GLOBAL BIFURCATION OF SOLUTIONS OF THE MEAN CURVATURE SPACELIKE EQUATION IN CERTAIN STANDARD STATIC SPACETIMES

. We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime with 0-Dirichlet boundary condition on the unit ball. According to the behavior of H near 0, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs.

interval of R, g the Riemannian metric on S obtained by restriction of the Lorentzian metric of M and a > 0 a smooth function on S [22,Prop. 12.38]. Moreover, aU is represented in this decomposition just as the coordinate vector field ∂/∂t. When this decomposition is global [1], then M is called a standard static spacetime [22,Def. 12.36]. In other words, a standard static spacetime is a product manifold I ×S, endowed the Lorentzian metric g = −a 2 dt 2 + g, (1.1) where I is an open interval of the real line, g is a Riemannian metric on the manifold S and a is a smooth and positive function on S. In the terminology of [22,Def. 7.33], the manifold with the Lorentzian metric (1.1) is a warped product with base (S, g), fiber (I, −dt 2 ) and warping function a. Let us consider here the case S = Ω an open domain of R N , N ≥ 1, with its canonical metric g and a invariant by certain isometry of Ω. Taking into account that an isometry φ of the base such that a • φ = a clearly induces an isometry of the standard static spacetime, this mathematical assumption may be interpreted as a natural symmetry of the spacetime. This is the case when Ω = I × S N −1 , I ⊂ R + an open interval (Ω may be seen as an open domain in R N ), where E > 0 is defined on I, dσ 2  is the unit timelike normal vector field on Σ u in M in the same time-orientation as ∂/∂ t .
If H is the mean curvature of the spacelike graph Σ u with respect to the unit timelike normal vector field given in (1.3), the function u may be seen as a solution of the mean curvature spacelike hypersurface equation (derived in Appendix, for the sake of completeness) (1.4) When a ≡ 1, H ≡ 0 and Ω = R N , equation (1.4) reduces to the well-known maximal hypersurface equation in (N +1)-dimensional Lorentz-Minkowski spacetime. In this case, Calabi [6] proved that the only entire (i.e., defined on all R N ) solutions are the affine functions defining spacelike hyperplanes for N ≤ 4. Further, Cheng and Yau [8] extended this result for all N . This uniqueness result contrasts with the behaviour of the entire solutions of the classical minimal hypersurface equation whose entire solutions are affine functions only for N ≤ 7 and counter-examples exist for each N > 7, [23]. When a ≡ 1 and H is a non-zero constant, some celebrated results for equation (1.4) were obtained by Treibergs [28]. If a ≡ 1, Ω is a bounded domain and H is a bounded function defined on Ω × R, Bartnik and Simon [2] proved that the equation (1.4) with Dirichlet boundary condition has a strictly spacelike solution. By topological degree or critical point theory, the authors of [5,9] studied the nonexistence, existence and multiplicity of positive solutions for it in the case of a ≡ 1 and Ω being a bounded domain. When a ≡ 1 and Ω = B R := B R (0) = x ∈ R N : |x| < R with R > 0, Bereanu, Jebelean and Torres [3,4] obtained some existence results for positive radial solutions of equation (1.4) with u = 0 on ∂Ω. Recently, when a ≡ 1, the first author [10] studied the nonexistence, existence and multiplicity of positive radial solutions of equation (1.4) with u = 0 on ∂Ω and N H = −λf (x, s) on the unit ball via bifurcation method, which were extended to the general domain in [13,14].
The main objective of this paper is to investigate the existence/nonexistence of one-sign radially symmetric spacelike solutions for equation (1.4) on the unit ball B mainly by bifurcation method in the same philosophy as in [15]. The solution is understood in the classical sense.
For this aim to be achieved, we consider the following 0-Dirichlet boundary value problem where, here and in what follows, the constrain |∇u| < 1/a is understood, λ is a nonnegative parameter which can represent in some sense the strength of mean curvature function, a : B → R is a smooth positive radially symmetric function, H : B × [−δ, δ] → R is a continuous function that is radially symmetric with respect to x, with some positive constant δ determined later. Taking g as in (1.2) and following [19], we have that the problem (1.5) can be reduced to the following boundary value problem where r = |x|, v(r) = u(|x|), a(r) = a(|x|) and f (r) = a(r)/E(r). Letting δ = max B (E(x)/a(x)), since the graph associated to v is spacelike, we deduce that v ∞ < δ. This is the reason we only require that H is defined on B × [−δ, δ]. Let λ 1 be the first eigenvalue of It is well known that λ 1 is simple, isolated and the associated eigenfunctions have one sign in [0, 1) (see for instance [29, p. 284]). Let It follows that the norm v is equivalent to the usual norm v ∞ + v ∞ . Let P + = {v ∈ X : v > 0 on [0, 1)} and P − = −P + . From now on, following [25], we add the point ∞ to our space R × X.
The main result of this paper is the following theorem.
The following result is concerning the nonexistence.
Assume that a(r) is nondecreasing and there exists a positive constant such that −H(r, s) s ≤ for any s = 0 and r ∈ (0, 1). Then, there exists * > 0 such that problem (1.6) has not any one-sign solution for λ ∈ (0, * ).
By arguments similar to those of Theorems 1.1-1.2, we can also show that the conclusions of Theorems 1.1-1.2 are valid for equation (1.4) on any annular domain with the Robin boundary condition. Concretely, let R 1 , R 2 ∈ R with 0 < R 1 < R 2 and A := x ∈ R N : R 1 ≤ |x| ≤ R 2 . Consider the following problem with the

