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

Boundary Value Problems20132013:74

DOI: 10.1186/1687-2770-2013-74

Received: 19 December 2012

Accepted: 14 March 2013

Published: 4 April 2013

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 ( n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq1_HTML.gif)
{ Δ u 1 + λ f 1 ( u 1 , u 2 , , u n ) = 0 , x Ω , Δ u 2 + λ f 2 ( u 1 , u 2 , , u n ) = 0 , x Ω , Δ u n + λ f n ( u 1 , u 2 , , u n ) = 0 , x Ω , u 1 ( x ) = = u n ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ1_HTML.gif
(1.1)

where λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq2_HTML.gif is a positive parameter, Ω is a bounded smooth domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq3_HTML.gif, and f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif ( i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq5_HTML.gif) are smooth real-valued functions defined on R + n = { ( u 1 , u 2 , , u n ) R n : u i > 0 , 1 i n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq6_HTML.gif satisfying the following.

Cooperativeness Define the Jacobian of the vector field ( f 1 , , f n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq7_HTML.gif as
H ( u 1 , , u n ) = ( f 1 u 1 f 1 u n f n u 1 f n u n ) = ( f 11 f 1 n f n 1 f n n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ2_HTML.gif
(1.2)

Then f i / u j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq8_HTML.gif ( i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq9_HTML.gif) for ( u 1 , , u n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq10_HTML.gif.

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 f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif.

The existence, uniqueness and stability of positive solutions to sublinear semilinear elliptic systems with two equations have been recently studied in [14]. 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:
{ Δ u + λ f ( u ) = 0 , x Ω , u ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ3_HTML.gif
(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
{ Δ u + λ f ( v ) = 0 , x Ω , Δ v + λ g ( u ) = 0 , x Ω , u ( x ) = v ( x ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ4_HTML.gif
(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 W 2 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq11_HTML.gif and W loc 2 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq12_HTML.gif for the standard Sobolev space, C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq13_HTML.gif for the space of continuous functions defined on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq14_HTML.gif, and C 0 ( Ω ¯ ) = { u C ( Ω ¯ ) : u ( x ) = 0 , x Ω } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq15_HTML.gif. We use N ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq16_HTML.gif and R ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq17_HTML.gif 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 U = ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq18_HTML.gif be a solution of (1.1). We shall denote the partial derivative of f i ( U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq19_HTML.gif with respect to u j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq20_HTML.gif by f i j ( U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq21_HTML.gif or f i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq22_HTML.gif. The stability of U is determined by the eigenvalue equation
{ Δ ξ 1 + λ f 11 ξ 1 + λ f 12 ξ 2 + + λ f 1 n ξ n = μ ξ 1 , x Ω , Δ ξ 2 + λ f 21 ξ 1 + λ f 22 ξ 2 + + λ f 2 n ξ n = μ ξ 2 , x Ω , Δ ξ n + λ f n 1 ξ 1 + λ f n 2 ξ 2 + + λ f n n ξ n = μ ξ n , x Ω , ξ 1 ( x ) = ξ 2 ( x ) = = ξ n ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ5_HTML.gif
(2.1)
which can be written as
L u = H u + μ u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ6_HTML.gif
(2.2)
where
u = ( ξ 1 ξ n ) , L u = ( Δ ξ 1 Δ ξ n ) , and H = ( f 11 f 1 n f n 1 f n n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ7_HTML.gif
(2.3)

Definition 2.1 [16]

An n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq23_HTML.gif matrix A is reducible if for some permutation matrix Q,
Q A Q T = ( B 0 C D ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equa_HTML.gif

where B and C are square matrices and Q T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq24_HTML.gif 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 2.2 Let X = [ W loc 2 , p ( Ω ) C 0 ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq25_HTML.gif, and let Y = [ L p ( Ω ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq26_HTML.gif for p > n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq27_HTML.gif. Suppose that L, H are given as in (2.3), and H = ( f i j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq28_HTML.gif is irreducible and satisfies f i j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq29_HTML.gif ( i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq9_HTML.gif) for ( u 1 , , u n ) R + n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq30_HTML.gif. Then we have the following.
  1. 1.

    μ 1 = inf { μ spt ( L H ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq31_HTML.gif is a real eigenvalue of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif, where spt ( L H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq33_HTML.gif is the spectrum of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif.

     
  2. 2.

    For μ = μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq34_HTML.gif, there exists a unique (up a constant multiple) eigenfunction u 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq35_HTML.gif, and u 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq36_HTML.gif in Ω.

     
  3. 3.

    For μ < μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq37_HTML.gif, the equation L u = H u + μ u + f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq38_HTML.gif is uniquely solvable for any f Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq39_HTML.gif, and u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq40_HTML.gif as long as f 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq41_HTML.gif.

     
  4. 4.

    (Maximum principle) For μ μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq42_HTML.gif, suppose that u [ W loc 2 , p ( Ω ) C ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq43_HTML.gif satisfies L u H u + μ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq44_HTML.gif in Ω, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq45_HTML.gif on Ω, then u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq45_HTML.gif in Ω.

     
  5. 5.

    If there exists u [ W loc 2 , p ( Ω ) C ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq46_HTML.gif satisfying L u H u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq47_HTML.gif in Ω, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq45_HTML.gif on Ω, and either u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq48_HTML.gif on Ω or L u H u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq49_HTML.gif in Ω, then μ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq50_HTML.gif.

