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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 284696, 11 pages
Research Article

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

1Departamento de AnΓ‘lisis MatemΓ‘tico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
2Departamento de DidΓ‘ctica de la MatemΓ‘tica, Facultad de Ciencias de la EducaciΓ³n, Campus de Cartuja, Universidad de Granada, 18071 Granada, Spain

Received 2 February 2012; Accepted 7 August 2012

Academic Editor: RuhanΒ Zhao

Copyright Β© 2012 JosΓ© L. GΓ‘mez 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.


This paper is devoted to two different but related tags: firstly, the side of the bifurcation from infinity at every eigenvalue of the problem βˆ’π‘’ξ…žξ…ž(𝑑)=πœ†π‘’(𝑑)+𝑔(𝑑,𝑒(𝑑)), π‘’βˆˆπ»10(0,πœ‹), 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 βˆ’π‘’ξ…žξ…ž(𝑑)=πœ†π‘’(𝑑)+𝑔(𝑑,𝑒(𝑑)),π‘‘βˆˆ(0,πœ‹)𝑒(0)=𝑒(πœ‹)=0,(1.1) we are interested in two different tags: firstly, the local behavior of bifurcations from infinity of (1.1) at every eigenvalue, πœŽπ‘˜, and, secondly how this behavior can help us to find solutions of the resonant problem βˆ’π‘’ξ…žξ…ž(𝑑)=πœŽπ‘˜π‘’(𝑑)+𝑔(𝑑,𝑒(𝑑)),π‘‘βˆˆ(0,πœ‹)𝑒(0)=𝑒(πœ‹)=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 πœ† of the connected set of solutions (πœ†,𝑒) 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 βˆ’π‘’ξ…žξ…ž(𝑑)=πœ†π‘’(𝑑)+𝑔(𝑒(𝑑)),π‘‘βˆˆ(0,πœ‹)𝑒(0)=𝑒(πœ‹)=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 ξ€œ0+βˆžπ‘”(𝑠)𝑠𝑑𝑠,(1.4) for positive solutions and ξ€œ0βˆ’βˆžπ‘”(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 Ξ©βŠ‚β„π‘ can be found in [2], where the interior of the domain Ξ© 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 GΓ‘mez [7] among other authors, and in a similar way of the case of the first eigenvalue 𝜎1 (see [1]), if the bifurcations from +∞ and βˆ’βˆž 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 [12–14], Habets et al. [15], Habets et al. [16] and CaΓ±ada 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 βˆ’βˆž.

In order to ensure that bifurcation occurs usually the hypotheses assumed on π‘”βˆΆ(0,πœ‹)×ℝ→ℝ are(𝐻)(i)π‘”βˆΆ[0,πœ‹]×ℝ→ℝ is a CarathΓ©odory function (i.e., continuous in π‘ βˆˆβ„β€‰β€‰for a.e. β€‰π‘‘βˆˆ(0,πœ‹) and measurable inπ‘‘βˆˆ(0,πœ‹),βˆ€π‘ βˆˆβ„),(ii)there exists π‘Ÿ> 1 and πΆβˆˆπΏπ‘Ÿ(0,πœ‹) such that |𝑔(𝑑,𝑠)|≀𝐢(𝑑)(1+|𝑠|), for all (𝑑,𝑠)∈(0,πœ‹)×ℝ,(iii)lim|𝑠|β†’βˆžπ‘”(𝑑,𝑠)/𝑠=0 uniformly in π‘‘βˆˆ[0,πœ‹]Considering the linearized problem, βˆ’π‘’ξ…žξ…ž(𝑑)=πœ†π‘’(𝑑),π‘‘βˆˆ(0,πœ‹)𝑒(0)=𝑒(πœ‹)=0(2.1) let π‘˜ be a fixed positive integer and denote by β€–β‹…β€– the usual norm in 𝐻10(0,πœ‹), that is ‖𝑒‖2=βˆ«πœ‹0(𝑒′(𝑑))2𝑑𝑑. Under (𝐻), every eigenvualue, πœŽπ‘˜ of (2.1) is a bifurcation point from infinity due to the fact that each πœŽπ‘˜ has algebraic and geometric multiplicity 1 (see [18]). That is, there exists a sequence (πœ†π‘›,𝑒𝑛) of solutions of (1.1) such that πœ†π‘›β†’πœŽπ‘˜ and ‖𝑒𝑛‖→+∞. Since the weak solutions of (1.1) lie in the space π‘Š2,π‘Ÿ(0,πœ‹) continuously embedded in 𝐢1([0,πœ‹]) (π‘Ÿ>1), 𝐢1([0,πœ‹]) will be the space to work.

