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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 284696, 11 pages
http://dx.doi.org/10.1155/2012/284696
Research Article

Bifurcation from Infinity and Resonance Results at High Eigenvalues in Dimension One

1Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
2Departamento de Didáctica de la Matemática, Facultad de Ciencias de la Educación, Campus de Cartuja, Universidad de Granada, 18071 Granada, Spain

Received 2 February 2012; Accepted 7 August 2012

Academic Editor: Ruhan Zhao

Copyright © 2012 José L. Gámez and Juan F. Ruiz-Hidalgo. 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.

Abstract

This paper is devoted to two different but related tags: firstly, the side of the bifurcation from infinity at every eigenvalue of the problem 𝑢(𝑡)=𝜆𝑢(𝑡)+𝑔(𝑡,𝑢(𝑡)), 𝑢𝐻10(0,𝜋), secondly, the solutions of the associated resonant problem at any eigenvalue. From the global shape of the nonlinearity g we obtain computable integral values which will decide the behavior of the bifurcations and, consequently, the possibility of finding solutions of the resonant problems.

1. Introduction

Considering the problem 𝑢(𝑡)=𝜆𝑢(𝑡)+𝑔(𝑡,𝑢(𝑡)),𝑡(0,𝜋)𝑢(0)=𝑢(𝜋)=0,(1.1) we are interested in two different tags: firstly, the local behavior of bifurcations from infinity of (1.1) at every eigenvalue, 𝜎𝑘, and, secondly how this behavior can help us to find solutions of the resonant problem 𝑢(𝑡)=𝜎𝑘𝑢(𝑡)+𝑔(𝑡,𝑢(𝑡)),𝑡(0,𝜋)𝑢(0)=𝑢(𝜋)=0.(1.2)

The first objective is the behavior of bifurcation. In Section 3, we determine if the branches are either subcritical or supercritical, that is, the parameters 𝜆 of the connected set of solutions (𝜆,𝑢) of the problem (1.1) lie either to the left of the eigenvalue or to the right. This question has just been studied by the authors, as in ([1]), where only the behaviour at first eigenvalue, 𝜎1, was treated. In particular, considering the problem 𝑢(𝑡)=𝜆𝑢(𝑡)+𝑔(𝑢(𝑡)),𝑡(0,𝜋)𝑢(0)=𝑢(𝜋)=0(1.3) the authors showed that the side of the bifurcation from infinity at 𝜎1 is determined by an integral value involving the nonlinearity 𝑔, concretely 0+𝑔(𝑠)𝑠𝑑𝑠,(1.4) for positive solutions and 0𝑔(𝑠)𝑠𝑑𝑠,(1.5) for negative ones. This result is consequence of a sharp estimate on the first eigenfunction values on a neighborhood of the boundary of the domain.

An extension of this result to the Laplacian operator in a bounded domain Ω𝑁 can be found in [2], where the interior of the domain Ω loses importance and the boundary joint with the integral of 𝑔(𝑦,𝑠)𝑠 is enough to decide the side of the bifurcation. The importance of the boundary lies in that the set of zeroes of the first eigenfunction coincides with the boundary. This is a first obstacle in order to adjust the result to higher eigenvalues. In fact, the zeroes of the eigenfunctions in dimensions greater than one form the called nodal lines (see, e.g., [3] or [4]), which does not allow us formulate a similar conclusion for dimensions 𝑁>1. In this way, a paper by Fleckinger et al. [5] could give new clues to make generalizations. A second hurdle is the lack of positivity of all eigenfunctions but the first one.

The second objective of this paper, in Section 4, is the solution of the resonant problem (1.2) as consequence of the study of laterality in (1.1). By using ideas put forward by Hess [6] and used by Arcoya and Gámez [7] among other authors, and in a similar way of the case of the first eigenvalue 𝜎1 (see [1]), if the bifurcations from + and are both subcritical (resp. supercritical), then problem (1.2) has, at least, one solution.

