Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions

  • Yibin Chang1Email author and

    Affiliated with

    • Haitao Yang1

      Affiliated with

      Boundary Value Problems20122012:33

      DOI: 10.1186/1687-2770-2012-33

      Received: 17 October 2011

      Accepted: 23 March 2012

      Published: 23 March 2012

      Abstract

      In this article, we mainly construct multiple blowing-up and concentrating solutions for a class of Liouville-type equations under mixed boundary conditions:

      - Δ v = ε 2 e v - 4 π i = 1 N α i δ p i , in Ω , ε ( 1 - t ) v ν + t b ( x ) v = 0 , on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equa_HTML.gif

      for ε small, where t ( 0 , 1 ] , N { 0 } , { α 1 , α 2 , , α N } ( - 1 , + ) \ ( { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq1_HTML.gif, Ω is a bounded, smooth domain in 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq2_HTML.gif, Γ := {p1, ..., p N } ⊂ Ω is the set of singular sources, δ p denotes the Dirac mass at p, ν denotes unit outward normal vector to Ω and b(x) > 0 is a smooth function on Ω.

      2000 Mathematics Subject Classification: 35B25; 35J25; 35B38.

      Keywords

      multiple blowing-up and concentrating solution Liouville-type equation singular source mixed boundary conditions finite dimensional reduction

      1 Introduction

      In this article, we mainly investigate the mixed boundary value problem:
      - Δ v = ε 2 e v - 4 π i = 1 N α i δ p i , in Ω , ε ( 1 - t ) v ν + t b ( x ) v = 0 , on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equa_HTML.gif
      (1)

      for ε small, where t ( 0 , 1 ] , N { 0 } , { α 1 , α 2 , , α N } ( - 1 , + ) \ ( { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq1_HTML.gif, Ω is a bounded, smooth domain in 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq2_HTML.gif, Γ:= {p1, ..., p N } ⊂ Ω is the set of singular sources, δ p denotes the Dirac mass at p, ν denotes unit outward normal vector to Ω and b(x) > 0 is a smooth function on Ω.

      Such problems occur in conformal geometry [1], statistical mechanics [24], Chern-Simons vortex theory [511] and several other fields of applied mathematics [1216]. In all these contexts, an interesting point is how to construct solutions which exactly "blow-up" and "concentrate" at some given points, whose location carries relevant information about the potentially geometrical or physical properties of the problem. However, the authors mainly consider the Dirichlet boundary value problem, and little is known for the problem with singular sources satisfying α i ∈ (-1, 0) for some i = 1, ..., N. The main purpose of this article is to study how to construct multiple blowing-up and concentrating solutions of the Equation (1) with the mixed boundary conditions and singular sources.

      Let G t,ε denotes the Green's function of -∆ with mixed boundary conditions on Ω, namely for any y ∈ Ω,
      - Δ x G t , ε ( x , y ) = 2 π δ y ( x ) , in Ω , ε ( 1 - t ) G t , ε ( x , y ) ν + t b ( x ) G t , ε ( x , y ) = 0 , on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ1_HTML.gif
      (2)
      and let H t,ε (x, y) = G t,ε (x, y) + log |x - y| be its regular part. Set G1 = G1,εand H1 = H1,ε. Since ε exactly disappears in the Equation (2)|t = 1, G1 and H1 don't depend on ε. The Equation (1) is equivalent to solving for u = v + 2 i = 1 N α i G t , ε ( x , p i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq3_HTML.gif, the regular part of v, the equation
      - Δ u = ε 2 x - p 1 2 α 1 x - p N 2 α N e - 2 i = 1 N α i H t , ε ( x , p i ) e u , in Ω , ε ( 1 - t ) u ν + t b ( x ) u = 0 , on Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equb_HTML.gif
      Thus, we consider the more general model problem:
      - Δ u = ε 2 x - p 1 2 α 1 x - p N 2 α N f ( x ) e u , in Ω , ε ( 1 - t ) u ν + t b ( x ) u = 0 , on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ2_HTML.gif
      (3)

      where f : Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq4_HTML.gif is a smooth function such that f(pi) > 0 for any i = 1, ..., N. Set Ω' = {x ∈ Ω: f(x) > 0}, S ( x ) = x - p 1 2 α 1 x - p N 2 α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq5_HTML.gif and Δ m = {p = (p1, ..., p m ) ∈ Ω m : p i = pj for some i ≠ j}.

      It is known that for { α 1 , , α N } ( 0 , + ) \ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq6_HTML.gif, or α i = 0 for any i = 1, ..., N, if u ε is a family of solutions of the Equation (3)|t = 1with inf Ω f > 0, which is not uniformly bounded from above for ε small, then u ε blows up at different points p k 1 , , p k n + m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq7_HTML.gif with n + m ≥ 1, 0 ≤ nN, p = ( p k n + 1 , , p k n + m ) ( Ω \ Γ ) m \ Δ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq8_HTML.gif and { p k 1 , , p k n } Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq9_HTML.gif, and satisfies the concentration property:
      ε 2 x - p 1 2 α 1 x - p N 2 α N f ( x ) e u ε 8 π i = 1 n ( 1 + α k i ) δ p k i + 8 π i = n + 1 n + m δ p k i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ3_HTML.gif
      (4)
      in the sense of measures in Ω ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq10_HTML.gif. Moreover, p = ( p k n + 1 , , p k n + m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq11_HTML.gif is a critical point of the function:
      φ n , m ( p ) = i = n + 1 n + m H 1 ( p k i , p k i ) + 1 2 log f ( p k i ) S ( p k i ) + i , j = n + 1 , i j n + m G 1 ( p k j , p k i ) + 2 i = 1 n j = n + 1 n + m ( 1 + α k i ) G 1 ( p k j , p k i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ4_HTML.gif
      (5)

      (see [7, 1723]). An obvious problem for the Equation (3) is the reciprocal, namely the existence of multiple blowing-up solutions with concentration points near critical points of φ n,m .

