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

Yibin Chang* and Haitao Yang

Author Affiliations

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

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Boundary Value Problems 2012, 2012:33  doi:10.1186/1687-2770-2012-33


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/33


Received:17 October 2011
Accepted:23 March 2012
Published:23 March 2012

© 2012 Chang and Yang; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1">View MathML</a>

for ε small, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2">View MathML</a>, Ω is a bounded, smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a>, Γ := {p1, ..., pN} ⊂ Ω is the set of singular sources, δp 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M1">View MathML</a>

(1)

for ε small, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M2">View MathML</a>, Ω is a bounded, smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a>, Γ:= {p1, ..., pN} ⊂ is the set of singular sources, δp 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 [2-4], Chern-Simons vortex theory [5-11] and several other fields of applied mathematics [12-16]. 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 αi ∈ (-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 Gt,ε denotes the Green's function of -∆ with mixed boundary conditions on Ω, namely for any y ∈ Ω,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M4">View MathML</a>

(2)

and let Ht,ε(x, y) = Gt,ε(x, y) + log |x - y| be its regular part. Set G1 = G1,ε and 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M5">View MathML</a>, the regular part of v, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M6">View MathML</a>

Thus, we consider the more general model problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M7">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M8">View MathML</a> is a smooth function such that f(pi) > 0 for any i = 1, ..., N. Set Ω' = {x ∈ Ω: f(x) > 0}, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M9">View MathML</a> and Δm = {p = (p1, ..., pm) ∈ m: pi = pj for some i ≠ j}.

It is known that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10">View MathML</a>, or αi = 0 for any i = 1, ..., N, if uε is a family of solutions of the Equation (3)|t = 1 with inff > 0, which is not uniformly bounded from above for ε small, then uε blows up at different points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M11">View MathML</a> with n + m ≥ 1, 0 ≤ n N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M13">View MathML</a>, and satisfies the concentration property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M14">View MathML</a>

(4)

in the sense of measures in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M15">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16">View MathML</a> is a critical point of the function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M17">View MathML</a>

(5)

(see [7,17-23]). 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 φn,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 αi = 0 for any i = 1,2, ..., N, they prove that any non-degenerate critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16">View MathML</a> of the function φn,m with n = 0 generates a family of the solutions uε which blow-up at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M18">View MathML</a>, 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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10">View MathML</a>. However, the asymptotic analysis method depends on the non-degenerate assumption of critical point of the function φn,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,m to get the existence of multiple blowing-up and concentrating solutions for the Equation (3)|t = 1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M19">View MathML</a>, namely αi = 0 for any i = 1,2, ..., N. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M10">View MathML</a>, a similar result for the Equation (3)|t = 1 under C0-stable assumption of critical point of φn,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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M20">View MathML</a> with some αi ∈ (-1, 0) and Robin boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M21">View MathML</a> 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 αi ∈ (-1, 0). Obviously, the linearized operator L easily produces the singularities at some singular sources with αi ∈ (-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, ..., αN}. On the other hand, the presence of the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M22">View MathML</a> in the Equation (3)|0<t<1 brings 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 φn,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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M23">View MathML</a>such that n + m ≥ 1. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M24">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M25">View MathML</a>is a C0-stable critical point for φn,m in (Ω' \ Γ)m \ Δm with m ≥ 1 (see Definition 4.1). Then there exists a family of solutions uε for the Equation (3) with the concentration property (4), which blow up at n-different points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M26">View MathML</a>in Γ, and m-points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16">View MathML</a>in (Ω' \ Γ)m \ Δm with φn,m(p*) = φn,m(p). Moreover, uε remains uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M27">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M28">View MathML</a>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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M29">View MathML</a>

(6)

where T Ω is a relatively closed subset and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M30">View MathML</a>. 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 uε for the Equation (6) with the concentration property (4), which blow up at n-different points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M26">View MathML</a>in Γ, and m-points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M16">View MathML</a>in (Ω' \ Γ)m \ Δm with φn,m(p*) = φn,m(p). Moreover, uε remains uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M27">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M28">View MathML</a>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):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M31">View MathML</a>

(7)

given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M32">View MathML</a>

(8)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M33">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M34">View MathML</a>, c = 0 if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M35">View MathML</a> (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 φn,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 μi, i = 1, ..., N + m, be positive numbers and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M36">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M37">View MathML</a>

