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

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)), 𝑢∈𝐻¹⁰(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)=𝜆𝑢𝑢((𝑡0))+=𝑔𝑢(𝑡(,𝜋𝑢)(=𝑡))0,,𝑡∈(0,𝜋)(1.1) we are interested in two different tags: firstly, the local behavior of bifurcations from infinity of (1.1) at every eigenvalue, 𝜎_𝑘, and, secondly how this behavior can help us to find solutions of the resonant problem −𝑢^(𝑡)=𝜎_𝑘𝑢𝑢((0𝑡))+=𝑔𝑢((𝑡𝜋,𝑢)=(𝑡)0).,𝑡∈(0,𝜋)(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)=𝜆𝑢𝑢((0𝑡))+=𝑔𝑢((𝑢𝜋()𝑡)=),0𝑡∈(0,𝜋)(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 ^0−∞𝑔(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 Ω⊂ℝ^𝑁 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 [12–14], 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)×ℝ→ℝ are(𝐻)(i)𝑔∶[0,𝜋]×ℝ→ℝ is a Carathéodory function (i.e., continuous in 𝑠∈ℝ  for a.e.  𝑡∈(0,𝜋) and measurable in𝑡∈(0,𝜋),∀𝑠∈ℝ),(ii)there exists 𝑟> 1 and 𝐶∈𝐿^𝑟(0,𝜋) such that |𝑔(𝑡,𝑠)|≤𝐶(𝑡)(1+|𝑠|), for all (𝑡,𝑠)∈(0,𝜋)×ℝ,(iii)lim_|𝑠|→∞𝑔(𝑡,𝑠)/𝑠=0 uniformly in 𝑡∈[0,𝜋]Considering the linearized problem, −𝑢^(𝑡𝑢)(=0)𝜆=𝑢(𝑢𝑡)(,𝜋)𝑡=∈0(0,𝜋)(2.1) let 𝑘 be a fixed positive integer and denote by ‖⋅‖ the usual norm in 𝐻¹⁰(0,𝜋), that is ‖𝑢‖²=∫^𝜋0(𝑢′(𝑡))²𝑑𝑡. Under (𝐻), every eigenvualue, 𝜎_𝑘 of (2.1) is a bifurcation point from infinity due to the fact that each 𝜎_𝑘 has algebraic and geometric multiplicity 1 (see [18]). That is, there exists a sequence (𝜆_𝑛,𝑢_𝑛) of solutions of (1.1) such that 𝜆_𝑛→𝜎_𝑘 and ‖𝑢_𝑛‖→+∞. Since the weak solutions of (1.1) lie in the space 𝑊^2,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, 𝜓_𝑘, is finite. Concretely, 𝜎_𝑘=𝑘² and the normalized eigenfunctions are as follows: 𝜓_𝑘(𝑡)=2𝑘²𝜋sin(𝑘𝑡),(2.2) and 𝜓^𝑘(0)=√2/𝜋>0, ∀𝑘∈ℕ.

It is well known that for any bifurcating sequence (𝜆_𝑛,𝑢_𝑛)→(𝜎_𝑘,∞) there exists a subsequence (denoted as the sequence) (𝜆_𝑛,𝑢_𝑛) such that 𝑢_𝑛‖‖𝑢_𝑛‖‖⟶𝜓^𝑘in𝐶¹([0,𝜋]),(2.3) where 𝜓_𝑘 is a eigenfunction associated to 𝜎_𝑘 with ‖𝜓_𝑘‖=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 (𝜎_𝑘,+∞)” to be the sequence of solutions (𝜆_𝑛,𝑢_𝑛) of (1.1) satisfying 𝑢_𝑛/‖𝑢_𝑛‖→𝜓_𝑘 (𝐶¹-convergence), where 𝜓^𝑘(0)>0. In a similar way, we mean by “bifurcation from (𝜎_𝑘,−∞)” the sequence of solutions (𝜆_𝑛,𝑢_𝑛) of (1.1) with 𝑢_𝑛/‖𝑢_𝑛‖→−𝜓_𝑘.

