Open Access

Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities

Boundary Value Problems20102011:715836

DOI: 10.1155/2011/715836

Received: 8 May 2010

Accepted: 23 September 2010

Published: 28 September 2010

Abstract

We consider the nonlinear eigenvalue problems https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq7_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq8_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq10_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq11_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq13_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq14_HTML.gif . We investigate the global structure of nodal solutions by using the Rabinowitz's global bifurcation theorem.

1. Introduction

We study the global structure of nodal solutions of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ2_HTML.gif
(1.2)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq16_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq17_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq18_HTML.gif is a positive parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq19_HTML.gif

In the case that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq20_HTML.gif the global structure of nodal solutions of nonlinear second-order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq21_HTML.gif -point eigenvalue problems (1.1), (1.2) have been extensively studied; see [15] and the references therein. However, relatively little is known about the global structure of solutions in the case that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq22_HTML.gif and few global results were found in the available literature when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq23_HTML.gif The likely reason is that the global bifurcation techniques cannot be used directly in the case. On the other hand, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq24_HTML.gif -point boundary value condition (1.2) is concerned, the discussion is more difficult since the problem is nonsymmetric and the corresponding operator is disconjugate. In [6], we discussed the global structure of positive solutions of (1.1), (1.2) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq25_HTML.gif However, to the best of our knowledge, there is no paper to discuss the global structure of nodal solutions of (1.1), (1.2) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq26_HTML.gif

In this paper, we obtain a complete description of the global structure of nodal solutions of (1.1), (1.2) under the following assumptions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq28_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq29_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq30_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq32_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq33_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq34_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq36_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq38_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq39_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ3_HTML.gif
(1.3)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ4_HTML.gif
(1.4)
with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ5_HTML.gif
(1.5)
respectively. Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq40_HTML.gif by setting
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ6_HTML.gif
(1.6)

Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq41_HTML.gif has a bounded inverse https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq42_HTML.gif and the restriction of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq43_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq44_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq45_HTML.gif is a compact and continuous operator; see [1, 2, 6].

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq46_HTML.gif function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq47_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq48_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq49_HTML.gif is a simple zero of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq50_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq51_HTML.gif For any integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq52_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq53_HTML.gif define sets https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq54_HTML.gif consisting of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq55_HTML.gif satisfying the following conditions:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq57_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq58_HTML.gif has only simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq59_HTML.gif and has exactly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq60_HTML.gif zeros in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq61_HTML.gif ;

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq64_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq65_HTML.gif has only simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq66_HTML.gif and has exactly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq67_HTML.gif zeros in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq68_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq69_HTML.gif has a zero strictly between each two consecutive zeros of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq70_HTML.gif .

Remark 1.1.

Obviously, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq71_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq72_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq73_HTML.gif The sets https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq74_HTML.gif are open in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq75_HTML.gif and disjoint.

Remark 1.2.

The nodal properties of solutions of nonlinear Sturm-Liouville problems with separated boundary conditions are usually described in terms of sets similar to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq76_HTML.gif see [7]. However, Rynne [1] stated that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq77_HTML.gif are more appropriate than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq78_HTML.gif when the multipoint boundary condition (1.2) is considered.

Next, we consider the eigenvalues of the linear problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ7_HTML.gif
(1.7)

We call the set of eigenvalues of (1.7) the spectrum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq79_HTML.gif and denote it by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq80_HTML.gif . The following lemmas or similar results can be found in [13].

Lemma 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq81_HTML.gif hold. The spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq82_HTML.gif consists of a strictly increasing positive sequence of eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq83_HTML.gif with corresponding eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq84_HTML.gif In addition,

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq85_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq86_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq88_HTML.gif is strictly positive on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq89_HTML.gif

We can regard the inverse operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq90_HTML.gif as an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq91_HTML.gif In this setting, each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq92_HTML.gif is a characteristic value of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq93_HTML.gif with algebraic multiplicity defined to be dim https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq94_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq95_HTML.gif denotes null-space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq96_HTML.gif is the identity on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq97_HTML.gif

Lemma 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq98_HTML.gif hold. For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq99_HTML.gif the algebraic multiplicity of the characteristic value https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq100_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq101_HTML.gif is equal to 1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq102_HTML.gif under the product topology. As in [7], we add the points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq103_HTML.gif to our space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq104_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq105_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq106_HTML.gif denote the closure of set of those solutions of (1.1), (1.2) which belong to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq107_HTML.gif The main results of this paper are the following.

Theorem 1.5.

Let (A1)–(A4) hold.

(a)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq108_HTML.gif then there exists a subcontinuum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq109_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq110_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq111_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ8_HTML.gif
(1.8)
(b)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq112_HTML.gif then there exists a subcontinuum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq113_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq114_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ9_HTML.gif
(1.9)

(c)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq115_HTML.gif then there exists a subcontinuum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq116_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq117_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq118_HTML.gif is a bounded closed interval, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq119_HTML.gif approaches https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq120_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq121_HTML.gif

Theorem 1.6.