Furthermore, the number of zeroes of the eigenfunctions, πœ“π‘˜, is finite. Concretely, πœŽπ‘˜=π‘˜2 and the normalized eigenfunctions are as follows: πœ“π‘˜ξ‚™(𝑑)=2π‘˜2πœ‹sin(π‘˜π‘‘),(2.2) and πœ“ξ…žπ‘˜βˆš(0)=2/πœ‹>0, βˆ€π‘˜βˆˆβ„•.

It is well known that for any bifurcating sequence (πœ†π‘›,𝑒𝑛)β†’(πœŽπ‘˜,∞) there exists a subsequence (denoted as the sequence) (πœ†π‘›,𝑒𝑛) such that π‘’π‘›β€–β€–π‘’π‘›β€–β€–βŸΆπœ“π‘˜in𝐢1([]0,πœ‹),(2.3) where πœ“π‘˜ is a eigenfunction associated to πœŽπ‘˜ with β€–πœ“π‘˜β€–=1. In the particular case of bifurcation at the principal eigenvalue 𝜎1, both πœ“1 and βˆ’πœ“1 have associated sequences as above. Since πœ“1 lies in the interior of the 𝐢1-cone of positive functions, we refer to such bifurcations as β€œbifurcation from (𝜎1,+∞),” and β€œbifurcations from (𝜎1,βˆ’βˆž)” 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 β€œbifurcation from (πœŽπ‘˜,+∞)” to be the sequence of solutions (πœ†π‘›,𝑒𝑛) of (1.1) satisfying 𝑒𝑛/β€–π‘’π‘›β€–β†’πœ“π‘˜ (𝐢1-convergence), where πœ“ξ…žπ‘˜(0)>0. In a similar way, we mean by β€œbifurcation from (πœŽπ‘˜,βˆ’βˆž)” the sequence of solutions (πœ†π‘›,𝑒𝑛) of (1.1) with 𝑒𝑛/β€–π‘’π‘›β€–β†’βˆ’πœ“π‘˜.

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 βˆ’π‘’ξ…žξ…ž(𝑑)=πœŽπ‘˜π‘’(𝑑)+𝑔(𝑒(𝑑)),π‘‘βˆˆ(0,πœ‹)𝑒(0)=𝑒(πœ‹)=0.(3.1)

Next hypothesis restricts the considered nonlinearities to a class of β€œsmall” functions with some technical properties in the boundary. (𝐺)(i)there exists π‘“βˆˆπΏ1(ℝ) with lim|𝑠|β†’+βˆžπ‘“(𝑠)𝑠=0 such that|𝑔(𝑠)𝑠|<𝑓(𝑠), for all π‘ βˆˆβ„,(ii)𝑔(𝑠) is continuous in ℝ.Observe that (G) is more restricted than (H).

For any function 𝑔 satisfying (G) and for every eigenfunction πœ“π‘˜, we define 𝐿+π‘˜ and πΏβˆ’π‘˜ as follows: (i) for even π‘˜πΏ+π‘˜=πΏβˆ’π‘˜ξ‚™βˆΆ=π‘˜πœ‹2ξ‚΅ξ€œ0+βˆžξ€œπ‘”(𝑠)𝑠𝑑𝑠+0βˆ’βˆžξ‚Άξ‚™π‘”(𝑠)𝑠𝑑𝑠=π‘˜πœ‹2ξ€œ+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠,(3.2)(ii) for odd π‘˜πΏ+π‘˜ξ‚™βˆΆ=πœ‹2ξ‚Έξ€œ(π‘˜+1)0+βˆžξ€œπ‘”(𝑠)𝑠𝑑𝑠+(π‘˜βˆ’1)0βˆ’βˆžξ‚Ή,𝐿𝑔(𝑠)π‘ π‘‘π‘ βˆ’π‘˜ξ‚™βˆΆ=πœ‹2ξ‚Έξ€œ(π‘˜βˆ’1)0+βˆžξ€œπ‘”(𝑠)𝑠𝑑𝑠+(π‘˜+1)0βˆ’βˆžξ‚Ή.𝑔(𝑠)𝑠𝑑𝑠(3.3) Observe that in previous expressions the term ∫0+βˆžπ‘”(𝑠)𝑠𝑑𝑠 appears twice the number of positive pieces πœ“π‘˜. Conversely, the term ∫0βˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠 appears twice the number of negative pieces of πœ“π‘˜.

