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

1. Introduction

We first consider the following problem with mean curvature operator in Minkowski space:

[mathematical expression not reproducible], (1)

where [lambda] [not equal to] 0 is a parameter, R is a positive constant, and [B.sub.R] (0) = {x [member of] [R.sup.n]: [absolute value of (x)] < R} is the standard open ball in the Euclidean space [R.sup.N] (N [greater than or equal to] 1), which is centered at the origin and has radius R. Here, the nonlinear function F [member of] C([bar.[B.sub.R] (0)] x [R.sup.2], R) and a([absolute value of (x)]) is a weighted function. Dirichlet problem (1) is associated to the mean curvature operator in the flat Minkowski space [L.sup.N+1] with ([x.sub.1], ..., [x.sub.N], t) and the Lorentzian metric [[SIGMA].sup.N.sub.i=1] [(d[x.sub.i]).sup.2] - [(dt).sup.2].

Some important and interesting results [1-3] 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 [lambda]av [equivalent to] F [equivalent to] 0 and [lambda]av + F [equivalent to] C, respectively. Bidaut-Veron and Ratto [6] and Lopez [7] studied problem (1) with a [equivalent to] 0, F = f(u) and a [equivalent to] 0, F = kv + [lambda], 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.

In 2016, Ma et al. [10] and Dai and Wang [11,12] studied the existence of radial positive solutions and radial nodal solutions for problem (1) (where [lambda]av + g = [lambda]f([absolute value of (x)], v)) by bifurcation techniques, respectively.

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) [equivalent to] 1, R [equivalent to] 1, Dai and Yang [16] have established a global bifurcation result from interval for the following problem with nondifferentiable nonlinearity:

[mathematical expression not reproducible]. (2)

It is clear that the radial solutions of (2) is equivalent to the solutions of the following problem:

[mathematical expression not reproducible], (3)

where [lambda] is a parameter, r = [absolute value of (x)] and x [member of] [B.sub.R](0), the nonlinear term F has the form F = f + g, where f, g [member of] C([0,R] x [R.sup.2]) are radially symmetric with respect to r, and a, f, and g satisfy the following conditions:

(H1) a [member of] C([0, R], (0, [infinity])) is radially symmetric

(H2) [absolute value of ((f(r,s, [lambda]))/s)] [less than or equal to] [M.sub.1], for all r [member of] (0, R), 0 < [absolute value of (s)] [less than or equal to] R and all X [member of] R, where [M.sub.1] is a positive constant

(H3) g(r, s, [lambda]) = o([absolute value of (s)]) near s = 0 uniformly in r [member of] (0, R) and [lambda] on bounded sets

Let C[0, R] is a real Banach space with the norm [[parallel]u[parallel].sub.[infinity]] = [max.sub.re[0,R]] [absolute value of (u(r))]. Let E = {u(r) [member of] [C.sup.1] [0, R]: u'(0) = u(R) = 0} with the usual norm [parallel]u[parallel] = max{[[parallel]u[parallel].sub.[infinity]], [[parallel]u'[parallel].sub.[infinity]]}. Let [S.sup.+.sub.k] denote the set of functions in E which have exactly k - 1 interior nodal (i.e., nondegenerate) zeros in (0, R) and are positive at r = 0, and set [S.sup.-.sub.k] = - [S.sup.+.sub.k] and [S.sub.k] = [S.sup.+.sub.k] [union] [S.sup.-.sub.k]. Let [[PHI].sup.[+ or -].sub.k] = R x [S.sup.[+ or -].sub.k] under the product topology. We use S to denote the closure of the nontrivial solutions set of problem (3) in R x E, and [S.sup.[+ or -].sub.k] to denote the subset of S with u [member of] [S.sup.[+ or -].sub.k] and [S.sub.k] = [S.sup.+.sub.k] [union] [S.sup.-.sub.k].

Using the same method to prove ([16, Theorem 1]) with obvious changes, we may get the following global bifurcation result about Lemma 1.

Lemma 1 (see [16, Theorem 1]). Let (H1), (H2), and (H3) hold. Let [I.sub.k] = [[[lambda].sub.k] - [M.sub.1], [[lambda].sub.k] + [M.sub.1]]. The component [C.sup.+.sub.k] of [S.sup.+.sub.k] [union] ([I.sub.k] x {0}), containing [I.sub.k] x {0}, is unbounded and lies in [S.sup.+.sub.k] [union] ([I.sub.k] x {0}) and the component [C.sup.-.sub.k] of [S.sup.-.sub.k] [union] ([I.sub.k] x {0}), containing [I.sub.k] x {0}, is unbounded and lies in [S.sup.-.sub.k] [union] ([I.sub.k] x {0}).

Let [[lambda].sub.k] be the kth eigenvalue of the following eigenvalue problem:

[mathematical expression not reproducible]. (4)

It is well known that [[lambda].sub.k] is positive and simple, and the eigenfunction corresponding to it has exactly k - 1 simple zeros in (0, R) (see [17, Theorem 2.1]).

Furthermore, Dai and Yang [16] (a(x) [equivalent to] 1, R [equivalent to] 1) established spectrum of the following problem:

[mathematical expression not reproducible], (5)

where a satisfies (H1), [u.sup.+] = max{u, 0}, [u.sup.-] = - min{u, 0}, a(r) and [beta](r) satisfy:

(H4) a(r), [beta](r) [member of] C[0, R] are radially symmetric

Problem (5) is called half linear because it is positively homogeneous and linear in the cones u > 0 and u < 0. Similar to [13], we say that [lambda] is a half-eigenvalue of problem (5), if there exists a nontrivial solution ([lambda], [u.sub.[lambda]]). The eigenvalue [lambda] is said to be simple if v = c[u.sub.[lambda]], [epsilon] > 0, for all solutions ([lambda], v) of problem (5).