Obviously, Qi(x) = S(x) for any i = N + 1, ..., N + m. Then the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M38">View MathML</a>

(9)

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M39">View MathML</a>

(10)

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

We hope to take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M40">View MathML</a> as an initial approximate solution of the problem (3). So we modify it to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M41">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M42">View MathML</a>(x) is the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M43">View MathML</a>

(12)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M44">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M45">View MathML</a>

(13)

Lemma 2.1 For t ∈ (0, 1] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M46">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M47">View MathML</a>

(14)

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49">View MathML</a>for ε small.

Proof. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M50">View MathML</a>. Then zt(x) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M51">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M52">View MathML</a>

For any t ∈ (0, 1], it is easy to check <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M275">View MathML</a>. If t = 1, from the maximum principle and smooth function b(x) > 0, it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M53">View MathML</a>

If 0 < t < 1, from the maximum principle with the Robin boundary condition (see [[33], Lemma 2.6]), it also follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M54">View MathML</a>

Thus using the interior estimate of derivative of harmonic function (see [[34], Theorem 2.10]), there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M55">View MathML</a>

for any compact subset K of , any t ∈ (0, 1] and any multi-index α with |α| ≤ 2, which derives (14) uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49">View MathML</a> for ε small. □

From this lemma we can construct the approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M56">View MathML</a>, which satisfies the mixed boundary conditions. On the other hand, we hope that the error <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M57">View MathML</a> is smaller near every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M58">View MathML</a>. In fact, we can realize this point by further choosing positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M59">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M60">View MathML</a>

(15)

Consider the scaling of the solution of the Equation (3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M61">View MathML</a>

then v(y) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M62">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M63">View MathML</a>. We also set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M64">View MathML</a> and define the new approximation in expanded variables as V(y) = U(εy) + 4 log ε. Furthermore, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M65">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M66">View MathML</a>

(18)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M67">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M68">View MathML</a>

Then given a small number δ > 0, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69">View MathML</a> for all i = 1, ..., n + m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M70">View MathML</a>

(19)

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71">View MathML</a> for some i,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M72">View MathML</a>

(20)

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69">View MathML</a> for all i = 1, ..., n + m, obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M73">View MathML</a>

(21)

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71">View MathML</a> for some i,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M74">View MathML</a>

Now from (14), (15) and (17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M75">View MathML</a>

(22)

In summary, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M76">View MathML</a>

(23)

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M69">View MathML</a> for all i = 1, ..., n + m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M77">View MathML</a>

(24)

while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M71">View MathML</a> for some i,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M78">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M79">View MathML</a>

(26)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M80">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M81">View MathML</a>

A basic fact to get the needed invertibility is that the linearized operator L formally approaches to the operator Li under suitable dilations and translations, which have some well-known properties that any bounded solution of Ltϕ = 0 is

- a linear combination of zi0 and zij for i = n + 1, ..., n + m, j = 1, 2 (see [24,35]);

- proportional to zi0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M82">View MathML</a> and i = 1, 2, ..., n (see [20,28,36]).

Remark 3.1 These properties of the operator Li have been discussed in the above papers only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M82">View MathML</a> for i = 1, ..., n, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M83">View MathML</a> for i = n + 1, ..., n + m. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M84">View MathML</a> for some i = 1, ..., n, the operator Li has also the corresponding properties.

Lemma 3.2 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M85">View MathML</a>, any bounded solution ϕ of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M86">View MathML</a>

(28)

is proportional to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M87">View MathML</a>.