     

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 { μ i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq51_HTML.gif of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif, and | μ i μ 1 | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq52_HTML.gif as i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq53_HTML.gif. We notice that μ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq54_HTML.gif is not necessarily real-valued. We call a solution U stable if μ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq50_HTML.gif; and otherwise, it is unstable ( μ 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq55_HTML.gif).

For our purpose, in this section, we also need to consider the adjoint operator of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif. Let the transpose matrix of the Jacobian be
H T = ( f 11 f n 1 f 1 n f n n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ8_HTML.gif
(2.4)
Then, evidently, the results in Lemma 2.2 also hold for the eigenvalue problem
L u = H T u + μ u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ9_HTML.gif
(2.5)
which is
{ Δ ξ 1 + λ f 11 ξ 1 + λ f 21 ξ 2 + + λ f n 1 ξ n = μ ξ 1 , x Ω , Δ ξ 2 + λ f 12 ξ 1 + λ f 22 ξ 2 + + λ f n 2 ξ n = μ ξ 2 , x Ω , Δ ξ n + λ f 1 n ξ 1 + λ f 2 n ξ 2 + + λ f n n ξ n = μ ξ n , x Ω , ξ 1 ( x ) = ξ 2 ( x ) = = ξ n ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ10_HTML.gif
(2.6)

where u = ( ξ 1 , ξ 2 , , ξ n ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq56_HTML.gif. It is easy to verify that L H T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq57_HTML.gif is the adjoint operator of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif, while both are considered as operators defined on subspaces of [ L 2 ( Ω ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq58_HTML.gif. By using the well-known functional analytic techniques (see [21, 22]), one can show the following.

Lemma 2.3 Let X, Y, L and H be the same as in Lemma  2.2. Then the principal eigenvalue μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq59_HTML.gif of L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq32_HTML.gif is also a real eigenvalue of L H T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq57_HTML.gif, μ 1 = inf { μ spt ( L H T ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq60_HTML.gif, and for μ = μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq34_HTML.gif, there exists a unique eigenfunction u 1 [ W loc 2 , p ( Ω ) C 0 ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq61_HTML.gif of L H T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq57_HTML.gif (up a constant multiple), and u 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq62_HTML.gif 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 2.4 Suppose that U = ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq18_HTML.gif is a positive solution of (1.1), f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif is cooperative and H = ( f i j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq28_HTML.gif is irreducible, then U is stable if f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif satisfies one of the following conditions: for any U R + n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq63_HTML.gif,

( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif)
f i ( U ) > j = 1 n f i j ( U ) u j , i = 1 , , n ; or http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ11_HTML.gif
(2.7)
( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif)
f i ( U ) > j = 1 n f j i ( U ) u j , i = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ12_HTML.gif
(2.8)

Proof Let U = ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq18_HTML.gif be a positive solution of (1.1), and let ( μ 1 , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq66_HTML.gif be the corresponding principal eigen-pair of (2.6) such that ξ i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq67_HTML.gif in Ω for any 1 i n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq68_HTML.gif.

We assume that f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif satisfies ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif). Multiplying the system (1.1) by u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq69_HTML.gif, the system (2.6) by U, integrating over Ω and subtracting, we obtain
μ 1 Ω i = 1 n u i ξ i d x = λ Ω i = 1 n f i ξ i d x λ Ω i = 1 n ξ i j = 1 n f i j u j d x = λ Ω i = 1 n ξ i ( f i j = 1 n f i j u j ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ13_HTML.gif
(2.9)

Hence μ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq50_HTML.gif if ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif) is satisfied.

Similar to the proof above, let ( μ 1 , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq70_HTML.gif be the corresponding principal eigen-pair of (2.1) such that ξ i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq71_HTML.gif in Ω for any i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq72_HTML.gif. We assume that f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif satisfies ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif). Multiplying the system (1.1) by u, the system (2.1) by U, integrating over Ω and subtracting, we can get
μ 1 Ω i = 1 n u i ξ i d x = λ Ω i = 1 n f i ξ i d x λ Ω i = 1 n ξ i j = 1 n f j i u j d x = λ Ω i = 1 n ξ i ( f i j = 1 n f j i u j ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ14_HTML.gif
(2.10)

Hence μ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq50_HTML.gif if ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif) is satisfied. □

On the other hand, the same proof also implies the following instability result under the opposite condition of ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif) and ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif).

Theorem 2.5 Suppose that U = ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq18_HTML.gif is a positive solution of (1.1), f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif is cooperative and H T = ( f j i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq73_HTML.gif is irreducible, then U is unstable if f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif satisfies one of the following conditions: for any U R + n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq63_HTML.gif,

( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq74_HTML.gif)
f i ( U ) < j = 1 n f i j ( U ) u j , i = 1 , , n ; or http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ15_HTML.gif
(2.11)
( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq75_HTML.gif)
f i ( U ) < j = 1 n f j i ( U ) u j , i = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ16_HTML.gif
(2.12)
Remark 2.6
  1. 1.
    In [8], for positive solutions of the scalar equation
    Δ u + λ h ( u ) = 0 , x Ω , u ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equb_HTML.gif

    the function h ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq76_HTML.gif is called a sublinear function if h ( u ) > u h ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq77_HTML.gif, and it is superlinear if h ( u ) < u h ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq78_HTML.gif. 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 ( H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq79_HTML.gif) for i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq80_HTML.gif is the generalization of sublinearity (or superlinearity) to n-variable vector fields.