      The earlier result concerning the existence of multiple blowing-up and concentrating solutions of the Equation (3) is given by Baraket and Pacard in [24]. When t = 1 and α i = 0 for any i = 1,2, ..., N, they prove that any non-degenerate critical point p = ( p k n + 1 , , p k n + m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq11_HTML.gif of the function φ n,m with n = 0 generates a family of the solutions u ε which blow-up at p k n + 1 , , p k n + m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq12_HTML.gif, and concentrate in the sense that (4) holds. Esposito [20] performs a similar asymptotic analysis and extends the previous result by allowing the presence of singular sources in the Equation (3)|t = 1, that is, { α 1 , , α N } ( 0 , + ) \ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq6_HTML.gif. However, the asymptotic analysis method depends on the non-degenerate assumption of critical point of the function φ n,m so much that it pays in return at a price of the very complicated and accurate control on the asymptotics of the solutions.

      In fact, the finite dimensional reduction method, used successfully in higher dimensional nonlinear elliptic equation involving critical Sobolev exponent (see [6, 25]), can avoid the technical difficulty in carrying out the asymptotic analysis method for the Equation (3). It is necessary to point out that the key step of the finite dimensional reduction is the analysis of the bounded invertibility of the corresponding linearized operator L of the Equation (3) at the suitable approximate solution. In [26, 27], the authors construct the approximate solution, carry out the finite dimensional reduction and use some stability assumptions of critical points of φ0,mto get the existence of multiple blowing-up and concentrating solutions for the Equation (3)|t = 1with Γ = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq13_HTML.gif, namely α i = 0 for any i = 1,2, ..., N. When { α 1 , , α N } ( 0 , + ) \ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq6_HTML.gif, a similar result for the Equation (3)|t = 1under C0-stable assumption of critical point of φ n,m (see Definition 4.1) is also established in [28].

      Here in the spirit of the finite dimensional reduction, we try to extend the result of the Equation (1) in [20, 28] by allowing the presence of singular sources 4 π i = 1 N α i δ p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq14_HTML.gif with some α i ∈ (-1, 0) and Robin boundary conditions ε ( 1 - t ) v ν + t b ( x ) v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq15_HTML.gif with t ∈ (0,1). When we carry out the finite dimensional reduction, we need to get the invertibility of the desired linearized operator L for the Equation (3) under some α i ∈ (-1, 0). Obviously, the linearized operator L easily produces the singularities at some singular sources with α i ∈ (-1, 0), which makes trouble for the analysis of the bounded invertibility of L. But we can successfully get rid of it by introducing a suitable L-weighted norm (see (30) below) related with a "gap interval" (- 1, α0), where α0 = min{0, α1, ..., α N }. On the other hand, the presence of the term ε ( 1 - t ) u ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq16_HTML.gif in the Equation (3)|0<t<1brings some new technical difficulties. A flexible approach exactly helps us overcome the difficulties by making use of the maximum principle. In addition, a weaker stable assumption of critical points of the function φ n,m also helps us construct multiple blowing-up and concentrating solutions of the Equation (3). As a consequence, we have the following result.

      Theorem 1.1 Let 0 ≤ n ≤ N and m { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq17_HTML.gif such that n + m ≥ 1. Assume that N { 0 } , { α 1 , α 2 , , α N } ( - 1 , + ) \ ( { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq18_HTML.gif and p * = ( p n + 1 * , , p n + m * ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq19_HTML.gif is a C0-stable critical point for φ n,m in (Ω' \ Γ) m \ Δ m with m ≥ 1 (see Definition 4.1). Then there exists a family of solutions u ε for the Equation (3) with the concentration property (4), which blow up at n-different points ( p k 1 , , p k n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq20_HTML.gif in Γ, and m-points p = ( p k n + 1 , , p k n + m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq11_HTML.gif in (Ω' \ Γ) m \ Δ m with φ n,m (p*) = φ n,m (p). Moreover, u ε remains uniformly bounded on Ω \ i = 1 n + m B λ ( p k i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq21_HTML.gif, and sup B λ ( p k i ) u ε + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq22_HTML.gif for any λ > 0.

      Let us point out that from the proof of Theorem 1.1 Robin boundary condition can be considered as a perturbation of Dirichlet boundary condition for the problem (3) in using perturbation techniques to construct multiple blowing-up and concentrating solutions. Based on this point, we also consider the Dirichlet-Robin boundary value problem:
      - Δ u = ε 2 x - p 1 2 α 1 x - p N 2 α N f ( x ) e u , in Ω , ε u ν + b ( x ) u = 0 , on T , u = 0 , on Ω \ T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ5_HTML.gif
      (6)

      where T Ω is a relatively closed subset and Ω \ T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq23_HTML.gif. This together with other similar mixed boundary value problems can be founded in [29, 30]. For the problem (6), we obtain the following result.

      Theorem 1.2 Under the assumption of Theorem 1.1, then there exists a family of solutions u ε for the Equation (6) with the concentration property (4), which blow up at n-different points ( p k 1 , , p k n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq20_HTML.gif in Γ, and m-points p = ( p k n + 1 , , p k n + m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq11_HTML.gif in (Ω' \ Γ) m \ Δ m with φn,m(p*) = φn,m(p). Moreover, u ε remains uniformly bounded on Ω \ i = 1 n + m B λ ( p k i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq21_HTML.gif, and sup B λ ( p k i ) u ε + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq22_HTML.gif for any λ > 0.

