Global bifurcation for nonlinear Dirac problems

In this paper we consider the nonlinear eigenvalue problems for the onedimensional Dirac equation. To exploit oscillatory properties of the components of the eigenvector-functions of linear one-dimensional Dirac system an appropriate family of sets is introduced. We show the existence of two families of continua of solutions contained in these sets and bifurcating from the intervals of the line of trivial solutions.

Because of the presence of the term h, problem (1.1)- (1.3) does not in general have a linearization about zero.For this reason, the set of bifurcation points for this problem with respect to the line of trivial solutions need not be discrete (cf. the example of [6, p. 381]).Therefore, to investigate the question of bifurcation for (1.1)-(1.3),one has to consider bifurcation from intervals rather than bifurcation points.We say that bifurcation occurs from an interval if this interval contains at least one bifurcation point [6].
In [21] the authors considered the nonlinear problem (1.1)-(1.3) in the case K + M < 1/2 and they show that there exists a natural number k 0 , such that their bifurcation intervals (which are the same as Berestycki's) do not overlap for every integer k, |k| ≥ k 0 , and corresponding global bifurcation Theorem 2.2 (from [21]) holds for this case.More precisely, for each k, |k| ≥ k 0 , the connected component D k of solutions of problem (1.1)-(1.3)emanating from bifurcation interval surrounding the k-th eigenvalue of the linear problem obtained from (1.1)-(1.3)by setting h ≡ 0 either is unbounded in C([0, π]; R 2 ), or meet another bifurcation interval.
Thanks to our recent work [4], which is devoted to the study of the oscillations of the linear problem, in this paper we study the structure of bifurcation points and completely investigate the behavior of two families of continua of solutions of problem (1.1)-(1.3)contained in the classes of vector-functions having the oscillation properties of the eigenvector-functions of the corresponding linear problem, and bifurcating from the points and intervals of the line of trivial solutions.Although the problem (1.1)-(1.3)does not have any linearization at the origin, but still can be related to some linear problems.The general idea is to approximate this equation by linearizable ones, for which we apply the global bifurcation results of Rabinowitz [19].Then, we pass to the limit using a priori bounds which are obtained with the aid of the asymptotic formulas for the eigenvalues of the linear Dirac systems.Note that in our case the bifurcation intervals may overlap, but the use of nodal properties ensures that this does not invalidate the global bifurcation results.

Preliminaries
If h ≡ 0, then (1.1)-(1.3) is a linear canonical one-dimensional Dirac system [12, Ch. 1, § 10] It is known (see [12,Ch. 1,§ 11]) that eigenvalues of the boundary value problem (2.1) are real, algebraically simple and the values range from −∞ to +∞ and can be numerated in increasing order.We consider a more general problem where P(x) = p(x) q(x) s(x) r(x) , q(x) and s(x) are real valued, continuous functions on the interval [0, π].The problem (2.2) is equivalent to the following eigenvalue problem for the system of two first-order ordinary differential equations Without loss of generality we can assume that s(x) ≡ q(x).Indeed, if s(x) ≡ q(x), then using the transformations x 0 ( q(t)− s(t))dt and z(x) = v(x)e − 1 2 x 0 ( q(t)− s(t))dt , we can rewrite the system (2.3) in the form z (x) − p (x)y(x) − q(x)z(x) = λy(x), (2.4) where q(x) ≡ s(x) ≡ 1 2 (q(x) + s(x)) .
Remark 2.2.If s(x) ≡ q(x), then the substitution where 2) into the following problem (which has of the form (2.1)) (see [12, Ch. 1, § 10]), ( Thus, the eigenvalues of the boundary value problem (2.2) are real, algebraically simple and the values range from −∞ to +∞ and can be numerated in increasing order.
One can readily show that there exists a unique solution w(x, λ) moreover, for each fixed x ∈ [0, π] the functions u(x, λ) and v(x, λ) are entire functions of the argument λ.The proof of this assertion reproduces that of Theorem 1.1 from [12, Ch. 1, § 1] with obvious modifications.We recall the Pr üfer angular variable (2.8) We recall that u, v have fixed initial values for x = 0, and all λ, given by (2.7).We define initially in view (2.7).For other x and λ, θ(x, λ) is given by (2.8) except for an arbitrary multiple of 2π, since u and v cannot vanish simultaneously.This multiple of 2π is to be fixed so that θ(x, λ) satisfies (2.9) and is continuous in x and λ.Since the (x, λ)-region, namely, 0 ≤ x ≤ π, −∞ < λ < +∞, is simply-connected, this defines θ(x, λ) uniquely.
Remark 2.3.From (2.8) it is obvious that the zeros of the functions u(x, λ) and v(x, λ) are the same as the occasions on which θ(x, λ) is an odd or even multiple of π/2, respectively.