The resonant problems have been studied by several authors. Chronologically, we can emphasize Dancer [8], Solimini [9], Ward [10], Mawhin and Schmitt [11], Schaff and Schmitt [1214], Habets et al. [15], Habets et al. [16] and Cañada and Ruiz [17]. Concretely, in [8, 14] asymptotics methods are used. Papers [9, 10], which are considered as classics in search of solution of resonant problems, are dedicated to periodic nonlinearities and use variational techniques. In paper [13], closer to this paper, authors use bifurcation to solve resonant problems with periodic nonlinearities in dimension one. Finally, in [17] variational techniques are managed, also with periodic nonlinearities.

The main contribution here is twofold: on the one hand, consider any eigenvalue not only the first one. On the other hand, periodic nonlinearities do not need to get solutions of (1.2).

The last section presents two examples which cannot be characterized in some of the papers cited.

2. Preliminaries

This section is devoted to present the hypotheses needed to use bifurcation tools and also to rewrite the definition of bifurcation from + and .

In order to ensure that bifurcation occurs usually the hypotheses assumed on 𝑔(0,𝜋)× are(𝐻)(i)𝑔[0,𝜋]× is a Carathéodory function (i.e., continuous in 𝑠  for a.e.  𝑡(0,𝜋) and measurable in𝑡(0,𝜋),𝑠),(ii)there exists 𝑟> 1 and 𝐶𝐿𝑟(0,𝜋) such that |𝑔(𝑡,𝑠)|𝐶(𝑡)(1+|𝑠|), for all (𝑡,𝑠)(0,𝜋)×,(iii)lim|𝑠|𝑔(𝑡,𝑠)/𝑠=0 uniformly in 𝑡[0,𝜋]Considering the linearized problem, 𝑢(𝑡)=𝜆𝑢(𝑡),𝑡(0,𝜋)𝑢(0)=𝑢(𝜋)=0(2.1) let 𝑘 be a fixed positive integer and denote by the usual norm in 𝐻10(0,𝜋), that is 𝑢2=𝜋0(𝑢(𝑡))2𝑑𝑡. Under (𝐻), every eigenvualue, 𝜎𝑘 of (2.1) is a bifurcation point from infinity due to the fact that each 𝜎𝑘 has algebraic and geometric multiplicity 1 (see [18]). That is, there exists a sequence (𝜆𝑛,𝑢𝑛) of solutions of (1.1) such that 𝜆𝑛𝜎𝑘 and 𝑢𝑛+. Since the weak solutions of (1.1) lie in the space 𝑊2,𝑟(0,𝜋) continuously embedded in 𝐶1([0,𝜋]) (𝑟>1), 𝐶1([0,𝜋]) will be the space to work.

Furthermore, the number of zeroes of the eigenfunctions, 𝜓𝑘, is finite. Concretely, 𝜎𝑘=𝑘2 and the normalized eigenfunctions are as follows: 𝜓𝑘(𝑡)=2𝑘2𝜋sin(𝑘𝑡),(2.2) and 𝜓𝑘(0)=2/𝜋>0, 𝑘.

It is well known that for any bifurcating sequence (𝜆𝑛,𝑢𝑛)(𝜎𝑘,) there exists a subsequence (denoted as the sequence) (𝜆𝑛,𝑢𝑛) such that 𝑢𝑛𝑢𝑛𝜓𝑘in𝐶1([]0,𝜋),(2.3) where 𝜓𝑘 is a eigenfunction associated to 𝜎𝑘 with 𝜓𝑘=1. In the particular case of bifurcation at the principal eigenvalue 𝜎1, both 𝜓1 and 𝜓1 have associated sequences as above. Since 𝜓1 lies in the interior of the 𝐶1-cone of positive functions, we refer to such bifurcations as “bifurcation from (𝜎1,+),” and “bifurcations from (𝜎1,)” respectively. One can also deduce from the above convergence that, near the bifurcation point, the solutions have constant sign. At higher eigenvalues the main difficulty revolves around the changes of sign of the eigenfunction. We overcome this trouble taking into account the existence of two branches of solutions (see [19]) bifurcating from infinity. We mean by “bifurcation from (𝜎𝑘,+)” to be the sequence of solutions (𝜆𝑛,𝑢𝑛) of (1.1) satisfying 𝑢𝑛/𝑢𝑛𝜓𝑘 (𝐶1-convergence), where 𝜓𝑘(0)>0. In a similar way, we mean by “bifurcation from (𝜎𝑘,)” the sequence of solutions (𝜆𝑛,𝑢𝑛) of (1.1) with 𝑢𝑛/𝑢𝑛𝜓𝑘.