Using the same method to prove ([16], Theorem 2) with obvious changes, we may get the following result about Lemma 2.

Lemma 2 (see [16, Theorem 2]). Let (H1) and (H4) hold. There exist two sequences of simple halfeigenvalues for problem (5), [[lambda].sup.+.sub.1] < [[lambda].sup.+.sub.2] ... < [[lambda].sup.+.sub.k] < ... and [[lambda].sup.- .sub.1] < [[lambda].sup.-.sub.2] ... < [[lambda].sup.-.sub.k] < .... The corresponding half-linear solutions are in {[[lambda].sup.+.sub.k]} x [S.sup.+.sub.k] and {[[lambda].sup.-.sub.k]} x [S.sup.-.sub.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 Theorem 2) for the following problem:

[mathematical expression not reproducible]. (6)

It is clear that the radial solutions of (6) is equivalent to the solutions of the following problem:

[mathematical expression not reproducible], (7)

where [lambda] [not equal to] 0 is a parameter, r = [absolute value of (x)] and u(r) = v([absolute value of (x)]), a satisfies (H1), [alpha], [beta] satisfy (H4). Assume g [member of] C([0, R] x [-R, R] x R) satisfies (H3) and the following condition:

(H5) There exists a function h(t,u) [member of] C([0,R] x [-R, R], R) with h(t, s)s > 0 for any r [member of] [0, R] and s [not equal to] 0, such that g(t, u, [lambda]) = [lambda]h(t, u)

By a solution to problem (7), we mean a function u = u(r) [member of] [C.sup.1] [0, R] with [[parallel]u'[parallel].sub.[infinity]] < 1, such that [r.sup.N-1] u'[square root of (1 - [u'.sup.2])] is differentiable and (7) is satisfied. Here, [[parallel]*[parallel].sub.[infinity]] denotes the usual sup-norm.

Furthermore, we shall investigate the existence of solutions for the following problems:

[mathematical expression not reproducible]. (8)

It is clear that the radial nodal solutions of (8) is equivalent to the solutions of the following problem:

[mathematical expression not reproducible], (9)

where [lambda] [not equal to] 0 is a parameter, a satisfies (H1) and [alpha] and [beta] satisfy (H4). Clearly, the nonlinear term of (9) is not necessarily differentiable at the origin because of the influence of the term [alpha][u.sup.+] + [beta][u.sup.-]. 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].

The rest of this paper is arranged as follows. In Section 2, we establish a unilateral global bifurcation result for half-linear perturbation problems (7). In Section 3, on the basis of the unilateral global bifurcation result (see Theorem 2), we shall investigate the existence of nodal solutions for a class of the half-linear perturbation problems (9).

2. Unilateral Global Bifurcation for Problem (7)

From Lemma 1 (or see Theorem 1 and its proof of [16]), we can easily get the following result.

Theorem 1. There exist two unbounded subcontinua [D.sup.+.sub.k] and [D.sup.-.sub.k] of solutions of (3) in R x E, bifurcating from [I.sub.k] x {0}, for v [member of] {+, -}, such that [D.sup.v.sub.k] [subset] ([[PHI].sup.v.sub.k] [union] ([I.sub.k] x {0})).

Similar to the process of ([11, (2.1)-p.60]) or ([12, (2.1)-p.469]) obtained, problem (7) is equivalent to

[mathematical expression not reproducible], (10)

where [H.sub.1](r, u, [lambda]) = [lambda]a(r)u + [alpha][u.sup.+] + [beta][u.sup.-] + g(r, u, [lambda]). If u is a solution of problem (7), then for any r [member of] (0, R), one has that

[mathematical expression not reproducible]. (11)

By (11), one obtains that -u" = [H.sub.1](r,u, [lambda])(1 - [u'.sup.2]) + N - 1/r u'(1 - [u'.sup.2]). (12)

Substituting (12) into (11), it follows that

-([r.sup.N-1] u')' = [r.sup.N-1] [H.sub.1] (r,u, [lambda])[(1 - [u'.sup.2]).sup.3/2] + (N - 1)[r.sup.N-2] [u'.sup.3]. (13)

Thus, problem (7) is equivalent to

[mathematical expression not reproducible], (14)

where

[mathematical expression not reproducible]. (15)

In Theorem 2, we shall prove that [K.sub.1] (r, u, [lambda]) satisfies (H3). Similar to ([13, (3.2)-p.383]), problem (14) is the perturbation of problem (5). In other words, problem (7) is the perturbation of problem (5).

Next, it is clear that problem (14) can be equivalently written as

u = [lambda][L.sub.N] u + [L.sub.N] [omicron] [K.sub.2] = F([lambda], u), (16)

where

[mathematical expression not reproducible], (17)

where G(r, s) be Green's function associated with the operator Lu: = - ([r.sup.N-1] u')' with the same boundary condition as in problem (14) (see [18]). L: E ----E is linear completely continuous (see ([18, (2.5)-p.502 to line 4-p.503])). L[omicron][K.sub.2]: R x E [right arrow] E is completely continuous (see [18]). Moreover, F is completely continuous from R x E [right arrow] E and F([lambda], 0) = 0, [for all][lambda] [member of] R.

Let

[mathematical expression not reproducible], (18)

on bounded sets,

then [bar.g] is nondecreasing and

[mathematical expression not reproducible], (19)

uniformly for r [member of] (0, R) and X on bounded sets.

Similar to Lemma 2.2 in [12], with obvious changes, we may get the following lemma.

Lemma 3. For fixed [lambda] > 0, if{[u.sub.k]} is a sequence of solutions of problem (7) satisfying [lim.sub.k[right arrow]+[infinity]] [u.sub.k] = 0 uniformly in r [member of] [0, R], then [lim.sub.k[right arrow]+[infinity]] [u'.sub.k] = 0 and [lim.sub.k[right arrow]+[infinity]]m [u".sub.k] = 0 uniformly in r [member of] [0, R].