      Finally, it is very interesting to mention that to prove the above results we need to choose the classification solutions of the following Liouville-type equation to construct concentrating solutions of the Equation (1) or (3):
      - Δ u = z 2 γ e u , in 2 , 2 z 2 γ e u < + , γ > - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ6_HTML.gif
      (7)
      given by
      u ( z ) = log 8 ( 1 + γ ) 2 μ 2 ( μ 2 + z γ + 1 - c 2 ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ7_HTML.gif
      (8)

      with μ > 0 , c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq24_HTML.gif if γ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq25_HTML.gif, c = 0 if γ ( - 1 , + ) \ ( { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq26_HTML.gif (see [5, 11, 31, 32]). Using these classification solutions scaled up and projected to satisfy the mixed boundary conditions up to a right order, the initial approximate solutions can be built up. Then through the finite dimensional reduction procedure and the notions of stability of critical points of the asymptotic reductional functional φ n,m , multiple blowing-up and concentrating solutions can be constructed as a small additive perturbation of the initial approximations.

      This article is organized as follows. In Section 2, we will construct the approximate solution and rewrite the Equation (3) in terms of a linearized operator L. In Section 3, we give the invertibility of the linearized operator L, carry out the finite dimensional reduction and get the asymptotical expansion of the functional of the Equation (3) with respect to the suitable approximate solution. In Section 4, we give the proofs of Theorems 1.1 and 1.2.

      2 Construction of the approximate solution

      In this section, we will construct the approximate solution for the Equation (3). Let μ i , i = 1, ..., N + m, be positive numbers and set
      α i = 0 , i = N + 1 , , N + m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equc_HTML.gif
      and
      Q i ( x ) = S ( x ) x - p i 2 α i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equd_HTML.gif
      Obviously, Q i (x) = S(x) for any i = N + 1, ..., N + m. Then the function
      u i ( x ) = log 8 μ i 2 ( 1 + α i ) 2 ( μ i 2 ε 2 + x - p i 2 ( 1 + α i ) ) 2 f ( p i ) Q i ( p i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ8_HTML.gif
      (9)
      satisfies
      - Δ u i = ε 2 x - p i 2 α i f ( p i ) Q i ( p i ) e u i , in 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ9_HTML.gif
      (10)

      Set {k1, ..., k n } ⊂ {1, ..., N} and k n+i = N + i for any i = 1, ..., m.

      We hope to take i = 1 n + m u k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq27_HTML.gif as an initial approximate solution of the problem (3). So we modify it to be
      U ( x ) : = i = 1 n + m U k i = i = 1 n + m ( u k i + H k i t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ10_HTML.gif
      (11)
      where H k i t ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq28_HTML.gif(x) is the solution of
      Δ H k i t = 0 in Ω , ε ( 1 - t ) ν H k i t + t b ( x ) H k i t = - ε ( 1 - t ) u k i ν + t b ( x ) u k i , on Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ11_HTML.gif
      (12)
      Then U k i : = u k i + H k i t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq29_HTML.gif satisfies
      - Δ U k i = ε 2 x - p k i 2 α k i f ( p k i ) Q k i ( p k i ) e u k i , in Ω , ε ( 1 - t ) ν U k i + t b ( x ) U k i = 0 , on Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ12_HTML.gif
      (13)
      Lemma 2.1 For t ∈ (0, 1] and p k i Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq30_HTML.gif,
      H k i t ( x ) = 4 ( 1 + α k i ) H t , ε ( x , p k i ) - log 8 μ k i 2 ( 1 + α k i ) 2 f ( p k i ) Q k i ( p k i ) + O ( ε 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ13_HTML.gif
      (14)

      uniformly in C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq31_HTML.gif and in C loc 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq32_HTML.giffor ε small.

      Proof. Set z t ( x ) = H k i t ( x ) - 4 ( 1 + α k i ) H t , ε ( x , p k i ) + log 8 μ k i 2 ( 1 + α k i ) 2 f ( p k i ) Q k i ( p k i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq33_HTML.gif. Then z t (x) satisfies
      Δ z t ( x ) = 0 , in Ω , ε ( 1 - t ) z t ( x ) ν + t b ( x ) z t ( x ) = F t ( x ) , on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Eque_HTML.gif
      where
      F t ( x ) = 4 ε ( 1 - t ) ( 1 + α k i ) x - p k i 2 α k i ν ( x ) ( x - p k i ) μ k i 2 ε 2 + x - p k i 2 ( 1 + α k i ) - ν ( x ) ( x - p k i ) x - p k i 2 + t b ( x ) 2 log μ k i 2 ε 2 + x - p k i 2 ( 1 + α k i ) - 4 ( 1 + α k i ) log x - p k i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equf_HTML.gif
      For any t ∈ (0, 1], it is easy to check F t x L Ω = O ε 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq34_HTML.gif. If t = 1, from the maximum principle and smooth function b(x) > 0, it follows
      max Ω ̄ z 1 ( x ) = max Ω z 1 ( x ) C ( b ) F 1 ( x ) L ( Ω ) = O ( ε 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equg_HTML.gif
      If 0 < t < 1, from the maximum principle with the Robin boundary condition (see [[33], Lemma 2.6]), it also follows
      max Ω ̄ z t ( x ) 1 t C ( b ) F t ( x ) L ( Ω ) = O ( ε 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equh_HTML.gif
      Thus using the interior estimate of derivative of harmonic function (see [[34], Theorem 2.10]), there holds
      max K D α z t ( x ) 2 α dist ( K , Ω ) α max Ω ̄ z t ( x ) = O ( ε 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equi_HTML.gif

      for any compact subset K of Ω, any t ∈ (0, 1] and any multi-index α with |α| ≤ 2, which derives (14) uniformly in C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq31_HTML.gif and in C loc 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq32_HTML.gif for ε small. □