Proof. If we express the bounded solution ϕ of the Equation (28) in Fourier expansion form as follow

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M88">View MathML</a>

un(r) is a bounded nontrivial solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M89">View MathML</a>

(29)

Since any solution of -∆u = eu in is given by the Liouville formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M91">View MathML</a>

for any meromorphic function F defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M92">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M93">View MathML</a>

with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M95">View MathML</a>, is the solution of -∆u = |z|2αeu in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M96">View MathML</a>. Moreover, its derivative with respect to a at a = 0

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M97">View MathML</a>

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

For |n| ≥ 1, since {ϕn(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 ϕn(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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M98">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M99">View MathML</a>

Then there exists a constant C > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M100">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M101">View MathML</a>

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 r R0, and χ(r) = 0 if r R0 + 1. Additionally, set α0 = min{0, α1, ..., αN}. For any α ∈ (-1, α0), we introduce the Banach space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M102">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M103">View MathML</a>

(30)

Now to get the invertibility of the linearized operator L, we only need to solve the following linear problems: given h of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M276">View MathML</a> with β ∈ (0,1), for m ≥ 1 and 0 ≤ n N, we find a function ϕ and scalars cij, i = n + 1, ..., n + m, j = 1, 2, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M104">View MathML</a>

(31)

and for m = 0 and 1 ≤ n ≤ N, we find a function ϕ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M105">View MathML</a>

(32)

Proposition 3.1 (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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106">View MathML</a>, l = n + 1, ..., n + m, in ', with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M107">View MathML</a>

(33)

there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M108">View MathML</a>, of the Equation (31), which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M109">View MathML</a>

(34)

for all ε < ε0 and t ∈ (0, 1]. Moreover, the map p' ϕ is C1 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M110">View MathML</a>

(35)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M111">View MathML</a>.

(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε0 and C such that there is a unique solution ϕ L(Ωε) of the Equation (32), which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M112">View MathML</a>

(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 Zi0, i = 1, ..., n + m, and Zij, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106">View MathML</a>, l = n + 1, ..., n + m, in ', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M113">View MathML</a>

(37)

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M114">View MathML</a>

(38)

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] under the orthogonality conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M115">View MathML</a>

(39)

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M116">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M117">View MathML</a> for ε small. After a translation and a rotation so that Ωε converges to the whole plan <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a>, the Equation (32) approaches Liϕ = 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a>. 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" <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M118">View MathML</a> and the "boundary norm" <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M119">View MathML</a>, we claim that there is a constant C > 0 such that if Lϕ = h in Ωε, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M120">View MathML</a>

(41)

We will establish it with the help of suitable barrier.

Consider that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M121">View MathML</a> is a radial solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M3">View MathML</a> of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M122">View MathML</a>

we define a bounded comparison function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M123">View MathML</a>

with a > 0. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M124">View MathML</a>. While <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M125">View MathML</a> for all i = 1, ..., n + m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M126">View MathML</a>

(42)

Moreover, according to (21) and (22), on the same region,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M127">View MathML</a>

(43)

So if a is small enough to satisfy (1 + α)2a-2(1+α) > C + 1, Ra is sufficiently large. As a result, by (42) and (43), for any R ≥ Ra, we have Z(y) > 0 and L(Z) < 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M128">View MathML</a>.

Let M be a large number such that for all i = 1, ..., n + m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M129">View MathML</a>. Consider now the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M130">View MathML</a>

A direct computation shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M131">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M132">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M133">View MathML</a>

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

Now we can construct the needed barrier:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M134">View MathML</a>

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M135">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M137">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M138">View MathML</a>. Since Z(y) > 0 and LZ(y) < 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136">View MathML</a>, from the maximum principle (see [[37], Theorem 10, Chap. 2 ]), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M137">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M139">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M136">View MathML</a>, which derives the estimate (41).

We prove the priori estimate (38) by contradiction. Assume that there exist a sequence εk → 0, points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M140">View MathML</a>, l = n + 1, ..., n + m, in Ω' which satisfy relation (33), functions hk with ║hkn,m → 0, solutions ϕk with ║ϕk= 1, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M141">View MathML</a>

Then from the estimate (41), ║ϕkl κ or ║ϕko κ for some κ > 0. Briefly set ε:= εk, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M142">View MathML</a>. If ║ϕkl κ, with no loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M143">View MathML</a> for some i Then if we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M145">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M146">View MathML</a>

for z BR(0). Obviously, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M147">View MathML</a> we easily get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M148">View MathML</a> in Lq(BR(0)). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M149">View MathML</a> is bounded in Lq(BR(0)) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M150">View MathML</a>, elliptic regularity theory readily implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M151">View MathML</a> converges uniformly over compact subsets near the origin to a bounded nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M153">View MathML</a>

From Lemma 3.2, this equation implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> is proportional to zi0 for i = 1, ..., n, or a linear combination of zi0 and zij for i = n + 1, ..., n + m, j = 1, 2. However, our assumed orthogonality conditions (37) on ϕk pass to limit and yield the corresponding conditions (37) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M154">View MathML</a>. Hence, it is absurd because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> is nontrivial.