     
  2. 2.

    The conditions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif) and ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif) can be written in a vector form F ( U ) > H ( U ) U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq81_HTML.gif and F ( U ) > H T ( U ) U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq82_HTML.gif, where F ( U ) = ( f 1 ( U ) , , f n ( U ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq83_HTML.gif, U = ( u 1 , , u n ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq84_HTML.gif, and H is the original Jacobian matrix of the vector field ( f 1 ( U ) , , f n ( U ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq85_HTML.gif, and H T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq86_HTML.gif is the transpose matrix of the Jacobian H. The condition ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif) is clearly more natural as the conditions for f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif are separate. Hence the sublinearity can be defined for a single n-variable function. The condition ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq65_HTML.gif) is defined for the whole vector field ( f 1 ( U ) , , f n ( U ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq85_HTML.gif.

     
  3. 3.

    If a solution U is stable, then it is necessarily a non-degenerate solution. That is, any eigenvalue μ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq54_HTML.gif 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:
{ Δ u 1 + λ ( g 11 ( u 1 ) + g 12 ( u 2 ) + + g 1 n ( u n ) ) = 0 , x Ω , Δ u 2 + λ ( g 21 ( u 1 ) + g 22 ( u 2 ) + + g 2 n ( u n ) ) = 0 , x Ω , Δ u n + λ ( g n 1 ( u 1 ) + g n 2 ( u 2 ) + + g n n ( u n ) ) = 0 , x Ω , u 1 ( x ) = u 2 ( x ) = = u n ( x ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ17_HTML.gif
(3.1)

Suppose that each g i j ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq87_HTML.gif ( i , j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq88_HTML.gif) is a smooth real function defined on R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq89_HTML.gif satisfying

( A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq90_HTML.gif) g i j ( 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq91_HTML.gif;

( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq92_HTML.gif) g i j ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq93_HTML.gif, ( g i j ( x ) / x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq94_HTML.gif, for all x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq95_HTML.gif.

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 n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq23_HTML.gif matrix A be a nonnegative irreducible matrix. Then ρ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq96_HTML.gif is a simple eigenvalue of A, associated to a positive eigenvector, where ρ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq96_HTML.gif denotes the spectral radius of A. Moreover, ρ ( A ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq97_HTML.gif.

Here let ( λ 1 , φ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq98_HTML.gif be the principal eigen-pair of
Δ φ = λ φ , x Ω , φ ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ18_HTML.gif
(3.2)

such that φ 1 ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq99_HTML.gif in Ω and φ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq100_HTML.gif. We have the following existence and uniqueness result for this sublinear problem.

Theorem 3.2 Assume that each of g i j ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq87_HTML.gif satisfies ( A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq90_HTML.gif), ( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq101_HTML.gif) and

( A 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq102_HTML.gif) lim x g i j ( x ) x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq103_HTML.gif.

  1. 1.

    If at least one of g i j ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq104_HTML.gif ( i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq5_HTML.gif) is positive and matrix G = ( g i j ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq105_HTML.gif is irreducible, then (3.1) has a unique positive solution U ( λ ) = ( u 1 ( λ ) , u 2 ( λ ) , , u n ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq106_HTML.gif for all λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq2_HTML.gif;

     
  2. 2.

    If g i j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq107_HTML.gif, g i j ( 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq108_HTML.gif for each i , j = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq109_HTML.gif and matrix G = ( g i j ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq110_HTML.gif is irreducible, then for some λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq111_HTML.gif, (3.1) has no positive solution when λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq112_HTML.gif, and (3.1) has a unique positive solution U ( λ ) = ( u 1 ( λ ) , u 2 ( λ ) , , u n ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq106_HTML.gif for λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq113_HTML.gif.

     

Moreover, { ( λ , u 1 ( λ ) , u 2 ( λ ) , , u n ( λ ) ) : λ > λ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq114_HTML.gif (in the first case, we assume λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq115_HTML.gif) is a smooth curve so that u i ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq116_HTML.gif is strictly increasing in λ, and u i ( λ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq117_HTML.gif as λ λ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq118_HTML.gif.

Proof Our proof follows that of Theorem 6.1 in [4]. Firstly, we extend g i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq119_HTML.gif to be defined on R and they are continuously differentiable on R. From ( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq92_HTML.gif), g i j ( x ) > g i j ( x ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq120_HTML.gif implies that j = 1 n g i j ( u j ) > j = 1 n g i j ( u j ) u j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq121_HTML.gif, so f i ( U ) = f i ( u 1 , , u n ) = j = 1 n g i j ( u j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq122_HTML.gif satisfies ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif). Hence from Theorem 2.4, any positive solution of (3.1) is stable.

