Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

Boundary Value Problems20102010:207649

DOI: 10.1155/2010/207649

Received: 6 March 2010

Accepted: 11 August 2010

Published: 17 August 2010

Abstract

An exact multiplicity result of positive solutions for the boundary value problems http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq4_HTML.gif is achieved, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq5_HTML.gif is a positive parameter. Here the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq6_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq7_HTML.gif and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq9_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq10_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq11_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq12_HTML.gif is asymptotically linear and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq13_HTML.gif can change sign only once. The weight function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq14_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq15_HTML.gif and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq17_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq18_HTML.gif . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq19_HTML.gif lying in various intervals in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq20_HTML.gif . Moreover, we indicate how to extend the result to the general case.

1. Introduction

Consider the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq21_HTML.gif is a parameter and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq22_HTML.gif is a weight function.

The existence and multiplicity of positive solutions for ordinary differential equations have been studied extensively in many literatures, see, for example, [13] and references therein. Several different approaches, such as the Leray-Schauder theory, the fixed-point theory, the lower and upper solutions theory, and the shooting method etc has been applied in these literatures. In [4, 5], Ma and Thompson obtained the multiplicity results for a class of second-order two-point boundary value problems depending on a positive parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq23_HTML.gif by using bifurcation theory.

Exact multiplicity of positive solutions have been studied by many authors. See, for example, the papers by Korman et al. [6], Ouyang and Shi [7, 8], Shi [9], Korman and Ouyang [10, 11], Korman [12], Rynne [13], Bari and Rynne [14] (for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq24_HTML.gif th-order problems), as well as Korman and Li [15]. In these papers, bifurcation techniques are used. The basic method of proving their results can be divided into three steps: proving positivity of solutions of the linearized problems; studying the direction of bifurcation; showing uniqueness of solution curves.

Ouyang and Shi [7] obtained the curves of positive solutions for the semilinear problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ2_HTML.gif
(1.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq25_HTML.gif is the unit ball in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq27_HTML.gif . In [7], the following two cases were considered:
  1. (i)

    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq28_HTML.gif does not change its sign on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq29_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq30_HTML.gif changes its sign only once on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq31_HTML.gif .

     
Korman and Ouyang [10] studied the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ3_HTML.gif
(1.3)
under the conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq32_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ4_HTML.gif
(1.4)

They obtained a full description of the positive solution set of (1.3) and proved that all positive solutions of (1.3) lie on a single smooth solution curve bifurcating from the point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq33_HTML.gif and tending to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq34_HTML.gif in the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq35_HTML.gif plane. Condition (1.4) is very important to conclude the direction of bifurcation curve.

Of course a natural question is how about the structure of the positive solution set of (1.3) when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq36_HTML.gif changes its sign only once on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq37_HTML.gif ?

It is extremely difficult to answer such a question in general. So we shift our study to the problem (1.1) in this paper. We are interested in discussing the exact multiplicity of positive solutions of (1.1) with a weight function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq38_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq39_HTML.gif changes its sign only once on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq40_HTML.gif .

Suppose the following.

(H1) One has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq41_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq42_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq44_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq45_HTML.gif .

(H2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq46_HTML.gif is concave convex that is, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq47_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ5_HTML.gif
(1.5)

(H3) The limits http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq49_HTML.gif .

(H4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq50_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq52_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq53_HTML.gif

In this paper, we obtain exactly two disjoint smooth curves of positive solutions of (1.1) under conditions (H1)–(H4). According to this, we can conclude the existence and exact numbers of positive solutions of (1.1) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq54_HTML.gif lying in various intervals in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq55_HTML.gif .

Remark 1.1.

Korman and Ouyang [10] obtained the unique positive solution curve of (1.3) under the condition (1.4). However they gave no information when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq56_HTML.gif can change sign. In [7], they did not treat the case that the equation contains a weight function.

On the other hand, suppose the following.

(H1′) One has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq58_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq59_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq60_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq62_HTML.gif .

Remark 1.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq63_HTML.gif , then we know from the proof in [4] that the assumptions (H1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq64_HTML.gif ) and (H3) imply that the component of positive solutions from the trivial solution and the component from infinity are coincident. However, these two components are disjoint under the assumptions (H1) and (H3) (see [5]). Hence, the essential role is played by the fact of whether http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq65_HTML.gif possesses zeros in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq66_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq67_HTML.gif . In Section 3, we prove that (1.1) has exactly two positive solution curves which are disjoint and have no turning point on them (Theorem 3.8) under Conditions (H1)–(H4). And (1.1) has a unique positive solution curve with only one turning point (Theorem 3.9) if (H1) is replaced by (H1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq68_HTML.gif ). The condition (H4) is used to prove the positivity of solutions of the linearized problems of (1.1) and the direction of bifurcation.