If ║ϕko ≥ κ and ║ϕkl → 0, there exists a point q Ω and a number R1 > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M155">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M156">View MathML</a>. Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M157">View MathML</a> and let us translate and rotate Ωε so that q' = 0 and Ωε approaches the upper half-plan <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M158">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M159">View MathML</a>for all i = 1, ..., n + m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M160">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M161">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M162">View MathML</a>. Moreover, we easily get hk(y) → 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M163">View MathML</a>. While t = 1, it is obvious to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M164">View MathML</a> on ε. So it is absurd because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M155">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M151">View MathML</a> converges uniformly on compact subsets near the origin to a bounded nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M165">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M166">View MathML</a>. It follows that its bounded solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> is zero. Hence, it is also absurd because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M152">View MathML</a> 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 Zij, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106">View MathML</a>, l = n + 1, ..., n + m, in ', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M167">View MathML</a>

(44)

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M109">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M112">View MathML</a>

(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 Zi0, i = 1, ..., n + m.

Let R > R0 + 1 be large and fixed, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M168">View MathML</a> be the solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M169">View MathML</a>

A simple computation shows that this solution is explicitly given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M170">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M171">View MathML</a>

where η1(r) and η2(r) are smooth cut-off functions with the properties: η1(r) = 1 for r < R, η1(r) = 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M172">View MathML</a>; η2(r) = 1 for r < δ, η2(r) = 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M173">View MathML</a>. We define a test function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M174">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M175">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M176">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M177">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M178">View MathML</a>, in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M177">View MathML</a> near ∂Ωɛ.

Now we modify ϕ to satisfy the orthogonality conditions with respect to Zi0, i = 1, ..., n+ m, and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M179">View MathML</a>

where the numbers di are chosen such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M180">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M181">View MathML</a>

(47)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M182">View MathML</a> satisfies all the orthogonality conditions in (37). From Lemma 3.3 (i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M183">View MathML</a>

(48)

In order to get the estimate (45) of ϕ, we need to give the sizes of di and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M184">View MathML</a> for any t ∈ (0, 1]. Multiplying the first equation of (47) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M185">View MathML</a>, integrating by parts and using the mixed boundary conditions of (47), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M186">View MathML</a>

(49)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M274">View MathML</a>. A simple computation shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M187">View MathML</a>, which in combination with (48) and (49) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M188">View MathML</a>

(50)

From some similar computations (see [[26], Lemma 3.2] and [[28], Lemma 3.3]), there exists a constant C > 0 independent of ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M189">View MathML</a>

(51)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M190">View MathML</a>

(52)

which combined with (50) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M191">View MathML</a>

(53)

Furthermore, from (48), (51), (53), and the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M192">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M193">View MathML</a>

(54)

Similarly, the proof of the case (ii) can also be done, we omit it. □

Proof of Proposition 3.1. Case (i): From Lemma 3.5 (i), and the Fredholm's alternative theory with Robin boundary condition instead of Dirichlet boundary condition if necessary (see [34,38] and the references therein), the proof can be similarly given through those in [[26], pp. 61-63].

Case (ii): Since the priori estimate (36) of the solution of the Equation (32) has been established in Lemma 3.5 (ii), we can use the Fredholm's alternative and obtain the unique solution of the Equation (32). □

Let us now introduce the auxiliary nonlinear problems: for m ≥ 1 and 0 ≤ n < N, we find the function ϕ and scalars cij, i = n + 1, ..., n + m, j = 1, 2, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M194">View MathML</a>

(55)

and for m = 0 and 1 ≤ n ≤ N, we find the solution ϕ of the nonlinear Equation (26).

The following result can be proved using standard arguments as in [26,28].

Proposition 3.2 (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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M106">View MathML</a>, l = n + 1, ..., n + m, in ' satisfying the relation (33), there is a unique solution for the Equation (55) which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M195">View MathML</a>

(56)

for all ε < ε0 and t ∈ (0, 1]. Moreover, the map p' ϕ is C1 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M196">View MathML</a>

(57)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M197">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M198">View MathML</a>.