Proof. Integrating the first equation of problem (7) from 0 to r for any r [member of] [0, R], we get that

[r.sup.N-1] [u'.sub.k]/[square root of (1 - [u'.sup.2.sub.k])] = - [[integral].sup.r.sub.0] [a[r.sup.N-1] [u.sup.+.sub.k] + [beta][r.sup.N-1] [u.sup.-.sub.k] + [r.sup.N-1] g(r, [u.sub.k], [lambda])]dr. (20)

By (H3), we have that [lim.sub.k[right arrow]+[infinity]] g(r, [u.sub.k], [lambda]) = 0 uniformly r [member of] [0, R] and [lambda] on bounded sets. It follows that [lim.sub.k[right arrow]+[infinity]][u".sub.k] = 0 uniformly in r [member of] [0, R].

By (12), for any r [member of] (0, R), one has that

[mathematical expression not reproducible]. (21)

By [lim.sub.k[right arrow]+[infinity]][u.sub.k] = 0 and [lim.sub.k[right arrow]+[infinity]][u'.sub.k] = 0 uniformly in r [member of] [0, R] together with (21), it follows that [lim.sub.k[right arrow]+[infinity]][u'.sub.k] = 0 uniformly in r [member of] (0, R).

Taking the limit r [right arrow] [0.sup.+] on both sides of equation (21), together with [mathematical expression not reproducible], by L'Hospital's rule, we have that

[mathematical expression not reproducible]. (22)

We obtain that [mathematical expression not reproducible].

By [mathematical expression not reproducible], taking the limit r [right arrow] [R.sup.-] on both sides of equation (21), we may get that [mathematical expression not reproducible].

Moreover, it follows that [lim.sub.k[right arrow]+[infinity]][u".sub.k] = 0 uniformly in r [member of] [0, R].

Lemma 4. For fixed [lambda] > 0, if u is a solutions of problem (7) satisfying u [right arrow] 0 uniformly in r [member of] [0, R], then

[mathematical expression not reproducible]. (23)

Proof. From Lemma 3, we have that [lim.sub.u[right arrow]0]u' = 0 and [lim.sub.u[right arrow]0]u" = 0 uniformly in r [member of] [0, R].

By L'Hospital's rule, we have

[mathematical expression not reproducible]. (24)

Using a similar method to prove ([14, Theorem 3.3]) or ([15, Theorem 3.1]) with obvious changes, we may obtain the following result.

Theorem 2. Let (H1) and (H3)-(H6) hold. For v [member of] {+, -}, ([[lambda].sup.v.sub.k], 0) is a bifurcation point for problem (7). Moreover, there exists a subcontinuum [D.sup.v.sub.k] of solutions of problem (7), for v [member of] {+, -}, such that

(i) [D.sup.v.sub.k] [subset] ([[PHI].sup.v.sub.k] [union] {([[lambda].sup.v.sub.k], 0)});

(ii) [D.sup.v.sub.k] is unbounded;

(iii) ([[lambda].sup.v.sub.k], 0) is the unique bifurcation point for problem (7);

(iv) [lim.sub.[lambda][right arrow]+[infinity]][parallel]u[parallel] = max{1, R} for ([lambda], u) [member of] [D.sup.v.sub.k]\{([[lambda].sup.v.sub.k], 0)};

(v) [lim.sub.k[right arrow]+[infinity]][parallel]u[parallel] = 0 for ([lambda],u) [member of] [D.sup.v.sub.k]\{( [[lambda].sup.v.sub.k],0)}.

Proof. By (19), one obtains that

[absolute value of (g(r,u, [lambda]))]/[absolute value of (u)] [less

than or equal to] [bar.g](r,u, [lambda])/ [absolute value of (u)] [less than or equal to] [bar.g](r, [absolute value of (u)], [lambda])/[absolute value of (u)] [right arrow] 0, as [absolute value of (u)] [right arrow] 0, (25)

uniformly for r [member of] (0, R) and [lambda] on bounded sets. It follows that

[mathematical expression not reproducible]. (26)

By Lemma 4, we can get that

[mathematical expression not reproducible]. (27)

By (26) and (27), we have that

[K.sub.1] (r, u, [lambda])/[absolute value of (u)] [right arrow] 0, as [absolute value of (u)] [right arrow] 0, (28)

uniformly for r [member of] (0, R) and [lambda] on bounded sets. Let [[alpha].sup.0] = [max.sub.r[member of][0,R]] [absolute value of ([alpha](r))] and [[beta].sup.0] = [max.sub.r[member of][0,R]] [absolute value of ([beta](r))]. For 0 < [absolute value of (u)] [less than or equal to] R, one get that [absolute value of ([alpha][u.sup.+] + [beta][u.sup.-])/u)] [less than or equal to] [[alpha].sup.0] + [[beta].sup.0]. Furthermore, let

[I.sup.0.sub.k] = [[[lambda].sub.k] - [[alpha].sup.0] + [[beta].sup.0]/[a.sub.0], [[lambda].sub.k] [[alpha].sup.0] + [[beta].sup.0]/[a.sub.0]]. (29)

Theorem 1 shows that there exist two unbounded subcontinua [D.sup.+.sub.k] and [D.sup.-.sub.k] of solutions of (14) in R x E, bifurcating from [I.sup.0.sub.k] x {0}, and [D.sup.v.sub.k] [subset] ([[PHI].sup.v.sub.k] [union] ([I.sup.0.sub.k] x {0})) for v = + and v = -, in other words, (i) and (ii) hold.