3. Laterality of the Bifurcation from Infinity at All Eigenvalues

For the sake of simplicity we firstly point out our attention on the autonomous problem (1.3) and on the suitable resonant problem 𝑢(𝑡)=𝜎𝑘𝑢(𝑡)+𝑔(𝑢(𝑡)),𝑡(0,𝜋)𝑢(0)=𝑢(𝜋)=0.(3.1)

Next hypothesis restricts the considered nonlinearities to a class of “small” functions with some technical properties in the boundary.(𝐺)(i)there exists 𝑓𝐿1() with lim|𝑠|+𝑓(𝑠)𝑠=0 such that|𝑔(𝑠)𝑠|<𝑓(𝑠), for all 𝑠,(ii)𝑔(𝑠) is continuous in .Observe that (G) is more restricted than (H).

For any function 𝑔 satisfying (G) and for every eigenfunction 𝜓𝑘, we define 𝐿+𝑘 and 𝐿𝑘 as follows: (i) for even 𝑘𝐿+𝑘=𝐿𝑘=𝑘𝜋20+𝑔(𝑠)𝑠𝑑𝑠+0𝑔(𝑠)𝑠𝑑𝑠=𝑘𝜋2+𝑔(𝑠)𝑠𝑑𝑠,(3.2)(ii) for odd 𝑘𝐿+𝑘=𝜋2(𝑘+1)0+𝑔(𝑠)𝑠𝑑𝑠+(𝑘1)0,𝐿𝑔(𝑠)𝑠𝑑𝑠𝑘=𝜋2(𝑘1)0+𝑔(𝑠)𝑠𝑑𝑠+(𝑘+1)0.𝑔(𝑠)𝑠𝑑𝑠(3.3) Observe that in previous expressions the term 0+𝑔(𝑠)𝑠𝑑𝑠 appears twice the number of positive pieces 𝜓𝑘. Conversely, the term 0𝑔(𝑠)𝑠𝑑𝑠 appears twice the number of negative pieces of 𝜓𝑘.

Theorem 3.1 (Asume (G)). Is true, (𝜆𝑛,𝑢𝑛) is a sequence of solutions of (1.3) bifurcating from (𝜎𝑘,+), then lim𝑛+𝑢𝑛3𝜎𝑘𝜆𝑛=𝜎𝑘𝐿+𝑘=𝑘2𝐿+𝑘.(3.4) If (𝜆𝑛,𝑢𝑛) is a sequence of solutions of (1.3) bifurcating from (𝜎𝑘,), then lim𝑛+𝑢𝑛3𝜎𝑘𝜆𝑛=𝜎𝑘𝐿𝑘=𝑘2𝐿𝑘.(3.5)