(ii) If m = 0 and 1≤n≤N, there exist positive numbers ε0 and C such that there is a unique solution for the Equation (26) which also satisfies the estimate (56) for all ε < ε0 and t ∈ (0, 1].

Now we only need to find a solution to the Equation (26) with m ≥ 1 and 0 ≤ n ≤ N, and hence to the Equation (55) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M199">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M200">View MathML</a>

(58)

Let us introduce the energy functional of the Equation (3), namely for t = 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M201">View MathML</a>

and for t ∈ (0,1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M202">View MathML</a>

Furthermore, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M203">View MathML</a>

(59)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M205">View MathML</a> with the solution ϕ of the Equation (55).

The finite dimensional variational reduction is meaningful in view of the following property.

Proposition 3.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M204">View MathML</a> satisfying the relation (33) is a critical point of Fε,t with t ∈ (0, 1], then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M206">View MathML</a>is a critical point of Jε,t, namely a solution of the Equation (3). Besides, on any compact subsets S of (Ω' \ Γ)m \ Δm the following expansion holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M207">View MathML</a>

(60)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M208">View MathML</a>

(61)

for ε small.

Proof. Step 1: Let us define for t = 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M209">View MathML</a>

and for t ∈ (0,1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M210">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M211">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M212">View MathML</a>. Moreover, for k = n + 1, ..., n + m, l = 1, 2, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M213">View MathML</a>

(62)

Since ϕ(p') is a solution of the Equation (55), v = V(p') + ϕ(p') satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M214">View MathML</a>

(63)

By (62), (63), DpFε,t(p) = 0 implies for t ∈ (0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M215">View MathML</a>

From the definition of V, it can be directly checked <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M216">View MathML</a>, where o(1) is in the sense of the L-norm for ε small. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M217">View MathML</a>. Hence it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M218">View MathML</a>

which is a strictly diagonal dominant system. This implies that cij = 0, ∀ i = n + 1, ..., n + m, j = 1, 2. By (63), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M206">View MathML</a> is a critical point of Jε,t, that is, a solution of the Equation (3).

Step 2: Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M219">View MathML</a>. Using DIε,t(V + ϕ)[ϕ] = 0, a Taylor expansion and an integration by parts, it follows that for t ∈ (0, 1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M220">View MathML</a>

(64)

and similarly, for t = 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M221">View MathML</a>

(65)

Note that ║ϕ= O(ρ|log ε|), ║N (ϕ)n,m = O(ρ2| log ε|2), ║R║n,m = O(ρ), and ║W║n,m = O(1). Then from (64), (65), it is easy to deduce for t ∈ (0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M222">View MathML</a>

(66)

Hence, from (66), the expansion (60) satisfies the property (61).□

Finally, we need to write the precisely asymptotical expansion of Jε,t(U). To realize it, we first establish the following result:

Lemma 3.6 Assume that points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M223">View MathML</a>, satisfy the relation (33), then fort ∈ (0,1) and i = 1, ..., n + m, there hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M224">View MathML</a>

(67)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M225">View MathML</a>

(68)

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49">View MathML</a> for ε small.

Proof. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M226">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M227">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M228">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M229">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M230">View MathML</a> has the gradient estimate on Ω (see [[26], p. 76])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M231">View MathML</a>

we can easily get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M232">View MathML</a>. Using the same technique with the proof of Lemma 2.1, we can also get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M233">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M48">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M49">View MathML</a> for ε small, which means (67). From the definition of the regular part of Green function, we can also derive (68). □

Proposition 3.4 The following asymptotical expansions hold for t = 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M234">View MathML</a>

(69)

and for t ∈ (0,1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M235">View MathML</a>

(70)

where φn,m(p) is defined by (5), and for ε small, Θε is a bounded, smooth function of p = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M236">View MathML</a>, uniformly on points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M223">View MathML</a> in ' satisfying the relation (33).

