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Positive solutions for a class of sublinear elliptic systems

Ruyun Ma*, Ruipeng Chen and Yanqiong Lu

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, PR China

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Boundary Value Problems 2014, 2014:28  doi:10.1186/1687-2770-2014-28


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/28


Received:17 October 2013
Accepted:9 January 2014
Published:30 January 2014

© 2014 Ma et al.; 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 paper, we are concerned with the existence of positive solutions of the semilinear elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2">View MathML</a> is a parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3">View MathML</a> is a continuous real function for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>. Under some appropriate assumptions, we show that the above system has at least one positive solution in certain interval of λ. The proofs of our main results are based upon bifurcation theory.

MSC: 34B15, 34B18.

Keywords:
sublinear elliptic systems; positive solutions; eigenvalues; bifurcation theory

1 Introduction

Let Ω be a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M6">View MathML</a>). In this paper, we study the existence of positive solutions of the semilinear elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M7">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2">View MathML</a> is a bifurcation parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3">View MathML</a> is a continuous real function for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>.

A solution of (1.1) is a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M11">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12">View MathML</a> is called a positive solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M13">View MathML</a> in Ω for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>. In the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M15">View MathML</a> also denotes the elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M16">View MathML</a>.

The following definitions will be used in the statement of our main results.

Definition 1.1[1]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>) be smooth real functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M19">View MathML</a>. Define the Jacobian of the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M20">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M21">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M22">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M23">View MathML</a>) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M24">View MathML</a>, then the semilinear elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M25">View MathML</a>

(1.2)

is said to be cooperative. Similarly, H is called a cooperative matrix.

Definition 1.2[2]

An <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M26">View MathML</a> matrix A is reducible if for some permutation matrix Q,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M27">View MathML</a>

where B and D are square matrices, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M28">View MathML</a> is the transpose of Q. Otherwise, A is irreducible.

In the past few years, the existence of positive solutions to sublinear semilinear elliptic systems with two equations have been extensively studied, see for example, [3-6] and the references therein. The sublinear condition plays an important role. Very recently, Wu and Cui [1] considered the existence, uniqueness and stability of positive solutions to the sublinear elliptic system (1.1). By using bifurcation theory and the continuation method, they proved the following.

Theorem AAssume that

(H1) Each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>) is a smooth real function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M31">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M32">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M34">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M35">View MathML</a>.

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M36">View MathML</a>.

(i) If at least one of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M37">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>) is positive and matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M39">View MathML</a>is irreducible, then (1.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M40">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M2">View MathML</a>;

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M43">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>and matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M45">View MathML</a>is irreducible, then for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M46">View MathML</a>, (1.1) has no positive solution when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M47">View MathML</a>, and (1.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M40">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M49">View MathML</a>.

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M50">View MathML</a> (in the first case, we assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M51">View MathML</a>) is a smooth curve so that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M52">View MathML</a>is strictly increasing inλ, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M53">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M54">View MathML</a>.

We are interested in the existence of positive solutions of (1.1) under weaker assumptions. More concretely, we consider the existence of positive solutions of (1.1) from the following two aspects: (a) To obtain the counterpart of Theorem A(ii) under the weaker assumptions than those of [1]. In other words, we will not assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>) are smooth functions any more. (b) Furthermore, we will also consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M57">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>) may not exist. More precisely, the following two theorems, which are the main results of the present paper, shall be proved.

Theorem 1.1Suppose that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M60">View MathML</a>) are continuous real functions satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M61">View MathML</a>

(A2) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M62">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M63">View MathML</a>

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M65">View MathML</a>.

If the matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M66">View MathML</a>is irreducible, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M67">View MathML</a>such that (1.1) has no positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M68">View MathML</a>and (1.1) has at least one positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M69">View MathML</a>.

Theorem 1.2Let (A1) and (A3) hold. Assume the following.

(A2)′ For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M70">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M71">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M72">View MathML</a>

(A4) The matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M73">View MathML</a>is irreducible, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M74">View MathML</a>.

Then for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M75">View MathML</a>, (1.1) has at least one positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M76">View MathML</a>.

Remark 1.1 It follows from (A2) and (A2)′ that the matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M78">View MathML</a> are all cooperative.

Remark 1.2 We note that our assumptions in Theorems 1.1 and 1.2 are weaker than those of Theorem A, and, accordingly, our results are weaker than Theorem A. Since we just suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> is continuous, we can only obtain the continua of positive solutions of (1.1) by applying bifurcation techniques, which are not necessarily curves of positive solutions, and thus the uniqueness and stability of positive solutions are not investigated. In [1], the authors obtained a smooth curve consisting of positive solutions of (1.1) by assuming stronger assumptions, under which the uniqueness and stability of positive solutions can be achieved.