Let (A1)–(A4) hold.

(a)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq122_HTML.gif , then (1.1), (1.2) has at least one solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq123_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq124_HTML.gif

(b)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq125_HTML.gif , then (1.1), (1.2) has at least one solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq126_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq127_HTML.gif

(c)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq128_HTML.gif then there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq129_HTML.gif such that (1.1), (1.2) has at least two solutions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq130_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq131_HTML.gif .

We will develop a bifurcation approach to treat the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq132_HTML.gif . Crucial to this approach is to construct a sequence of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq133_HTML.gif which is asymptotic linear at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq134_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ10_HTML.gif
(1.10)
By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq135_HTML.gif via Rabinowitz's global bifurcation theorem [8], and this enables us to find unbounded components https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq136_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ11_HTML.gif
(1.11)

The rest of the paper is organized as follows. Section 2 contains some preliminary propositions. In Section 3, we use the global bifurcation theorems to analyse the global behavior of the components of nodal solutions of (1.1), (1.2).

2. Preliminaries

Definition 2.1 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq137_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq138_HTML.gif a family of subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq139_HTML.gif Then the superior limit https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq140_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq141_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ12_HTML.gif
(2.1)

Lemma 2.2 (see [9]).

Each connected subset of metric space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq142_HTML.gif is contained in a component, and each connected component of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq143_HTML.gif is closed.

Lemma 2.3 (see [6]).

Assume that

(i)there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq145_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq146_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq147_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq148_HTML.gif ;

(iii)for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq149_HTML.gif is a relative compact set of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq150_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ13_HTML.gif
(2.2)

Then there exists an unbounded connected component https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq151_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq153_HTML.gif .

Define the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq154_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ14_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ15_HTML.gif
(2.4)

It is easy to verify that the following lemma holds.

Lemma 2.4.

Assume that (A1)-(A2) hold. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq155_HTML.gif is completely continuous.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq156_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ16_HTML.gif
(2.5)

Lemma 2.5.

Let (A1)-(A2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq157_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ17_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq158_HTML.gif

Proof.

The proof is similar to that of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq159_HTML.gif in [6]; we omit it.

Lemma 2.6.

Let (A1)-(A2) hold, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq160_HTML.gif is a sequence of solutions of (1.1), (1.2). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq161_HTML.gif for some constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq162_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq163_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ18_HTML.gif
(2.7)

Proof.

From the relation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq164_HTML.gif we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq165_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ19_HTML.gif
(2.8)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq166_HTML.gif is bounded whenever https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq167_HTML.gif is bounded.

3. Proof of the Main Results

For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq168_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq169_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ20_HTML.gif
(3.1)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq170_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ21_HTML.gif
(3.2)
By (A3), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ22_HTML.gif
(3.3)
Now let us consider the auxiliary family of the equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ23_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ24_HTML.gif
(3.5)

Lemma 3.1 (see [1, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq171_HTML.gif ]).

Let (A1), (A2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq172_HTML.gif is a nontrivial solution of (3.4), (3.5), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq173_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq174_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq175_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ25_HTML.gif
(3.6)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ26_HTML.gif
(3.7)
Let us consider
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ27_HTML.gif
(3.8)

as a bifurcation problem from the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq176_HTML.gif

Equation (3.8) can be converted to the equivalent equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ28_HTML.gif
(3.9)

Further we note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq177_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq178_HTML.gif near https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq179_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq180_HTML.gif

The results of Rabinowitz [8] for (3.8) can be stated as follows. For each integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq181_HTML.gif , there exists a continuum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq182_HTML.gif of solutions of (3.8) joining https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq183_HTML.gif to infinity in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq184_HTML.gif Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq185_HTML.gif

Proof of Theorem 1.5.

Let us verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq186_HTML.gif satisfies all of the conditions of Lemma 2.3.

Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ29_HTML.gif
(3.10)
condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq187_HTML.gif in Lemma 2.3 is satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq188_HTML.gif . Obviously
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ30_HTML.gif
(3.11)

and accordingly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq189_HTML.gif holds. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq190_HTML.gif can be deduced directly from the Arzela-Ascoli Theorem and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq191_HTML.gif . Therefore, the superior limit of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq192_HTML.gif , contains an unbounded connected component https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq193_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq194_HTML.gif .

From the condition (A2), applying Lemma 2.2 with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq195_HTML.gif in [10], we can show that the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ31_HTML.gif
(3.12)

has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq196_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq198_HTML.gif . Therefore, any nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq199_HTML.gif of (1.1), (1.2) has only simple zeros in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq201_HTML.gif . Meanwhile, (A1) implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq202_HTML.gif [1, proposition 4.1]. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq203_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq204_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq205_HTML.gif by (1.1) and (1.2).

We divide the proof into three cases.

Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq206_HTML.gif ).

In this case, we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq207_HTML.gif .