Robin boundary condition
is a continuous function that is radially symmetric with respect to x, ∂v/∂ν is the outward normal derivative of v and a : A → R is a smooth positive radially symmetric function. As that of (1.6), the problem (1.8) is reduced to the following one where f (r) = a(r)/E(r). Let λ 1 be the first eigenvalue (see [20]) of = 0 with the norm u := (a/E)u ∞ . Then, in particular, we have the following consequence.
The rest of this paper is arranged as follows. In Section 2, we introduce an approximation problem of (1.6) and investigate its global bifurcation phenomenon. Section 3 is devoted to the study of the convergence of solutions of our approximation problem as → 0 + . The proofs of Theorems 1.1 and 1.2 will be given in Section 4. In Section 5, on the basis of Theorem 1.1 (or Corollary 1.1) and the standard prolongability theorem of ordinary differential equations, we study the existence of entire radially spacelike graphs of functions satisfying equation (1.4) for Ω = R N (or exterior region). Two examples are also given in this section. Finally, the derivation of equation (1.4) is given in Appendix.

Bifurcation for an approximation problem.
If v is a solution of problem (1.6), then we have that where f (r) = a(r)/E(r). Furthermore, we obtain that Note that the above equation is singular at r = 0. To overcome this singularity, we consider the following approximation equation for any ∈ (0, 1]. It follows that So, we have that Now, we consider the following problem (2.1) Using the above expanding process, we can see that if v is a solution of the following approximation problem Conversely, the following lemma implies that v is a solution of problem (2.2) if v is a solution of problem (2.1).
Proof. Suppose, by contradiction, that (a |v |) /E can achieve 1 on [0, 1]. Let r * be the first such kind of point. Since v (0) = 0, one has that r * ∈ (0, 1]. Note that v satisfies Since a (r * ) |v (r * )| = E (r * ), there exists r * ∈ (0, r * ) such that f |v | > 1/2 for all r ∈ (r * , r * ). It follows that for r ∈ (r * , r * ). Integrating this equality from r * to r ∈ (r * , r * ), we obtain that Letting r → r * , we see that the left side term tends to infinity while the right one is bounded, which is a contradiction. Due to Lemma 2.1, problem (2.2) is equivalent to problem (2.1). Furthermore, we have the following monotonicity result.
Proof. We only prove the case of v ≥ 0 because the proof of v ≤ 0 is similar. Note that By the variation of constants formula, we have that It follows that Next, we give a Dancer's type unilateral global bifurcation result which will be used later. Let E be a real Banach space with the norm · and O be an open subset of R × E. Consider the following operator equation Let r(L) be the characteristic value set of L. By an arguments similar to that of [17, Theorem 2] (or [18,Theorem]) with obvious changes, we obtain the following result, which is the local version of Theorem 2 of [17].
