Open Access Research

Existence, uniqueness and stability of positive solutions to a general sublinear elliptic systems

Boying Wu1 and Renhao Cui12*

Author Affiliations

1 Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, P.R. China

2 Y.Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, P.R. China

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Boundary Value Problems 2013, 2013:74  doi:10.1186/1687-2770-2013-74


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/74


Received:19 December 2012
Accepted:14 March 2013
Published:4 April 2013

© 2013 Wu and Cui; 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 make use of a new stability result and bifurcation theory to study the existence and uniqueness of positive solutions to semilinear elliptic systems with some general sublinear conditions. Moreover, we obtain the precise global bifurcation diagrams of the system in a single monotone solution curve.

MSC: 35J55, 35B32.

Keywords:
semilinear elliptic systems; positive solution; stability; existence; uniqueness

1 Introduction

We consider positive solutions of a semilinear elliptic system with n equations (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M1">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3">View MathML</a> is a positive parameter, Ω is a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M4">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6">View MathML</a>) are smooth real-valued functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M7">View MathML</a> satisfying the following.

Cooperativeness Define the Jacobian of the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M8">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M9">View MathML</a>

(1.2)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M11">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M12">View MathML</a>.

The purpose of this paper is to study the existence, uniqueness and stability of positive solutions of such cooperative system (1.1) under certain conditions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a>.

The existence, uniqueness and stability of positive solutions to sublinear semilinear elliptic systems with two equations have been recently studied in [1-4]. The sublinear condition plays an important role. In this paper, we continue the effort in [3] to prove the stability of a positive solution to (1.1) under some reasonable sublinear conditions, and the stability implies the uniqueness of the positive solution. We also prove corresponding existence results using bifurcation theory and the continuation method. This is motivated by the existence study of exact multiplicity (and uniqueness) of positive solutions to the scalar semilinear elliptic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M14">View MathML</a>

(1.3)

starting from Korman et al.[5,6]. In Ouyang and Shi [7,8], their result classified the different global exact multiplicity of (1.3) for more general nonlinearity f. There are more results on the existence and uniqueness of solution to the semilinear cyclic elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M15">View MathML</a>

(1.4)

The notation of sublinearity and superlinearity of the nonlinear vector field or the ones in higher dimension was considered in Sirakov [9]. Our definition of sublinear nonlinearity is quite different, and ours is similar to the one in Ouyang and Shi [8] for the scalar case. Dalmasso [10] obtained the existence and uniqueness result for a more special sublinear system, and it was extended by Shi and Shivaji [4]. The uniqueness of a positive solution for large λ was proved in Hai [11,12], Hai and Shivaji [13]. If Ω is a finite ball or the whole space, then the positive solutions of systems are radically symmetric and decreasing in radical direction by the result of Troy [14]. Hence the system can be converted into a system of ODEs. Several authors have taken that approach for the existence of the solutions, see Serrin and Zou [15,16], and much success has been achieved for Lane-Emden systems. Using the scaling invariant in the Lane-Emden system, the uniqueness of the radial positive solution was shown in Dalmasso [10], Korman and Shi [17]. Cui et al.[18,19] considered cyclic systems with three equations, and the uniqueness and existence of positive solutions were obtained. For the Lane-Emden systems with n equations, Maniwa [20] obtained the uniqueness and existence of positive solutions to systems under the sublinear conditions.

We organize the rest of this paper in the following way. In Section 2, we recall the maximum principle and prove the main stability result. In Section 3, we use the stability result and bifurcation theory to prove the existence and uniqueness of a positive solution. We also obtain the precise global bifurcation diagrams of the system (the bifurcation diagram is a single monotone solution curve in all cases) and give some examples. In Section 4, we consider the similar question for merely Hölder continuous nonlinearities, and we use monotone methods for existence. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M17">View MathML</a> for the standard Sobolev space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M18">View MathML</a> for the space of continuous functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M19">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M20">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M22">View MathML</a> to denote the null space and the range space of a linear operator L.