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)=𝜎𝑢_𝑘𝑢(0(𝑡))=+𝑢𝑔((𝜋𝑢)(𝑡=))0,.𝑡∈(0,𝜋)(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 𝑓∈𝐿¹(ℝ) with lim_{|𝑠|→+∞}𝑓(𝑠)𝑠=0 such that|𝑔(𝑠)𝑠|<𝑓(𝑠), for all 𝑠∈ℝ,(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 𝜓_𝑘, we define 𝐿^+𝑘 and 𝐿^−𝑘 as follows: (i) for even 𝑘𝐿^+𝑘=𝐿^−𝑘∶=𝑘𝜋2^0+∞𝑔(𝑠)𝑠𝑑𝑠+^0−∞𝑔(𝑠)𝑠𝑑𝑠=𝑘𝜋2^+∞−∞𝑔(𝑠)𝑠𝑑𝑠,(3.2)(ii) for odd 𝑘𝐿^+𝑘∶=𝜋2(𝑘+1)^0+∞𝐿𝑔(𝑠)𝑠𝑑𝑠+(𝑘−1)^0−∞𝑔(𝑠)𝑠𝑑𝑠_−𝑘,∶=𝜋2(𝑘−1)^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 𝜓_𝑘. Conversely, the term ∫^0−∞𝑔(𝑠)𝑠𝑑𝑠 appears twice the number of negative pieces of 𝜓_𝑘.