      From this lemma we can construct the approximate solution U ( x ) = i = 1 n + m ( u k i + H k i t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq35_HTML.gif, which satisfies the mixed boundary conditions. On the other hand, we hope that the error U ( x ) - u k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq36_HTML.gif is smaller near every p k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq37_HTML.gif. In fact, we can realize this point by further choosing positive number μ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq38_HTML.gif such that
      log 8 μ k i 2 ( 1 + α k i ) 2 f ( p k i ) Q k i ( p k i ) = 4 ( 1 + α k i ) H t , ε ( p k i , p k i ) + 4 j = 1 , j i j = n + m ( 1 + α k j ) G t , ε ( p k i , p k j ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ14_HTML.gif
      (15)
      Consider the scaling of the solution of the Equation (3)
      v ( y ) = u ( ε y ) + 4 log ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equj_HTML.gif
      then v(y) satisfies
      - Δ v = S ( ε y ) f ( ε y ) e v , in Ω ε , ( 1 - t ) v ν + t b ( ε y ) v = 4 t b ( ε y ) log ε , on Ω ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ15_HTML.gif
      (16)
      where Ω ε = 1 ε Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq39_HTML.gif. We also set p k i = 1 ε p k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq40_HTML.gif and define the new approximation in expanded variables as V(y) = U(εy) + 4 log ε. Furthermore, set
      ρ k i = ε 1 1 + α k i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ16_HTML.gif
      (17)
      and
      W ( y ) = S ( ε y ) f ( ε y ) e V ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ17_HTML.gif
      (18)

      Obviously, ρ k n + i = ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq41_HTML.gif for all i = 1, ..., m.

      Here, we want to see how well -∆V(y) match with W(y) through V(y). A simple computation shows
      - Δ V ( y ) = - ε 2 Δ x U ( x ) = - ε 2 i = 1 n + m Δ x u k i ( x ) + H k i t ( x ) = i = 1 n + m ε ρ k i 2 8 μ k i 2 ( 1 + α k i ) 2 ε y - p k i ρ k i 2 α k i μ k i 2 + ε y - p k i ρ k i 2 ( 1 + α k i ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equk_HTML.gif
      Then given a small number δ > 0, if ε y - p k i ρ k i > δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq42_HTML.gif for all i = 1, ..., n + m,
      - Δ V ( y ) = O ( ε 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ18_HTML.gif
      (19)
      and if ε y - p k i ρ k i δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq43_HTML.gif for some i,
      - Δ V ( y ) = ε ρ k i 2 8 μ k i 2 ( 1 + α k i ) 2 ε y - p k i ρ k i 2 α k i μ k i 2 + ε y - p k i ρ k i 2 ( 1 + α k i ) 2 + O ( ε 4 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ19_HTML.gif
      (20)
      On the other hand, if ε y - p k i ρ k i > δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq42_HTML.gif for all i = 1, ..., n + m, obviously,
      W ( y ) = O ( ε 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ20_HTML.gif
      (21)
      and if ε y - p k i ρ k i δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq43_HTML.gif for some i,
      W ( y ) = ε y - p k i 2 α k i Q k i ( ε y ) f ( ε y ) e V ( y ) = ε 4 8 μ k i 2 ( 1 + α k i ) 2 ε y - p k i 2 α k i ε 2 μ k i 2 + ε y - p k i 2 ( 1 + α k i ) 2 f ( ε y ) Q k i ( ε y ) f ( p k i ) Q k i ( p k i ) × exp H k i t ( ε y ) + j = 1 , j i n + m log 8 μ k j 2 ( 1 + α k j ) 2 ε 2 μ k j 2 + ε y - p k j 2 ( 1 + α k j ) 2 f ( p k j ) Q k j ( p k j ) + H k j t ( ε y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equl_HTML.gif
      Now from (14), (15) and (17), we have
      W ( y ) = ε ρ k i 2 8 μ k i 2 ( 1 + α k i ) 2 ε y - p k i ρ k i 2 α k i μ k i 2 + ε y - p k i ρ k i 2 ( 1 + α k i ) 2 1 + O ρ k i ε y - p k i ρ k i + O ( ε 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ21_HTML.gif
      (22)
      In summary, we set
      R ( y ) = Δ V ( y ) + W ( y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ22_HTML.gif
      (23)
      and if ε y - p k i ρ k i > δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq42_HTML.gif for all i = 1, ..., n + m,
      R ( y ) = O ( ε 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ23_HTML.gif
      (24)
      while ε y - p k i ρ k i δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq43_HTML.gif for some i,
      R ( y ) = ε ρ k i 2 8 μ k i 2 ( 1 + α k i ) 2 ε y - p k i ρ k i 2 α k i μ k i 2 + ε y - p k i ρ k i 2 ( 1 + α k i ) 2 O ρ k i ε y - p k i ρ k i + O ( ε 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ24_HTML.gif
      (25)
      In the rest of this article, we try to find a solution v of the form v = V + ϕ of the Equation (16). In terms of ϕ, the problem (3) becomes
      L ϕ = Δ ϕ + W ϕ = - [ R + N ( ϕ ) ] , in Ω ε , ( 1 - t ) ϕ ν + t b ( ε y ) ϕ = 0 , on Ω ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ25_HTML.gif
      (26)
      where
      N ( ϕ ) = W [ e ϕ - 1 - ϕ ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ26_HTML.gif
      (27)