2 Stability and linearized equations

In this section, we study the stability result about a positive solution. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23">View MathML</a> be a solution of (1.1). We shall denote the partial derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M24">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M25">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M26">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M27">View MathML</a>. The stability of U is determined by the eigenvalue equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M28">View MathML</a>

(2.1)

which can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M29">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M30">View MathML</a>

(2.3)

Definition 2.1[16]

An <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M31">View MathML</a> matrix A is reducible if for some permutation matrix Q,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M32">View MathML</a>

where B and C are square matrices and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M33">View MathML</a> is the transpose of Q. Otherwise, A is irreducible.

Throughout this paper, H is assumed to be irreducible, since if not the case, the linearized system (2.1) can be reduced to two subsystems with one being not coupled with the other. If we assume that H is cooperative and irreducible, then the maximum principles hold for such systems. Before stating our results, we recall some known results as required.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M34">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M35">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M36">View MathML</a>. Suppose thatL, Hare given as in (2.3), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M37">View MathML</a>is irreducible and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M38">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M11">View MathML</a>) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M40">View MathML</a>. Then we have the following.

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M41">View MathML</a>is a real eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M43">View MathML</a>is the spectrum of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>.

2. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M45">View MathML</a>, there exists a unique (up a constant multiple) eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M46">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M47">View MathML</a>in Ω.

3. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M48">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M49">View MathML</a>is uniquely solvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M50">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M51">View MathML</a>as long as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M52">View MathML</a>.

4. (Maximum principle) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M53">View MathML</a>, suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M54">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M55">View MathML</a>in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56">View MathML</a>onΩ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56">View MathML</a>in Ω.

5. If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M58">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M59">View MathML</a>in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M56">View MathML</a>onΩ, and either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M61">View MathML</a>onΩ or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M62">View MathML</a>in Ω, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63">View MathML</a>.

For the result and proofs, see Sweers [21], Proposition 3.1 and Theorem 1.1. Moreover, from a standard compactness argument, there are countably many eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M64">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M66">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M67">View MathML</a>. We notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M68">View MathML</a> is not necessarily real-valued. We call a solution Ustable if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63">View MathML</a>; and otherwise, it is unstable (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M70">View MathML</a>).

For our purpose, in this section, we also need to consider the adjoint operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>. Let the transpose matrix of the Jacobian be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M72">View MathML</a>

(2.4)

Then, evidently, the results in Lemma 2.2 also hold for the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M73">View MathML</a>

(2.5)

which is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M74">View MathML</a>

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M75">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76">View MathML</a> is the adjoint operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>, while both are considered as operators defined on subspaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M78">View MathML</a>. By using the well-known functional analytic techniques (see [21,22]), one can show the following.

Lemma 2.3LetX, Y, LandHbe the same as in Lemma 2.2. Then the principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M79">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M42">View MathML</a>is also a real eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M82">View MathML</a>, and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M45">View MathML</a>, there exists a unique eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M84">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M76">View MathML</a> (up a constant multiple), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M86">View MathML</a>in Ω.

Cui et al.[3] obtained the stability result of a positive solution for the system with two equations. We give the following stability result about a positive solution of (1.1).

Theorem 2.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23">View MathML</a>is a positive solution of (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a>is cooperative and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M37">View MathML</a>is irreducible, thenUis stable if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a>satisfies one of the following conditions: for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M91">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M93">View MathML</a>

(2.7)

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M95">View MathML</a>

(2.8)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23">View MathML</a> be a positive solution of (1.1), and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M97">View MathML</a> be the corresponding principal eigen-pair of (2.6) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M98">View MathML</a> in Ω for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M99">View MathML</a>.

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>). Multiplying the system (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M102">View MathML</a>, the system (2.6) by U, integrating over Ω and subtracting, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M103">View MathML</a>

(2.9)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63">View MathML</a> if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>) is satisfied.

Similar to the proof above, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M106">View MathML</a> be the corresponding principal eigen-pair of (2.1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M107">View MathML</a> in Ω for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>). Multiplying the system (1.1) by u, the system (2.1) by U, integrating over Ω and subtracting, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M111">View MathML</a>

(2.10)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M63">View MathML</a> if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>) is satisfied. □

On the other hand, the same proof also implies the following instability result under the opposite condition of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>).

