Bifurcation in nonlinearizable eigenvalue problems for ordinary differential equations of fourth order with indefinite weight

We consider a nonlinearizable eigenvalue problem for the beam equation with an indefinite weight function. We investigate the structure of bifurcation set and study the behavior of connected components of the solution set bifurcating from the line of trivial solutions and contained in the classes of positive and negative functions.

It is well known that fourth-order problems arise in many applications (see [7,21]) and the references therein); problem (1.1)-(1.2) in particular, is often used to describe the deformation of an elastic beam, which is subject to axial forces (see [7]).Problems with sign-changing weight arise from population modeling.In this model, weight function g changes sign corresponding to the fact that the intrinsic population growth rate is positive at same points and is negative at others, for details, see [9,14].
The purpose of this work is to study the global bifurcation of solutions of problem (1.1)-(1.2) in the classes of positive and negative functions, bifurcating from the intervals of the line of trivial solutions.
The problem (1.1)-(1.2) for the case of f ≡ 0 is studied in [16].In the case of f ≡ 0 the linearization of (1.1)-(1.2) at u = 0 is the linear eigenvalue problem (p(t)u (t)) − (q(t)u (t)) = λr(t)u(t), t ∈ (0, 1), where by B.C. we denote the set of boundary conditions (1.2).In [16] it was shown that there exist two positive and negative principal eigenvalues (i.e., eigenvalues corresponding to eigenfunctions which have no zeros in (0, 1)), λ + 1 and λ − 1 , of problem (1.6).Moreover, in [16] it was also proved that for each σ ∈ { + , − } and each ν ∈ { + , − } there exists a continuum 1 .Because of the presence of the term f , problem (1.1)-(1.2) does not in general have a linearization about zero.For this reason, the set of bifurcation points for (1.1)-(1.2) with respect to the line of trivial solutions need not be discrete (cf. the example in [6, p. 381]).Therefore, to investigate bifurcation for (1.1)-(1.2),one has to consider bifurcation from intervals rather than from bifurcation points.We say that bifurcation occurs from an interval if this interval contains at least one bifurcation point [6].
The problem (1.1)-(1.2) with r > 0 was considered in a recent paper [3] where, in particular, it was shown that for each k ∈ N and ν = + or −, there exists a connected component (maximal connected subset) D ν k of the set of solutions that emanating from the bifurcation interval λ k − K r 0 , λ k − K r 0 × {0} (r 0 = min t∈[0,1] r(t)) of the line of trivial solutions, has the standard oscillation properties (the number of zeros of a function is equal to the index of the eigenvalue of the corresponding linear problem minus one), is unbounded in R × C 3 , and lim t→0 ν sgn u(t) = 1 for each (λ, u) ∈ D ν k .Similar results on global bifurcation of solutions of nonlinear Sturm-Liouville problems obtained before by Rabinowitz [22], Berestycki [6], Schmitt and Smith [24], Chiappinelli [10], Aliyev and Mamedova [4], Rynne [23] and Dai [12].
It should be noted that to study the global bifurcation of solutions of problem (1.1)-(1.2) in the classes of positive and negative functions the method of [3] cannot be applied.This is due to the fact that the weight function r(x) changes sign in the interval (0, 1) and the eigenfunctions of linear problem (1.6) corresponding to the principal eigenvalues have no zeros in the interval (0, 1).Therefore, in investigating global bifurcation in the nonlinear problem (1.1)-(1.2) the following questions must be addressed: using new approaches to finding bifurcation intervals of solutions to (1.1)-(1.2) and to the study of the behavior of the connected components of the set of solutions emanating from these intervals.
The structure of this paper is as follows.
In Section 2, a family of sets to exploit oscillatory properties of eigenfunctions of problem (1.6) and their derivatives is introduced.Although problem (1.1)-(1.2) is not linearizable in a neighborhood of the origin (when f ≡ 0), it is nevertheless related to a linear problem which is perturbation of problem (1.6).In Section 3, we estimate the distance between the principal eigenvalues of the perturbed and unperturbed problem.Using this estimation in Section 4 we find the bifurcation intervals.We show the existence of two pair of unbounded continua of solutions emanating from the bifurcation intervals and contained in the classes of positive and negative functions.

Principal eigenvalues of perturbation linear problem
For the linear eigenvalue problem (1.6) we have the following result.
Note that the proof of Theorem 3.1 is based on a method used by Brown and Lin [8].Now we analyze the existence of principal eigenvalues using the method of Hess and Kato [15] (see also [1]).This is due to the fact that we will need further reasoning in order to find the bifurcation intervals of problem (1.1)-(1.2) corresponding to the principal eigenvalues of (1.6).
Define the linear differential operator L : D(L) → L 2 (0, 1) by It is known that the differential operator L is a densely defined self-adjoint operator on H whose spectrum contains only positive eigenvalues [5] (see also Remark 3.2).
For fixed λ ∈ R we consider the following eigenvalue problem By [3, Theorem 1.2] the problem has a sequence of real and simple eigenvalues Moreover, for each k ∈ N the eigenfunction u k (t, λ) corresponding to the eigenvalue µ k (λ) has k − 1 simple zeros in the interval (0, 1) (it should be noted that u 1 (t, λ) ∈ S 1 ).Let where It is clear that T λ is bounded below.It is shown in Courant and Hilbert [11] by variational arguments that µ 1 (λ) = min T λ .Moreover, it follows by the above argument that the eigenfunction u 1 (t, λ) corresponding to µ 1 (λ) does not vanish on (0, 1).Thus, clearly, λ is a principal eigenvalue of (1.6) if and only if µ 1 (λ) = 0.For fixed u ∈ D(L) the mapping is an affine and therefore a concave function.Since the minimum of any collection of concave functions is concave, it follows that λ → µ 1 (λ) is a concave function.Besides, by considering test functions u 1 , u 2 such that 1 0 r(t)|u 1 (t)| 2 dt > 0 and 1 0 r(t)|u 2 (t)| 2 dt < 0, it is easy to see that µ 1 (λ) → − ∞ as λ → ± ∞.Thus µ 1 (λ) is an increasing function until it attains its maximum, and is a decreasing function thereafter.
Then, as can be seen from the variational characterization of µ 1 (λ) or the fact that L has a positive principal eigenvalue, µ 1 (0) > 0 and thus µ 1 (λ) must has a graph which intersects the real axis in two points first of which is to the left, and second to the right from origin of coordinates.Hence, problem (1.6) has exactly two simple principal eigenvalues, one positive and one negative, which coincide with the λ + 1 and λ − 1 , respectively.Moreover, we have Lemma 3.3.For each σ ∈ {+ , −} the following relation is true: Proof.By (3.1) we have Multiplying (3.4) by u 1 (t, λ) and integrating this relation from 0 to 1 while taking into account the self-adjointness of the operator L we obtain which implies (3.2).The proof of this lemma is complete.
Remark 3.4.Since λ → μ1 (λ) is also a concave function on R and μ1 (λ) where λ+ 1 and λ− 1 are the positive and negative principal eigenvalues of problem (3.5), respectively.We need the following result which is basic in the sequel.Lemma 3.5.For each σ ∈ {+, −} the following relation is true: Proof.Let i.e. l σ is the line which tangent to the graph of the function µ 1 (λ) at point λ σ 1 .We introduce the following notation: , where |AB| is the distance between the points A and B.
Let L denote the closure of the set of nontrivial solutions of (1.1)-(1.2).
Remark 3.2.The problem (1.6) with r > 0 is a completely regular Sturmian system as defined by S. A. Janczewsky (see[17, p. 523]) provided that the excluded the cases α