      3 The finite dimensional reduction

      In this section, we will carry out the finite dimensional reduction to solve the Equation (26). First of all, we need to get the desired invertibility of linearized operation L. Set
      z i 0 ( z ) = z 2 ( 1 + α k i ) - μ k i 2 z 2 ( 1 + α k i ) + μ k i 2 , for i = 1 , 2 , , n + m , z i j ( z ) = 4 z j z 2 + μ k i 2 , for i = n + 1 , , n + m , j = 1 , 2 , L i ϕ = Δ ϕ + 8 μ k i 2 ( 1 + α k i ) 2 z 2 α k i μ k i 2 + z 2 ( 1 + α k i ) 2 ϕ , for i = 1 , , n + m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equm_HTML.gif
      A basic fact to get the needed invertibility is that the linearized operator L formally approaches to the operator L i under suitable dilations and translations, which have some well-known properties that any bounded solution of L t ϕ = 0 is
      • a linear combination of zi 0and z ij for i = n + 1, ..., n + m, j = 1, 2 (see [24, 35]);

      • proportional to z i0 for 0 < α k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq44_HTML.gif and i = 1, 2, ..., n (see [20, 28, 36]).

      Remark 3.1 These properties of the operator L i have been discussed in the above papers only if 0 < α k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq44_HTML.gif for i = 1, ..., n, or α k i = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq45_HTML.gif for i = n + 1, ..., n + m. In fact, if - 1 < α k i < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq46_HTML.gif for some i = 1, ..., n, the operator L i has also the corresponding properties.

      Lemma 3.2 For - 1 < α { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq47_HTML.gif, any bounded solution ϕ of
      Δ ϕ + 8 μ 2 ( 1 + α ) 2 z 2 α μ 2 + z 2 ( 1 + α ) 2 ϕ = 0 , z \ { 0 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ27_HTML.gif
      (28)

      is proportional to z 2 ( 1 + α ) - μ 2 z 2 ( 1 + α ) + μ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq48_HTML.gif.

      Proof. If we express the bounded solution ϕ of the Equation (28) in Fourier expansion form as follow
      ϕ ( z ) = n = - + u n ( r ) e - i n θ , z = r e i θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equn_HTML.gif
      u n (r) is a bounded nontrivial solution of the equation
      u ( r ) + 1 r u ( r ) - n 2 r 2 u + 8 μ 2 ( 1 + α ) 2 r 2 α ( μ 2 + r 2 ( 1 + α ) ) 2 u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ28_HTML.gif
      (29)
      Since any solution of -∆u = e u in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq49_HTML.gif is given by the Liouville formula
      ln 8 F ( z ) 2 1 + F ( z ) 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equo_HTML.gif
      for any meromorphic function F defined on { z : F ( z ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq50_HTML.gif, the function
      ln 8 μ 2 ( 1 + α ) 2 1 + n + α + 1 α + 1 a z n 2 ( μ 2 + z 2 ( 1 + α ) 1 + a z n 2 ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equp_HTML.gif
      with any n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq51_HTML.gif and a < α + 1 n + α + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq52_HTML.gif, is the solution of -∆u = |z|2αe u in \ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq53_HTML.gif. Moreover, its derivative with respect to a at a = 0
      ϕ n ( z ) = 1 α + 1 ( n + α + 1 ) μ 2 + ( n - α - 1 ) z 2 ( 1 + α ) μ 2 + z 2 ( 1 + α ) z n , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equq_HTML.gif

      is a solution of the Equation (29) with r = |z|.

      For |n| ≥ 1, since {ϕ n (r), ϕ -n (r)} is a set of linearly independent solutions of the second order linear homogeneous ODE (29), any bounded solution is a linear combination of ϕ n (r) and ϕ -n (r). However, ϕ|n|(r) ( resp. ϕ -n (r) ) tends to 0 ( resp. ∞ ) as r ↦ 0 and ϕ|n|(r) ( resp. ϕ -|n| (r) ) tends to ∞ ( resp. 0 ) as r ↦ + ∞, which implies that the Equation (29) ||n|≥1 has no bounded nontrivial solution.

      For n = 0, ϕ 0 ( z ) = - z 2 ( 1 + α ) - μ 2 z 2 ( 1 + α ) + μ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq54_HTML.gif is a bounded solution of the Equation (29)|n = 0, that is, of the Equation (28). We claim that there does not exist the second linearly independent bounded solution of the Equation (29)|n = 0. Otherwise, let ω be another linearly independent bounded solution of (29)|n = 0. Writing ω(r) = c(r)ϕ0(r), we get that
      c ( r ) ϕ 0 + c ( r ) 2 ϕ 0 + 1 r ϕ 0 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equr_HTML.gif
      Then there exists a constant C > 0 such that
      c ( r ) = C r ϕ 0 2 ( r ) = C ( r 2 ( 1 + α ) + μ 2 ) 2 r ( r 2 ( 1 + α ) - μ 2 ) 2 ~ C r for r small, c ( r ) ~ C log r for r small . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equs_HTML.gif

      Hence, ω(r) ~ C log r for r small, which implies ω(r) is unbounded on (0, + ∞). It contradicts the assumption that ω is bounded. □