Theorem 2.4 ([4, Theorem 2.1]).
The following properties of the angular function θ(x, λ) are true: (i) θ(x, λ) satisfies the differential equation, with respect to x, then as x increases, θ cannot tend to a multiple of π/2 from above, and as x decreases, θ cannot tend to a multiple of π/2 from below; if λ + p(x) < 0, λ + r(x) < 0 for x ∈ [0, π] , then as x increases, θ cannot tend to a multiple of π/2 from below, and as x decreases, θ cannot tend to a multiple of π/2 from above; (iii) as λ increases, for fixed x, θ is increasing; in particular, θ(π, λ) is a strictly increasing function of λ.
We have the following oscillation theorem.
2) can be numbered in ascending order on the real axis The eigenvector-functions w k have, with a suitable interpretation, the following oscillation properties: if k > 0 and k = 0, α ≥ β (except the cases α = β = 0 and ) where s(g) the number of zeros of the function g ∈ C([0, π] ; R) in the interval (0, π) and is true.
We define E to be the Banach space For each w = ( u v ) ∈ S we define θ(w, •) to be continuous function on [0, π] satisfying where w k (x) is an eigenvector-function corresponding to the eigenvalue λ k of problem (2.2).Let S + k be set of w ∈ S which satisfy the conditions: ), then for fixed w, as x increases from 0 to π, the function θ cannot tend to a multiple of π/2 from above, and as x decreases, the function θ cannot tend to a multiple of π/2 from below; if k < 0 or k = 0, α < β, then for fixed w, as x increases, the function θ cannot tend to a multiple of π/2 from below, and as x decreases, the function θ cannot tend to a multiple of π/2 from above. Let ∈ S k , k ∈ Z, then the number of zeros of functions u(x) and v(x) are determined by (2.12)-(2.13)and there functions have only nodal zeros in (0, π).
where c 0 is a positive constant.Integrating both sides of the inequality (2.17 Assume that λ = 0 is not an eigenvalue of (2.1).Then the problem (1.1)-(1.3)can be converted to the equivalent integral equation where K(x, t) = K(x, t, 0) is the appropriate Green's matrix (see [12, Ch. 1, formula (13.8)]).Define L : E → E by The Green matrix K(x, t) is continuous in [0, π; 0, π] everywhere except on the diagonal x = t, where it has a jump K(x, x + 0) − K(x, x − 0) = B. Then L is completely continuous in E.
The operators F and G can be represented as a compositions of a operator L and the superposition operators f(λ, w(x)) = f (x, w(x), λ) and g(λ, w(x)) = g(x, w(x), λ), respectively.Since and therefore, it is enough to investigate the structure of the set of solutions of (1.1)-