We divide the rest of proofs into the following several steps:

(iii) Let us show that [D.sup.v.sub.k] [intersection] (R x {0}) = ([[lambda].sup.v.sub.k], 0), i.e., ([[lambda].sup.v.sub.k], 0) is the unique bifurcation point for problem (7). Indeed, if there exists ([[lambda].sub.n], [u.sub.n]) be a sequence of solutions of problem (7) converging to ([lambda], 0). By [parallel][u.sub.n][parallel] for the two side of (14), let [w.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel], then [w.sub.n] should be a solution of problem:

[mathematical expression not reproducible]. (30)

By (19), it follows that

[mathematical expression not reproducible], (31)

uniformly for r [member of] (0, R) and [lambda] on bounded sets.

By proof of Theorem 1.1 of [11] or proof of Theorem 1.1 of [12], we have that

[mathematical expression not reproducible]. (32)

By (31) and (32), we have that

[K.sub.1] (r, [u.sub.n], [[lambda].sub.n])/[parallel][u.sub.n][parallel]

[right arrow] 0 as [parallel][u.sub.n][parallel] [right arrow] 0, (33)

uniformly for r [member of] (0, R) and [[lambda].sub.n] on bounded sets.

By (33) and the compactness of L, we obtain that, for some convenient subsequence [w.sub.n] [right arrow] [w.sub.0] as n [right arrow] +[infinity].

Now, [w.sub.0] verifies the following equation:

- ([r.sup.N-1] [w'.sub.0])' = [lambda]a(r)[r.sup.N-1] [w.sub.0] + [alpha](r)[r.sup.N-1] [w.sup.+.sub.0] + [beta](r)[r.sup.N-1] [w.sup.-.ssub.0], (34)

and [parallel][w.sub.0][parallel] = 1. This implies that [lambda] = [[lambda].sup.v.sub.k] for v [member of] {+, -}.

(iv) Now, we shall prove that [lim.sub.[lambda][right arrow]+[infinity]] [parallel][u.sub.n][parallel] = max{1, R}.

The idea is similar to the proof of Theorem 1.1 of [12], but we give a rough sketch of the proof for readers' convenience. We only prove the case of v = + since the case of v = - is similar. For any ([[lambda].sub.n], [u.sub.n]) [member of] [D.sup.v.sub.k]\{([[lambda].sup.v.sub.k],0)} with [[lambda].sub.n] [right arrow] +[infnity] as n [right arrow] +[infnity]. Let [t.sup.n.sub.0] = 0 < [t.sup.n.sub.1] < [t.sup.n.sub.2] < ... < [t.sup.n.sub.k-1] < [t.sup.n.sub.k] = R denote the zeros of [u.sub.n] and [r.sup.n.sub.0] [member of] [[t.sup.n.sub.j], [t.sup.n.sub.j+1]] (j [member of] {0,..., k - 1}) such that [absolute value of ([u'.sub.n] ([r.sup.n.sub.0]))] = [[parallel][u'.sub.n][parallel].sub.[infinity]]. Let [r.sup.n.sub.1] = min {r [member of] ([t.sup.n.sub.j], [t.sup.n.sub.j+1]); [u'.sub.n] (r) = 0} (see [12, lines 1-4 in p.472]). Letting [[rho].sub.n] [member of] ([r.sup.n.sub.1], [r.sup.n.sub.0]) (or [[rho].sub.n] [member of] ([r.sup.n.sub.0], [r.sup.n.sub.1])), and [rho]* = [liminf.sub.n [right arrow] +[infinity]] [[rho].sub.n], [r.sup.[infinity].sub.1] = [liminf.sub.n [right arrow] +[infinity]], [r.sup.[infinity].sub.1], by [12, line 23: p.472-line 25: p.473], one may obtain that [u.sub.n] ([r.sup.[infinity].sub.1] + p) [greater than or equal to] [[sigma].sub.0] for n large enough and constant [[sigma].sub.0] [greater than or equal to] 0, where [rho] [member of] (0, [rho]* - [r.sup.[infinity].sub.1]). By (H5), letting [mathematical expression not reproducible]. It follows that g(r, [u.sub.n], [[lambda].sub.n]) [greater than or equal to] [[lambda].sub.n][h.sub.1]. Now, integrating the first equation of problem (7) from [r.sup.[infinity].sub.1] + [rho]/4 to r for any r [member of] [[r.sup.[infinity].sub.1] + [rho]/2, [r.sup.[infinity].sub.1] + [rho]] and n large enough, we get that

[mathematical expression not reproducible]. (35)

Set [a.sub.0] = [min.sub.t[member of][0, R]] a(t), [[alpha].sup.0] = [max.sub.t[member of][0, R]] [absolute value of ([alpha](t))] and [[beta].sup.0] := [max.sub.t[member of][0, R]] [absolute value of ([beta](t))]. By simple computation, we can show that

[mathematical expression not reproducible]. (36)

Moreover, we have that

[mathematical expression not reproducible]. (37)

By [lim.sub.n[right arrow]+[infinity]] [[lambda].sub.n] = + [infinity], it follows that [lim.sub.n[right arrow]+[infinity]] [[parallel][u'.sub.n][parallel].sub.[infinity]] = 1. Noting that

[absolute value of ([u.sub.n] (r))] = [absolute value of ([[integral].sup.r.sub.R] [u.sub.n] (t)dt)] [less than or equal to] [absolute value of ([u.sub.n] (t)')]dt [less than or equal to] [[parallel][u'.sub.n][parallel].sub.[infinity]]R. (38)

Thus, one may obtain that [lim.sub.[lambda][right arrow]+[infinity]][[parallel]u[parallel] = max{1, R}.

(v) Finally, we show that

[mathematical expression not reproducible]. (39)

By Lemma 4, we only show that [lim.sub.k[right arrow]+[infinity]] [[parallel][u.sub.k][parallel].sub.[infinity]] = 0. On the contrary, one supposes [lim.sub.k[right arrow]+[infinity]] [[parallel][u.sub.k][parallel].sub.[infinity]] = [rho] > 0.