      Let us denote
      Z i 0 ( y ) = z i 0 ε y - p k i ρ k i , for i = 1 , 2 , , n + m , Z i j ( y ) = z i j ε y - p k i ρ k i , for i = n + 1 , , n + m , j = 1 , 2 , χ ι ( y ) = χ ε y - p k i ρ k i , for i = 1 , 2 , , n + m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equt_HTML.gif
      where χ(r) is a smooth, non-increasing cut-off function such that for a large but fixed number R0 > 0, χ(r) = 1 if rR0, and χ(r) = 0 if rR0 + 1. Additionally, set α0 = min{0, α1, ..., α N }. For any α ∈ (-1, α0), we introduce the Banach space
      C n , m : = { ψ L ( Ω ε ) : ψ n , m < + } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equu_HTML.gif
      with the norm
      ψ n , m = sup y Ω ε ψ ( y ) ε 2 + i = 1 , α i < 0 n ε ρ i 2 ε y - p i ρ i 2 α i 1 + ε y - p i ρ i 4 + 2 α + 2 α i + i = 1 , α i 0 n + m ε ρ i 2 1 + ε y - p i ρ i 4 + 2 α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ29_HTML.gif
      (30)
      Now to get the invertibility of the linearized operator L, we only need to solve the following linear problems: given h of class C n , m C 0 , β Ω ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq55_HTML.gif with β ∈ (0,1), for m ≥ 1 and 0 ≤ nN, we find a function ϕ and scalars c ij , i = n + 1, ..., n + m, j = 1, 2, such that
      L ϕ = Δ ϕ + W ϕ = h + j = 1 2 i = n + 1 n + m c i j χ i Z i j , in Ω ε , ( 1 - t ) ν ϕ + t b ( ε y ) ϕ = 0 , on Ω ε , Ω ε χ i Z i j ϕ d y = 0 , i = n + 1 , , n + m , j = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ30_HTML.gif
      (31)
      and for m = 0 and 1 ≤ n ≤ N, we find a function ϕ such that
      L ϕ = Δ ϕ + W ϕ = h , in Ω ε , ( 1 - t ) ν ϕ + t b ( ε y ) ϕ = 0 , on Ω ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ31_HTML.gif
      (32)
      Proposition 3.1 (i) If m ≥ 1 and 0 ≤ nN, given a fixed number δ > 0, there exist positive numbers ε0 and C such that for any points p k l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq56_HTML.gif, l = n+ 1, ..., n + m, in Ω', with
      dist( p k l , Ω ) δ , p k l - p k i δ , f o r l i a n d i = 1 , , n + m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ32_HTML.gif
      (33)
      there is a unique solution ϕ L ( Ω ε ) , c n + 1 , , c n + m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq57_HTML.gif, of the Equation (31), which satisfies
      ϕ C log 1 ε h n , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ33_HTML.gif
      (34)
      for all ε < ε0 and t ∈ (0, 1]. Moreover, the map p'ϕ is C1 and
      D p ϕ C log 1 ε 2 h n , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ34_HTML.gif
      (35)

      where p : = 1 ε p k n + 1 , , 1 ε p k n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq58_HTML.gif.

      (ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε0 and C such that there is a unique solution ϕL ε ) of the Equation (32), which satisfies
      ϕ C log 1 ε h n , 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ35_HTML.gif
      (36)

      for all ε < ε0 and t ∈ (0, 1].

      These results can be established through some technical lemmas. First for the linear Equation (32) under the additional orthogonality conditions with respect to Zi 0, i = 1, ..., n + m, and Z ij , i = n + 1, ..., n + m, j = 1, 2, we prove the following priori estimates.

      Lemma 3.3 (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε0 and C such that for any points p k l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq56_HTML.gif, l = n+ 1, ..., n + m, in Ω', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
      Ω ε χ i Z i 0 ϕ d y = 0 , i = 1 , , n + m , Ω ε χ i Z i j ϕ d y = 0 , i = n + 1 , , n + m , j = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ36_HTML.gif
      (37)
      one has
      ϕ C h n , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ37_HTML.gif
      (38)

      for all ε < ε0.

      (ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε 0 and C such that for any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
      Ω ε χ i Z i 0 ϕ d y = 0 , i = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ38_HTML.gif
      (39)
      one has
      ϕ C h n , 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ39_HTML.gif
      (40)

      for all ε < ε0.

      Remark 3.4 The idea behind these estimates partly comes from observing the linear Equation (32) with h = 0 on bounded set B i , R : = y Ω ε : ε y - p k i ρ k i < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq59_HTML.gif for ε small. After a translation and a rotation so that Ω ε converges to the whole plan 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq2_HTML.gif, the Equation (32) approaches L i ϕ = 0 in 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq2_HTML.gif. As a result, the solution of the Equation (32) under the additional orthogonality conditions (37) should be zero.

      Proof. Case (i): First consider the "inner norm" ϕ l = sup i = 1 n + m B i , R ̄ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq60_HTML.gif and the "boundary norm" ϕ o = sup Ω ε ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq61_HTML.gif, we claim that there is a constant C > 0 such that if Lϕ = h in Ω ε , then
      ϕ C ϕ l + ϕ o + h n , m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ40_HTML.gif
      (41)

      We will establish it with the help of suitable barrier.