Theorem 3.1 (Asume (G)). Is true, (πœ†π‘›,𝑒𝑛) is a sequence of solutions of (1.3) bifurcating from (πœŽπ‘˜,+∞), then lim𝑛→+βˆžβ€–β€–π‘’π‘›β€–β€–3ξ€·πœŽπ‘˜βˆ’πœ†π‘›ξ€Έ=πœŽπ‘˜πΏ+π‘˜=π‘˜2𝐿+π‘˜.(3.4) If (πœ†π‘›,𝑒𝑛) is a sequence of solutions of (1.3) bifurcating from (πœŽπ‘˜,βˆ’βˆž), then lim𝑛→+βˆžβ€–β€–π‘’π‘›β€–β€–3ξ€·πœŽπ‘˜βˆ’πœ†π‘›ξ€Έ=πœŽπ‘˜πΏβˆ’π‘˜=π‘˜2πΏβˆ’π‘˜.(3.5)

Proof. We consider the bifurcation from +∞. The bifurcation from βˆ’βˆž can be proved by using similar steps. Firstly, we remark that the eigenfunction associated to (2.1) πœ“π‘˜ has, exactly, π‘˜βˆ’1 zeroes in the interval (0,πœ‹). These zeroes coincide with the points π‘–πœ‹/π‘˜, where 𝑖=1,…,π‘˜βˆ’1.
Taking a sequence (πœ†π‘›,𝑒𝑛) of solutions of (1.3) bifurcating from (πœŽπ‘˜,+∞), for any π‘›βˆˆβ„•, there exist π‘˜+1 zeroes of 𝑒𝑛, named 𝑧𝑛,𝑖, such that 𝑒𝑛(𝑧𝑛,𝑖)=0, βˆ€π‘›βˆˆβ„• and limπ‘›β†’βˆžπ‘§π‘›,𝑖=π‘–πœ‹/π‘˜. Observe that 𝑧𝑛,0=0 and 𝑧𝑛,π‘˜=πœ‹ for all π‘›βˆˆβ„•.
For every π‘˜, let πœ“π‘˜ be a function test in the problem (1.3) obtaining ξ€·πœŽπ‘˜βˆ’πœ†π‘›ξ€Έξ€œπœ‹0π‘’π‘›πœ“π‘˜=ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜.(3.6) Taking into account that ξ€œπœ‹0πœ“2π‘˜=1πœŽπ‘˜ξ€œπœ‹0||βˆ‡πœ“π‘˜||2=1πœŽπ‘˜=1π‘˜2>0,(3.7) and for 𝑛 large enough ξ€œπœ‹0π‘’π‘›πœ“π‘˜=β€–β€–π‘’π‘›β€–β€–ξ€œπœ‹0π‘’π‘›β€–β€–π‘’π‘›β€–β€–πœ“π‘˜βŽ§βŽͺ⎨βŽͺβŽ©π‘’>0,ifπ‘›β€–β€–π‘’π‘›β€–β€–βŸΆπœ“π‘˜,𝑒<0,ifπ‘›β€–β€–π‘’π‘›β€–β€–βŸΆβˆ’πœ“π‘˜,(3.8) and, following from the equality (3.6), the sign of lim𝑛→+βˆžξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜(3.9) will decide the side of the bifurcation.
Therefore, the question is reduced to prove that 𝐿+π‘˜=lim𝑛→+βˆžβ€–β€–π‘’π‘›β€–β€–2ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜.(3.10) We divide the integral βˆ«πœ‹0𝑔(𝑒𝑛)πœ“π‘˜ in 3π‘˜ integrals (see Figure 1), ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜=π‘˜βˆ’1𝑖=0ξƒ¬ξ€œπ‘§π‘›,𝑖+𝑑0𝑧𝑛,𝑖+ξ€œπ‘§π‘›,𝑖+1βˆ’π‘‘0𝑧𝑛,𝑖+𝑑0+ξ€œπ‘§π‘›,𝑖+1𝑧𝑛,𝑖+1βˆ’π‘‘0ξƒ­π‘”ξ€·π‘’π‘›ξ€Έπœ“(𝑑)π‘˜(𝑑)𝑑𝑑=π‘˜βˆ’1𝑖=0𝐼1,𝑖+𝐼2,𝑖+𝐼3,𝑖,(3.11) where 𝑑0∈(0,πœ‹/2π‘˜) is β€œsmall enough” (see Lemma 1 in [1]). By using the hypothesis (𝐺) and the same arguments as in [1], limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2𝐼2,𝑖=0.(3.12)
The rest of the integrals 𝐼1,𝑖 and 𝐼3,𝑖 should be considered depending on the value of 𝑒′(𝑧𝑛,𝑖), which depend on the parity of 𝑖. By changing variables 𝑒𝑛(𝑑)=𝑠, with 𝑑𝑑=𝑑𝑠/π‘’ξ…žπ‘›(π‘’π‘›βˆ’1(𝑠)), one can see, as in Lemma 1 in [1], (i) for 𝑖 even, ‖𝑒𝑛‖/π‘’ξ…žπ‘›(π‘’π‘›βˆ’1(𝑠))β†’βˆ’1/πœ“ξ…žπ‘˜βˆš(π‘–πœ‹/π‘˜)=πœ‹/2, (ii) for 𝑖 odd, ‖𝑒𝑛‖/π‘’ξ…žπ‘›(π‘’π‘›βˆ’1(𝑠))β†’1/πœ“ξ…žπ‘˜βˆš(π‘–πœ‹/π‘˜)=βˆ’πœ‹/2.
Moreover, taking the suitable limits and by using the convergence theorem as in cited lemma, one can see that if 𝑖=0,…,π‘˜βˆ’1, limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2𝐼3,𝑖+𝐼1,𝑖+1ξ€Έ=ξ‚™πœ‹2ξ€œ+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠(3.13)
and also that 𝐼1,0=ξ‚™πœ‹2ξ€œ0+βˆžπΌπ‘”(𝑠)𝑠𝑑𝑠,3,π‘˜=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ‚™πœ‹2ξ€œ0βˆ’βˆžξ‚™π‘”(𝑠)𝑠𝑑𝑠,whenπ‘˜iseven,πœ‹2ξ€œ0+βˆžπ‘”(𝑠)𝑠𝑑𝑠,whenπ‘˜isodd.(3.14) Therefore, the value of limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜(3.15) depends on the parity of π‘˜. Concretely, when π‘˜ is even, limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜ξ‚™=π‘˜πœ‹2ξ€œ0+βˆžξ‚™π‘”(𝑠)𝑠𝑑𝑠+π‘˜πœ‹2ξ€œ0βˆ’βˆžξ‚™π‘”(𝑠)𝑠𝑑𝑠=π‘˜πœ‹2ξ€œ+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠(3.16) and when π‘˜ is odd, limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–2ξ€œπœ‹0π‘”ξ€·π‘’π‘›ξ€Έπœ“π‘˜ξ‚™=(π‘˜+1)πœ‹2ξ€œ0+βˆžξ‚™π‘”(𝑠)𝑠𝑑𝑠+(π‘˜βˆ’1)πœ‹2ξ€œ0βˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠.(3.17) This completes the proof of Theorem 3.1.

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 ∫0βˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠 and ∫0+βˆžπ‘”(𝑠)𝑠𝑑𝑠 as follows.