Remark 1.3 For related results, established via bifurcation techniques, for other kind of problems, we refer the readers to [7-9] and the references therein.

The rest of the paper is arranged as follows. In Section 2, we recall some basic knowledges on the maximum principle of cooperative systems as well as the eigenvalues of cooperative matrices. Finally in Section 3, we prove our main results Theorems 1.1 and 1.2 by applying bifurcation theory.

2 Preliminaries

We shall essentially work in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M80">View MathML</a>, here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M81">View MathML</a>

The norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M82">View MathML</a> will be defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M83">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M84">View MathML</a> denotes the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M85">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M87">View MathML</a> for the standard Sobolev space. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M89">View MathML</a> to denote the null and the range space of a linear operator L, respectively.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M90">View MathML</a> be a solution of (1.1). Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>) are smooth real functions. Then we can deduce the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M93">View MathML</a>

(2.1)

which can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M94">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M95">View MathML</a>

(2.3)

Lemma 2.1[1,4]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M97">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M98">View MathML</a>. Suppose thatL, Hare given as in (2.3), andHis cooperative and irreducible. Then we have the following:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M99">View MathML</a>is a real eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M101">View MathML</a>is the spectrum of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>.

(b) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M103">View MathML</a>, there exists a unique (up a constant multiple) eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M104">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M105">View MathML</a>in Ω.

(c) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M106">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M107">View MathML</a>is uniquely solvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M108">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M109">View MathML</a>as long as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M110">View MathML</a>.

(d) (Maximum principle) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M106">View MathML</a>, assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M112">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M113">View MathML</a>in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114">View MathML</a>onΩ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114">View MathML</a>in Ω.

(e) If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M112">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M117">View MathML</a>in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M114">View MathML</a>onΩ, and either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M119">View MathML</a>onΩ or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M120">View MathML</a>in Ω, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M121">View MathML</a>.

For the results and proofs, see Proposition 3.1 and Theorem 1.1 of Sweers [10]. Moreover, from a standard compactness argument, there are countably many eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M122">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M124">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M125">View MathML</a>. We notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M126">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M127">View MathML</a>) are not necessarily real-valued.

In this section, we also need to consider the adjoint operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>. Let the transpose matrix of H be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M129">View MathML</a>

Then it is clear that the results in Lemma 2.1 are also true for the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M130">View MathML</a>

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M131">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M132">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133">View MathML</a> is the adjoint operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>, while both are considered as operators defined on subspaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M135">View MathML</a>.

The following lemmas are crucial in the proof of our main results.

Lemma 2.2[1]

LetY, Z, LandHbe the same as in Lemma 2.1. Then the principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M136">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M100">View MathML</a>is also a real eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M139">View MathML</a>, and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M103">View MathML</a>, there exists a unique eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M141">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M133">View MathML</a> (up a constant multiple), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M143">View MathML</a>in Ω.

Lemma 2.3 [[11], Theorem 5.3.1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M26">View MathML</a>matrixAbe a nonnegative irreducible matrix. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M145">View MathML</a>is a simple eigenvalue ofA, associated to a positive eigenvector, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M145">View MathML</a>denotes the spectral radius ofA. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M147">View MathML</a>.

Lemma 2.4[12]

LetVbe a real Banach space. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M148">View MathML</a>

is completely continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M149">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M150">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M151">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M152">View MathML</a>) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M153">View MathML</a>is the isolated solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M154">View MathML</a>

(2.5)

Furthermore, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M155">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M156">View MathML</a>is an isolated neighborhood of trivial solutions. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M157">View MathML</a>

Then there exists a continuum (i.e., a closed connected set) ofcontaining<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M160">View MathML</a>, and either

(i) is unbounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M162">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M163">View MathML</a>.

Finally, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M164">View MathML</a>be the principal eigen-pair of the linear eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M165">View MathML</a>

(2.6)

such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M166">View MathML</a>in Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M167">View MathML</a>.

3 Proof of the main results

Proof of Theorem 1.1. We extend each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> to be a nonnegative continuous function, which is still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a>, defined on ℝ in the following way: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M170">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M171">View MathML</a>.

Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M172">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M150">View MathML</a>. Then it follows from (A1) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M174">View MathML</a> is continuous, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M175">View MathML</a> is always a solution of (1.1). Moreover, (A2) implies that F is differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M175">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M177">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M178">View MathML</a>. By (A2), all entries of J are positive. Therefore Lemma 2.3 yields the result that J has a positive principal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M179">View MathML</a>, the corresponding eigenvector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M180">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M181">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>). Moreover, it is not difficult to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M183">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M184">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M185">View MathML</a> is a positive eigenvector of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M186">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M187">View MathML</a> has the same principal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M179">View MathML</a> and the corresponding eigenvector is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M189">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M190">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>) is a positive constant. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M192">View MathML</a>

Hence when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M186">View MathML</a> is not invertible and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M195">View MathML</a> is a potential bifurcation point. More precisely, the null space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M196">View MathML</a>

is one dimensional. In addition, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M198">View MathML</a> exist for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M199">View MathML</a>.

We divide the rest of the proof into two steps.

Step 1. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200">View MathML</a> is actually a bifurcation point.

Indeed, the proof of this is similar to the proof of Theorem A(ii), we state it here for the readers’ convenience.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M201">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M202">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M203">View MathML</a>

(3.4)

Let us consider the adjoint eigenvalue equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M204">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>. Multiplying the system (3.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M207">View MathML</a>, multiplying the system (3.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M208">View MathML</a>, integrating on Ω and subtracting, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M209">View MathML</a>

(3.6)

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M201">View MathML</a> if and only if (3.6) holds, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M211">View MathML</a> is one dimensional.

Next, we verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M212">View MathML</a>

(3.7)

Otherwise, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M213">View MathML</a>

(3.8)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M214">View MathML</a>

(3.9)

multiplying the system (3.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M215">View MathML</a> and using (3.8), we can get a contradiction that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M216">View MathML</a>

By using [[13], Theorem 1.7], we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200">View MathML</a> is a bifurcation point. Furthermore, by the Rabinowitz global bifurcation theorem [14], there exists a continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218">View MathML</a> of positive solutions of (1.1), which joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M200">View MathML</a> to infinity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M220">View MathML</a>. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M221">View MathML</a>

(3.10)

since (1.1) has only the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M222">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M223">View MathML</a>.

Step 2: We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218">View MathML</a> cannot blow up at some finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M225">View MathML</a>.

Otherwise, a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M226">View MathML</a> can be taken such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M227">View MathML</a>

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M228">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M229">View MathML</a> be the Green operator of −Δ subject to Dirichlet boundary conditions, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M230">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M231">View MathML</a>

By the elliptic regularity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M232">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M233">View MathML</a>

(3.12)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> also denotes the Nemytski operator generated by itself. Clearly, (3.12) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M235">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M236">View MathML</a>. It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M229">View MathML</a> is continuous and compact, and so is continuous and compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M239">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M240">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M242">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M243">View MathML</a>. Dividing both sides of (3.13) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M244">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M245">View MathML</a>

(3.14)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M4">View MathML</a>, from (A2) and (A3) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M247">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M248">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M249">View MathML</a>

(3.15)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M250">View MathML</a>

is bounded in X. This together with the compactness of implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M252">View MathML</a> has a subsequence, denoted by itself, satisfying, in X,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M253">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M254">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255">View MathML</a>) in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256">View MathML</a>. In addition, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M257">View MathML</a>. Or else, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M258">View MathML</a>, then by (3.14) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M259">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255">View MathML</a>) in Ω, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256">View MathML</a>.

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M262">View MathML</a>

(3.16)

Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M255">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M264">View MathML</a>

by Lebesgue control convergence theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M265">View MathML</a>

which together with (3.15) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M266">View MathML</a>

(3.17)

On the other hand, we know from (A2) and (3.15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M267">View MathML</a>

(3.18)

Hence we conclude from (3.17) and (3.18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M268">View MathML</a>

(3.19)

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M258">View MathML</a> in (3.14), using (3.19) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M257">View MathML</a> we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M271">View MathML</a>

which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M256">View MathML</a>.

Finally, by (3.10), the connectness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M218">View MathML</a> and above arguments, we can find some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M67">View MathML</a> such that (1.1) has no positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M68">View MathML</a>, and (1.1) has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M69">View MathML</a>. □

To prove Theorem 1.2, we need the following lemmas as required.

