Global bifurcation from intervals for Sturm – Liouville problems which are not linearizable ∗

In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm–Liouville problems of the form  − (pu′)′ + qu = λau+ af (x, u, u′, λ) + g (x, u, u′, λ) for x ∈ (0, 1), b0u(0) + c0u ′(0) = 0, b1u(1) + c1u ′(1) = 0, where a ∈ C([0, 1], [0,+∞)) and a(x) 6≡ 0 on any subinterval of [0, 1], f, g ∈ C([0, 1]× R3,R). Suppose that f and g satisfy |f(x, ξ, η, λ)| ≤M0|ξ|+M1|η|, ∀x ∈ [0, 1] and λ ∈ R, g(x, ξ, η, λ) = o(|ξ|+ |η|), uniformly in x ∈ [0, 1] and λ ∈ Λ, as either |ξ| + |η| → 0 or |ξ| + |η| → +∞, for some constants M0, M1, and any bounded interval Λ.

The aim of this paper is to improve or extend the corresponding results of [9] and [16] under weaker assumptions.In order to introduce our main results, next, we give some notations.
Let Lu := − (pu ) + qu.It is well known (see [4] or [20, p. 269]) that the linear Sturm-Liouville problem possesses infinitely many eigenvalues The eigenfunction ϕ k corresponding to λ k has exactly k − 1 simple zeros in (0, 1).Let k denote the set of functions in E which have exactly k − 1 simple zeros in (0,1) and are positive near x = 0, and set S − k = −S + k , and under the product topology.Finally, we use S to denote the closure in R × E of the set of nontrivial solutions of (1.1), and S ± k to denote the subset of S with u ∈ S ± k and The first main result of this paper is the following theorem.
for every k ∈ N * and some constants c 1 k and c 2 k which only depend on k.And assume that (A0), (A1) and (A3) hold.Then the component Use T to denote the closure in R × E of the set of nontrivial solutions of (1.1) under conditions (A0), (A2) and (A4).Our second main result is the following theorem.
for every k ∈ N * and some constants d 1 k and d 2 k which only depend on k.For every ν ∈ {+, −}, there exists a component and M is a neighborhood of I k × {∞} whose projection on R lies in Λ and whose projection on E is bounded away from 0, then either If 2 o occurs and D ν k −M has a bounded projection on R, then D ν k −M meets I j ×{∞} for some j = k.In addition, there exists a neighborhood The rest of this paper is arranged as follows.In Section 2, we give the proof of Theorem 1.1.In Section 3, we present the proof of Theorem 1.2 and give some remarks.

Proof of Theorem 1.1
Firstly, by an argument similar to that of [9, Lemma 2.2], we can show the following lemma.(2.1) The next lemma will play a key role in this paper which provides uniform a priori bounds for the solutions of problem (2.1) near the trivial solutions and will also ensure that Lemma 2.2.Let n , 0 ≤ n ≤ 1, be a sequence converging to 0. If there exists a sequence Proof.Without loss of generality, we may assume that u n ≤ 1.Let w n = u n / u n , then w n satisfies the problem where It follows from (A3) that g n (x) → 0 uniformly in x ∈ [0, 1].Furthermore, (A1) implies that 2), we know that w n is bounded in C 2 .By the Arzelà-Ascoli theorem, we may assume that w n → v in C 1 with w = 1.Clearly, we have w ∈ S ν k .We claim that w ∈ S ν k .On the contrary, suppose that w ∈ ∂S ν k , then w has at least one double zero x * ∈ [0, 1].It follows that w n (x * ) → 0 and w n (x * ) → 0 as n → +∞.Then by the argument of [2, p. 379], we can deduce w n → 0 in C 1 , which is a contradiction with w n = 1.Now, we deduce the boundedness of λ.Let ϕ ν k ∈ S ν k be an eigenfunction of problem (1.2) corresponding to λ k and [α, β] ⊆ [0, 1].Integrating by parts and taking the limit as n → +∞, we can obtain that It was shown in [2] that there are two intervals (ξ 1 , η 1 ) and (ξ 2 , η 2 ) in (0, 1) where w n and ψ ν k do not vanish and have the same sign and such that EJQTDE, 2013 No. 65, p. 4 So we have that Furthermore, one has that We choose c k ≥ 1 and c k ≥ 1 such that It follows that and where Therefore, we have that λ ∈ I k .
Proof of Theorem 1.1.By Lemma 2.1, 2.2 and an argument similar to that of [10, Theorem 2.1], we can obtain the desired conclusion.
3 Proof of Theorem 1.2 We add the points {(λ, ∞) λ ∈ R} to the space R × E. Note that if M 0 1 = 0, Theorem 1.1 degenerates to Theorem 2.1 of [9], and if M ∞ 1 = 0, Theorem 1.2 degenerates to Theorem 2.2 and 2.3 of [9].In fact, even in these special cases, the bifurcation intervals in this paper are smaller than the corresponding ones of [9].Remark 3.2.Note that our assumption on a is weaker than any mentioned paper (in introduction) dealing with this kind of problems.