Corollary 3.2. Under hypothesis (G), (a) if π‘˜ is even and ∫+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcations from (πœŽπ‘˜,+∞) and from (πœŽπ‘˜,βˆ’βˆž) of solutions of (1.3) are both subcritical (resp., supercritical),(b.1)if π‘˜ is odd and ∫(π‘˜βˆ’1)0βˆ’βˆžβˆ«π‘”(𝑠)𝑠𝑑𝑠+(π‘˜+1)0+βˆžπ‘”(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (πœŽπ‘˜,+∞) of solutions of (1.3) is subcritical (resp., supercritical), (b.2)If π‘˜ is odd and ∫(π‘˜+1)0βˆ’βˆžβˆ«π‘”(𝑠)𝑠𝑑𝑠+(π‘˜βˆ’1)0+βˆžπ‘”(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (πœŽπ‘˜,βˆ’βˆž) of solutions of (1.3) is subcritical (resp., supercritical).

Remark 3.3. The nonautonomous case. By including the dependence on π‘‘βˆˆ[0,πœ‹] and under hypothesis (πΊξ…ž) (πΊξ…ž)(i)there exists π‘“βˆˆπΏ1(ℝ) with lim|𝑠|β†’+βˆžπ‘“(𝑠)𝑠=0 such that|𝑔(𝑑,𝑠)𝑠|<𝑓(𝑠),  for π‘‘βˆˆ(0,πœ‹) and for all π‘ βˆˆβ„, (ii)𝑔(𝑑,𝑠) is continuous in [0,πœ‹]×ℝ.one can alter the integrals πΏπ‘˜, given by (3.2) 𝑦 (3.3), as follows. (i) If π‘˜ is even, 𝐿+π‘˜=ξ‚™πœ‹2π‘˜βˆ’1𝑖=0ξ‚Έξ€œ0+βˆžπ‘”ξ‚€π‘–πœ‹π‘˜ξ‚ξ€œ,𝑠𝑠𝑑𝑠+0βˆ’βˆžπ‘”ξ‚΅(𝑖+1)πœ‹π‘˜ξ‚Άξ‚Ή,𝐿,π‘ π‘ π‘‘π‘ βˆ’π‘˜=ξ‚™πœ‹2π‘˜βˆ’1𝑖=0ξ‚Έξ€œ0+βˆžπ‘”ξ‚΅(𝑖+1)πœ‹π‘˜ξ‚Άξ€œ,𝑠𝑠𝑑𝑠+0βˆ’βˆžπ‘”ξ‚€π‘–πœ‹π‘˜ξ‚ξ‚Ή.,𝑠𝑠𝑑𝑠(3.18)(ii)If π‘˜ is odd, 𝐿+π‘˜=ξ‚™πœ‹2ξƒ¬π‘˜ξ“π‘–=0ξ€œ0+βˆžπ‘”ξ‚€π‘–πœ‹π‘˜ξ‚,𝑠𝑠𝑑𝑠+π‘˜βˆ’2𝑖=0ξ€œ0βˆ’βˆžπ‘”ξ‚΅(𝑖+1)πœ‹π‘˜ξ‚Άξƒ­,𝐿,π‘ π‘ π‘‘π‘ βˆ’π‘˜=ξ‚™πœ‹2ξƒ¬π‘˜βˆ’2𝑖=0ξ€œ0+βˆžπ‘”ξ‚΅(𝑖+1)πœ‹π‘˜ξ‚Ά,𝑠𝑠𝑑𝑠+π‘˜ξ“π‘–=0ξ€œ0βˆ’βˆžπ‘”ξ‚€π‘–πœ‹π‘˜ξ‚ξƒ­.,𝑠𝑠𝑑𝑠(3.19)Theorem 3.1 can be rewritten. Assume (G') holds, if (πœ†π‘›,𝑒𝑛) is a sequence of solutions of (1.1) bifurcating from (πœŽπ‘˜,+∞), then lim𝑛→+βˆžβ€–β€–π‘’π‘›β€–β€–3ξ€·πœŽπ‘˜βˆ’πœ†π‘›ξ€Έ=πœŽπ‘˜πΏ+π‘˜=π‘˜2𝐿+π‘˜.(3.20) If (πœ†π‘›,𝑒𝑛) is a sequence of solutions of (1.1) bifurcating from (πœŽπ‘˜,βˆ’βˆž), then lim𝑛→+βˆžβ€–β€–π‘’π‘›β€–β€–3ξ€·πœŽπ‘˜βˆ’πœ†π‘›ξ€Έ=πœŽπ‘˜πΏβˆ’π‘˜=π‘˜2πΏβˆ’π‘˜.(3.21)