By Remark 1.1 and Lemma 2.3, the matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M278">View MathML</a> have the principal eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M280">View MathML</a>, respectively, and the corresponding positive eigenvectors are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M282">View MathML</a>. Moreover, it is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M283">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M284">View MathML</a>

(3.21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M285">View MathML</a> is given as in (2.6). By Lemma 2.2, the matrices<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M287">View MathML</a> have principal eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M289">View MathML</a>, respectively, the associated positive eigenvectors are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M291">View MathML</a>. We can easily verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M292">View MathML</a>

(3.22)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M293">View MathML</a>

(3.23)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M294">View MathML</a> be the closure of the set of positive solutions to (1.1). We extend each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M29">View MathML</a> to be a function defined on ℝ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M296">View MathML</a>

(3.24)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M297">View MathML</a> on ℝ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M298">View MathML</a> be a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M299">View MathML</a>

(3.25)

Then by (3.24), for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M301">View MathML</a>

where K is given as in the proof of Theorem 1.1. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12">View MathML</a> is a nonnegative solution of (3.25). Moreover, from (3.24) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M12">View MathML</a> is a solution of (1.1). On the other hand, (3.25) has no half-trivial solutions. Otherwise, U must have trivial and nontrivial components, and so there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M304">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M305">View MathML</a> in Ω, and by the maximum principle of elliptic boundary value problems, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M306">View MathML</a>

which is a contradiction. Therefore, the closure of the set of nontrivial solutions of (3.25) is exactly Σ.

In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M307">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M308">View MathML</a>

(3.26)

where is given as in (3.13), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M310">View MathML</a>

is the associated Nemytski operator. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M311">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M312">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M313">View MathML</a> denote the degree of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M314">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M315">View MathML</a> with respect to 0.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M316">View MathML</a>be a compact interval with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M317">View MathML</a>. Then there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M318">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M319">View MathML</a>

Proof Suppose on the contrary that there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M320">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M321">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M322">View MathML</a>

(3.27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M323">View MathML</a>

(3.28)

Apparently, (3.27) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M324">View MathML</a>

(3.29)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M325">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>) in Ω, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M328">View MathML</a>. Furthermore, it follows from (A2)′ that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M329">View MathML</a> sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M330">View MathML</a>. This together with (3.28) implies that, for k large enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M331">View MathML</a>

(3.30)

Multiplying (3.29) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M332">View MathML</a>, multiplying (3.22) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M333">View MathML</a>, integrating on Ω and adding, using (3.30) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M328">View MathML</a>, we know that, for k large enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M336">View MathML</a>

(3.31)

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M337">View MathML</a> for k sufficiently large. Similarly, by (3.23) and (3.30), we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M338">View MathML</a> for k large enough. Consequently, for k sufficiently large we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M339">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M320">View MathML</a>. □

Corollary 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M341">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M343">View MathML</a>.

Proof Lemma 3.1, applied to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M344">View MathML</a>, guarantees the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M318">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M347">View MathML</a>

Hence for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M342">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M349">View MathML</a>

 □

Lemma 3.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M351">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M352">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M353">View MathML</a>.

Proof Suppose on the contrary that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M354">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M355">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M357">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M358">View MathML</a>

which is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M359">View MathML</a>

(3.32)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M325">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M14">View MathML</a>) in Ω. Multiplying (3.22) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M333">View MathML</a>, multiplying (3.32) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M363">View MathML</a>, integrating over Ω and adding, then by (3.30) we know that, for k large enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M364">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M365">View MathML</a> for k sufficiently large, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350">View MathML</a>. □

Corollary 3.4For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M350">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M369">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M370">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M371">View MathML</a> is the constant given as in Lemma 3.3. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M314">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M373">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M374">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M376">View MathML</a>. By Lemma 3.3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M377">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M378">View MathML</a>

 □

Proof of Theorem 1.2. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M379">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M380">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M383">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M384">View MathML</a>, it is easy to see that the assumptions of Lemma 2.4 are all satisfied. Therefore there exists a continuum of solutions of (3.25) containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M386">View MathML</a>, and either

(i) is unbounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M220">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M389">View MathML</a>.

By Lemma 3.1, the case (ii) cannot occur, and hence is unbounded bifurcated from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M386">View MathML</a>. Note that (3.25) has only trivial solutions when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M223">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M393">View MathML</a>. Moreover, from Lemma 3.1 it follows that for a closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M394">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M395">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M396">View MathML</a> in X is impossible. Thus must be bifurcated from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M398">View MathML</a>. Finally, applying similar methods to the proof of Step 2 of Theorem 1.1, we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M399">View MathML</a>

Consequently, (1.1) has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/28/mathml/M400">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RM and RC completed the main study, carried out the results of this article and drafted the manuscript. YL checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP(No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).

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