UNILATERAL GLOBAL INTERVAL BIFURCATION FOR KIRCHHOFF TYPE PROBLEMS AND ITS APPLICATIONS

. In this paper, we establish a unilateral global bifurcation result from interval for a class of Kirchhoﬀ type problems with nondiﬀerentiable non- linearity. By applying the above result, we shall prove the existence of one-sign solutions for the following Kirchhoﬀ type problems. where Ω is a bounded domain in R N with a smooth boundary ∂ Ω, M is a continuous function, r is a parameter, a ( x ) ∈ C (Ω) is positive, u + = max { u, 0 } ,u − = − min { u, 0 } , α,β ∈ C (Ω); f ∈ C ( R , R ), sf ( s ) > 0 for s ∈ R + , and f 0 ∈ (0 , ∞ ) and f ∞ ∈ (0 , ∞ ] or f 0 = ∞ and f ∞ ∈ [0 , ∞ ] , where f 0 = lim | s |→ 0 f ( s ) /s,f = lim | s |→ + ∞ f ( s ) /s. We use unilateral glo- bal bifurcation techniques and the approximation of connected components to prove our main results.


1.
Introduction. In 1883, Kirchhoff [19] proposed the following problem as an extension of the classical d'Alembert's wave equation for free vibrations of elastic strings. The model studied is where u denotes the displacement, f is the external force, b represents the initial tension, and a is related to the intrinsic properties of the string. Kirchhoff's model (1) takes into account the changes in length of the string produced by transverse vibrations. Consider the following Kirchhoff type problem    −M ( Ω |∇u| 2 dx)∆u = λb(x)u + g(x, u, λ), in Ω, where Ω is a bounded domain in R N with a smooth boundary ∂Ω, M is a continuous function, λ is a parameter, b(x) ∈ L ∞ (Ω) is positive and the perturbation function g(x, s, λ) s = 0 uniformly for a.e. x ∈ Ω and λ on bounded sets. The problem (2) is nonlocal as the appearance of the term Ω |∇u| 2 dx which implies that it is not a pointwise identity. This causes some mathematical difficulties which make the study for the problem (2) particularly interesting. The main difficulties when dealing with this problem lie in the presence of the nonlocal terms which arises in nonlinear vibrations and the analogous to the stationary case of equations that arise in the study of string or membrane vibrations. After the famous article by Lions [23], Eq. (2) received much attention, and some important and interesting results have been obtained, for example, see [14,10,15,2,6,9]. In recent years, there has been considerable interest in the above problem (2) by variational method, see [28,26,7,20,14,27]. We refer to [21,17,8,4] for Kirchhoff models with critical exponents. Meanwhile, by applying the bifurcation techniques, there are few papers to study Kirchhoff-type problems, see for example [22,18].
Dai et al. [12] also assume that g and M satisfies the following conditions: (A0) There exist c > 0 and p ∈ (1, 2 * ) such that |g(x, s, λ)| ≤ c(1 + |s| (p−1) ), for a.e. x ∈ Ω and λ on bounded sets, where However, among the above papers, the nonlinearities are differentiable at the origin. In [5], Berestycki established an important global bifurcation theorem from intervals for a class of second-order problems involving non-differentiable nonlinearity. Recently, Ma and Dai [25] improved Berestycki's result(in [5]) to show a unilateral global bifurcation result for a class of second-order problems involving non-differentiable nonlinearity. Later, Dai and Ma [11] also considered a class of high-dimensional p-Laplacian problems involving non-differentiable nonlinearity.
Motivated by above papers, in this paper, we shall establish a Dancer-type unilateral global bifurcation result from interval for the Kirchhoff type problems where F : Ω × R 2 → R is a continuous function. Moreover, the nonlinear term F has the form F = f + g, where b, f and g satisfy the following conditions: (H2) f (x,s,λ) s ≤ M 1 , for all x ∈ Ω, 0 < |s| ≤ 1 and all λ ∈ R, where M 1 is a positive constant.
Obviously, by (H4), one may get that Furthermore, we shall investigate the existence of one-sign solutions for the following Kirchhoff- We assume that a, f satisfies the following assumptions: (H5) a(x) ∈ C(Ω) is positive. (H6) sf (s) > 0 for s = 0.