Proof. We consider the bifurcation from +. The bifurcation from can be proved by using similar steps. Firstly, we remark that the eigenfunction associated to (2.1) 𝜓𝑘 has, exactly, 𝑘1 zeroes in the interval (0,𝜋). These zeroes coincide with the points 𝑖𝜋/𝑘, where 𝑖=1,,𝑘1.
Taking a sequence (𝜆𝑛,𝑢𝑛) of solutions of (1.3) bifurcating from (𝜎𝑘,+), for any 𝑛, there exist 𝑘+1 zeroes of 𝑢𝑛, named 𝑧𝑛,𝑖, such that 𝑢𝑛(𝑧𝑛,𝑖)=0, 𝑛 and lim𝑛𝑧𝑛,𝑖=𝑖𝜋/𝑘. Observe that 𝑧𝑛,0=0 and 𝑧𝑛,𝑘=𝜋 for all 𝑛.
For every 𝑘, let 𝜓𝑘 be a function test in the problem (1.3) obtaining 𝜎𝑘𝜆𝑛𝜋0𝑢𝑛𝜓𝑘=𝜋0𝑔𝑢𝑛𝜓𝑘.(3.6) Taking into account that 𝜋0𝜓2𝑘=1𝜎𝑘𝜋0||𝜓𝑘||2=1𝜎𝑘=1𝑘2>0,(3.7) and for 𝑛 large enough 𝜋0𝑢𝑛𝜓𝑘=𝑢𝑛𝜋0𝑢𝑛𝑢𝑛𝜓𝑘𝑢>0,if𝑛𝑢𝑛𝜓𝑘,𝑢<0,if𝑛𝑢𝑛𝜓𝑘,(3.8) and, following from the equality (3.6), the sign of lim𝑛+𝜋0𝑔𝑢𝑛𝜓𝑘(3.9) will decide the side of the bifurcation.
Therefore, the question is reduced to prove that 𝐿+𝑘=lim𝑛+𝑢𝑛2𝜋0𝑔𝑢𝑛𝜓𝑘.(3.10) We divide the integral 𝜋0𝑔(𝑢𝑛)𝜓𝑘 in 3𝑘 integrals (see Figure 1), 𝜋0𝑔𝑢𝑛𝜓𝑘=𝑘1𝑖=0𝑧𝑛,𝑖+𝑡0𝑧𝑛,𝑖+𝑧𝑛,𝑖+1𝑡0𝑧𝑛,𝑖+𝑡0+𝑧𝑛,𝑖+1𝑧𝑛,𝑖+1𝑡0𝑔𝑢𝑛𝜓(𝑡)𝑘(𝑡)𝑑𝑡=𝑘1𝑖=0𝐼1,𝑖+𝐼2,𝑖+𝐼3,𝑖,(3.11) where 𝑡0(0,𝜋/2𝑘) is “small enough” (see Lemma 1 in [1]). By using the hypothesis (𝐺) and the same arguments as in [1], lim𝑛𝑢𝑛2𝐼2,𝑖=0.(3.12)
The rest of the integrals 𝐼1,𝑖 and 𝐼3,𝑖 should be considered depending on the value of 𝑢(𝑧𝑛,𝑖), which depend on the parity of 𝑖. By changing variables 𝑢𝑛(𝑡)=𝑠, with 𝑑𝑡=𝑑𝑠/𝑢𝑛(𝑢𝑛1(𝑠)), one can see, as in Lemma 1 in [1], (i) for 𝑖 even, 𝑢𝑛/𝑢𝑛(𝑢𝑛1(𝑠))1/𝜓𝑘(𝑖𝜋/𝑘)=𝜋/2, (ii) for 𝑖 odd, 𝑢𝑛/𝑢𝑛(𝑢𝑛1(𝑠))1/𝜓𝑘(𝑖𝜋/𝑘)=𝜋/2.
Moreover, taking the suitable limits and by using the convergence theorem as in cited lemma, one can see that if 𝑖=0,,𝑘1, lim𝑛𝑢𝑛2𝐼3,𝑖+𝐼1,𝑖+1=𝜋2+𝑔(𝑠)𝑠𝑑𝑠(3.13)
and also that 𝐼1,0=𝜋20+𝐼𝑔(𝑠)𝑠𝑑𝑠,3,𝑘=𝜋20𝑔(𝑠)𝑠𝑑𝑠,when𝑘iseven,𝜋20+𝑔(𝑠)𝑠𝑑𝑠,when𝑘isodd.(3.14) Therefore, the value of lim𝑛𝑢𝑛2𝜋0𝑔𝑢𝑛𝜓𝑘(3.15) depends on the parity of 𝑘. Concretely, when 𝑘 is even, lim𝑛𝑢𝑛2𝜋0𝑔𝑢𝑛𝜓𝑘=𝑘𝜋20+𝑔(𝑠)𝑠𝑑𝑠+𝑘𝜋20𝑔(𝑠)𝑠𝑑𝑠=𝑘𝜋2+𝑔(𝑠)𝑠𝑑𝑠(3.16) and when 𝑘 is odd, lim𝑛𝑢𝑛2𝜋0𝑔𝑢𝑛𝜓𝑘=(𝑘+1)𝜋20+𝑔(𝑠)𝑠𝑑𝑠+(𝑘1)𝜋20𝑔(𝑠)𝑠𝑑𝑠.(3.17) This completes the proof of Theorem 3.1.

