Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin

positive constant, and BR(0) � x ∈ R: |x|<R 􏼈 􏼉 is the standard open ball in the Euclidean space R(N≥ 1) which is centered at the origin and has radius R. a(|x|) ∈C[0, R] is positive, v �max{v, 0}, v � − min{v, 0}, α(|x|), β(|x|) ∈C[0, R]; f ∈ C(R,R), s f (s) > 0 for s≠ 0, and f0 ∈ [0,∞], where f0 � lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.


Introduction
We first consider the following problem with mean curvature operator in Minkowski space: where λ ≠ 0 is a parameter, R is a positive constant, and B R (0) � x ∈ R N : |x| < R is the standard open ball in the Euclidean space R N (N ≥ 1), which is centered at the origin and has radius R. Here, the nonlinear function F ∈ C(B R (0) × R 2 , R) and a(|x|) is a weighted function. Dirichlet problem (1) is associated to the mean curvature operator in the flat Minkowski space L N+1 with (x 1 , . . ., x N , t) and the Lorentzian metric N i�1 (dx i ) 2 − (dt) 2 . Some important and interesting results [1][2][3] for this type of problems have been obtained. Some specialists have studied problem (1); for example, Cheng and Yau [4] and Treibergs [5] studied problem (1) with λav ≡ F ≡ 0 and λav + F ≡ C, respectively. Bidaut-Véron and Ratto [6] and López [7] studied problem (1) with a ≡ 0, F � f(u) and a ≡ 0, F � kv + λ, respectively.
Recently, Bereanu et al. [8,9] have proved existence of classical positive radial solutions for problem (1) by Leray-Schauder degree argument and critical point theory.
On the contrary, among the abovementioned papers, the nonlinearities are differentiable at the origin. In [13], Berestycki established an important global bifurcation theorem from intervals for a class of second-order problems involving nondifferentiable nonlinearity.
Recently, Dai and Ma [14,15] considered interval bifurcation problem for second-order and high-dimensional p-Laplacian problems involving nondifferentiable nonlinearity, respectively.
In 2016, when a(x) ≡ 1, R ≡ 1, Dai and Yang [16] have established a global bifurcation result from interval for the following problem with nondifferentiable nonlinearity: It is clear that the radial solutions of (2) is equivalent to the solutions of the following problem: where λ is a parameter, r � |x| and x ∈ B R (0), the nonlinear term F has the form are radially symmetric with respect to r, and a, f, and g satisfy the following conditions: interior nodal (i.e., nondegenerate) zeros in (0, R) and are positive at r � 0, and set S − k under the product topology. We use S to denote the closure of the nontrivial solutions set of problem (3) in R × E, and S ± k to denote the subset of S with u ∈ S ± k and S k � S + k ∪ S − k . Using the same method to prove ([16, eorem 1]) with obvious changes, we may get the following global bifurcation result about Lemma 1.
Using the same method to prove ( [16], eorem 2) with obvious changes, we may get the following result about Lemma 2.
ere exist two sequences of simple halfeigenvalues for problem (5) . e corresponding half-linear solutions are in λ + k × S + k and λ − k × S − k . Furthermore, aside from these solutions and the trivial one, there are no other solutions of problem (5).
Motivated by the abovementioned papers, in this paper, we shall firstly establish a Dancer-type unilateral global bifurcation result (see eorem 2) for the following problem: It is clear that the radial solutions of (6) is equivalent to the solutions of the following problem: where λ ≠ 0 is a parameter, r � |x| and u(r) � v(|x|), a satisfies (H1), α, β satisfy (H4).
satisfies (H3) and the following condition: By a solution to problem (7), we mean a function (7) is satisfied. Here, ‖·‖ ∞ denotes the usual sup-norm.
Furthermore, we shall investigate the existence of solutions for the following problems: It is clear that the radial nodal solutions of (8) is equivalent to the solutions of the following problem: where λ ≠ 0 is a parameter, a satisfies (H1) and α and β satisfy (H4). Clearly, the nonlinear term of (9) is not necessarily differentiable at the origin because of the influence of the term αu + + βu − . So, the bifurcation theory of [11,12] cannot be applied directly to obtain our results. Fortunately, using the global interval bifurcation, we can obtain some results of the existence of radial solutions which extend the corresponding results of [12]. e rest of this paper is arranged as follows. In Section 2, we establish a unilateral global bifurcation result for halflinear perturbation problems (7). In Section 3, on the basis of the unilateral global bifurcation result (see eorem 2), we shall investigate the existence of nodal solutions for a class of the half-linear perturbation problems (9).