Let [t.sup.0.sub.k] = 0 < [t.sup.1.sub.k] < [t.sup.2.sub.k] < ... < [t.sup.k-1.sub.k] < [t.sup.k.sub.k] = R denote the zeros of [u.sub.k]. By [12, p. 475], there exists [r.sub.k] [member of] [[t.sup.j.sub.k], [t.sup.j+1.sub.k]] (j [member of] {0, ..., k - 1}) such that [absolute value of ([u.sub.k] ([r.sub.k])| = [[parallel][u.sub.k][parallel].sub.[infinity]] = [[rho].sub.k], where [[rho].sub.k] [greater than or equal to] [rho]/2 for any k [greater than or equal to] [k.sub.0], [k.sub.0] [member of] N. Let [r.sup.k.sub.0] [member of] ([r.sub.k], [t.sup.j+1.sub.k]) satisfy [u.sub.k] ([r.sup.k.sub.0]) = [theta][[rho].sub.k], where [theta] [member of] (0, 1). Set [a.sup.0] = [max.sub.t[member of][0, R]] a(t), where [h.sup.1] := [max.sub.(t,u) [member of][0,R]x[-R,R]] [absolute value of (h(t, [u.sub.k]))]. By [12, p. 475], one may obtain that

[mathematical expression not reproducible]. (40)

Furthermore, we have that

[mathematical expression not reproducible]. (41)

Therefore, we may obtain that

[t.sup.j+1.sub.k] - [t.sup.j.sub.k] > [r.sup.k.sub.0] - [r.sub.k] [greater than or equal to] N(1 - [theta])[rho]/2[[lambda]([Ra.sup.0] + [h.sup.1]) + R([[alpha].sup.0] + [[beta].sup.0])]R > 0. (42)

However, by an argument similar to that of [19, Proposition 3.7], one obtains that [t.sup.j+1.sub.k] - [t.sup.j.sub.k] [less than or equal to] C/k [right arrow] 0, as k [right arrow] +0, and we get a contradiction.

Thus, we have that [limsup.sub.k[right arrow]+[infinity]] [[parallel][u.sub.k][parallel].sub.[infinity]] = [rho] = 0.

Remark 1. Theorem 2 indicates that the bifurcation interval [I.sup.0.sub.k] = {[[lambda].sup.+.sub.k], [[lambda].sup.-.sub.k]}, i.e., for problem (7), the bifurcation interval [I.sup.0.sub.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 Theorem 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, Theorem 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.

Lemma 5 (see [20, Lemma 2.5]). Let X be a normal space and let {[C.sub.n]|n = 1, 2, ...} be a sequence of unbounded connected subsets of X. Assume that

(i) There exists z* [member of] lim [inf.sub.n[right arrow]+[infinity]] [C.sub.n] with [parallel]z*[parallel] < +[infinity].

(ii) For every R > 0, ([[union].sup.+[infinity].sub.n=1] [C.sub.n]) [intersection] [[bar.B].sub.R] is a relatively compact set of X, where

[B.sub.R] ={x [member of] X |[parallel]x[parallel] [less than or equal to] R}. (43)

Then, D: = [limsup.sub.n[right arrow]+[infinity]] [C.sub.n] is unbounded, closed, and connected.

In order to treat the problems with nonasymptotic nonlinearity at [infinity], we shall need the following lemmas.

Lemma 6 (see [20, Corollary 2.1]). Let X be a normal vector space and let {[C.sub.n]\n = 1, 2, ...} be a sequence of unbounded connected subsets of X. Assume that

(i) There exists z* [member of] lim [inf.sub.n[right arrow]+[infinity]] [C.sub.n] with [parallel]z*[parallel] = +[infinity].

(ii) There exists a homeomorphism T: X [right arrow] X such that [parallel]T(z*)[parallel] < + [infinity] = and {T([C.sub.n])} be a sequence of unbounded connected subsets in X.

(iii) For every R > 0, ([[union].sup.+[infinity].sub.n=1] [C.sub.n]) [intersection] [[bar.B].sub.R] is a relatively compact set of X, where

[B.sub.R] ={x [member of] X |[parallel]x[parallel] [less than or equal to] R}. (44)

Then, D: = [limsup.sub.n[right arrow]+[infinity]] [C.sub.n] is unbounded, closed, and connected.

Lemma 7 (see [20, Lemma 2.6]). Let (X, [rho]) be a metric space. If [{[C.sub.i]}.sub.i[member of]N] is a sequence of sets whose limit superior is L and there exists a homeomorphism T: X [right arrow] X such that, for every R > 0, ([[union].sup.+[infinity].sub.i=1] T ([C.sub.i])) [intersection] [[bar.B].sub.R] is a relatively compact set, then for each [epsilon] > 0 there exists an m such that, for every n > m, [C.sub.n] [subset] [V.sub.[epsilon]] (L), where [V.sub.[epsilon]] (L) denotes the set of all points p with p(p, x) < [epsilon] for any x [member of] L.

3. Nodal Solutions for Problem (9)

Following Theorem 2, we shall investigate the existence of nodal solutions for problem (9), where a(t), [alpha](t), and [beta](t) satisfy conditions (H1) and (H4), respectively. We assume that f satisfies the following assumptions:

(H6) sf(s) > 0 for s [not equal to] 0.

(H7) [f.sub.0] [member of] (0, [infinity]).

(H8) [f.sub.0] = [infinity].

(H9) [f.sub.0] = 0, where

[mathematical expression not reproducible]. (45)

Applying Theorem 2 to problem (1.10), we have the following results.