Our main tool is the following bifurcation theorem of Crandall and Rabinowitz.

Theorem 1.3 (see [16]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq70_HTML.gif be Banach spaces. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq71_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq72_HTML.gif be a continuously differentiable mapping of an open neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq73_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq74_HTML.gif . Let the null-space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq75_HTML.gif be one dimensional and codim http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq76_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq77_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq78_HTML.gif is a complement of span http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq79_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq80_HTML.gif , then the solution of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq81_HTML.gif near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq82_HTML.gif forms a curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq83_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq84_HTML.gif is a continuously differentiable function near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq86_HTML.gif .

2. Notations and Preliminaries

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq87_HTML.gif with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ6_HTML.gif
(2.1)
and let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ7_HTML.gif
(2.2)
with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ8_HTML.gif
(2.3)
Set
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ9_HTML.gif
(2.4)
equipped with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ10_HTML.gif
(2.5)
Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq88_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ11_HTML.gif
(2.6)

Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq89_HTML.gif is a completely continuous operator.

Definition 2.1.

For a nontrivial solution of (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq90_HTML.gif is degenerate if the linearized problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ12_HTML.gif
(2.7)

has a nontrivial solution; otherwise, it is nondegenerate.

Lemma 2.2.

Let (H1) and (H4) hold. For any degenerate positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq91_HTML.gif of (1.1), the nontrivial solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq92_HTML.gif of (2.7) can be chosen as positive.

Proof.

The proof is motivated by Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq93_HTML.gif in [11].

Suppose to the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq94_HTML.gif has zeros on (0,1). Without loss of generality, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq95_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq97_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ13_HTML.gif
(2.8)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ14_HTML.gif
(2.9)
respectively. We claim that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq98_HTML.gif has at most one zero in (0,1). Otherwise, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq99_HTML.gif be the first two zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq100_HTML.gif . Then,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ15_HTML.gif
(2.10)
Multiplying (2.9) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq101_HTML.gif and (2.8) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq102_HTML.gif , subtracting, and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq103_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ16_HTML.gif
(2.11)
with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq104_HTML.gif to be specified. We denote the left side of (2.11) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq105_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq106_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq107_HTML.gif . Integrating by parts,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ17_HTML.gif
(2.12)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ18_HTML.gif
(2.13)
on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq108_HTML.gif From (2.10), (2.13), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq109_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ19_HTML.gif
(2.14)

Note that the right side of (2.11) is zero, which is a contradiction.

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq110_HTML.gif has at most one zero in (0,1). Suppose that there is one point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq111_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq112_HTML.gif . Then,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ20_HTML.gif
(2.15)

Repeating the above proof on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq113_HTML.gif , we can get similar contradiction.

Finally, integrating the differential equation in (2.13), we can choose
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ21_HTML.gif
(2.16)

In view of (H4), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq114_HTML.gif . So, the auxiliary function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq115_HTML.gif exists.

The following lemma is an important result in this paper.

Lemma 2.3.

Let (H1) and (H4) hold. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq116_HTML.gif is a degenerate positive solution of (1.1). Then, the following are considered.

(i) All solutions of (1.1) near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq117_HTML.gif have the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq118_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq119_HTML.gif and some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq120_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq121_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq122_HTML.gif .

(ii) One has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq123_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq124_HTML.gif is concave convex; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq125_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq126_HTML.gif is convex concave.

Proof.
  1. (i)

    The proof is standard. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq127_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq128_HTML.gif . We will show that the conditions of Theorem 1.3 hold.

     

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq129_HTML.gif is a degenerate positive solution of (1.1), we denote the corresponding solution of (2.7) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq130_HTML.gif . From Lemma 2.2 and the theory of compact disturbing of a Fredholm operator, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq131_HTML.gif is one dimensional and codim http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq132_HTML.gif .

