Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq4_HTML.gif is achieved, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq6_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq7_HTML.gif and satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq9_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq10_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq11_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq12_HTML.gif is asymptotically linear and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq14_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq15_HTML.gif and satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq17_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq19_HTML.gif lying in various intervals in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq21_HTML.gif is a parameter and https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq25_HTML.gif is the unit ball in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq26_HTML.gif and https://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)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq28_HTML.gif does not change its sign on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq29_HTML.gif ; (ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq30_HTML.gif changes its sign only once on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ3_HTML.gif
(1.3)
under the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq32_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq33_HTML.gif and tending to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq34_HTML.gif in the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq36_HTML.gif changes its sign only once on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq38_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq39_HTML.gif changes its sign only once on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq40_HTML.gif .

Suppose the following.

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

(H2) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq47_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ5_HTML.gif
(1.5)

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

(H4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq50_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq52_HTML.gif if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq54_HTML.gif lying in various intervals in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq58_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq59_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq60_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq62_HTML.gif .

Remark 1.2.

If https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq65_HTML.gif possesses zeros in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq66_HTML.gif https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq70_HTML.gif be Banach spaces. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq71_HTML.gif and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq73_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq74_HTML.gif . Let the null-space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq75_HTML.gif be one dimensional and codim https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq76_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq77_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq78_HTML.gif is a complement of span https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq79_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq80_HTML.gif , then the solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq81_HTML.gif near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq82_HTML.gif forms a curve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq83_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq84_HTML.gif is a continuously differentiable function near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq86_HTML.gif .

2. Notations and Preliminaries

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

Then, https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq90_HTML.gif is degenerate if the linearized problem
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq91_HTML.gif of (1.1), the nontrivial solution https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq95_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq97_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ13_HTML.gif
(2.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ14_HTML.gif
(2.9)
respectively. We claim that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq99_HTML.gif be the first two zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq100_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ15_HTML.gif
(2.10)
Multiplying (2.9) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq101_HTML.gif and (2.8) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq102_HTML.gif , subtracting, and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq103_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ16_HTML.gif
(2.11)
with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq105_HTML.gif and a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq106_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq107_HTML.gif . Integrating by parts,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ17_HTML.gif
(2.12)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ18_HTML.gif
(2.13)
on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq108_HTML.gif From (2.10), (2.13), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq109_HTML.gif , we have
https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq111_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq112_HTML.gif . Then,
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ21_HTML.gif
(2.16)

In view of (H4), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq114_HTML.gif . So, the auxiliary function https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq117_HTML.gif have the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq118_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq119_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq120_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq121_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq122_HTML.gif .

(ii) One has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq123_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq124_HTML.gif is concave convex; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq125_HTML.gif if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq127_HTML.gif be such that https://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 https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq131_HTML.gif is one dimensional and codim https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq132_HTML.gif .

Now, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq133_HTML.gif . Suppose to the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq134_HTML.gif . Then, there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq135_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ22_HTML.gif
(2.17)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ23_HTML.gif
(2.18)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq136_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ24_HTML.gif
(2.19)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ25_HTML.gif
(2.20)
Multiplying (2.17) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq137_HTML.gif and (2.19) by https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq140_HTML.gif into (1.1), we obtain
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ27_HTML.gif
    (2.22)
     
Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq142_HTML.gif is also https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq144_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ28_HTML.gif
(2.23)
Evaluating at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq145_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ29_HTML.gif
(2.24)
Multiplying (2.24) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq146_HTML.gif and (2.19) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq147_HTML.gif , subtracting, and integrating, we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq148_HTML.gif . Next, for the sign of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq149_HTML.gif , we consider the sign of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq150_HTML.gif .

We first prove that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq151_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ32_HTML.gif
(2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ33_HTML.gif
(2.28)
Multiplying, (2.27) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq152_HTML.gif and (2.28) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq153_HTML.gif , subtracting, and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq154_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ34_HTML.gif
(2.29)
with https://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),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ35_HTML.gif
(2.30)
Let
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ37_HTML.gif
(2.32)
Solving the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq156_HTML.gif , we can choose the auxiliary function
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq157_HTML.gif in [8].

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

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq176_HTML.gif is a subsolution and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq179_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq180_HTML.gif . This contradicts https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq182_HTML.gif . Multiplying (2.36) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq183_HTML.gif and (2.37) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq184_HTML.gif , subtracting, and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq185_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ46_HTML.gif
(2.41)

since https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq188_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq190_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ47_HTML.gif
(2.42)

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

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq192_HTML.gif is convex concave, then https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ48_HTML.gif
(3.1)
where https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ50_HTML.gif
(3.3)

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq195_HTML.gif , algebraic multiplicity of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq197_HTML.gif has exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq198_HTML.gif simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq199_HTML.gif .

Definition 3.3 (see [7]).

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

Lemma 3.4.
  1. (i)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq212_HTML.gif and (H4) hold. Suppose that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq215_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq216_HTML.gif is superlinear (resp., sublinear) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq217_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq219_HTML.gif .