Let us define
F ( λ , U ) = ( Δ u 1 + λ [ g 11 ( u 1 ) + g 12 ( u 2 ) + + g 1 n ( u n ) ] Δ u 2 + λ [ g 21 ( u 1 ) + g 22 ( u 2 ) + + g 2 n ( u n ) ] Δ u n + λ [ g n 1 ( u 1 ) + g n 2 ( u 2 ) + + g n n ( u n ) ] ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ19_HTML.gif
(3.3)

where λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq123_HTML.gif and u 1 , u 2 , , u n C 0 2 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq124_HTML.gif. Here g i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq119_HTML.gif are at least C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq125_HTML.gif, then F : R × X Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq126_HTML.gif is continuously differentiable, where X = [ C 0 2 , α ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq127_HTML.gif and Y = [ C α ( Ω ¯ ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq128_HTML.gif. For weak solutions U = ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq18_HTML.gif, one can also consider X = [ W 2 , p ( Ω ) W 0 1 , p ( Ω ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq129_HTML.gif and Y = [ L p ( Ω ) ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq26_HTML.gif where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq130_HTML.gif is properly chosen.

It is easy to see that ( λ , U ) = ( 0 , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq131_HTML.gif is a solution of (3.1). We apply the implicit function theorem at ( λ , U ) = ( 0 , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq131_HTML.gif. The Fréchet derivative of F is given by
F U ( λ , U ) ( ϕ 1 ϕ 2 ϕ n ) = ( Δ ϕ 1 + λ [ g 11 ( u 1 ) ϕ 1 + g 12 ( u 2 ) ϕ 2 + + g 1 n ( u n ) ϕ n ] Δ ϕ 2 + λ [ g 21 ( u 1 ) ϕ 1 + g 22 ( u 2 ) ϕ 2 + + g 2 n ( u n ) ϕ n ] Δ ϕ n + λ [ g n 1 ( u 1 ) ϕ 1 + g n 2 ( u 2 ) ϕ 2 + + g n n ( u n ) ϕ n ] ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ20_HTML.gif
(3.4)
Then F U ( 0 , 0 , , 0 ) ( ϕ 1 , , ϕ n ) T = ( Δ ϕ 1 , , Δ ϕ n ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq132_HTML.gif is an isomorphism from X to Y, and the implicit function theorem implies that F ( λ , U ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq133_HTML.gif has a unique solution ( λ , U ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq134_HTML.gif for λ ( 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq135_HTML.gif for some small δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq136_HTML.gif, and ( u 1 ( 0 ) , , u n ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq137_HTML.gif is the unique solution of
{ Δ ϕ 1 + g 11 ( 0 ) + g 12 ( 0 ) + + g 1 n ( 0 ) = 0 , x Ω , Δ ϕ 2 + g 21 ( 0 ) + g 22 ( 0 ) + + g 2 n ( 0 ) = 0 , x Ω , Δ ϕ n + g n 1 ( 0 ) + g n 2 ( 0 ) + + g n n ( 0 ) = 0 , x Ω , ϕ 1 ( x ) = = ϕ n ( x ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ21_HTML.gif
(3.5)
Therefore,
{ u 1 ( 0 ) = ( g 11 ( 0 ) + g 12 ( 0 ) + + g 1 n ( 0 ) ) e u 2 ( 0 ) = ( g 21 ( 0 ) + g 22 ( 0 ) + + g 2 n ( 0 ) ) e u n ( 0 ) = ( g n 1 ( 0 ) + g n 2 ( 0 ) + + g n n ( 0 ) ) e , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equc_HTML.gif
where e is the unique positive solution of
Δ e + 1 = 0 , x Ω , e ( x ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ22_HTML.gif
(3.6)

If j = 1 n g i j ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq138_HTML.gif for any i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq5_HTML.gif, then U ( λ ) = ( u 1 ( λ ) , , u n ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq139_HTML.gif is positive for λ ( 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq140_HTML.gif. If there exists i such that g i 1 ( 0 ) + g i 2 ( 0 ) + + g i n ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq141_HTML.gif, then u i ( λ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq142_HTML.gif for λ ( 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq143_HTML.gif. For k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq144_HTML.gif, Δ u k ( λ ) = λ [ g k 1 ( λ ) + + g k n ( λ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq145_HTML.gif and g i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq119_HTML.gif is positive, hence u k ( λ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq146_HTML.gif as well.

Next we assume that g i j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq107_HTML.gif and g i j ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq147_HTML.gif ( i , j = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq148_HTML.gif). The linearized operator is
F U ( λ , 0 , , 0 ) ( Φ 1 Φ 2 Φ n ) = ( Δ Φ 1 + λ [ g 11 ( 0 ) Φ 1 + g 12 ( 0 ) Φ 2 + + g 1 n ( 0 ) Φ n ] Δ Φ 2 + λ [ g 21 ( 0 ) Φ 1 + g 22 ( 0 ) Φ 2 + + g 2 n ( 0 ) Φ n ] Δ Φ n + λ [ g n 1 ( 0 ) Φ 1 + g n 2 ( 0 ) Φ 2 + + g n n ( 0 ) Φ n ] ) = ( Δ Φ 1 Δ Φ 2 Δ Φ n ) + λ ( g 11 ( 0 ) g 12 ( 0 ) g 1 n ( 0 ) g 21 ( 0 ) g 22 ( 0 ) g 2 n ( 0 ) g n 1 ( 0 ) g n 2 ( 0 ) g n n ( 0 ) ) ( Φ 1 Φ 2 Φ n ) = ( Δ Φ 1 Δ Φ 2 Δ Φ n ) + λ J ( Φ 1 Φ 2 Φ n ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ23_HTML.gif
(3.7)
where
J = ( g 11 ( 0 ) g 1 n ( 0 ) g n 1 ( 0 ) g n n ( 0 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equd_HTML.gif