Proof. According to [[26,28], Lemma 6.1], it only remains to discuss the asymptotical expansion of the energy Jε,t(U) with respect to t ∈ (0,1). By (11), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M237">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M238">View MathML</a>

which together with the Equation (13) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M239">View MathML</a>

Furthermore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M240">View MathML</a>

(71)

From (9), (13), and (14), it implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M241">View MathML</a>

Then if i ≠ j for any i and j,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M242">View MathML</a>

(72)

and if i = j for any i,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M243">View MathML</a>

(73)

On the other hand, from (21) and (22), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M244">View MathML</a>

As a result, it derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M245">View MathML</a>

(74)

Now using the choice for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M246">View MathML</a>'s by (15), together with (71)-(74), it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M247">View MathML</a>

which derives the asymptotical expansion (70) by (5), (67), and (68). □

4 Proofs of theorems

In this section, we carry out the proofs of Theorems 1.1 and 1.2 basing on the finite dimensional reduction. Now we introduce the definition of C0-stable critical point of the function φn,m just like in [28,36,39].

Definition 4.1. We say that p is a C0-stable critical point of φn,m in (Ω' \ Γ)m \ Δm, which says that if for any sequence of the functions ψj such that ψj φn,m uniformly on the compact subsets of (Ω' \ Γ)m \ Δm, ψj has a critical point ξj such that ψj(ξj) → + φn,m.

In particular, if p is a strict local maximum or minimum point of φn,m, p is a C0-stable critical point of φn,m.

Proof of Theorem 1.1. Case (i): m ≥ 1 and 0 ≤ n ≤ N. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M248">View MathML</a>

where ϕ is the unique solution of the problem (55), which is established in Proposition 3.2. From Proposition 3.3, v(y) is a solution of the Equation (16), namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M249">View MathML</a> is a solution of the Equation (3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M250">View MathML</a> satisfying the relation (33) is a critical point of the function Fε,t(p) with t ∈ (0, 1]. This implies that we only need to find a critical point pε of the following function in (Ω' \ Γ)m \ Δm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M251">View MathML</a>

(75)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M252">View MathML</a>

From Propositions 3.3 and 3.4, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M253">View MathML</a>

(76)

where θε,t(p) = O(ρ2| log ε|) for any t ∈ (0, 1], and Θε(p) is uniformly bounded on any compact subset S of (Ω' \ Γ)m\ Δm for ε small. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M254">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M255">View MathML</a> for any t ∈ (0,1), uniformly on for ε small. By Definition 4.1, there exists a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M250">View MathML</a> of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M257">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M258">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M259">View MathML</a> for any t ∈ (0,1). Moreover, up to a subsequence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M260">View MathML</a> such that pε p for ε small, and φn,m(p*) = φn,m(p). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M261">View MathML</a> is a family of solutions of the Equation (3). As a consequence, from the related properties of U(pε) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M262">View MathML</a>, we easily know that for any λ > 0, uε is uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M263">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M264">View MathML</a> for ε small.

Finally, we show that uε satisfies the concentration property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M265">View MathML</a>

(77)

for ε small. In fact, using the inequality |es - 1| e|s||s| for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M266">View MathML</a> and the estimate (56), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M267">View MathML</a>

Then from the asymptotical expression (21) and (22) of W(pε)(y), we can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M268">View MathML</a>

which implies (77).

Case (ii): m = 0 and 1 ≤ n ≤ N. From Proposition 3.2 (ii), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M269">View MathML</a> is a family of solutions of the Equation (3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M270">View MathML</a>. Then we can get the needed multiple blowing-up and concentrating properties of uε through the similar proof of Case (i). □

In order to give the proof of Theorem 1.2, we need a version of the maximum principle under Dirichlet-Robin boundary conditions, which is the extension of the corresponding one with respect to Dirichlet or Robin boundary condition only.

Lemma 4.2 Assume that T is a relatively closed subset, b > 0 is a smooth function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M271">View MathML</a>is a smooth function. If u is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M272">View MathML</a>

where λ > 0, there exists a constant C(b) > 0 only depending on b(x) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/33/mathml/M273">View MathML</a>

Proof. The proof is similar to that of Lemma 2.6 in [33]. □

Proof of Theorem 1.2. Using the maximum principle with Dirichlet-Robin boundary conditions instead of Robin boundary condition if necessary (see Lemma 4.2), the proof can be similarly given out through that of Theorem 1.1. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

This study was supported by N. N. S. F. C. (11171214).

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