Bifurcation for a class of linearizable problems
We suppose that f ≡ 0 (3.1) (in effect, we suppose that the nonlinearity h itself satisfies (1.5)).Then, by (2.24), problem (1.1)-(1.3) is equivalent to the following problem Note that problem (3.2) is of the form (0.1) of [19].The linearization of this problem at w = 0 is the spectral problem w = λLw.
Obviously, the problem (3.3) is equivalent to the spectral problem (2.1).We denote by Y the closure in R × E of the set of nontrivial solutions of (2.24) (i.e. of (1.1)-(1.3)).
In the following, we will denote by w , k ∈ Z, the unique eigenvectorfunction of linear problem (2.1) associated to eigenvalue λ k such that lim x→0+ sgn u + k (x) = 1 and w + k (x) = 1.The linear existence theory for the problem (2.1) (or problem (3.3)) can be stated as: for each integer k and each ν, there exists a half line of solutions of problem (3.3) An analogous result holds for problem (3.2).
Theorem 3.1.Suppose that (3.1) holds.Then for each integer k and each ν, there exists a continuum of solutions The proof of this theorem is similar to that of Theorem 2.3 of [19] (see also [10]), using the above arguments from Section 2 and relation (2.23).
We say that the point (λ, 0) is a bifurcation point of problem (1.1)-(1.3)with respect to the set R × S ν k , k ∈ Z, if in every small neighborhood of this point there is solution to this problem which contained in R × S ν k (see [3]).
Corollary 4.2.The set of bifurcation points of problem (4.2) is nonempty, and if (λ, 0) is a bifurcation point of (4.2) with respect to the set R × S ν k , then λ ∈ J k .
For each k ∈ Z and each ν, we define the set D ν k ⊂ Y to be the union of all the components D ν k, λ of Y which bifurcating from the bifurcation points (λ, 0) of (4.2) with respect to the set R × S ν k .By Lemma 4.1 and Corollary 4.2 the set Proof.Suppose that (λ, w(x)) ∈ R × E is a solution of problem (4.2).Let Then (λ, w) is a solution of the following eigenvalue problem  Let w(x) ∈ S k for some k ∈ Z.According to Theorem 2.5 λ is a k-th eigenvalue of problem (4.12).Taking into account (1.4) where ε ∈ (0, 1].By (1.4) the function f (x, |w| ε w, λ) satisfies the condition (4.5).Then, by Theorem 3.1, for each integer k and each ν there exists an unbounded continuum A ν k,ε of solutions of (5.1) such that Hence, it follows that for any ε ∈ (0, 1] there exists a solution (λ τ, ε , w τ, ε ) of problem (5.1) such that w τ, ε ∈ S ν k and w τ, ε = τ.It is obvious that (λ τ, ε , w τ, ε ) is a solution of the nonlinear problem where and the functions ϕ ε (x), ψ ε (x), φ ε (x) and τ ε (x) are determined of right hand sides of (4.6) with (λ τ, ε , w τ, ε ) instead of (λ ε , w ε ).
Corollary 5.2.The set of bifurcation points of problem (1.1)-(1.3)with respect to the set R × S ν k is nonempty.

Theorem 4 . 3 .Lemma 4 . 4 .
For each k ∈ Z and each ν,the connected component D ν k of Y lies in (R × S ν k ) ∪ (J k × {0}) and is unbounded in R × E.Proof.By Lemma 4.1, Corollary 4.2 and an argument similar to that of [13, Theorem 2.1], we can obtain the desired conclusion.Assume that the function f (x, w, λ) satisfies the condition (1.4) for all x ∈ [0, π] and (w, λ) ∈ R 2 × R. Thus we have the following result.Let (λ, w) ∈ R × E be a solution of problem (4.2).Then w ∈ ∞ k =−∞ S k , and if w ∈ S k , then λ ∈ J k .

Theorem 5 . 5 .
For each k ∈ Z and each ν, the connected componentT ν k of Y lies in (R × S ν k ) ∪ (J k × {0}) and is unbounded in R × E.The proof of Theorem 5.5 is similar to that of [13, Theorem 2.1] using Lemmas 5.1, 5.3 and Corollaries 5.2, 5.4.
follows by (2.16), Remark 2.3 and Theorems 2.4, 2.5 that w k ∈ S k , k ∈ Z, i.e. the sets S − k , S + k and S k are nonempty.Moreover, if w
Proof.Alongside with the problem (1.1)-(1.3)we shall consider the following approximate problem w