Theorem 3. Let (H1), (H4), (H6), and (H7) hold. For v [member of] {+, -}, ([[lambda].sup.v.sub.k]/[f.sub.0], 0) is a bifurcation point for problem (9). Moreover, there exists an unbounded continuum [D.sup.v] of solutions of problem (9), for v [member of] {+, -}such that [D.sup.v.sub.k] joins ([[lambda].sup.v.sub.k]/[f.sub.0],0) to ([infinity], max{1, R}), [D.sup.v.sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {[[lambda].sup.v.sub.k]/[f.sub.0], 0}), and [D.sup.v.sub.k] [intersection] (R x {0}) = ([[lambda].sup.v.sub.k]/[f.sub.0],0). Furthermore, one obtains that [lim.sub.k[right arrow]+[infinity]] [parallel]u[parallel] = 0 for ([lambda], [u.sub.[lambda]]) [member of] [D.sup.v.sub.k]\{(([[lambda].sup.v.sub.k]/[f.sub.0]), 0)}.

Theorem 4. Let (H1), (H4), (H6), and (H8) hold. For v [member of] {+, -}, (0, 0) is a bifurcation point for problem (9). Moreover, there exists an unbounded continuum [D.sup.v.sub.k] of solutions of problem (9), for v [member of] {+, -} such that [D.sup.v.sub.k] joins (0, 0) to ([infinity], max {1, R}), [D.sup.v.sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {(0,0)}), and [D.sup.v.sub.k] [intersection] (R x {0}) = {(0,0)}. Furthermore, one obtains that [lim.sub.k[right arrow]+[infinity]] [parallel]u[parallel] = 0 for ([lambda], u) [member of] [D.sup.v.sub.k]\{(0,0)}.

Theorem 5. Let (H1), (H4), (H6), and (H9) hold. ([infinity], 0) is a bifurcation point for problem (9). Moreover, there exists an unbounded continuum [D.sup.v.sub.k] of solutions of problem (9), for v [member of] {+, -} and such that [D.sup.v.sub.k] joins ([infinity], 0) to ([infinity], max{1, R}), [D.sup.v.sub.k] [subset] (R x [S.sup.v.sub.k]), and [Proj.sub.R] ([D.sup.v.sub.k]) [not equal to] 0. Furthermore, one obtain that [lim.sub.k[right arrow]+[infinity]] [parallel]u[parallel] = 0 for ([lambda], u) [member of] [D.sup.v.sub.k].

From Theorems 3-5, we can easily derive the following corollary, which give the ranges of parameter guaranteeing problem (9) has zero, two, or four radial nodal solutions.

Corollary 1. Assume that (H1), (H2), (H6), and (H7) hold. We may get the following results:

(i) If [[lambda].sup.v.sub.k] > 0 for v [member of] {+, -}, then there exists [[mu].sub.1] [member of] (0, [[lambda].sup.v.sub.k]/[f.sub.0]) such that problem (9) has no radial solution for all [lambda] [member of] (0, [[mu].sub.1]) and has at least two radial nodal solutions for all [lambda] [member of] ([[mu].sub.1], +[infinity])

(ii) If v[[lambda].sup.v.sub.k] > 0 for v [member of] {+, -}, then there exist [[mu].sup.+.sub.1] [member of] (0, [[lambda].sup.+.sub.k]/[f.sub.0]) and [[mu].sup.-.sub.1] [member of] ([[lambda].sup.-.sub.k]/[f.sub.0],0) such that problem (9) has no radial solution for all [lambda] [member of] ([[mu].sup.-.sub.1], 0) [union] (0, [[mu].sup.+.sub.1]) and has at least two radial nodal solutions for all [lambda] [member of] (-[infinity], [[mu].sup.-.sub.1])) [union] ([[mu].sup.+.sub.1], +[infinity])

(iii) If v[[lambda].sup.v.sub.k] < 0 for v [member of] {+, -}, then there exist [[mu].sup.-.sub.1] [member of] (0, [[lambda].sup.+.sub.k]/[f.sub.0]) and [[mu].sup.-.sub.1] [member of] ([[lambda].sup.-.sub.k]/[f.sub.0]) such that problem (9) has no radial solution for all [lambda] [member of] ([[mu].sup.+.sub.1], 0) [union] (0, [[mu].sup.-.sub.1])) and has at least two radial nodal solutions for all [lambda] [member of] (-[infinity], [[mu].sup.+.sub.1]) [union] ([[mu].sup.-.sub.1], +[infinity])

(iv) If [[lambda].sup.v.sub.k] < 0 for v [member of] {+, - }, then there exists [[mu].sub.1] [member of] ([[lambda].sup.v.sub.k]/[f.sub.0],0) such that problem (9) has no radial solution for all [lambda] [member of] ([[mu].sub.1], 0) and has at least two radial nodal solutions for all [lambda] [member of] (-[infinity], [[mu].sub.1])

Corollary 2. Assume that (H1), (H2), (H6), and (H8) hold. We may get the following results:

(i) If [[lambda].sup.v.sub.k] > 0 for v e {+, - }, then problem (9) has at least two radial nodal solutions for all [lambda] [member of] (0, +[infinity])

(ii) If v[[lambda].sup.v.sub.k] [not equal to] 0 for v [member of] {+, -}, then problem (9) has at least two radial nodal solutions for all [lambda] [member of] (-[infinity], 0) [union] (0, +[infinity])

(iii) If [[lambda].sup.v.sub.k] < 0 for v [member of] {+, -}, then problem (9) has at least two radial nodal solutions for all [lambda] [member of] (-[infinity], 0)

Corollary 3. Assume that (H1), (H2), (H6), and (H9) hold. We may get the following results:

(i) If [[lambda].sup.v.sub.k] > 0 for v [member of] {+, -}, then there exists 0 < [[mu].sub.2] [less than or equal to] [[mu].sub.3] such that problem (9) has no radial solution for all [lambda] [member of] (0, [[mu].sub.2]) and has at least four radial nodal solutions for all [lambda] [member of] ([[mu].sub.3], +[infinity])