Remark 3.
For the abstract unilateral global bifurcation theory, we refer the reader to [29,5,25,13] and the references therein.
The rest of this paper is arranged as follows. In Section 2, we given some Preliminaries. In Section 3, we establish the unilateral global bifurcation result from the interval for the problem (5). In Section 4, on the basis of the unilateral global interval bifurcation result, we shall investigate the existence of one-sign solutions for the Kirchhoff-type problems (8). 24
From [12, p.733], it is clear that the problem (2) can be equivalently written as , where H(λ, ·) denotes the usual Nemitsky operator associated with g.
Next, we give an important lemma which will be used later. Lemma 2.3. (see [1]). Let u, v ∈ C 1 (Ω), v = 0 in Ω. Then we have the following identity: Remark 4. (see [1]). By Young's inequality, we get and the equality holds if and only if ∇( u v ) = 0, a.e. Ω, i.e., u = kv for some constant k in each component of Ω.
By Lemma 2.3 and Remark 4,we have the following result: Next, we summarize following Lemma from [25] which will be used later.
. There exist two simple half-eigenvalues λ + and λ − for the following problem The corresponding half-linear solutions are in {λ + } × P + and {λ − } × P − . Furthermore, aside from λ + and λ − , there is no other half-eigenvalue with positive or negative eigenfunction.Where the definitions of half-eigenvalue problem (10) and principal half-eigenvalue λ ± are given by [25,In Section 3].
Furthermore, by Lemma 2.5 and Remark 2, it follows that The corresponding half-linear solutions are in are given in Remark 2 by Remark 1.
Now, we also give an lemma which will be used later.
be two weight functions satisfying the conditions given in Section 2.1 (see [28] Then any solution v of In order to treat the problems with non-asymptotic nonlinearity at 0 and ∞, we shall need the following definition and lemma. Definition 2.8. (see [31]). Let X be a Banach space and let {C n |n = 1, 2, ...} be a certain infinite collection of subsets of X. Then the superior limit D of {C n } is defined by Lemma 2.9. (see [24]). Let X be a Banach space and let {C n |n = 1, 2, ...} be a family of closed connected subsets of X. Assume that (i) there exist z n ∈ C n , n = 1, 2, ... and z * ∈ X, such that z n → z * ; Then there exists an unbounded component C in D and z * ∈ C.
3. Unilateral global bifurcation. Let S ± denote the closure in K ± of the set of nontrivial solutions of (5). The first main result for (5) is the following theorem.
To prove Theorem 3.1, we introduce the following auxiliary approximate problem: To prove Theorem 3.1, the next lemma will play a key role.
Proof. Without loss of generality, we may assume that u n ≤ 1. Let where where H(λ n , ·) denotes the usual Nemitsky operator associated with g. By (A1), (H3) and [12, p.773-774], it follows that uniformly on bounded λ intervals. Furthermore, it follows that lim un →0 uniformly on bounded λ sets.
Since u n / u n is bounded in X, so |u n / u n | ≤ c 0 for some positive constant c 0 . Furthermore, (H2) implies that for a.e. x ∈ Ω and n large enough. Using this fact with (13) and (14), we have that un is bounded in L ∞ (Ω) for n large enough. By the Arzelà-Ascoli theorem, the completely continuous of L −1 implies that w n is strong convergence in C 1 (Ω). Without loss of generality, we may assume that w n → w in E. Clearly, we have w ∈ P ν . Now, we deduce the boundedness of λM (0). Let ψ ∈ P ν be an eigenfunction of problem (9) corresponding to λ 1 M (0).
Proof of Theorem 3.1. We only prove the case of C + since the case of C − is similar.
We divide the rest of proofs into two steps.
Step 2. We prove that C + is unbounded. Suppose on the contrary that C + is bounded. By the similar method to prove [5, Theorem 1] with obvious changes, we can find a neighborhood O of C + such that ∂O ∩ S + = ∅.
In order to complete the proof of this theorem, we consider the problem (12). For > 0, it is easy to show that nonlinear term f (x, u|u| , λ) + g(x, u, λ) satisfies the condition (3) So there exists (λ , u ) ∈ C + ∩ ∂O for all > 0. Since O is bounded in R × P + , Eq. (12) shows that (λ , u ) is bounded in R × X independently of . By the compactness of L −1 , one can find a sequence n → 0 such that (λ n , u n ) converges to a solution (λM (0), u) of (5). So u ∈ P + . We claim that u ≡ 0. Suppose on the contrary that u ≡ 0. By Lemma 3.2, λM (0) ∈ I, which contradicts the definition of O. Using a similar way employed in Step 1, we can show that u > 0 in Ω. Then we have that (λM (0), u) ∈ S + ∩ ∂O which contradicts S + ∩ ∂O = ∅.
From Theorem 3.1 and its proof, we can easily get the following two corollaries.  On the basis of Theorem 3.1 and Corollary 1, we shall obtain the following result for the Kirchhoff type problem (7). Proof. Let α 0 := max x∈Ω |α(x)|, β 0 := max x∈Ω |β(x)| and Corollary 1 shows that there exist two unbounded sub-continua D + and D − of solutions of (7) in R×X, bifurcating from I 0 ×{0}, and D ν ⊂ (R×P ν )∪(I 0 ×{0}) for ν = + and ν = −. Let us show that is a bifurcation point for problem (7). Indeed, if there exists (λ n , u n ) be a sequence of solutions of the problem (7) converging to (λM (0), 0), let v n = un un , then v n should be a solution of problem where A(x) = α(x)/M (0) and B(x) = β(x)/M (0) are given in Remark 2 by Remark 1.