Theorem 2.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23">View MathML</a>is a positive solution of (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a>is cooperative and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M118">View MathML</a>is irreducible, thenUis unstable if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a>satisfies one of the following conditions: for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M91">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M121">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M122">View MathML</a>

(2.11)

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M123">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M124">View MathML</a>

(2.12)

Remark 2.6

1. In [8], for positive solutions of the scalar equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M125">View MathML</a>

the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M126">View MathML</a> is called a sublinear function if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M127">View MathML</a>, and it is superlinear if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M128">View MathML</a>. It was proved in Proposition 3.14 of [8] that a positive solution u is stable if h is sublinear, and u is unstable if h is superlinear. Now our conclusions, Theorems 2.4 and 2.5, are generalizations of the corresponding results. The condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M129">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M130">View MathML</a> is the generalization of sublinearity (or superlinearity) to n-variable vector fields.

2. The conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>) can be written in a vector form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M134">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M136">View MathML</a>, and H is the original Jacobian matrix of the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M137">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M138">View MathML</a> is the transpose matrix of the Jacobian H. The condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>) is clearly more natural as the conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> are separate. Hence the sublinearity can be defined for a single n-variable function. The condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M94">View MathML</a>) is defined for the whole vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M137">View MathML</a>.

3. If a solution U is stable, then it is necessarily a non-degenerate solution. That is, any eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M68">View MathML</a> of (2.1) has a positive real part. But when a solution is proved to be unstable, it can be a degenerate one with zero or pure imaginary eigenvalues.

3 Existence and uniqueness

In this section, we consider the uniqueness and existence of positive solutions for the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M144">View MathML</a>

(3.1)

Suppose that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M145">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M146">View MathML</a>) is a smooth real function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M147">View MathML</a> satisfying

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M149">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M152">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M153">View MathML</a>.

The Perron-Frobenius theorem plays a critical role in our main result.

Lemma 3.1 (Perron-Frobenius theorem: strong form [[23], Theorem 5.3.1])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M31">View MathML</a>matrixAbe a nonnegative irreducible matrix. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M155">View MathML</a>is a simple eigenvalue ofA, associated to a positive eigenvector, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M155">View MathML</a>denotes the spectral radius ofA. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M157">View MathML</a>.

Here let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M158">View MathML</a> be the principal eigen-pair of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M159">View MathML</a>

(3.2)

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M160">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161">View MathML</a>. We have the following existence and uniqueness result for this sublinear problem.

Theorem 3.2Assume that each of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M145">View MathML</a>satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M164">View MathML</a>) and

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M165">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M166">View MathML</a>.

1. If at least one of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M167">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6">View MathML</a>) is positive and matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M169">View MathML</a>is irreducible, then (3.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M170">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3">View MathML</a>;

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M173">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M174">View MathML</a>and matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M175">View MathML</a>is irreducible, then for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M176">View MathML</a>, (3.1) has no positive solution when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M177">View MathML</a>, and (3.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M170">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179">View MathML</a>.

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M180">View MathML</a> (in the first case, we assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M181">View MathML</a>) is a smooth curve so that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M182">View MathML</a>is strictly increasing inλ, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M183">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M184">View MathML</a>.

Proof Our proof follows that of Theorem 6.1 in [4]. Firstly, we extend <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185">View MathML</a> to be defined on R and they are continuously differentiable on R. From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M187">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M188">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M189">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>). Hence from Theorem 2.4, any positive solution of (3.1) is stable.

Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M191">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M192">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M193">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185">View MathML</a> are at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M195">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M196">View MathML</a> is continuously differentiable, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M198">View MathML</a>. For weak solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M23">View MathML</a>, one can also consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M35">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M202">View MathML</a> is properly chosen.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M203">View MathML</a> is a solution of (3.1). We apply the implicit function theorem at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M203">View MathML</a>. The Fréchet derivative of F is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M205">View MathML</a>

(3.4)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M206">View MathML</a> is an isomorphism from X to Y, and the implicit function theorem implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M207">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M209">View MathML</a> for some small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M210">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M211">View MathML</a> is the unique solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M212">View MathML</a>

(3.5)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M213">View MathML</a>

where e is the unique positive solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M214">View MathML</a>

(3.6)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M215">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M6">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M217">View MathML</a> is positive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M218">View MathML</a>. If there exists i such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M219">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M220">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M221">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M185">View MathML</a> is positive, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M225">View MathML</a> as well.