4. The β€œStrongly” Resonant Problems at High Eigenvalues

Arguments used in [1, 7] or [6], under hypotheses either (𝐺) or (πΊξ…ž), ensure that if the bifurcation from (πœŽπ‘˜,+∞) and bifurcation from (πœŽπ‘˜,βˆ’βˆž) 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 π‘˜, if sign(𝐿+π‘˜)=sign(πΏβˆ’π‘˜), then the resonant problem (1.2) has, at least, one solution. More concretely, (a) for any even π‘˜, if ∫+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠≠0, resonant problem (3.1) has, at least, one solution (see Corollary 3.2(π‘Ž)), (b)for odd π‘˜, if ∫+βˆžβˆ’βˆžπ‘”(𝑠)𝑠𝑑𝑠≠0, there exists π‘˜0βˆˆβ„• such that every resonant problem (3.1) has, at least, one solution βˆ€π‘˜β‰₯π‘˜0 (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 π‘‘βˆˆ[0,πœ‹].

5.1. Autonomous Example

Consider problem (1.3) with 𝑔(𝑠)=π‘’βˆ’(π‘ βˆ’0.15)2. In this case ξ€œ0+βˆžξ€œπ‘”(𝑠)π‘ π‘‘π‘ β‰ˆ0.664,0βˆ’βˆžξ€œπ‘”(𝑠)π‘ π‘‘π‘ β‰ˆβˆ’0.378,+βˆžβˆ’βˆžπ‘”(𝑠)π‘ π‘‘π‘ β‰ˆ0.286(5.1) and then we have the following. (i) For even π‘˜, 𝐿+π‘˜=πΏβˆ’π‘˜>0 and the bifurcations from +∞ and from βˆ’βˆž are both subcritical. (ii)For every odd π‘˜, 𝐿+π‘˜>0. Moreover, πΏβˆ’1<0, πΏβˆ’3<0, and πΏβˆ’π‘˜>0 for all π‘˜β‰₯5, and the bifurcations have variety of behaviors (see Figure 2). Consequently, we can ensure that if either π‘˜=2 or π‘˜β‰₯4, the resonant problem (3.1) with 𝑔(𝑠)=π‘’βˆ’(π‘ βˆ’0.15)2 has, at least, one solution.

Figure 2: Bifurcation diagram for 𝑔(𝑠)=π‘’βˆ’(π‘ βˆ’0.15)2.
5.2. Nonautonomous Example

Consider a function 𝑔(π‘₯,𝑠)=sin(7π‘₯)𝑔1(𝑠), βˆ€π‘₯∈(0,πœ‹) where 𝑔1 satisfies ξ€œ0+βˆžπ‘”1ξ€œ(𝑠)𝑠𝑑𝑠=0βˆ’βˆžπ‘”1(𝑠)𝑠𝑑𝑠=1.(5.2) In this case, the values πΏπ‘˜ have easily computable forms (see Remark 3.3).

Since 𝐿+π‘˜=πΏβˆ’π‘˜ and by watching carefully the πΏπ‘˜ values represented in Figure 3, one can see that

Figure 3: Values 𝐿+π‘˜=πΏβˆ’π‘˜ for β„Ž(π‘₯)=sin(7π‘₯).

(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 (πœŽπ‘˜,±∞) for π‘˜=4,5,6 are supercritical, (v)bifurcations from (𝜎7,±∞): nothing can be concluded, (vi)bifurcations from (πœŽπ‘˜,±∞) for π‘˜β‰₯8 are subcritical. Furthermore, for π‘˜=2,3,4,5,6 and π‘˜β‰₯8, every resonant problem (1.2) has, at least, one solution.


This research was supported by FEDER and D.G.I. Project AnΓ‘lisis no Lineal y Ecuaciones en Derivadas Parciales ElΓ­pticas, MTM2006-09282 and FQM116 of Junta de AndalucΓ­a.


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