Assume on the contrary that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ32_HTML.gif
(3.13)
then there exists a sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq208_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ33_HTML.gif
(3.14)
for some positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq209_HTML.gif depending not on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq210_HTML.gif . From Lemma 2.6, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ34_HTML.gif
(3.15)
Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq211_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq212_HTML.gif Now, choosing a subsequence and relabelling if necessary, it follows that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq213_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ35_HTML.gif
(3.16)
such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ36_HTML.gif
(3.17)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq214_HTML.gif , we can show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ37_HTML.gif
(3.18)
The proof is similar to that of the step 1 of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq215_HTML.gif in [7]; we omit it. So, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ38_HTML.gif
(3.19)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ39_HTML.gif
(3.20)
and subsequently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq216_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq217_HTML.gif . This contradicts (3.16). Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ40_HTML.gif
(3.21)

Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq218_HTML.gif ).

In this case, we can show easily that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq219_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq220_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq221_HTML.gif by using the same method used to prove Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq222_HTML.gif in [2].

Case 3 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq223_HTML.gif ).

In this case, we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq224_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq225_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq226_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq227_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ41_HTML.gif
(3.22)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq228_HTML.gif is bounded, say, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq229_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq230_HTML.gif depending not on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq231_HTML.gif , then we may assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ42_HTML.gif
(3.23)

Taking subsequences again if necessary, we still denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq232_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq233_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq234_HTML.gif , all the following proofs are similar.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ43_HTML.gif
(3.24)

denote the zeros of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq235_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq236_HTML.gif . Then, after taking a subsequence if necessary, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq237_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq238_HTML.gif . Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq239_HTML.gif . We can choose at least one subinterval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq240_HTML.gif which is of length at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq241_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq242_HTML.gif . Then, for this https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq243_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq244_HTML.gif is large enough. Put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq245_HTML.gif .

Obviously, for the above given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq246_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq247_HTML.gif have the same sign on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq248_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq249_HTML.gif . Without loss of generality, we assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ44_HTML.gif
(3.25)
Moreover, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ45_HTML.gif
(3.26)
Combining this with the fact
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ46_HTML.gif
(3.27)
and using the relation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ47_HTML.gif
(3.28)
we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq250_HTML.gif must change its sign on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq251_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq252_HTML.gif is large enough. This is a contradiction. Hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq253_HTML.gif is unbounded. From Lemma 2.6, we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ48_HTML.gif
(3.29)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq254_HTML.gif satisfies the autonomous equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ49_HTML.gif
(3.30)

We see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq255_HTML.gif consists of a sequence of positive and negative bumps, together with a truncated bump at the right end of the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq256_HTML.gif with the following properties (ignoring the truncated bump) (see, [1]):

(i)all the positive (resp., negative) bumps have the same shape (the shapes of the positive and negative bumps may be different);

(ii)each bump contains a single zero of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq257_HTML.gif , and there is exactly one zero of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq258_HTML.gif between consecutive zeros of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq259_HTML.gif ;

(iii)all the positive (negative) bumps attain the same maximum (minimum) value.

Armed with this information on the shape of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq260_HTML.gif it is easy to show that for the above given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq261_HTML.gif is an unbounded sequence. That is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ50_HTML.gif
(3.31)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq262_HTML.gif is concave on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq263_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq264_HTML.gif small enough,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ51_HTML.gif
(3.32)
This together with (3.31) implies that there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq265_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq266_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ52_HTML.gif
(3.33)
Hence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ53_HTML.gif
(3.34)

Now, we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq267_HTML.gif .

Suppose on the contrary that, choosing a subsequence and relabeling if necessary, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq268_HTML.gif for some constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq269_HTML.gif . This implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ54_HTML.gif
(3.35)

From (3.28) we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq270_HTML.gif must change its sign on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq271_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq272_HTML.gif is large enough. This is a contradiction. Therefore https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq273_HTML.gif .

Proof of Theorem 1.6.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq275_HTML.gif are immediate consequence of Theorem 1.5 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq277_HTML.gif , respectively.

To prove https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq278_HTML.gif , we rewrite (1.1), (1.2) to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ55_HTML.gif
(3.36)
By Lemma 2.5, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq280_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ56_HTML.gif
(3.37)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq281_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq282_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ57_HTML.gif
(3.38)
Then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq284_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ58_HTML.gif
(3.39)
This means that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_Equ59_HTML.gif
(3.40)

By Lemma 2.6 and Theorem 1.5, it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq285_HTML.gif is also an unbounded component joining https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq287_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq288_HTML.gif . Thus, (3.40) implies that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq289_HTML.gif (1.1), (1.2) has at least two solutions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F715836/MediaObjects/13661_2010_Article_53_IEq290_HTML.gif .

Declarations

Acknowledgments

The author is very grateful to the anonymous referees for their valuable suggestions. This paper was supported by NSFC (no.10671158), 11YZ225, YJ2009-16 (no.A06/1020K096019).

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Institute of Technology

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© Yulian An. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.