Since g i j ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq149_HTML.gif, all entries of matrix J are positive. Therefore, by using Lemma 3.1, there exist a positive principal eigenvalue χ J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq150_HTML.gif and the corresponding eigenvector ( k 1 , k 2 , , k n ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq151_HTML.gif of J for some k i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq152_HTML.gif such that ( k 1 φ 1 , k 2 φ 1 , , k n φ 1 ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq153_HTML.gif is a positive eigenvector of F ( u 1 , , u n ) ( λ , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq154_HTML.gif, where λ = λ 1 / χ J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq155_HTML.gif. Similarly, J T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq156_HTML.gif has the same principal eigenvalue χ J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq150_HTML.gif, and the corresponding eigenvector ( k 1 φ 1 , k 2 φ 1 , , k n φ 1 ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq157_HTML.gif, where k i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq158_HTML.gif ( i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq72_HTML.gif) is a positive constant.

Hence when λ = λ = λ 1 / χ J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq159_HTML.gif, F U ( λ , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq160_HTML.gif is not invertible and λ = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq161_HTML.gif is a potential bifurcation point. More precisely, the null space N ( F U ( λ , 0 , , 0 ) ) = span { ( k 1 φ 1 , , k n φ n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq162_HTML.gif is one-dimensional.

Suppose that ( h 1 , , h n ) T R ( F U ( λ , 0 , , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq163_HTML.gif, then there exist ( ψ 1 , , ψ n ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq164_HTML.gif such that
F U ( λ , 0 , , 0 ) ( ψ 1 ψ n ) = ( Δ ψ 1 Δ ψ n ) + λ ( g 11 ( 0 ) g 1 n ( 0 ) g n 1 ( 0 ) g n n ( 0 ) ) ( ψ 1 ψ n ) = ( h 1 h n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ24_HTML.gif
(3.8)
Consider the adjoint eigenvalue equation
( Δ ψ 1 Δ ψ n ) + λ J T ( ψ 1 ψ n ) = ( Δ ψ 1 Δ ψ n ) + λ ( g 11 ( 0 ) g n 1 ( 0 ) g 1 n ( 0 ) g n n ( 0 ) ) ( ψ 1 ψ n ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ25_HTML.gif
(3.9)

where ( ψ 1 , , ψ n ) T = ( k 1 φ 1 , , k n φ n ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq165_HTML.gif.

Multiplying the system (3.8) by ( ψ 1 , , ψ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq166_HTML.gif, the system (3.9) by ( ψ 1 , , ψ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq167_HTML.gif, integrating over Ω and subtracting, we get
Ω ( h 1 ψ 1 + + h n ψ n ) d x = Ω ( k 1 h 1 φ 1 + + k n h n φ 1 ) d x = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ26_HTML.gif
(3.10)

Hence ( h 1 , , h n ) T R ( F U ( λ , 0 , , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq168_HTML.gif if and only if (3.9) holds, which implies that the codimension of R ( F U ( λ , 0 , , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq169_HTML.gif is one.

Next we verify that F λ U ( λ , 0 , , 0 ) [ k 1 φ 1 , , k n φ 1 ] T R ( F U ( λ , 0 , , 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq170_HTML.gif. Indeed,
F λ U ( λ , 0 , , 0 ) ( k 1 φ 1 k n φ 1 ) = J ( k 1 k n ) φ 1 = χ J ( k 1 k n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ27_HTML.gif
(3.11)
But this contradicts with
0 = χ J Ω ( k 1 k 1 + + k n k n ) φ 1 2 d x > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ28_HTML.gif
(3.12)
Hence
F λ U ( λ , 0 , , 0 ) [ k 1 φ 1 , , k n φ 1 ] T R ( F U ( λ , 0 , , 0 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Eque_HTML.gif

By using a bifurcation from a simple eigenvalue theorem of Crandall-Rabinowitz [24], we conclude that ( λ , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq171_HTML.gif is a bifurcation point. The nontrivial solutions of F ( λ , u 1 , , u n ) = ( 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq172_HTML.gif near ( λ , 0 , , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq173_HTML.gif are in the form of { ( λ ( s ) , u 1 ( s ) , , u n ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq174_HTML.gif for s ( δ , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq175_HTML.gif, where u i ( s ) = k i s φ 1 + o ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq176_HTML.gif ( i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq72_HTML.gif). From the stability of positive solutions, each positive solution is non-degenerate.

Next we claim that (1.1) has no positive solution when λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq177_HTML.gif. Let ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq178_HTML.gif is a positive solution of (1.1) and ( k 1 φ 1 , , k n φ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq179_HTML.gif satisfy
( Δ k 1 φ 1 Δ k n φ 1 ) + λ ( g 11 ( 0 ) g n 1 ( 0 ) g 1 n ( 0 ) g n n ( 0 ) ) ( k 1 φ 1 k n φ 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ29_HTML.gif
(3.13)
Multiplying the system (3.1) by ( k 1 φ 1 , k 2 φ 1 , , k n φ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq180_HTML.gif, the system (3.13) by ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq178_HTML.gif, integrating over Ω and subtracting, by ( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq92_HTML.gif) and j = 1 n g i j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq181_HTML.gif, we obtain
λ Ω i = 1 n [ j = 1 n k i g i j ( 0 ) u j φ 1 ] d x = λ Ω j = 1 n [ i = 1 n k i g i j ( u j ) φ 1 ] d x < λ Ω j = 1 n [ i = 1 n k i g i j ( 0 ) u j φ 1 ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ30_HTML.gif
(3.14)

Hence (3.1) has no positive solution when λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq177_HTML.gif. And the solution { ( λ ( s ) , u 1 ( s ) , , u n ( s ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq174_HTML.gif can also be parameterized as ( λ , u 1 ( λ ) , , u n ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq182_HTML.gif for λ ( λ , λ + δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq183_HTML.gif. With the implicit function theorem, we can extend this curve to the largest λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq184_HTML.gif.