Theorem 3.1 (Asume (G)). Is true, (𝜆_𝑛,𝑢_𝑛) is a sequence of solutions of (1.3) bifurcating from (𝜎_𝑘,+∞), then lim_𝑛→+∞‖‖𝑢_𝑛‖‖³𝜎_𝑘−𝜆_𝑛=𝜎_𝑘𝐿^+𝑘=𝑘²𝐿^+𝑘.(3.4) If (𝜆_𝑛,𝑢_𝑛) is a sequence of solutions of (1.3) bifurcating from (𝜎_𝑘,−∞), then lim_𝑛→+∞‖‖𝑢_𝑛‖‖³𝜎_𝑘−𝜆_𝑛=𝜎_𝑘𝐿^−𝑘=𝑘²𝐿^−𝑘.(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) 𝜓_𝑘 has, exactly, 𝑘−1 zeroes in the interval (0,𝜋). These zeroes coincide with the points 𝑖𝜋/𝑘, where 𝑖=1,…,𝑘−1.
Taking a sequence (𝜆_𝑛,𝑢_𝑛) of solutions of (1.3) bifurcating from (𝜎_𝑘,+∞), for any 𝑛∈ℕ, there exist 𝑘+1 zeroes of 𝑢_𝑛, named 𝑧_𝑛,𝑖, such that 𝑢_𝑛(𝑧_𝑛,𝑖)=0, ∀𝑛∈ℕ and lim_𝑛→∞𝑧_𝑛,𝑖=𝑖𝜋/𝑘. Observe that 𝑧_𝑛,0=0 and 𝑧_𝑛,𝑘=𝜋 for all 𝑛∈ℕ.
For every 𝑘, let 𝜓_𝑘 be a function test in the problem (1.3) obtaining 𝜎_𝑘−𝜆_𝑛^𝜋0𝑢_𝑛𝜓_𝑘=^𝜋0𝑔𝑢_𝑛𝜓_𝑘.(3.6) Taking into account that ^𝜋0𝜓^2𝑘=1𝜎_𝑘^𝜋0||∇𝜓_𝑘||²=1𝜎_𝑘=1𝑘²>0,(3.7) and for 𝑛 large enough ^𝜋0𝑢_𝑛𝜓_𝑘=‖‖𝑢_𝑛‖‖^𝜋0𝑢_𝑛‖‖𝑢_𝑛‖‖𝜓^𝑘⎧⎪⎨⎪⎩>0,if𝑢^𝑛‖‖𝑢_𝑛‖‖⟶𝜓^𝑘<0,if𝑢_𝑛,‖‖𝑢_𝑛‖‖⟶−𝜓^𝑘,(3.8) and, following from the equality (3.6), the sign of lim_𝑛→+∞^𝜋0𝑔𝑢_𝑛𝜓_𝑘(3.9) will decide the side of the bifurcation.
Therefore, the question is reduced to prove that 𝐿^+𝑘=lim_𝑛→+∞‖‖𝑢_𝑛‖‖²^𝜋0𝑔𝑢_𝑛𝜓_𝑘.(3.10) We divide the integral ∫^𝜋0𝑔(𝑢_𝑛)𝜓_𝑘 in 3𝑘 integrals (see Figure 1), ^𝜋0𝑔𝑢_𝑛𝜓_𝑘=^𝑘−1_𝑖=0^{𝑧𝑛,𝑖+𝑡0}_{𝑧𝑛,𝑖}+^{𝑧𝑛,𝑖+1−𝑡0}_{𝑧𝑛,𝑖+𝑡0}+^{𝑧𝑛,𝑖+1}_{𝑧𝑛,𝑖+1−𝑡0}𝑔𝑢^𝑛(𝑡)𝜓_𝑘(𝑡)𝑑𝑡=^𝑘−1_𝑖=0𝐼_1,𝑖+𝐼_2,𝑖+𝐼_3,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], lim_𝑛→∞‖‖𝑢_𝑛‖‖²𝐼_2,𝑖=0.(3.12)
The rest of the integrals 𝐼_1,𝑖 and 𝐼_3,𝑖 should be considered depending on the value of 𝑢′(𝑧_𝑛,𝑖), which depend on the parity of 𝑖. By changing variables 𝑢_𝑛(𝑡)=𝑠, with 𝑑𝑡=𝑑𝑠/𝑢^𝑛(𝑢^𝑛−1(𝑠)), one can see, as in Lemma 1 in [1], (i) for 𝑖 even, ‖𝑢_𝑛‖/𝑢^𝑛(𝑢^𝑛−1(𝑠))→−1/𝜓^𝑘(𝑖𝜋/𝑘)=√𝜋/2, (ii) for 𝑖 odd, ‖𝑢_𝑛‖/𝑢^𝑛(𝑢^𝑛−1(𝑠))→1/𝜓^𝑘(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,…,𝑘−1, lim_𝑛→∞‖‖𝑢_𝑛‖‖²𝐼_3,𝑖+𝐼_1,𝑖+1=𝜋2^+∞−∞𝑔(𝑠)𝑠𝑑𝑠(3.13)
and also that 𝐼_1,0=𝜋2^0+∞𝐼𝑔(𝑠)𝑠𝑑𝑠,_3,𝑘=⎧⎪⎪⎨⎪⎪⎩𝜋2^0−∞𝑔(𝑠)𝑠𝑑𝑠,when𝑘iseven,𝜋2^0+∞𝑔(𝑠)𝑠𝑑𝑠,when𝑘isodd.(3.14) Therefore, the value of lim_𝑛→∞‖‖𝑢_𝑛‖‖²^𝜋0𝑔𝑢_𝑛𝜓_𝑘(3.15) depends on the parity of 𝑘. Concretely, when 𝑘 is even, lim_𝑛→∞‖‖𝑢_𝑛‖‖²^𝜋0𝑔𝑢_𝑛𝜓_𝑘=𝑘𝜋2^0+∞𝑔(𝑠)𝑠𝑑𝑠+𝑘𝜋2^0−∞=𝑘𝑔(𝑠)𝑠𝑑𝑠𝜋2^+∞−∞𝑔(𝑠)𝑠𝑑𝑠(3.16) and when 𝑘 is odd, lim_𝑛→∞‖‖𝑢_𝑛‖‖²^𝜋0𝑔𝑢_𝑛𝜓_𝑘=(𝑘+1)𝜋2^0+∞𝑔(𝑠)𝑠𝑑𝑠+(𝑘−1)𝜋2^0−∞𝑔(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 ∫^0−∞𝑔(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 (𝜎_𝑘,+∞) and from (𝜎_𝑘,−∞) of solutions of (1.3) are both subcritical (resp., supercritical),(b.1)if 𝑘 is odd and (𝑘−1)∫^0−∞𝑔(𝑠)𝑠𝑑𝑠+(𝑘+1)∫^0+∞𝑔(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (𝜎_𝑘,+∞) of solutions of (1.3) is subcritical (resp., supercritical), (b.2)If 