      Consider that the function g ( z ) = z 2 ( 1 + α ) - 1 z 2 ( 1 + α ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq62_HTML.gif is a radial solution in 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq2_HTML.gif of
      Δ g ( z ) + 8 ( 1 + α ) 2 z 2 α 1 + z 2 ( 1 + α ) 2 g ( z ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equv_HTML.gif
      we define a bounded comparison function
      Z ( y ) = i = 1 n + m g a ε y - p k i ρ k i , y Ω ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equw_HTML.gif
      with a > 0. Set R a = 1 a 3 1 2 ( 1 + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq63_HTML.gif. While ε y - p k i ρ k i R a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq64_HTML.gif for all i = 1, ..., n + m,
      - Δ Z ( y ) = i = 1 n + m ε ρ k i 2 a 2 8 ( 1 + α ) 2 a ε y - p k i ρ k i 2 α 1 + a ε y - p k i ρ k i 2 ( 1 + α ) 2 g a ε y - p k i ρ k i i = 1 n + m ε ρ k i 2 a 2 4 ( 1 + α ) 2 a ε y - p k i ρ k i 2 α 1 + a ε y - p k i ρ k i 2 ( 1 + α ) 2 > i = 1 n + m ε ρ k i 2 a - 2 ( 1 + α ) ( 1 + α ) 2 ε y - p k i ρ k i 2 α + 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ41_HTML.gif
      (42)
      Moreover, according to (21) and (22), on the same region,
      W ( y ) Z ( y ) C i = 1 n + m ε ρ k i 2 1 ε y - p k i ρ k i 2 α k i + 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ42_HTML.gif
      (43)

      So if a is small enough to satisfy (1 + α)2a-2(1+α) > C + 1, R a is sufficiently large. As a result, by (42) and (43), for any R ≥ R a , we have Z(y) > 0 and L(Z) < 0 in Ω R , ε c : = Ω ε \ i = 1 n + m B i , R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq65_HTML.gif.

      Let M be a large number such that for all i = 1, ..., n + m, Ω B ( p k i , M ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq66_HTML.gif. Consider now the solution of the problem
      - Δ ψ i = ε ρ k i 2 4 ε y - p k i ρ k i 2 α + 4 + 4 ε 2 , R < ε y - p k i ρ k i < M ρ k i , ψ i ( y ) = 0 , for ε y - p k i ρ k i = R and ε y - p k i ρ k i = M ρ k i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equx_HTML.gif
      A direct computation shows
      ψ i ( y ) = φ i ( r i ) - φ i M ρ k i log r i R log M R ρ k i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equy_HTML.gif
      where
      r i = ε y - p k i ρ k i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equz_HTML.gif
      and
      φ i ( t ) = 1 ( 1 + α ) 2 1 R 2 ( 1 + α ) - 1 t 2 ( 1 + α ) + ρ k i 2 ( R 2 - t 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equaa_HTML.gif

      For the sake of the convenience, we choose R larger if necessary. Then it easily see that these functions ψ i , i = 1, ..., n + m, have a uniform bound independent of ε.

      Now we can construct the needed barrier:
      ϕ ̃ ( y ) = 2 ( ϕ l + ϕ o ) Z ( y ) + h n , m i = 1 n + m ψ i ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equab_HTML.gif

      It is easy to check that L ϕ ̃ < h = L ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq67_HTML.gif in Ω R , ε c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq68_HTML.gif, and ϕ ̃ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq69_HTML.gif on Ω R , ε c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq70_HTML.gif. Since Z(y) > 0 and LZ(y) < 0 in Ω R , ε c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq68_HTML.gif, from the maximum principle (see [[37], Theorem 10, Chap. 2 ]), it follows that ϕ ̃ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq69_HTML.gif in Ω R , ε c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq68_HTML.gif. Similarly, - ϕ ̃ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq71_HTML.gif in Ω R , ε c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq68_HTML.gif, which derives the estimate (41).

      We prove the priori estimate (38) by contradiction. Assume that there exist a sequence ε k → 0, points p k l k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq72_HTML.gif, l = n + 1, ..., n + m, in Ω' which satisfy relation (33), functions h k with ║h k n,m → 0, solutions ϕ k with ║ϕ k = 1, such that
      L ϕ k = Δ ϕ k + W ϕ k = h k , in Ω ε , ( 1 - t ) ν ϕ k + t b ( ε y ) ϕ k = 0 , on Ω ε , Ω ε χ i Z i 0 ϕ k d y = 0 , i = 1 , , n + m , Ω ε χ i Z i 0 ϕ k d y = 0 , i = n + 1 , , n + m , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equac_HTML.gif
      Then from the estimate (41), ║ϕ k l κ or ║ϕ k o κ for some κ > 0. Briefly set ε:= ε k , p k i : p k i k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq73_HTML.gif. If ║ϕ k l κ, with no loss of generality, we assume that sup B i , R ϕ k k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq74_HTML.gif for some i Then if we set ϕ ^ k ( z ) = ϕ k ρ k i ε z + p k i ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq75_HTML.gif and ĥ k ( z ) = h k ρ k i ε z + p k i ε , ϕ ^ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq76_HTML.gif satisfies
      Δ ϕ ^ k + 8 μ k i 2 ( 1 + α k i ) 2 z 2 α k i [ μ k i 2 + z 2 ( 1 + α k i ) ] 2 [ 1 + O ( ρ k i z ) + O ( ε 2 ) ] ϕ ^ k = ρ k i ε 2 ĥ k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equad_HTML.gif
      for zB R (0). Obviously, for any q 1 , - 1 α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq77_HTML.gif we easily get ρ k i ε 2 ĥ k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq78_HTML.gif in L q (B R (0)). Since 8 μ k i 2 ( 1 + α k i ) 2 z 2 α k i [ μ k i 2 + z 2 ( 1 + α k i ) ] 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq79_HTML.gif is bounded in L q (B R (0)) and ϕ ^ k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq80_HTML.gif, elliptic regularity theory readily implies that ϕ ^ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq81_HTML.gif converges uniformly over compact subsets near the origin to a bounded nontrivial solution ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif of the equation
      L i ϕ ^ = Δ ϕ ^ + 8 μ k i 2 ( 1 + α k i ) 2 z 2 α k i [ μ k i 2 + z 2 ( 1 + α k i ) ] 2 ϕ ^ = 0 , in 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equae_HTML.gif

      From Lemma 3.2, this equation implies that ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif is proportional to zi 0for i = 1, ..., n, or a linear combination of zi 0and z ij for i = n + 1, ..., n + m, j = 1, 2. However, our assumed orthogonality conditions (37) on ϕ k pass to limit and yield the corresponding conditions (37) on ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif, which means ϕ ^ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq83_HTML.gif. Hence, it is absurd because ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif is nontrivial.