Let Γ = { ( λ , u 1 ( λ ) , , u n ( λ ) ) : λ < λ < λ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq185_HTML.gif. We show that ( u 1 ( λ ) , , u n ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq186_HTML.gif is strictly increasing in λ for λ ( λ , λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq187_HTML.gif. In fact, ( u 1 ( λ ) / λ , , u n ( λ ) / λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq188_HTML.gif satisfies the equation
F U ( λ , U ) ( u 1 ( λ ) λ u n ( λ ) λ ) = ( Δ u 1 ( λ ) λ Δ u n ( λ ) λ ) + λ ( g 11 ( u 1 ( λ ) ) g 1 n ( u n ( λ ) ) g n 1 ( u 1 ( λ ) ) g n n ( u n ( λ ) ) ) ( u 1 ( λ ) λ u n ( λ ) λ ) = ( g 11 ( u 1 ( λ ) ) + + g 1 n ( u n ( λ ) ) g n 1 ( u 1 ( λ ) ) + + g n n ( u n ( λ ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ31_HTML.gif
(3.15)

Hence ( u 1 ( λ ) / λ , , u n ( λ ) / λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq189_HTML.gif from the maximum principle (Lemma 2.2 part 3) and the fact that μ 1 ( ( u 1 ( λ ) , , u n ( λ ) ) ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq190_HTML.gif from the stability of positive solutions. We claim that λ = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq191_HTML.gif. Suppose not, then λ < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq192_HTML.gif and ( u 1 ( λ ) , , u n ( λ ) ) X < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq193_HTML.gif. Then one can show that the curve Γ can be extended to λ = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq194_HTML.gif from some standard elliptic estimates, then from the implicit function theorem, Γ can be extended beyond λ = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq194_HTML.gif, which is a contradiction; if λ < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq192_HTML.gif and ( u 1 ( λ ) , , u n ( λ ) ) X = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq195_HTML.gif, a contradiction can be derived with the solution curve which cannot blow up at finite λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq184_HTML.gif (see similar arguments for the scalar equation in [25]). Hence we must have λ = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq191_HTML.gif.

Finally, we claim the uniqueness. If there is another positive solution for some λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq113_HTML.gif, then the arguments above show that this solution also belongs to a solution curve defined for λ ( λ , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq196_HTML.gif, and the solutions on the curve are increasing in λ, but the nonexistence of positive solutions for λ < λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq197_HTML.gif and the local bifurcation at λ = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq161_HTML.gif exclude the possibility of another solution curve. Hence the positive solution is unique for all λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq113_HTML.gif. This completes the proof. □

Example 3.3

We consider the following cyclic system:
{ Δ u 1 + λ f 1 ( u 2 ) = 0 , x Ω , Δ u 2 + λ f 2 ( u 3 ) = 0 , x Ω , Δ u n + λ f n ( u 1 ) = 0 , x Ω , u 1 ( x ) = = u n ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ32_HTML.gif
(3.16)

where f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif are smooth real functions defined on R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq89_HTML.gif satisfying ( A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq90_HTML.gif), ( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq92_HTML.gif) and ( A 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq102_HTML.gif). 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
{ Δ u 1 + λ b 1 ( a 11 + u 1 ) p 11 ( a 1 n + u n ) p 1 n = 0 , x Ω , Δ u i + λ b i ( a i 1 + u 1 ) p i 1 ( a i n + u n ) p i n = 0 , x Ω , Δ u i + λ b n ( a n 1 + u 1 ) p n 1 ( a n n + u n ) p n n = 0 , x Ω , u 1 ( x ) = = u n ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ33_HTML.gif
(3.17)

where λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq2_HTML.gif is a positive parameter, b i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq198_HTML.gif, a i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq199_HTML.gif ( 1 i , j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq200_HTML.gif) are positive constants, p i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq201_HTML.gif ( 1 i , j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq200_HTML.gif) are nonnegative constants satisfying i = 1 n p i j < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq202_HTML.gif and Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq203_HTML.gif denotes a bounded domain of class C 2 , α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq204_HTML.gif, α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq205_HTML.gif. For the case N = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq206_HTML.gif, (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 N = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq206_HTML.gif, p 11 = p 22 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq207_HTML.gif, 0 < p 12 p 21 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq208_HTML.gif.