     
  2. (ii)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq220_HTML.gif and (H4) hold. Suppose that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq223_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq224_HTML.gif is superlinear (resp., sublinear) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq226_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq227_HTML.gif ) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq228_HTML.gif , then https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq233_HTML.gif . Suppose that https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq235_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq236_HTML.gif ,

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq239_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ51_HTML.gif
(3.4)
Clearly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ52_HTML.gif
(3.5)
Let us consider
https://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 https://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
https://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 https://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 https://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,
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq243_HTML.gif . Note that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ57_HTML.gif
(3.10)

has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq245_HTML.gif . This contradicts https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ58_HTML.gif
(3.11)
In fact, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq249_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ59_HTML.gif
(3.12)

since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq250_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq251_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq253_HTML.gif , if https://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 https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq256_HTML.gif ).

On the contrary, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq257_HTML.gif is a degenerate solution with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq258_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq259_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq260_HTML.gif . By (3.11), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ60_HTML.gif
(3.13)
Multiplying (1.1) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq261_HTML.gif and (2.7) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq262_HTML.gif , subtracting, and integrating, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq263_HTML.gif for all https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq265_HTML.gif ).

On the contrary, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq266_HTML.gif is a degenerate solution with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq268_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq269_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq270_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq271_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq272_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq273_HTML.gif . It follows that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq274_HTML.gif close to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq275_HTML.gif we have two solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq277_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq278_HTML.gif strictly increasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq280_HTML.gif with strictly decreasing in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq282_HTML.gif is strictly increasing for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq283_HTML.gif .

Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq284_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq285_HTML.gif close to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq286_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq287_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq288_HTML.gif be the largest https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq290_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq291_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq292_HTML.gif . Differentiating (1.1) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq293_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ62_HTML.gif
(3.15)
We can extend evenly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq294_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq295_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq296_HTML.gif , then we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq297_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq298_HTML.gif . This contradicts that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq299_HTML.gif .

By Lemma 2.3, we get https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq302_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq303_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq304_HTML.gif lying on https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq308_HTML.gif .

So, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq310_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq311_HTML.gif without intersection. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq312_HTML.gif bifurcates from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq313_HTML.gif to infinity and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq314_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq315_HTML.gif bifurcates from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq316_HTML.gif to infinity and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq319_HTML.gif , and for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq320_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq322_HTML.gif has exactly one positive solution for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq324_HTML.gif (see Figure 1).

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

Figure 1

Proof.
  1. (i)
    Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq325_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq326_HTML.gif , then https://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], https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq329_HTML.gif and continue https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq331_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq332_HTML.gif . So, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq333_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq334_HTML.gif . From (1.1), we have
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ64_HTML.gif
    (3.17)
     

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq335_HTML.gif . Obviously, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq336_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq337_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq338_HTML.gif is bounded. Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq340_HTML.gif cannot stop at a finite point https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq345_HTML.gif is greater than https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq347_HTML.gif by Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq348_HTML.gif and Corollary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq349_HTML.gif in [18], there exists a subcontinuum https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq351_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq352_HTML.gif as an interval such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq353_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq354_HTML.gif as a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq355_HTML.gif whose projection on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq356_HTML.gif lies in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq357_HTML.gif and whose projection on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq358_HTML.gif is bounded away from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq359_HTML.gif . Then, there exists a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq360_HTML.gif such that any positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq361_HTML.gif of (1.1) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq362_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq363_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq364_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq365_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq366_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq368_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq369_HTML.gif

Hence, https://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 https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq373_HTML.gif axis. We claim that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq374_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq375_HTML.gif cannot blow up if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq377_HTML.gif of (1.1) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq378_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq379_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq380_HTML.gif . Then, by Lemma 3.5(ii), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq382_HTML.gif cannot stop at a finite point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq383_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq384_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq385_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq386_HTML.gif .

Finally, we show that both curves https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq387_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq390_HTML.gif . Without loss of generality, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq391_HTML.gif . Note that https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ65_HTML.gif
(3.18)

It follows that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq396_HTML.gif can be parameterized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq397_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq398_HTML.gif and some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq399_HTML.gif ; thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq400_HTML.gif . This contradicts that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq402_HTML.gif of which is less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq403_HTML.gif lies on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq406_HTML.gif . And https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq407_HTML.gif bifurcates from https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq413_HTML.gif (see Figure 2).

https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ66_HTML.gif
(3.19)

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

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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq423_HTML.gif ), (H2), (H3), and (H4 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq425_HTML.gif . And https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq426_HTML.gif bifurcates from https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq434_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq435_HTML.gif without intersection. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq436_HTML.gif bifurcates from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq437_HTML.gif to infinity and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq438_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq439_HTML.gif bifurcates from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq440_HTML.gif to infinity and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq442_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq443_HTML.gif , and for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq444_HTML.gif , https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq449_HTML.gif in [10], where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq450_HTML.gif for https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ68_HTML.gif
(4.1)

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

Example 4.2.

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

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

Example 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq461_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq462_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq463_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq464_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_IEq465_HTML.gif is a large enough constant. Then, https://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 https://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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