Now, we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq133_HTML.gif . Suppose to the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq134_HTML.gif . Then, there is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq135_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ22_HTML.gif
(2.17)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ23_HTML.gif
(2.18)
Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq136_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ24_HTML.gif
(2.19)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ25_HTML.gif
(2.20)
Multiplying (2.17) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq137_HTML.gif and (2.19) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq138_HTML.gif , subtracting, and integrating on both sides, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ26_HTML.gif
(2.21)
However, the left side of (2.21) is equal to zero according to boundary conditions (2.18) and (2.20). This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq139_HTML.gif . According to Theorem 1.3, the result (i) holds.
  1. (ii)
    Substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq140_HTML.gif into (1.1), we obtain
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ27_HTML.gif
    (2.22)
     
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq141_HTML.gif then, by the implicit function theorem, the solution curve near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq142_HTML.gif is also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq143_HTML.gif Differentiating (2.22) twice with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq144_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ28_HTML.gif
(2.23)
Evaluating at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq145_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ29_HTML.gif
(2.24)
Multiplying (2.24) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq146_HTML.gif and (2.19) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq147_HTML.gif , subtracting, and integrating, we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ30_HTML.gif
(2.25)

According to (H1), (H4), and Lemma 2.2, we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq148_HTML.gif . Next, for the sign of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq149_HTML.gif , we consider the sign of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq150_HTML.gif .

We first prove that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ31_HTML.gif
(2.26)
Differentiating (1.1) and (2.19) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq151_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ32_HTML.gif
(2.27)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ33_HTML.gif
(2.28)
Multiplying, (2.27) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq152_HTML.gif and (2.28) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq153_HTML.gif , subtracting, and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq154_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ34_HTML.gif
(2.29)
with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq155_HTML.gif to be specified. Integrating by parts on the left side of (2.29),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ35_HTML.gif
(2.30)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ36_HTML.gif
(2.31)
From (2.29), we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ37_HTML.gif
(2.32)
Solving the equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq156_HTML.gif , we can choose the auxiliary function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ38_HTML.gif
(2.33)

Combining with (2.32), we obtain (2.26).

The following proof is motivated by the proof of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq157_HTML.gif in [8].

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq158_HTML.gif , (2.26) implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq159_HTML.gif must change sign. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq160_HTML.gif is concave convex, then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq161_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ39_HTML.gif
(2.34)
Next, we claim that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq162_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ40_HTML.gif
(2.35)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq163_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq164_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq165_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq166_HTML.gif has at least one zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq167_HTML.gif . Moreover, we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq168_HTML.gif has only one zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq169_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq170_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ41_HTML.gif
(2.36)
We get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ42_HTML.gif
(2.37)
since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq172_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq173_HTML.gif has more than one zero in (0, 1). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq174_HTML.gif be the last two zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq175_HTML.gif , then we say that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ43_HTML.gif
(2.38)
We first prove the above statement. On the contrary, suppose that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ44_HTML.gif
(2.39)
Consider the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ45_HTML.gif
(2.40)

Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq176_HTML.gif is a subsolution and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq177_HTML.gif is a supersolution of (2.40), respectively. Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq178_HTML.gif . By the strong maximum principle, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq179_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq180_HTML.gif . This contradicts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq181_HTML.gif . Hence, the statement holds.

Now let us consider the claim related to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq182_HTML.gif . Multiplying (2.36) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq183_HTML.gif and (2.37) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq184_HTML.gif , subtracting, and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq185_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ46_HTML.gif
(2.41)

since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq186_HTML.gif . Note that the left side is nonnegative. Such a contradiction implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq187_HTML.gif has only one zero in (0,1). By varying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq188_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq189_HTML.gif we can conclude the claim.

From the claim and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq190_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ47_HTML.gif
(2.42)

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq191_HTML.gif from (2.25).

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq192_HTML.gif is convex concave, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq193_HTML.gif with a similar proof.

3. The Main Results and the Proofs

In this section we state our main results and proofs.

Definition 3.1.

Define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ48_HTML.gif
(3.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq194_HTML.gif is the first eigenvalue of the corresponding linear problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ49_HTML.gif
(3.2)

Remark 3.2.

It is well known that the eigenvalues of (3.2) are given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ50_HTML.gif
(3.3)

For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq195_HTML.gif , algebraic multiplicity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq196_HTML.gif is equal to 1, and the corresponding eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq197_HTML.gif has exactly http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq198_HTML.gif simple zeros in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq199_HTML.gif .

Definition 3.3 (see [7]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq200_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq201_HTML.gif is said to be superlinear (resp., sublinear) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq202_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq203_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq204_HTML.gif ) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq205_HTML.gif . And http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq206_HTML.gif is said to be sup-sub (resp., sub-sup) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq207_HTML.gif if there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq208_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq209_HTML.gif is superlinear (resp., sublinear) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq210_HTML.gif , and superlinear (resp., sublinear) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq211_HTML.gif .