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 e ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq209_HTML.gif be the unique positive solution of
Δ e + 1 = 0 , x Ω , e ( x ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equf_HTML.gif
We construct a super-solution ( u ¯ 1 , , u ¯ n ) = M ( e ( x ) , , e ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq210_HTML.gif. There exists a suitable positive constant M such that
Δ ( M e ) + λ b i ( a i 1 + M e ) p i 1 ( a i n + M e ) p i n M + λ b i ( a ¯ i + M e ) p i 1 + + p i n 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equg_HTML.gif
where a ¯ i = max { a i 1 , , a i n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq211_HTML.gif. We construct a sub-solution in the form of ( u ̲ 1 , , u ̲ n ) = ( ε φ 1 , , ε φ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq212_HTML.gif, where ε will be specified later. Recall that φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq213_HTML.gif is the positive principal eigenfunction with φ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq100_HTML.gif. Now, for the i th equation of (3.17), we have
Δ ( ε φ 1 ) + λ b i ( a i 1 + ε φ 1 ) p i 1 ( a i n + ε φ 1 ) p i n ε λ 1 φ 1 + λ b i ( a ̲ i + ε φ 1 ) p i 1 + + p i n ε λ 1 φ 1 + λ b i ( ε φ 1 ) p i 1 + + p i n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equh_HTML.gif

where a ̲ i = min { a i 1 , , a i n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq214_HTML.gif. Hence ( u ̲ 1 , , u ̲ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq215_HTML.gif is a subsolution of (3.17), if we choose ε smaller, and it satisfies ε < ( λ b i / λ 1 ) 1 1 ( p i 1 + + p i n ) φ 1 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq216_HTML.gif. Hence if we choose ε smaller so that ( u ̲ 1 , , u ̲ n ) ( u ¯ 1 , , u ¯ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq217_HTML.gif, then ( u ̲ , v ̲ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq218_HTML.gif is a subsolution of (4.1). Therefore, there exists a positive solution ( u 1 , , u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq178_HTML.gif of (3.17) between the supersolution and the subsolution.

Next, we show the solution is stable. Letting f i ( U ) = b i j = 1 n ( a i j + u j ) p i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq219_HTML.gif, by simple calculation, we get
f i j ( U ) = b i p i j l = 1 , l j n ( a i l + u l ) p i l ( a i j + u i j ) p i j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equi_HTML.gif
and
f i ( U ) j = 1 n f i j ( U ) u j = b i j = 1 n ( a i j + u j ) p i j j = 1 n b i p i j l = 1 , l j n ( a i l + u l ) p i l ( a i j + u j ) p i j 1 = b i l = 1 n ( a i l + u l ) p i l [ 1 j = 1 n p i j a i j + u j u j ] > b i l = 1 n ( a i l + u l ) p i l [ 1 j = 1 n p i j ] > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equj_HTML.gif

So, f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq4_HTML.gif satisfies ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq64_HTML.gif), hence from Theorem 2.4, any positive solution of (3.17) is stable.

4 Application: Hölder continuous case

In this section, we consider that f i C α ( R ¯ + n ) C 1 ( R + n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq220_HTML.gif for α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq205_HTML.gif.

Example 4.1

Consider
{ Δ u + λ ( a u u 2 + u v ) = 0 , x Ω , Δ v + λ ( b v v 2 + u v ) = 0 , x Ω , u ( x ) = v ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equ34_HTML.gif
(4.1)

where λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq2_HTML.gif is a positive parameter, a , b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq221_HTML.gif.

We use the monotone method to prove the existence of a solution. For the supersolution, we choose ( u ¯ , v ¯ ) = ( a + b + 1 , a + b + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq222_HTML.gif. Then
Δ u ¯ + λ ( a u ¯ u ¯ 2 + u ¯ v ¯ ) λ [ a ( a + b + 1 ) ( a + b + 1 ) 2 + ( a + b + 1 ) ] λ ( a + b + 1 ) [ ( a + 1 ) ( a + b + 1 ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equk_HTML.gif
Similarly, we have 0 < v < ( a + b + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq223_HTML.gif. Then it is clear that ( u ¯ , v ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq224_HTML.gif is a supersolution of (4.1). We construct a sub-solution in the form of ( u ̲ , v ̲ ) = ( ε φ 1 , ε φ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq225_HTML.gif, where ε will be specified later. Recall that φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq213_HTML.gif is the positive principal eigenfunction with φ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq100_HTML.gif. Now, for the equation of u or the equation of v, we have
Δ ( ε φ 1 ) + λ ( a ε φ 1 ( ε φ 1 ) 2 + ε φ 1 ) = ε λ 1 φ 1 + λ ( a ε φ 1 ( ε φ 1 ) 2 + ε φ 1 ) = ε φ 1 [ λ ( a + 1 ) λ 1 λ ε φ 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_Equl_HTML.gif

Hence if we choose ε smaller and it satisfies ε < min { λ ( a + 1 ) λ 1 λ φ 1 , λ ( b + 1 ) λ 1 λ φ 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq226_HTML.gif, so that ( u ̲ , v ̲ ) ( u ¯ , v ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq227_HTML.gif, then ( u ̲ , v ̲ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq218_HTML.gif is a subsolution of (4.1). Therefore, there exists a positive solution ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq228_HTML.gif of (4.1) between the supersolution and the subsolution when λ > max { λ 1 / ( a + 1 ) , λ 1 / ( b + 1 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq229_HTML.gif. We remark that the stability defined in Section 2 can still be established for a nonlinear function f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq230_HTML.gif to become ∞ near Ω by using Remark 3.1 in [28]. Thus the positive solution ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq228_HTML.gif 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 u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq231_HTML.gif as proposed here, and they do not satisfy the conditions of Theorem 3.2, thus, the solution may not be unique.