      If ║ϕ k o ≥ κ and ║ϕ k l → 0, there exists a point q Ω and a number R1 > 0 such that sup Ω ε B R 1 ( q ) ϕ k ( y ) k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq84_HTML.gif with q = 1 ε q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq85_HTML.gif. Consider ϕ ^ k ( y ) = ϕ k ( y - q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq86_HTML.gif and let us translate and rotate Ω ε so that q' = 0 and Ω ε approaches the upper half-plan + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq87_HTML.gif. Since ε q - p k i ρ k i > δ ρ k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq88_HTML.giffor all i = 1, ..., n + m, ϕ ^ k ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq89_HTML.gif satisfies
      Δ ϕ ^ k ( y ) + O ( ε 4 ) ϕ ^ k ( y ) = h k ( y ) , in Ω ε \ i = 1 n + m B i , δ ρ k i - 1 , ( 1 - t ) ν ϕ ^ k + t b ( ε y ) ϕ ^ k = 0 , on Ω ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equaf_HTML.gif
      with ( 1 - t ) Ω ε ϕ ^ k 2 + t Ω ε b ( ε y ) ϕ ^ k 2 < C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq90_HTML.gif. Moreover, we easily get h k (y) → 0 in Ω ε \ i = 1 n + m B i , δ ρ k i - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq91_HTML.gif. While t = 1, it is obvious to see that ϕ ^ k ( y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq92_HTML.gif on Ω ε . So it is absurd because of sup Ω ε B R 1 ( q ) ϕ k ( y ) k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq84_HTML.gif. On the other hand, for any t ∈ (0,1), elliptic regularity theory with the Robin boundary condition (see [30, 34, 38] and the references therein) implies that ϕ ^ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq81_HTML.gif converges uniformly on compact subsets near the origin to a bounded nontrivial solution ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif of the equation
      Δ ϕ ^ = 0 , in + 2 , ( 1 - t ) ν ϕ ^ + t b ( 0 ) ϕ ^ = 0 , on + 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equag_HTML.gif

      with ( 1 - t ) + 2 ϕ ^ 2 + t b ( 0 ) + 2 ϕ ^ 2 < C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq93_HTML.gif. It follows that its bounded solution ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif is zero. Hence, it is also absurd because ϕ ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq82_HTML.gif is nontrivial, which derives the priori estimate (38) of the case (i). Since the proof of the case (ii) is similar to that of the case (i), we omit it.□

      We will give next the priori estimate for the solution of the Equation (32) that satisfies orthogonality conditions with respect to Z ij , i = n + 1, ..., n + m, j = 1, 2, only.

      Lemma 3.5. (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε0 and C such that for any points p k l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq56_HTML.gif, l = n + 1, ..., n + m, in Ω', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
      Ω ε χ i Z i j ϕ d y = 0 , i = n + 1 , , n + m , j = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ43_HTML.gif
      (44)
      one has
      ϕ C log 1 ε h n , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ33_HTML.gif
      (45)

      for all ε < ε0.

      (ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε0 and C such that for any solution ϕ of the Equation (32) with t ∈ (0, 1], one has
      ϕ C log 1 ε h n , 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equ35_HTML.gif
      (46)

      for all ε < ε0.

      Proof. Case (i): Let ϕ satisfy the Equation (32) under the orthogonality conditions (44). We will modify ϕ to satisfy the orthogonality conditions (37). To realize this point, we consider some related modifications with compact support of the functions Zi 0, i = 1, ..., n + m.

      Let R > R0 + 1 be large and fixed, and let i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq94_HTML.gif be the solution of the equation
      Δ i 0 + 8 μ k i 2 ( 1 + α k i ) 2 z 2 α k i ( μ k i 2 + z 2 ( 1 + α k i ) ) 2 i 0 = 0 , for R < z < δ 3 ρ k i , i 0 ( z ) = z i 0 ( R ) on z = R , i 0 ( z ) = 0 on z = δ 3 ρ k i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equah_HTML.gif
      A simple computation shows that this solution is explicitly given by
      i 0 ( z ) = z i 0 ( r ) 1 - R r d s s z i 0 2 ( s ) R δ 3 ρ k i d s s z i 0 2 ( s ) , r = z . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equai_HTML.gif
      Set
      i 0 ( y ) = i 0 ε y - p k i ρ k i , η 1 i ( y ) = η 1 ε y - p k i ρ k i , η 2 i ( y ) = η 2 4 ρ k i ε y - p k i ρ k i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equaj_HTML.gif
      where η1(r) and η2(r) are smooth cut-off functions with the properties: η1(r) = 1 for r < R, η1(r) = 0 for r > R + 1 , η 1 ( r ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq95_HTML.gif; η2(r) = 1 for r < δ, η2(r) = 0 for r > 4 δ 3 , η 2 ( r ) C η 2 ( r ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq96_HTML.gif. We define a test function
      Z ̃ i 0 = η 1 i Z i 0 + ( 1 - η 1 i ) η 2 i i 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_Equak_HTML.gif

      Obviously, Z ̃ i 0 ( y ) = Z i 0 ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq97_HTML.gif if ε y - p k i ρ k i < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq98_HTML.gif, and Z ̃ i 0 ( y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-33/MediaObjects/13661_2011_Article_158_IEq99_HTML.gif if ε y - p k i