Lemma 3.4.
  1. (i)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq212_HTML.gif and (H4) hold. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq213_HTML.gif is a point where a bifurcation from the trivial solutions occurs and that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq214_HTML.gif is the corresponding positive solution bifurcation curve of (1.1). If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq215_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq216_HTML.gif is superlinear (resp., sublinear) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq217_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq218_HTML.gif tends to the left (resp., the right) near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq219_HTML.gif .

     
  2. (ii)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq220_HTML.gif and (H4) hold. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq221_HTML.gif is a point where a bifurcation from infinity occurs and that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq222_HTML.gif is the corresponding positive solution bifurcation curve of (1.1). If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq223_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq224_HTML.gif is superlinear (resp., sublinear) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq226_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq227_HTML.gif ) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq228_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq229_HTML.gif tends to the right (resp., the left) near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq230_HTML.gif .

     

Proof.

The proof is similar to that of Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq231_HTML.gif in [7], so we omit it.

Lemma 3.5.

Let (H1)–(H4) hold, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq232_HTML.gif be a bounded and closed interval, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq233_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq234_HTML.gif are positive solutions of (1.1). Then,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq235_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq236_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq237_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq238_HTML.gif .

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq239_HTML.gif be such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ51_HTML.gif
(3.4)
Clearly,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ52_HTML.gif
(3.5)
Let us consider
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ53_HTML.gif
(3.6)

as a bifurcation problem from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq240_HTML.gif . Note that (3.6) is the same as to (1.1). From Remark 3.2 and the standard bifurcation theorem from simple eigenvalues [17], we have (i).

Let us consider
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ54_HTML.gif
(3.7)

as a bifurcation problem from infinity. Note that (3.7) is also the same as to (1.1). The proof of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq241_HTML.gif in [5] ensures that (ii) is correct.

Lemma 3.6.

Let (H1), (H4) hold. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq242_HTML.gif is a positive solution of (1.1). Then,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ55_HTML.gif
(3.8)

Proof.

Suppose to the contrary that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ56_HTML.gif
(3.9)
By (1.1) and (H1), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq243_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq244_HTML.gif . By the uniqueness of solutions of initial value problem, the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ57_HTML.gif
(3.10)

has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq245_HTML.gif . This contradicts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq246_HTML.gif .

The following Lemma is an interesting and important result.

Lemma 3.7.

Let (H1)–(H4) hold. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq247_HTML.gif is a positive solution of (1.1), then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq248_HTML.gif is nondegenerate.

Proof.

From conditions (H1)–(H3), we can check easily that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ58_HTML.gif
(3.11)
In fact, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq249_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ59_HTML.gif
(3.12)

since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq250_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq251_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq252_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq253_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq254_HTML.gif . This together with (3.12) implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq255_HTML.gif and (3.11).

Now, we give the proof in two cases.

Case I ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq256_HTML.gif ).

On the contrary, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq257_HTML.gif is a degenerate solution with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq258_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq259_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq260_HTML.gif . By (3.11), we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ60_HTML.gif
(3.13)
Multiplying (1.1) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq261_HTML.gif and (2.7) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq262_HTML.gif , subtracting, and integrating, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ61_HTML.gif
(3.14)

By Lemma 2.2, (3.13), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq263_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq264_HTML.gif , the right side of (3.14) is negative. This is a contradiction.

Case II ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq265_HTML.gif ).

On the contrary, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq266_HTML.gif is a degenerate solution with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq267_HTML.gif . According to Lemmas 2.2 and 2.3, we know that all solutions of (1.1) near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq268_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq269_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq270_HTML.gif and some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq271_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq272_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq273_HTML.gif . It follows that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq274_HTML.gif close to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq275_HTML.gif we have two solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq277_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq278_HTML.gif strictly increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq280_HTML.gif with strictly decreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq281_HTML.gif . We will show that the lower branch http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq282_HTML.gif is strictly increasing for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq283_HTML.gif .

Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq284_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq285_HTML.gif close to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq286_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq287_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq288_HTML.gif be the largest http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq289_HTML.gif where this inequality is violated; that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq290_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq291_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq292_HTML.gif . Differentiating (1.1) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq293_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ62_HTML.gif
(3.15)
We can extend evenly http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq294_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq295_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq296_HTML.gif , then we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ63_HTML.gif
(3.16)

By the strong maximum principle, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq297_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq298_HTML.gif . This contradicts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq299_HTML.gif .