284696.fig.001
Figure 1: Solution of (1.3) near to the bifurcation point (𝜎2,+).

Previous theorem enables us to describe the side of bifurcations in terms of 0𝑔(𝑠)𝑠𝑑𝑠 and 0+𝑔(𝑠)𝑠𝑑𝑠 as follows.

Corollary 3.2. Under hypothesis (G), (a) if 𝑘 is even and +𝑔(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcations from (𝜎𝑘,+) and from (𝜎𝑘,) of solutions of (1.3) are both subcritical (resp., supercritical),(b.1)if 𝑘 is odd and (𝑘1)0𝑔(𝑠)𝑠𝑑𝑠+(𝑘+1)0+𝑔(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (𝜎𝑘,+) of solutions of (1.3) is subcritical (resp., supercritical), (b.2)If 𝑘 is odd and (𝑘+1)0𝑔(𝑠)𝑠𝑑𝑠+(𝑘1)0+𝑔(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (𝜎𝑘,) of solutions of (1.3) is subcritical (resp., supercritical).

Remark 3.3. The nonautonomous case. By including the dependence on 𝑡[0,𝜋] and under hypothesis (𝐺)(𝐺)(i)there exists 𝑓𝐿1() with lim|𝑠|+𝑓(𝑠)𝑠=0 such that|𝑔(𝑡,𝑠)𝑠|<𝑓(𝑠),  for 𝑡(0,𝜋) and for all 𝑠, (ii)𝑔(𝑡,𝑠) is continuous in [0,𝜋]×.one can alter the integrals 𝐿𝑘, given by (3.2) 𝑦 (3.3), as follows. (i) If 𝑘 is even, 𝐿+𝑘=𝜋2𝑘1𝑖=00+𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠+0𝑔(𝑖+1)𝜋𝑘,𝐿,𝑠𝑠𝑑𝑠𝑘=𝜋2𝑘1𝑖=00+𝑔(𝑖+1)𝜋𝑘,𝑠𝑠𝑑𝑠+0𝑔𝑖𝜋𝑘.,𝑠𝑠𝑑𝑠(3.18)(ii)If 𝑘 is odd, 𝐿+𝑘=𝜋2𝑘𝑖=00+𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠+𝑘2𝑖=00𝑔(𝑖+1)𝜋𝑘,𝐿,𝑠𝑠𝑑𝑠𝑘=𝜋2𝑘2𝑖=00+𝑔(𝑖+1)𝜋𝑘,𝑠𝑠𝑑𝑠+𝑘𝑖=00𝑔𝑖𝜋𝑘.,𝑠𝑠𝑑𝑠(3.19)Theorem 3.1 can be rewritten. Assume (G') holds, if (𝜆𝑛,𝑢𝑛) is a sequence of solutions of (1.1) bifurcating from (𝜎𝑘,+), then lim𝑛+𝑢𝑛3𝜎𝑘𝜆𝑛=𝜎𝑘𝐿+𝑘=𝑘2𝐿+𝑘.(3.20) If (𝜆𝑛,𝑢𝑛) is a sequence of solutions of (1.1) bifurcating from (𝜎𝑘,), then lim𝑛+𝑢𝑛3𝜎𝑘𝜆𝑛=𝜎𝑘𝐿𝑘=𝑘2𝐿𝑘.(3.21)

4. The “Strongly” Resonant Problems at High Eigenvalues

Arguments used in [1, 7] or [6], under hypotheses either (𝐺) or (𝐺), ensure that if the bifurcation from (𝜎𝑘,+) and bifurcation from (𝜎𝑘,) are either both subcritical or both supercritical, then the problem (1.2) has, at least, one solution. Consequently, by using Corollary 3.2 one can determine the laterality of bifurcations from ± and deduce the existence of solutions of the resonant problems (3.1) and (1.2).