(ii) If v[[lambda].sup.v] [not equal to] 0 for v [member of] {+ , -}, then there exists 0 [not equal to] [[mu].sup.v.sub.2] < [[mu].sup.v.sub.3] such that problem (9) has no radial solution for all [lambda] [member of] ([[mu].sup.-v.sub.2], 0) [union] (0, [[mu].sup.v.sub.2]) and has at least four radial nodal solutions for all [lambda] [member of] (-[infinity], [[mu].sup.-v.sub.3]) [union] ([[mu].sup.-v.sub.3], +[infinity])

(iii) If [[lambda].sup.v.sub.k] < 0 for v [member of] {+, -}, then there exists [[mu].sub.3] < [[mu].sub.2] < 0 such that problem (9) has no radial solution for all [lambda] [member of] ([[mu].sub.2], 0)and has at least four radial nodal solutions for all [lambda] [member of] (-[infinity], [[mu].sub.3])

Proof of Theorem 3. Let [zeta] [member of] C(R, R) be such that

f (u) = [f.sub.0] u + [zeta](u), (46)

with

[mathematical expression not reproducible]. (47)

By [H, (2A)-p. 60] or [12, (2A)-p. 469], problem (9) is equivalent to

[mathematical expression not reproducible], (48)

where

[mathematical expression not reproducible]. (49)

Using a similar method to prove (28) with obvious changes, we have that

[K.sub.3] (r, u, [lambda])/[absolute value of (u)] [right arrow] 0, as [absolute value of (u)] [right arrow] 0, (50)

uniformly for r [member of] (0, R) and [lambda] on bounded sets.

Let us consider problem (48) as a bifurcation problem from the trivial solution axis.

Applying Theorem 2 to problem (48), we have the following result.

For v [member of] {+, -}, ([[lambda].sup.v.sub.k]/[f.sub.0], 0) is a bifurcation point for problem (48). Moreover, there exists an unbounded continuum [D.sup.v.sub.k] of solutions of problem (48), such that [D.sup.v.sub.k] joins ([[lambda].sup.v.sub.k]/[f.sub.0],0) to ([infinity], max{1, R}), [D.sup.v.sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {([[lambda].sup.v.sub.k]/[f.sub.0], 0)}), [D.sup.v.sub.k] [intersection] (R x {0}) = ([[lambda].sup.v.sub.k]/[f.sub.0],0), and [lim.sub.k[right arrow]+[infinity]] [parallel]u[parallel] = 0 for ([lambda], [u.sub.[lambda]]) [member of] [D.sup.v.sub.k]\{[[lambda].sup.v.sub.k]/[f.sub.0], 0)}.

Proof of Theorem 4. Inspired by the idea of [22], we define the cut-off function of f as the following:

[mathematical expression not reproducible]. (51)

We consider the following problem:

[mathematical expression not reproducible]. (52)

Clearly, we can see that [lim.sub.n[right arrow]+[infinity]] [f.sup.[n]] (s) = f(s), [([f.sup.[n]]).sub.0] = n.

Similar to the proof of Theorem 3, there exists an unbounded continuum [D.sup.v[n].sub.k] of solutions of problem (52) emanating from ([[lambda].sup.v.sub.k]/n, 0) and joining to ([infinity], max{1, R}) such that [D.sup.v[n].sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {([[lambda].sup.v.sub.k]/n, 0)}).

Taking [z.sup.*] = (0, 0), we easily obtain that [z.sup.*] [member of] [liminf.sub.n[right arrow]+[infinity]] [D.sup.v[n].sub.k]. So, condition (i) in Lemma 5 is satisfied with [z.sup.*] = (0, 0).

Since F is completely continuous from R * E [right arrow] E, we have that ([[union].sup.[infinity].sub.n=1] [D.sup.v[n].sub.k]) [intersection] [[bar.B].sub.R] is precompact, and accordingly (ii) in Lemma 5 holds.

Therefore, by Lemma 5, [D.sup.v.sub.k] = [limsup.sub.n[right arrow]+[infinity]] [D.sup.v[n].sub.k] is un bounded closed connected such that z* [member of] [D.sup.v.sub.k], and ([infinity], max{1,R}) [member of] [D.sup.v.sub.k]. Clearly, u is the solution of problem (9) for any ([lambda], u) [member of] [D.sup.v.sub.k]. From the definition of superior limit (see [21, P. 7]), we can easily see that [D.sup.v.sub.k] [??] [[union].sup.[infinity].sub.n=1] [D.sup.v[n].sub.k]. So, one has that [D.sup.v.sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {(0,0)}).

We may claim that [D.sup.v.sub.k] [intersection] (R x {0}) = {(0,0)}, i.e., [z.sup.*] = (0, 0), is the unique bifurcation point of [D.sup.v.sub.k]. Suppose on the contrary that there exists a sequence ([[lambda].sub.n], [u.sub.n]) [member of] [D.sup.v.sub.k]\{(0,0)} = [limsup.sub.n[right arrow]+[infinity]] [D.sup.v[n].sub.k]\{(0,0)} such that [lim.sub.n[right arrow]+[infinity]][[lambda].sub.n] [not equal to] 0 and [lim.sub.n[right arrow]+[infinity]][[lambda].sub.n] = 0. Hence, for any [N.sub.0] [member of] N, there exists [n.sub.0] [greater than or equal to] [N.sub.0] such that [mathematical expression not reproducible]. By (52), it follows that [Mathematical expression not reproducible.] for [n.sub.0] [greater than or equal to] [N.sub.0]. From the arbitrary of [N.sub.0], it implies that [n.sub.0] [right arrow] [infinity], i.e., [mu] = 0, which contradicts the assumption of [mu] [not equal to] 0.