By Lemma 2.3, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq300_HTML.gif at every degenerate positive solution. Hence, there is no degenerate positive solution on the lower branch http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq301_HTML.gif . However, the lower branch has no place to go. In fact, there must exist some positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq302_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq303_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq304_HTML.gif lying on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq305_HTML.gif . Hence, the lower branch cannot go to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq306_HTML.gif axis. And it also cannot go to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq307_HTML.gif axis, since (1.1) has only the trivial solution at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq308_HTML.gif .

So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq309_HTML.gif is nondegenerate.

Our main result is the following.

Theorem 3.8.

Let (H1)–(H4) hold. Then the following are considered.

(i) All positive solutions of (1.1) lie on two continuous curves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq310_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq311_HTML.gif without intersection. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq312_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq313_HTML.gif to infinity and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq314_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq315_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq316_HTML.gif to infinity and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq317_HTML.gif . There is no degenerate positive solution on these curves. For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq318_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq319_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq320_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq321_HTML.gif .

(ii) Equation (1.1) has no positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq322_HTML.gif has exactly one positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq323_HTML.gif but and has exactly two positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq324_HTML.gif (see Figure 1).

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Fig1_HTML.jpg

Figure 1

Proof.
  1. (i)
    Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq325_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq326_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq327_HTML.gif . From Lemma 3.5(i) and the standard Crandall and Rabinowitz theorem on local bifurcation from simple eigenvalues [17], http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq328_HTML.gif is the unique point where a bifurcation from the trivial solution occurs. Moreover, by Lemma 3.4, the curve bifurcates to the right. We denote this local curve by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq329_HTML.gif and continue http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq330_HTML.gif to the right as long as it is possible. Meanwhile, by Lemma 3.6, there is no positive solution of (1.1) which has the maximum value http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq331_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq332_HTML.gif . So, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq333_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq334_HTML.gif . From (1.1), we have
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ64_HTML.gif
    (3.17)
     

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq335_HTML.gif . Obviously, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq336_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq337_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq338_HTML.gif is bounded. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq339_HTML.gif cannot blow up.

On the other hand, Lemma 3.7 and the implicit function theorem ensure that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq340_HTML.gif cannot stop at a finite point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq341_HTML.gif .

From the above discussion, we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq342_HTML.gif can be extended continuously to infinity and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq343_HTML.gif . Meanwhile, the maximum values of all positive solutions of (1.1) are less than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq344_HTML.gif .

Now, we consider positive solutions of (1.1), for which the maximum value on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq345_HTML.gif is greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq346_HTML.gif .

Let us return to consider (3.6) as the bifurcation problem from infinity. Note that (3.6) is also the same as to (1.1). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq347_HTML.gif by Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq348_HTML.gif and Corollary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq349_HTML.gif in [18], there exists a subcontinuum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq350_HTML.gif of positive solutions of (3.6) which meets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq351_HTML.gif . Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq352_HTML.gif as an interval such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq353_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq354_HTML.gif as a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq355_HTML.gif whose projection on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq356_HTML.gif lies in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq357_HTML.gif and whose projection on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq358_HTML.gif is bounded away from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq359_HTML.gif . Then, there exists a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq360_HTML.gif such that any positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq361_HTML.gif of (1.1) satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq362_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq363_HTML.gif and some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq364_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq365_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq366_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq367_HTML.gif denotes the normalized eigenvector of (3.2) corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq368_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq369_HTML.gif

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq370_HTML.gif is a continuous curve, and we denote it by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq371_HTML.gif . It tends to the right from Lemma 3.4(ii). From Lemma 3.7 and the implicit function theorem, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq372_HTML.gif can be continued to a maximal interval of definition over the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq373_HTML.gif axis. We claim that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq374_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq375_HTML.gif cannot blow up if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq376_HTML.gif is bounded. In fact, suppose that there exists a positive solutions sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq377_HTML.gif of (1.1) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq378_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq379_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq380_HTML.gif . Then, by Lemma 3.5(ii), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq381_HTML.gif . This is a contradiction. On the other hand, the implicit function theorem implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq382_HTML.gif cannot stop at a finite point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq383_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq384_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq385_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq386_HTML.gif .