Next we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M227">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M228">View MathML</a>). The linearized operator is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M229">View MathML</a>

(3.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M230">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M231">View MathML</a>, all entries of matrix J are positive. Therefore, by using Lemma 3.1, there exist a positive principal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M232">View MathML</a> and the corresponding eigenvector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M233">View MathML</a> of J for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M234">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M235">View MathML</a> is a positive eigenvector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M236">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M237">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M238">View MathML</a> has the same principal eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M232">View MathML</a>, and the corresponding eigenvector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M240">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M241">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108">View MathML</a>) is a positive constant.

Hence when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M244">View MathML</a> is not invertible and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M245">View MathML</a> is a potential bifurcation point. More precisely, the null space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M246">View MathML</a> is one-dimensional.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M247">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M248">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M249">View MathML</a>

(3.8)

Consider the adjoint eigenvalue equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M250">View MathML</a>

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M251">View MathML</a>.

Multiplying the system (3.8) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M252">View MathML</a>, the system (3.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M253">View MathML</a>, integrating over Ω and subtracting, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M254">View MathML</a>

(3.10)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M255">View MathML</a> if and only if (3.9) holds, which implies that the codimension of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M256">View MathML</a> is one.

Next we verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M257">View MathML</a>. Indeed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M258">View MathML</a>

(3.11)

But this contradicts with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M259">View MathML</a>

(3.12)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M260">View MathML</a>

By using a bifurcation from a simple eigenvalue theorem of Crandall-Rabinowitz [24], we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M261">View MathML</a> is a bifurcation point. The nontrivial solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M262">View MathML</a> near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M263">View MathML</a> are in the form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M264">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M265">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M266">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M108">View MathML</a>). From the stability of positive solutions, each positive solution is non-degenerate.

Next we claim that (1.1) has no positive solution when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M268">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269">View MathML</a> is a positive solution of (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M270">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M271">View MathML</a>

(3.13)

Multiplying the system (3.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M272">View MathML</a>, the system (3.13) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269">View MathML</a>, integrating over Ω and subtracting, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M275">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M276">View MathML</a>

(3.14)

Hence (3.1) has no positive solution when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M268">View MathML</a>. And the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M264">View MathML</a> can also be parameterized as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M279">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M280">View MathML</a>. With the implicit function theorem, we can extend this curve to the largest <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M281">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M282">View MathML</a>. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M283">View MathML</a> is strictly increasing in λ for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M284">View MathML</a>. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M285">View MathML</a> satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M286">View MathML</a>

(3.15)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M287">View MathML</a> from the maximum principle (Lemma 2.2 part 3) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M288">View MathML</a> from the stability of positive solutions. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M289">View MathML</a>. Suppose not, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M291">View MathML</a>. Then one can show that the curve Γ can be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M292">View MathML</a> from some standard elliptic estimates, then from the implicit function theorem, Γ can be extended beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M292">View MathML</a>, which is a contradiction; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M295">View MathML</a>, a contradiction can be derived with the solution curve which cannot blow up at finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M281">View MathML</a> (see similar arguments for the scalar equation in [25]). Hence we must have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M289">View MathML</a>.

