Global bifurcation result and nodal solutions for Kirchho ﬀ -type equation

: We investigate the global structure of nodal solutions for the Kirchho ﬀ -type problem , where a > 0 , b > 0 are real constants, λ is a real parameter. f ∈ C ( R , R ) and there exist four constants s 1 ≤ s 2 < 0 < s 3 ≤ s 4 such that f (0) = f ( s i ) = 0 , i = 1 , 2 , 3 , 4, f ( s ) > 0 for s ∈ ( s 1 , s 2 ) ∪ (0 , s 3 ) ∪ ( s 4 , + ∞ ) , f ( s ) < 0 for s ∈ ( −∞ , s 1 ) ∪ ( s 2 , 0) ∪ ( s 3 , s 4 ). Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of nodal solutions of this problem which bifurcate from the line of trivial solutions or from inﬁnity,


Introduction
This paper is devoted to the Kirchhoff-type problem where a > 0, b > 0 are real constants, λ is a real parameter. In recent years, a lot of classical results have been concerned on a bounded domain for Kirchhoff equation. For example, the existence of solutions can be founded in [1][2][3][4][5][6][7][8][9] and the references therein.
When a = 1, b = 0 in problem (1.1), it reduces to the classic second-order semilinear problem. The conclusions of global bifurcation of such problems are well known, see [10][11][12][13][14] for details. In particular, Ma [10], Ma and Han [12] discussed the existence of nodal solutions when the nonlinear term of the problem (1.1) has two non-zero zeros.
In this article, we are interested in studying nodal solutions of problem (1.1) with the nonlinear term f has some zeros in R \ {0}. This work is motivated by the recent results of Cao and Dai [1] who concerned with determining values of λ for which there exist nodal solutions of the Kirchhoff-type problem (1.2) (1.2) is often used to describe the stationary problem of a model introduced by Kirchhoff to describe the transversal oscillations of a stretched string. Where f satisfies the following assumptions: (A1) f ∈ C((0, 1) × R, R) with s f (x, s) > 0 for all x ∈ (0, 1) and any s 0.
Using the bifurcation results of [1], the authors further established the following result: problem (1.2) possesses at least two solutions u + k and u − k such that u + k has exactly k − 1 simple zeros in (0, 1) and is positive near 0, and u − k has exactly k − 1 simple zeros in (0, 1) and is negative near 0. Based on the above works, of course the natural question is what would happen if f is allowed to have some zeros in R \ {0}? In this paper, we will establish the global bifurcation results about the components nodal solutions for the Kirchhoff-type problem (1.1). In order to obtain our main results, let us make the assumptions as follows: (H1) f ∈ C(R, R) and there exist s 1 ≤ s 2 < 0 such that f (0) = f (s 1 ) = f (s 2 ) = 0, and f (s) > 0 for s ∈ (s 1 , s 2 ), f (s) < 0 for s ∈ (−∞, s 1 ) ∪ (s 2 , 0).
s 3 uniformly with respect to all x ∈ (0, 1). The paper is organized as follows. In Section 2, we state some notations and preliminary results. Sections 3 and 4 are devoted to study the bifurcation from the trivial solution and infinity of problem (1.1), and we show the optimal intervals of λ for which the nodal solutions exist.

Preliminary
In this section, we introduce some lemmas and well-known results which will be used in the subsequent section.
Definition 2.1. Let X be a Banach space, {C n |n = 1, 2, 3, · · · } be a family of subsets of X. Then the superior D of C n is defined by ) Let X be a Banach space, C n is a component of X, assume that (i) There exists z n ∈ C n (n = 1, 2, · · · ) and z * ∈ X, such that z n → z * ; (ii) lim n→∞ r n = ∞, where r n = sup{ x : x ∈ C n }; Then D := lim sup n→∞ C n contains an unbounded component C such that z * ∈ C.
Let S + k denote the set of functions in E which have exactly k − 1 interior nodal (i.e. non-degenerate) zeros in (0, 1) and are positive near t = 0, and set S − When considering Kirchhoff-type problem, Dancer-type unilateral global bifurcation theorem is established in [1], which can be applied to similar problems.
and λ on bounded sets. Moreover, there are two distinct unbounded continua in R×H 1 0 (0, 1), C + k and C − k , consisting of the bifurcation branch C k emanating from (aλ k , 0), Obviously, as a bifurcation problem from the trivial solution u ≡ 0, and as a bifurcation problem from infinity. (2.1) and (2.2) are equivalent to the problem (1.1).
Let us discuss (2.1). According to Lemma 2.4, we can see that for each integer k ≥ 1 and v ∈ {+, −}, there exists a continuum C v k of solutions of (2.1) joining ( aλ k f 0 , 0) to infinity. In addition, Let us discuss (2.2). According to the proof of Theorem 1.3 of [1], we can see that for each integer k ≥ 1 and v ∈ {+, −}, there exists a continuum D v k of solutions of (2.2) meeting ( bµ k f ∞ , ∞). In addition, We note that when λ = 0, (1.1) has only trivial solution. Therefore, C + k and C − k are separated by the hyperplane λ = 0. Furthermore, we know that C + k and C − k are both unbounded.

Global bifurcation results for
In this section, we will provide more details about the connected components of nodal solutions under the assumptions that f has some zeros.  u(x) < s 1 .
Noting (3.4), we obtain that It is straightforward to see from s 3 > 0 that By virtue of the strong maximum principle [16], we can show that s 3 > u(x), x ∈ [τ j , τ j+1 ]. This contradicts (3.3). If (3.2) holds, then there exists j ∈ {0, · · · , l − 1} such that and Similarly, we claim that there exists a constant m > 0 such that Noting (3.6), we obtain that It is straightforward to see from s 2 < 0 that By virtue of the strong maximum principle [16], we can show that s 2 < u(x), x ∈ [τ j , τ j+1 ]. This contradicts (3.5).
The argument of (ii) is similar to that of (i).

Global bifurcation results for f ∞ = +∞
Theorem 4.1. Let (H1), (H3) and (H5) hold. Then, (i) if λ ∈ (0, aλ k f 0 ), then problem (1.1) has at least two solutions u − k,∞ and u + k such that u + k has exactly k − 1 zeros in (0,1) and is positive near 0, u − k,∞ has exactly k − 1 zeros in (0,1) and is negative near 0; (ii) if λ = aλ k f 0 , then problem (1.1) has at least one solution u − k,∞ ; (iii) if λ ∈ ( aλ k f 0 , +∞), then problem (1.1) has at least two solutions u − k,∞ , u − k,0 . Proof. For any n ∈ N + and n > −s 1 . Define the function f [n] : R → R as follows We can see from (H5) that lim n→∞ ( f [n] ) ∞ = +∞. Consider the following auxiliary problem Then lim |u|→∞ η [n] (u) as a bifurcation problem from infinity. It is easy to see from [17, Theorem 1.6 and Corollary 1.8] that for each integer k ≥ 1 and n ∈ N + with n > −s 1 , there exists a continuum D [n],− k,∞ of solutions of (4.2) meeting . Similar to the proof of Theorem 3.1, for any (λ, u) ∈ D [n],− k,∞ , we obtain that u(x 0 ) < s 1 for some x 0 ∈ (0, 1). Further, it is direct to check that We claim that u l ∞ → ∞ as l → ∞. (4.7) In fact, it is straightforward to see that (κ l , u l ) satisfies Thus, we know that there exists x l ∈ (0, 1) such that u l (x l ) = 0 and There is a positive constant N such that u l ∞ ≤ N for each l. Further, combining the definition of f [n] and (4.7), gives u l ∞ ≤ N for some N > 0 and all l.