Finally, we show that both curves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq387_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq388_HTML.gif are the only two positive solutions curves of (1.1). On the contrary, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq389_HTML.gif is a positive solution of (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq390_HTML.gif . Without loss of generality, assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq391_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq392_HTML.gif is nondegenerate, so we can extend it to form a curve. We denote this curve by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq393_HTML.gif and the corresponding maximal interval of definition by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq394_HTML.gif . Since all positive solutions of (1.1) are nondegenerate, according to the implicit function theorem, we must have that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ65_HTML.gif
(3.18)

It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq395_HTML.gif from Lemma 3.5(ii). But all solutions near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq396_HTML.gif can be parameterized by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq397_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq398_HTML.gif and some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq399_HTML.gif ; thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq400_HTML.gif . This contradicts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq401_HTML.gif .

Similarly, we can show that every positive solution of (1.1), the maximum value on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq402_HTML.gif of which is less than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq403_HTML.gif lies on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq404_HTML.gif .
  1. (ii)

    The result (ii) is a corollary of (i).

     

Next, we will give directly other theorems. Their proofs are similar to that of Theorem 3.8. So, we omit them.

Theorem 3.9.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq405_HTML.gif and (H2)–(H4) hold. Then, the following are considered.

(i) All positive solutions of (1.1) lie on a single continuous curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq406_HTML.gif . And http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq407_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq408_HTML.gif to the right to a unique degenerate positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq409_HTML.gif of (1.1), then it tends to the left to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq410_HTML.gif .

(ii) Equation (1.1) has no positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq411_HTML.gif , and has exactly one positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq412_HTML.gif , and has exactly two positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq413_HTML.gif (see Figure 2).

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Fig2_HTML.jpg

Figure 2

Remark 3.10.

In fact, if we reverse the inequalities in (H1), (H1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq414_HTML.gif ), (H2), we will obtain corresponding results similar to Theorems 3.8 and 3.9.

Also using the method in this paper, we can obtain the exact numbers of positive solutions for the Dirichlet problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ66_HTML.gif
(3.19)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq415_HTML.gif is a parameter. We assume that

(H) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq417_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq418_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq419_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq420_HTML.gif .

Definition 3.11 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq421_HTML.gif .

Define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ67_HTML.gif
(3.20)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq422_HTML.gif is the first eigenvalue of the corresponding linear problem of (3.19).

Theorem 3.12.

Let (H1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq423_HTML.gif ), (H2), (H3), and (H4 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq424_HTML.gif ) hold. Then, the following are considered.

(i) All positive solutions of (3.19) lie on a single continuous curve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq425_HTML.gif . And http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq426_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq427_HTML.gif to the right to a unique degenerate positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq428_HTML.gif of (3.19), then it tends to the left to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq429_HTML.gif .

(ii) Equation (1.1) has no positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq430_HTML.gif but has exactly one positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq431_HTML.gif and has exactly two positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq432_HTML.gif .

Theorem 3.13.

Let (H1), (H2), (H3), (H4 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq433_HTML.gif ) hold. Then

(i) All positive solutions of (3.19) lie on two continuous curves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq434_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq435_HTML.gif without intersection. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq436_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq437_HTML.gif to infinity and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq438_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq439_HTML.gif bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq440_HTML.gif to infinity and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq441_HTML.gif . There is no degenerate positive solution on these curves. For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq442_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq443_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq444_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq445_HTML.gif .

(ii) Equation (3.19) has no positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq446_HTML.gif , and has exactly one positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq447_HTML.gif , and has exactly two positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq448_HTML.gif .

Remark 3.14.

Theorems 3.12 and 3.13 extend the main result Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq449_HTML.gif in [10], where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq450_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq451_HTML.gif .

4. Examples

In this section, we give some examples.

Example 4.1.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ68_HTML.gif
(4.1)

Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq452_HTML.gif satisfies (H1), (H2), and (H3). Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq453_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq454_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq455_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq456_HTML.gif .

Example 4.2.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ69_HTML.gif
(4.2)

Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq457_HTML.gif satisfies (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq458_HTML.gif ), (H2), and (H3). Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq459_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq460_HTML.gif .

Example 4.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq461_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq462_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq463_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq464_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq465_HTML.gif is a large enough constant. Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq466_HTML.gif satisfies (H4). On the other hand, functions which satisfy (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq467_HTML.gif ) can be found easily.

Declarations

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. An is supported by SRFDP (no. 20060736001), YJ2009-16 A06/1020K096019, 11YZ225. Luo is supported by grant no. L09DJY065.

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Institute of Technology
(2)
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics

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© Yulian An and Hua Luo. 2010

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