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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 284696, 11 pages
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.
This paper is devoted to two different but related tags: firstly, the side of the bifurcation from infinity at every eigenvalue of the problem , , 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.
Considering the problem we are interested in two different tags: firstly, the local behavior of bifurcations from infinity of (1.1) at every eigenvalue, , and, secondly how this behavior can help us to find solutions of the resonant problem
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 of the connected set of solutions 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 (), where only the behaviour at first eigenvalue, , was treated. In particular, considering the problem the authors showed that the side of the bifurcation from infinity at is determined by an integral value involving the nonlinearity , concretely for positive solutions and 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 can be found in , where the interior of the domain loses importance and the boundary joint with the integral of 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.,  or ), which does not allow us formulate a similar conclusion for dimensions . In this way, a paper by Fleckinger et al.  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  and used by Arcoya and Gámez  among other authors, and in a similar way of the case of the first eigenvalue (see ), 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 , Solimini , Ward , Mawhin and Schmitt , Schaff and Schmitt [12–14], Habets et al. , Habets et al.  and Cañada and Ruiz . 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 , closer to this paper, authors use bifurcation to solve resonant problems with periodic nonlinearities in dimension one. Finally, in  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.
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 are(i) is a Carathéodory function (i.e., continuous in for a.e. and measurable in(ii)there exists > 1 and such that for all ,(iii) uniformly in Considering the linearized problem, let be a fixed positive integer and denote by the usual norm in , that is . Under (), every eigenvualue, of (2.1) is a bifurcation point from infinity due to the fact that each has algebraic and geometric multiplicity 1 (see ). That is, there exists a sequence of solutions of (1.1) such that and . Since the weak solutions of (1.1) lie in the space continuously embedded in (), will be the space to work.
Furthermore, the number of zeroes of the eigenfunctions, , is finite. Concretely, and the normalized eigenfunctions are as follows: and , .
It is well known that for any bifurcating sequence there exists a subsequence (denoted as the sequence) such that where is a eigenfunction associated to with . In the particular case of bifurcation at the principal eigenvalue , both and have associated sequences as above. Since lies in the interior of the -cone of positive functions, we refer to such bifurcations as “bifurcation from ,” and “bifurcations from ” 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 ) bifurcating from infinity. We mean by “bifurcation from ” to be the sequence of solutions of (1.1) satisfying (-convergence), where . In a similar way, we mean by “bifurcation from ” the sequence of solutions of (1.1) with .
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
Next hypothesis restricts the considered nonlinearities to a class of “small” functions with some technical properties in the boundary. (i)there exists with such that for all ,(ii) is continuous in .Observe that (G) is more restricted than (H).
For any function satisfying (G) and for every eigenfunction , we define and as follows: (i) for even (ii) for odd Observe that in previous expressions the term appears twice the number of positive pieces . Conversely, the term appears twice the number of negative pieces of .
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) has, exactly, zeroes in the interval . These zeroes coincide with the points , where .
Taking a sequence of solutions of (1.3) bifurcating from , for any , there exist zeroes of , named , such that , and . Observe that and for all .
For every , let be a function test in the problem (1.3) obtaining Taking into account that and for large enough and, following from the equality (3.6), the sign of will decide the side of the bifurcation.
Therefore, the question is reduced to prove that We divide the integral in integrals (see Figure 1), where is “small enough” (see Lemma 1 in ). By using the hypothesis () and the same arguments as in ,
The rest of the integrals and should be considered depending on the value of , which depend on the parity of . By changing variables , with , one can see, as in Lemma 1 in , (i) for even, , (ii) for odd, .
Moreover, taking the suitable limits and by using the convergence theorem as in cited lemma, one can see that if ,
and also that Therefore, the value of depends on the parity of . Concretely, when is even, and when is odd, This completes the proof of Theorem 3.1.
Previous theorem enables us to describe the side of bifurcations in terms of and as follows.
Corollary 3.2. Under hypothesis (G), (a) if is even and (resp., ), the bifurcations from and from of solutions of (1.3) are both subcritical (resp., supercritical),(b.1)if is odd and (resp., ), the bifurcation from of solutions of (1.3) is subcritical (resp., supercritical), (b.2)If is odd and (resp., ), the bifurcation from of solutions of (1.3) is subcritical (resp., supercritical).
Remark 3.3. The nonautonomous case. By including the dependence on and under hypothesis () (i)there exists with such that for and for all , (ii) is continuous in .one can alter the integrals , given by (3.2) (3.3), as follows. (i) If is even, (ii)If is odd, Theorem 3.1 can be rewritten. Assume (G') holds, if is a sequence of solutions of (1.1) bifurcating from , then If is a sequence of solutions of (1.1) bifurcating from , then
4. The “Strongly” Resonant Problems at High Eigenvalues
Arguments used in [1, 7] or , under hypotheses either () or (), ensure that if the bifurcation from and bifurcation from 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 , if , then the resonant problem (1.2) has, at least, one solution. More concretely, (a) for any even , if , resonant problem (3.1) has, at least, one solution (see Corollary 3.2), (b)for odd , if , there exists such that every resonant problem (3.1) has, at least, one solution (see Corollary 3.2 and ).
Two examples are given. The first one requires Theorem 3.1 and the second one add a the dependence on .
5.1. Autonomous Example
Consider problem (1.3) with . In this case and then we have the following. (i) For even , and the bifurcations from and from are both subcritical. (ii)For every odd , . Moreover, , , and for all , and the bifurcations have variety of behaviors (see Figure 2). Consequently, we can ensure that if either or , the resonant problem (3.1) with has, at least, one solution.
5.2. Nonautonomous Example
Consider a function , where satisfies In this case, the values have easily computable forms (see Remark 3.3).
Since and by watching carefully the values represented in Figure 3, one can see that
(i) bifurcation from : no conclusion can be extracted by using Remark 3.3, (ii)bifurcations from are both supercritical, (iii)bifurcations from are both subcritical, (iv)bifurcations from for are supercritical, (v)bifurcations from : nothing can be concluded, (vi)bifurcations from for are subcritical. Furthermore, for and , every resonant problem (1.2) has, at least, one solution.
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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