Corollary 4.1. Under (G ) and for any 𝑘, if sign(𝐿+𝑘)=sign(𝐿𝑘), then the resonant problem (1.2) has, at least, one solution. More concretely, (a) for any even 𝑘, if +𝑔(𝑠)𝑠𝑑𝑠0, resonant problem (3.1) has, at least, one solution (see Corollary 3.2(𝑎)), (b)for odd 𝑘, if +𝑔(𝑠)𝑠𝑑𝑠0, there exists 𝑘0 such that every resonant problem (3.1) has, at least, one solution 𝑘𝑘0 (see Corollary 3.2(𝑏.1) and (𝑏.2)).

5. Examples

Two examples are given. The first one requires Theorem 3.1 and the second one add a the dependence on 𝑡[0,𝜋].

5.1. Autonomous Example

Consider problem (1.3) with 𝑔(𝑠)=𝑒(𝑠0.15)2. In this case 0+𝑔(𝑠)𝑠𝑑𝑠0.664,0𝑔(𝑠)𝑠𝑑𝑠0.378,+𝑔(𝑠)𝑠𝑑𝑠0.286(5.1) and then we have the following. (i) For even 𝑘, 𝐿+𝑘=𝐿𝑘>0 and the bifurcations from + and from are both subcritical. (ii)For every odd 𝑘, 𝐿+𝑘>0. Moreover, 𝐿1<0, 𝐿3<0, and 𝐿𝑘>0 for all 𝑘5, and the bifurcations have variety of behaviors (see Figure 2). Consequently, we can ensure that if either 𝑘=2 or 𝑘4, the resonant problem (3.1) with 𝑔(𝑠)=𝑒(𝑠0.15)2 has, at least, one solution.

284696.fig.002
Figure 2: Bifurcation diagram for 𝑔(𝑠)=𝑒(𝑠0.15)2.
5.2. Nonautonomous Example

Consider a function 𝑔(𝑥,𝑠)=sin(7𝑥)𝑔1(𝑠), 𝑥(0,𝜋) where 𝑔1 satisfies 0+𝑔1(𝑠)𝑠𝑑𝑠=0𝑔1(𝑠)𝑠𝑑𝑠=1.(5.2) In this case, the values 𝐿𝑘 have easily computable forms (see Remark 3.3).

Since 𝐿+𝑘=𝐿𝑘 and by watching carefully the 𝐿𝑘 values represented in Figure 3, one can see that

284696.fig.003
Figure 3: Values 𝐿+𝑘=𝐿𝑘 for (𝑥)=sin(7𝑥).

(i) bifurcation from (𝜎1,±): no conclusion can be extracted by using Remark 3.3, (ii)bifurcations from (𝜎2,±) are both supercritical, (iii)bifurcations from (𝜎3,±) are both subcritical, (iv)bifurcations from (𝜎𝑘,±) for 𝑘=4,5,6 are supercritical, (v)bifurcations from (𝜎7,±): nothing can be concluded, (vi)bifurcations from (𝜎𝑘,±) for 𝑘8 are subcritical. Furthermore, for 𝑘=2,3,4,5,6 and 𝑘8, every resonant problem (1.2) has, at least, one solution.

Acknowledgments

This research was supported by FEDER and D.G.I. Project Análisis no Lineal y Ecuaciones en Derivadas Parciales Elípticas, MTM2006-09282 and FQM116 of Junta de Andalucía.

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