𝑘 is odd and (𝑘+1)∫^0−∞𝑔(𝑠)𝑠𝑑𝑠+(𝑘−1)∫^0+∞𝑔(𝑠)𝑠𝑑𝑠>0 (resp., <0), the bifurcation from (𝜎_𝑘,−∞) 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 𝑓∈𝐿¹(ℝ) with lim_{|𝑠|→+∞}𝑓(𝑠)𝑠=0 such that|𝑔(𝑡,𝑠)𝑠|<𝑓(𝑠),  for 𝑡∈(0,𝜋) and for all 𝑠∈ℝ, (ii)𝑔(𝑡,𝑠) is continuous in [0,𝜋]×ℝ.one can alter the integrals 𝐿_𝑘, given by (3.2) 𝑦 (3.3), as follows. (i) If 𝑘 is even, 𝐿^+𝑘=𝜋2^𝑘−1_𝑖=0^0+∞𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠+^0−∞𝑔(𝑖+1)𝜋𝐿𝑘,𝑠𝑠𝑑𝑠_−𝑘,=𝜋2^𝑘−1_𝑖=0^0+∞𝑔(𝑖+1)𝜋𝑘,𝑠𝑠𝑑𝑠+^0−∞𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠.(3.18)(ii)If 𝑘 is odd, 𝐿^+𝑘=𝜋2^𝑘_𝑖=0^0+∞𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠+^𝑘−2_𝑖=0^0−∞𝑔(𝑖+1)𝜋𝐿𝑘,𝑠𝑠𝑑𝑠_−𝑘,=𝜋2^𝑘−2_𝑖=0^0+∞𝑔(𝑖+1)𝜋𝑘,𝑠𝑠𝑑𝑠+^𝑘_𝑖=0^0−∞𝑔𝑖𝜋𝑘,𝑠𝑠𝑑𝑠.(3.19)Theorem 3.1 can be rewritten. Assume (G') holds, if (𝜆_𝑛,𝑢_𝑛) is a sequence of solutions of (1.1) bifurcating from (𝜎_𝑘,+∞), then lim_𝑛→+∞‖‖𝑢_𝑛‖‖³𝜎_𝑘−𝜆_𝑛=𝜎_𝑘𝐿^+𝑘=𝑘²𝐿^+𝑘.(3.20) If (𝜆_𝑛,𝑢_𝑛) is a sequence of solutions of (1.1) bifurcating from (𝜎_𝑘,−∞), then lim_𝑛→+∞‖‖𝑢_𝑛‖‖³𝜎_𝑘−𝜆_𝑛=𝜎_𝑘𝐿^−𝑘=𝑘²𝐿^−𝑘.(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 (𝜎_𝑘,+∞) and bifurcation from (𝜎_𝑘,−∞) 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(𝐿^−𝑘), then the resonant problem (1.2) has, at least, one solution. More concretely, (a) for any even 𝑘, if ∫^+∞−∞𝑔(𝑠)𝑠𝑑𝑠≠0, resonant problem (3.1) has, at least, one solution (see Corollary 3.2(𝑎)), (b)for odd 𝑘, if ∫^+∞−∞𝑔(𝑠)𝑠𝑑𝑠≠0, there exists 𝑘₀∈ℕ such that every resonant problem (3.1) has, at least, one solution ∀𝑘≥𝑘₀ (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^{−(𝑠−0.15)2}. In this case ^0+∞𝑔(𝑠)𝑠𝑑𝑠≈0.664,^0−∞𝑔(𝑠)𝑠𝑑𝑠≈−0.378,^+∞−∞𝑔(𝑠)𝑠𝑑𝑠≈0.286(5.1) and then we have the following. (i) For even 𝑘, 𝐿^+𝑘=𝐿^−𝑘>0 and the bifurcations from +∞ and from −∞ are both subcritical. (ii)For every odd 𝑘, 𝐿^+𝑘>0. Moreover, 𝐿⁻¹<0, 𝐿⁻³<0, and 𝐿^−𝑘>0 for all 𝑘≥5, and the bifurcations have variety of behaviors (see Figure 2). Consequently, we can ensure that if either 𝑘=2 or 𝑘≥4, the resonant problem (3.1) with 𝑔(𝑠)=𝑒^{−(𝑠−0.15)2} has, at least, one solution.

Figure 2 

Bifurcation diagram for 𝑔(𝑠)=𝑒−(𝑠−0.15)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=^0−∞𝑔₁(𝑠)𝑠𝑑𝑠=1.(5.2) In this case, the values 𝐿_𝑘 have easily computable forms (see Remark 3.3).

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

Figure 3 

Values 𝐿+𝑘=𝐿−𝑘 for ℎ(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 (𝜎_𝑘,±∞) for 𝑘=4,5,6 are supercritical, (v)bifurcations from (𝜎₇,±∞): nothing can be concluded, (vi)bifurcations from (𝜎_𝑘,±∞) for 𝑘≥8 are subcritical. Furthermore, for 𝑘=2,3,4,5,6 and 𝑘≥8, 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.