4.
One-sign solutions for Kirchhoff type problems. Let Σ ± denote the closure in K ± of the set of nontrivial solutions of (8).
We divide the rest proof into two steps.
Let ξ ∈ C(R, R) be such that We divide the equation , in Ω, by u n and set v n = un un . Since v n is bounded in X, choosing a subsequence and relabeling if necessary, we have that v n → v for some v ∈ X and v n → v in L 2 (Ω).
It follows from (24) that for any ε > 0, there exists a constant C such that |ξ(u n )| ≤ C + ε|u n |.

WENGUO SHEN
By the compactness of L −1 , we obtain that where µ := lim n→∞ µ n , again choosing a subsequence and relabeling if necessary.
Step 2. We show that there exists a constant number M > 0 such that |µ n | ∈ (0, M ], for n ∈ N large enough. On the contrary, we suppose that lim n→+∞ |µ n | = +∞. Since (µ n , u n ) ∈ D ν , it follows from the compactness of L −1 that Let ψ ν be an eigenfunction corresponding to λ ν and [ λ ν ra(x) But if lim n→+∞ f 1n (x) = −∞, applying Lemma 2.7 to u n and ψ ν we have that ψ ν must change sign for n large enough, which is impossible. So lim n→+∞ f 1n (x) = +∞. By Lemma 2.7, we get that u n must change sign for n large enough, and this contradicts the fact that u n ∈ P ν . The result follows. (i) r ∈ (0, λ ν f0 M (0)) for λ ν > 0.
Proof. We only prove the case of (i) since the proof of (ii)-(iv) can be given similarly.
Inspired by the idea of [3], we define the cut-off function of f as the following We consider the following problem  (26) can be converted to the equivalent equation (18). Since D ν[n] ⊂ (R × P ν ), we conclude D ν ⊂ (R × P ν ). Moreover, D ν ⊂ ν by (8).
Proof. We shall only prove the case of (i) since the proofs of the cases for (ii), (iii) and (iv) are completely analogous.
Proof. We shall only prove the case of (i) since the proofs of the cases for (ii), (iii) and (iv) are completely analogous. Define