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology
(2)
Y.Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University

References

  1. Chen ZY, Chern JL, Shi J, Tang YL: On the uniqueness and structure of solutions to a coupled elliptic system. J. Differ. Equ. 2010, 249: 3419-3442.MathSciNetView Article
  2. Chern JL, Tang YL, Lin CS, Shi J: Existence, uniqueness and stability of positive solutions to sublinear elliptic systems. Proc. R. Soc. Edinb. A 2011, 141: 45-64.MathSciNetView Article
  3. Cui, RH, Li, P, Shi, J, Wang, YW: Existence, uniqueness and stability of positive solutions for a class of semilinear elliptic systems. Topol. Methods Nonlinear Anal. (to appear)
  4. Shi, J, Shivaji, R: Exact multiplicity of positive solutions to cooperative elliptic systems. Preprint (2012)
  5. Korman P, Li Y, Ouyang T: Exact multiplicity results for boundary value problems with nonlinearities generalising cubic. Proc. R. Soc. Edinb. A 1996, 126: 599-616.MathSciNetView Article
  6. Korman P, Li Y, Ouyang T: An exact multiplicity result for a class of semilinear equations. Commun. Partial Differ. Equ. 1997, 22: 661-684.MathSciNetView Article
  7. Ouyang T, Shi J: Exact multiplicity of positive solutions for a class of semilinear problem. J. Differ. Equ. 1998, 146: 121-156.MathSciNetView Article
  8. Ouyang T, Shi J: Exact multiplicity of positive solutions for a class of semilinear problem, II. J. Differ. Equ. 1999, 158: 94-151.MathSciNetView Article
  9. Sirakov B: Notions of sublinearity and superlinearity for nonvariational elliptic systems. Discrete Contin. Dyn. Syst. 2005, 13: 163-174.MathSciNetView Article
  10. Dalmasso R: Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal. 2000, 39: 559-568.MathSciNetView Article
  11. Hai DD: Existence and uniqueness of solutions for quasilinear elliptic systems. Proc. Am. Math. Soc. 2005, 133: 223-228.MathSciNetView Article
  12. Hai DD: Uniqueness of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl. 2006, 313: 761-767.MathSciNetView Article
  13. Hai DD, Shivaji R: An existence result on positive solutions for a class of semilinear elliptic systems. Proc. R. Soc. Edinb. A 2004, 134: 137-141.MathSciNetView Article
  14. Troy WC: Symmetry properties in systems of semilinear elliptic equations. J. Differ. Equ. 1981, 42: 400-413.MathSciNetView Article
  15. Serrin J, Zou H: Existence of positive solutions of the Lane-Emden system. Atti Semin. Mat. Fis. Univ. Modena 1998, 46: 369-380. suppl.MathSciNet
  16. Serrin J, Zou H: Existence of positive entire solutions of elliptic Hamiltonian systems. Commun. Partial Differ. Equ. 1998, 23: 577-599.MathSciNet
  17. Korman P, Shi J: On Lane-Emden type systems. Discrete Contin. Dyn. Syst. 2005, suppl.: 510-517.MathSciNet
  18. Cui RH, Wang YW, Shi J: Uniqueness of the positive solution for a class of semilinear elliptic systems. Nonlinear Anal. 2007, 67: 1710-1714.MathSciNetView Article
  19. Cui RH, Shi J, Wang YW: Existence and uniqueness of positive solutions for a class of semilinear elliptic systems. Acta Math. Sin. Engl. Ser. 2011, 27: 1079-1090.MathSciNetView Article
  20. Masaaki M: Uniqueness and existence of positive solutions for some semilinear elliptic systems. Nonlinear Anal. 2004, 59: 993-999.MathSciNetView Article
  21. Sweers G:Strong positivity in C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq232_HTML.gif for elliptic systems. Math. Z. 1992, 209: 251-271.MathSciNetView Article
  22. Kato T Classics in Mathematics. In Perturbation Theory for Linear Operators. Springer, Berlin; 1995. Reprint of the 1980 edition.
  23. Serre D Graduate Texts in Mathematics 216. In Matrices: Theory and Applications. Springer, Berlin; 2002.
  24. Crandall MG, Rabinowitz PH: Bifurcation from simple eigenvalues. J. Funct. Anal. 1971, 8: 321-340.MathSciNetView Article
  25. Shi J: Blow up points of solution curves for a semilinear problem. Topol. Methods Nonlinear Anal. 2000, 15: 251-266.MathSciNet
  26. Ali J, Brown K, Shivaji R:Positive solutions for n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq23_HTML.gif elliptic systems with combined nonlinear effects. Differ. Integral Equ. 2011, 24(3-4):307-324.MathSciNet
  27. Lee EK, Shivaji R, Ye J:Classes of infinite semipositone n × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-74/MediaObjects/13661_2012_Article_324_IEq23_HTML.gif systems. Differ. Integral Equ. 2011, 24(3-4):361-370.MathSciNet
  28. Ambrosetti A, Brézis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994, 122: 519-543.MathSciNetView Article

Copyright

© Wu and Cui; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. 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.