Finally, we claim the uniqueness. If there is another positive solution for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179">View MathML</a>, then the arguments above show that this solution also belongs to a solution curve defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M299">View MathML</a>, and the solutions on the curve are increasing in λ, but the nonexistence of positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M300">View MathML</a> and the local bifurcation at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M245">View MathML</a> exclude the possibility of another solution curve. Hence the positive solution is unique for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M179">View MathML</a>. This completes the proof. □

Example 3.3

We consider the following cyclic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M303">View MathML</a>

(3.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> are smooth real functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M147">View MathML</a> satisfying (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M148">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M150">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M165">View MathML</a>). Then Theorem 3.2 implies the existence and uniqueness of a positive solution of problem (3.16), and the bifurcation diagram is a single monotone solution curve. The n-dimensional cyclic positone and semipositone system was considered in [26] and [27]. They got the existence and multiplicity of positive solutions result for some combined sublinear condition by the method of sub-super solutions.

Example 3.4

Consider the general Lane-Emden system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M309">View MathML</a>

(3.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3">View MathML</a> is a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M312">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M313">View MathML</a>) are positive constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M314">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M313">View MathML</a>) are nonnegative constants satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M316">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M317">View MathML</a> denotes a bounded domain of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M319">View MathML</a>. For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M320">View MathML</a>, (3.17) has been studied by many authors. Especially, Dalmasso [10] proved the uniqueness and existence of positive solutions to (3.17) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M323">View MathML</a>.

In this section, we show the uniqueness and existence of positive solutions for (3.17) by using super-subsolution methods and the stability of positive solutions by using Theorem 2.4.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M324">View MathML</a> be the unique positive solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M325">View MathML</a>

We construct a super-solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M326">View MathML</a>. There exists a suitable positive constant M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M327">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M328">View MathML</a>. We construct a sub-solution in the form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M329">View MathML</a>, where ε will be specified later. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M330">View MathML</a> is the positive principal eigenfunction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161">View MathML</a>. Now, for the ith equation of (3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M332">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M333">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M334">View MathML</a> is a subsolution of (3.17), if we choose ε smaller, and it satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M335">View MathML</a>. Hence if we choose ε smaller so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M336">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M337">View MathML</a> is a subsolution of (4.1). Therefore, there exists a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M269">View MathML</a> of (3.17) between the supersolution and the subsolution.

Next, we show the solution is stable. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M339">View MathML</a>, by simple calculation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M340">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M341">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M5">View MathML</a> satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M92">View MathML</a>), hence from Theorem 2.4, any positive solution of (3.17) is stable.

4 Application: Hölder continuous case

In this section, we consider that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M344">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M319">View MathML</a>.

Example 4.1

Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M346">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M3">View MathML</a> is a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M348">View MathML</a>.

We use the monotone method to prove the existence of a solution. For the supersolution, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M349">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M350">View MathML</a>

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M351">View MathML</a>. Then it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M352">View MathML</a> is a supersolution of (4.1). We construct a sub-solution in the form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M353">View MathML</a>, where ε will be specified later. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M330">View MathML</a> is the positive principal eigenfunction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M161">View MathML</a>. Now, for the equation of u or the equation of v, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M356">View MathML</a>

Hence if we choose ε smaller and it satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M357">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M358">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M337">View MathML</a> is a subsolution of (4.1). Therefore, there exists a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M360">View MathML</a> of (4.1) between the supersolution and the subsolution when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M361">View MathML</a>. We remark that the stability defined in Section 2 can still be established for a nonlinear function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M362">View MathML</a> to become ∞ near Ω by using Remark 3.1 in [28]. Thus the positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M360">View MathML</a> of (4.1) is stable.

Remark 4.2 Since (4.1) is a cooperative model from ecology with logistic growth rate and sublinear interaction term, we can get the stable result. When the interaction terms are uv (Lotka-Volterra type) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/74/mathml/M364">View MathML</a> as proposed here, and they do not satisfy the conditions of Theorem 3.2, thus, the solution may not be unique.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

BW and RC carried out the proof of the main part of this article, RC corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.

Acknowledgements

Partially supported by the National Natural Science Foundation of China (No. 11071051, 11271100 and 11101110), the Aerospace Supported Fund, China, under Contract (Grant 2011-HT-HGD-06), Science Research Foundation of the Education Department of Heilongjiang Province (Grant No. 12521153), Science Foundation of Heilongjiang Province (Grant A201009) and Harbin Normal University advanced research Foundation (11xyg-02).

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