0 δ1 ), R = −α0 δ1 δ2 and u where δ2 ∈ (0, 1) is a sufficiently small number. 2 References [1] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) 253–294. [2] A. Bahri, Y. Li, O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 (1995) 67–93. [3] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991) 18–24. [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983) 437–477. [5] H. Brezis, L.A. Peletier, Asymptotics for elliptic equations involving critical growth, in: Partial Differential Equations and the Calculus of Variations, vol. I, in: Progr. Nonlinear Differential Equations Appl., vol. 1, Birkhäuser, Boston, 1989, pp. 149–192. [6] D. Cao, S. Peng, S. Yan, Infinity many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal. 262 (2012) 2861–2902. [7] W. Chen, C. Li, On Nirenberg and the related problems – a necessary and sufficient condition, Comm. Pure Appl. Math. 48 (1995) 657–667. [8] C.C. Chen, C.S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom. 49 (1998) 115–178. [9] K.S. Chou, C.W. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. 48 (1993) 137–151. [10] W.Y. Ding, Positive solutions of Δu + u(n+2)/(n−2) = 0 on contractible domains, J. Partial Differ. Equ. 2 (1989) 83–88. [11] F. Gladiali, M. Grossi, On the spectrum of a nonlinear planar problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 191–222. [12] F. Gladiali, M. Grossi, H. Ohtsuka, On the number of peaks of the eigenfunctions of the linearized Gel’fand problem, Ann. Mat. Pura Appl. 195 (2016) 79–93.

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