By [D.sup.v.sub.k] [??][[union].sup.[infinity].sub.n=1] [D.sup.v[n].sub.k] and Theorem 3, one obtains that [lim.sub.k[right arrow]+[infinity]][parallel]u[parallel] = 0 for ([lambda], u) [member of] [D.sup.v.sub.k]\{(0,0)}.

Proof of Theorem 5. Inspired by the idea of [22], we define the cut-off function of f as the following:

[mathematical expression not reproducible]. (53)

We consider the following problem:

[mathematical expression not reproducible]. (54)

Clearly, we can see that [lim.sub.n[right arrow]+[infinity]][f.sup.[n]] (s) = f(s), [([f.sup.[n]]).sub.0] = 1/n.

Similar to the proof of Theorem 3, there exists an unbounded continuum [D.sup.v[n].sub.k] of solutions of the problem (54) emanating from ([[lambda].sup.v.sub.k]n, 0), such that [D.sup.v[n].sub.k] [subset] ((R x [S.sup.v.sub.k]) [union] {([[lambda].sup.v] n, 0)}) and ([infinity], max{1,R}) [member of] [D.sup.v[n].sub.k].

Taking [z.sup.*] = ([infinity], 0), we easily obtain that [z.sup.*] [member of] [liminf.sub.n[right arrow]+[infinity]] [D.sup.v[n]] with [[parallel] [z.sup.*] [parallel].sub.RxE] =+[infinity] So, condition (ii) in Lemma 6 is satisfied with [z.sup.*] = (+[infinity], 0).

Define a mapping T: R * X [right arrow] R * X such that

[mathematical expression not reproducible]. (55)

It is easy to verify that T is a homeomorphism and [[parallel]T([z.sup.*])[parallel].sub.RxX] = 0. Obviously, {T([D.sup.v[n].sub.k])} be a sequence of unbounded connected subsets in X. So, (ii) in Lemma 6 holds. Since F is completely continuous from R * E [right arrow] E, we have that ([[union].sup.[infinity].sub.n=1]T [D.sup.v[n]])) [intersection] [[bar.B].sub.R] is precompact, and accordingly (iii) in Lemma 6 holds.

Therefore, by Lemma 6, [D.sup.v.sub.k] = [limsup.sub.n[right arrow]+[infinity]][D.sup.v[n].sub.k]] is un bounded closed connected such that [z.sup.*] [member of] [D.sup.v.sub.k] and ([infinity], max{1, R}) [member of] [D.sup.v.sub.k]. Obviously, u is the solution of problem (9) for any ([lambda], u) [member of] [D.sup.v.sub.k]. From the definition of superior limit (see [21, P. 7]), we can easily see that [D.sup.v.sub.k] [??][[union].sup.[infinity].sub.n=1] [D.sup.v[n].sub.k]]. So, one has that [D.sup.v.sub.k] [subset] (R * ([S.sup.v.sub.k] [union] {0})). Next, we show that [D.sup.v.sub.k] [intersection] (R * {0}) = 0. Suppose on the contrary that there exists a sequence {([[lambda].sub.n], [u.sub.n])} [??][D.sup.v.sub.k]] such that [lim.sub.n[right arrow]+[infinity]] [[lambda].sub.n] = [mu] and [lim.sub.n[right arrow]+[infinity]][parallel][u.sub.n][parallel] = 0. From (9), we can easily get that

[mathematical expression not reproducible]. (56)

Letting [w.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel], we have that

[mathematical expression not reproducible]. (57)

Similar to (31), we can show that

[mathematical expression not reproducible]. (58)

By (32), (57), and (58), we have that

[absolute value of ([w.sub.n])] [less than or equal to] M [[integral].sup.R.sub.0] G(r, s)[s.sup.N-1] [absolute value of ([w.sub.n])]dr, (59)

where M = max{[[alpha].sup.0], [[beta].sup.0]}. By the Gronwall-Bellman inequality ([23, Lemma 2.1]), we obtain that [parallel]w[parallel] = 0. This contradicts the fact of [parallel]w[parallel] = 1. Hence, we have that [D.sup.v] [subset] (R x [P.sup.v]]).

Finally, we show that [Proj.sub.R] ([D.sup.v.sub.k]) [not equal to] 0. From the argument of Theorem 3, we have known that [D.sup.v[n].sub.k] has unbounded projection on R for any fixed n [member of] N. By Lemma 7, for each fixed [epsilon] > 0 there exists an m such that, for every n > m, [D.sup.v[n].sub.k] [subset] [V.sub.[epsilon]] ([D.sup.v.sub.k]). This implies that

([[lambda].sup.v] n +[infinity]) [??] [Proj.sub.R] ([D.sup.v[n].sub.k]) [??] [Proj.sub.R] ([V.sub.[epsilon]] ([D.sup.v.sub.k])), (60)

where [Proj.sub.R] ([D.sup.v.sub.k]) denotes the projection of [D.sup.v.sub.k] on R. It follows that the projection of [D.sup.v.sub.k] is nonempty on R.

By an argument similar to that of Theorem 4, we can get the asymptotic behavior of [lim.sub.k[right arrow]+[infinity]][parallel]u[parallel] = 0 for ([lambda], u) [member of] [D.sup.v.sub.k].

https://doi.org/10.1155/2020/9801931

Data Availability

Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

The author conceived the study, drafted the manuscript, and approved the final manuscript.

Acknowledgments

This research was supported by the NSFC (no. 11561038) and "Kaiwu" Innovation Team Support Project of Lanzhou Institute of Technology (no. 2018KW-03).

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Wenguo Shen [ID]

Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, China

Correspondence should be addressed to Wenguo Shen; shenwg369@163.com

Received 14 December 2019; Accepted 4 February 2020; Published 13 April 2020

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