Unilateral Global Bifurcation for Problem (7)
From Lemma 1 (or see eorem 1 and its proof of [16]), we can easily get the following result.
Next, it is clear that problem (14) can be equivalently written as where K 2 (r, u, λ) � αu + + βu − + K 1 (r, u, λ), where G(r, s) be Green's function associated with the operator Lu: � − (r N− 1 u ′ ) ′ with the same boundary condition as in problem (14) (see [18] then g is nondecreasing and uniformly for r ∈ (0, R) and λ on bounded sets. Similar to Lemma 2.2 in [12], with obvious changes, we may get the following lemma. Proof. Integrating the first equation of problem (7) from 0 to r for any r ∈ [0, R], we get that By (H3), we have that lim k⟶+∞ g(r, u k , λ) � 0 uniformly r ∈ [0, R] and λ on bounded sets. It follows that lim k⟶+∞ u k ″ � 0 uniformly in r ∈ [0, R].
Taking the limit r ⟶ 0 + on both sides of equation (21), together with lim r⟶0 + u k (r) � 0 and lim r⟶0 + u k ′ (r) � 0, by L'Hospital's rule, we have that We obtain that lim r⟶0 + u k ″ (r) � 0. By lim r⟶R − u k (r) � lim r⟶R − u k ′ (r) � 0, taking the limit r ⟶ R − on both sides of equation (21), we may get that lim r⟶R − u k ″ (r) � 0.

Theorem 2. Let (H1) and (H3)-(H6) hold. For
is a bifurcation point for problem (7). Moreover, there exists a subcontinuum D ] k of solutions of problem (7), for v ∈ {+, − }, such that Proof. By (19), one obtains that uniformly for r ∈ (0, R) and λ on bounded sets. It follows that By Lemma 4, we can get that By (26) and (27), we have that uniformly for r ∈ (0, R) and λ on bounded sets. Let (29) eorem 1 shows that there exist two unbounded subcontinua D + k and D − k of solutions of (14) for v � + and v � − , in other words, (i) and (ii) hold.
(v) Finally, we show that By Lemma 4, we only show that lim k⟶∞ ‖u k ‖ ∞ � 0. On the contrary, one supposes lim k⟶∞ ‖u k ‖ ∞ � ρ > 0. 6 Journal of Function Spaces Let t 0 k � 0 < t 1 k < t 2 k < · · · < t k− 1 k < t k k � R denote the zeros of u k . By [12, p. 475 R] |h(t, u k )|. By [12, p. 475], one may obtain that Furthermore, we have that erefore, we may obtain that However, by an argument similar to that of [19, Proposition 3.7], one obtains that t j+1 k − t j k ≤ C/k ⟶ 0, as k ⟶ +0, and we get a contradiction.

Remark 1. eorem 2 indicates that the bifurcation interval
i.e., for problem (7), the bifurcation interval I 0 k is a finite point set. What conditions can ensure that the component indeed bifurcating from an interval is still an open problem for the problems with the mean curvature operator in Minkowski space.
To prove eorem 4, we use Whyburn type superior limit theorems. From [20], if the collection of the infinite sequence of sets is unbounded, Whyburn's limit theorem ([21, eorem 9.1]) cannot be used directly because the collection may not be relatively compact (where the definitions of superior limit and inferior limit, see [20, line 11 to line 16]). Dai [20] overcomed this difficulty and established the following results.
(ii) For every R > 0, (∪ +∞ n�1 C n ) ∩ B R is a relatively compact set of X, where en, D: � limsup n⟶∞ C n is unbounded, closed, and connected.
In order to treat the problems with nonasymptotic nonlinearity at ∞, we shall need the following lemmas.
Lemma 6 (see [20,Corollary 2.1]). Let X be a normal vector space and let {C n |n � 1, 2, . . .} be a sequence of unbounded connected subsets of X. Assume that (i) ere exists z * ∈ lim inf n⟶+∞ C n with ‖z * ‖ � +∞. (ii) ere exists a homeomorphism T: X ⟶ X such that ‖T(z * )‖ < + ∞ and {T(C n )} be a sequence of unbounded connected subsets in X. (iii) For every R > 0, (∪ +∞ n�1 C n ) ∩ B R is a relatively compact set of X, where en, D: � limsup n⟶∞ C n is unbounded, closed, and connected.
Lemma 7 (see [20,Lemma 2.6]). Let (X, ρ) be a metric space. If C i i∈N is a sequence of sets whose limit superior is L and there exists a homeomorphism T: X ⟶ X such that, for every R > 0, (∪ +∞ i�1 T(C i )) ∩ B R is a relatively compact set, then for each ϵ > 0 there exists an m such that, for every n > m, C n ⊂ V ϵ (L), where V ϵ (L) denotes the set of all points p with ρ(p, x) < ϵ for any x ∈ L.