is geometric multiplicity 1 and odd algebraic multiplicity, then S possesses two maximal continua C + µ , C − µ ⊂ O such that (µ, 0) ∈ C ± µ and one of the following three properties is satisfied by C ± µ : To investigate the bifurcation phenomenon of problem (2.1), we consider the following eigenvalue problem From [29, p. 269], we know that problem (2.4) has a principal eigenvalue λ 1 ( ), which is simple and isolated. Then, we have the following unilateral global bifurcation result for problem (2.1).
Let G(r, s) be the Green's function associated to the operator L v with the same boundary condition as problem (2.5). Then problem (2.5) can be equivalently writ- It is well known that L : X −→ X is linear compact (see [24]). It follows from f |v | < 1 on [0, 1] that v < 1. By the conclusion of [24], we know that H : Then ξ is nondecreasing with respect to u and It follows from (2.6) that uniformly in r ∈ (0, 1). So, we have that uniformly in r ∈ (0, 1). Clearly, one has that 1/(r + ) ≤ 1/ for any r ∈ [0, 1] and uniformly in r ∈ (0, 1). It follows the claim as desired. Applying Lemma 2.3 with O = R × B, there exist two continua, C + and C − , of solution set of problem (2.5) emanating from (λ 1 ( ), 0) which satisfy one of the following three properties: ( Thus, the second alternative does not occur. From the variation of constants formula, we can see that problem (2.2) has only trivial solution if λ = 0. Since 0 is not an eigenvalue of problem (2.4), we have that C ± ∩ ({0} × X) = ∅.
Our assumptions of H imply that there exists a positive constant ρ 0 such that m(r) ≥ ρ 0 for any r ∈ (0, 1). By some elementary calculations, we find that problem (2.7) is equivalent to Multiplying the first equation of problem (2.8) by ϕ 1 , we obtain after integration by parts that It follows that λ n ≤ λ 1 ( )/ (N δ 0 ρ 0 ), which is a contradiction.
By the conclusion of (a), there exist two sequences unbounded continua C + ,n and C − ,n of one-sign solutions set of problem (2.10) in R × X emanating from (λ 1 ( )n, 0) for any n ∈ N and joining to (+∞, 1) := z * .

R×X
For fixed n ∈ N, we claim that C ± ,n ∩ S = ∅. Suppose, by contradiction, that there exists a sequence (λ m , v m ) ∈ C ± ,n such that (λ m , v m ) → (+∞, v * ) ∈ S with v * ∈ (0, 1). Then by the argument as that of (a), we obtain that λ m ≤ c n for some positive constant c n , which is a contradiction. It follows that ∪ +∞ n=1 C ± ,n ∩S = ∪ +∞ n=1 C ± ,n ∩ S = ∅. Letting C ± = lim sup n→+∞ C ± ,n , since C ± ⊆ ∪ +∞ n=1 C ± ,n , we have that C ± ∩ S = ∅. Therefore, we have that C ± ∩ {∞} = {z * , z * }. Now we show that C ± \ {∞} = ∅. It is enough to show that the projection of C ± on R is nonempty. From the argument of (a), we have known that C ± ,n has unbounded projection on R for any fixed n ∈ N. By Proposition 2 of [12], for any fixed σ > 0 there exists an N 1 > 0 such that for every n > N 1 , C ± ,n ⊂ V σ (C ± ), where V σ (C ± ) denotes the σ-neighborhood of C ± in R × X. It follows that (λ 1 ( )n, +∞) ⊆ pr R C ± ,n ⊆ pr R V σ C ± .
Note that the monotonicity of a is only used to obtain the asymptotic behavior of v λ as λ → +∞ for (λ, v λ ) ∈ C ν \ {(λ 1 ( ), 0)}. If we don't care the asymptotic behavior, this condition can be removed.
3. Convergence of solutions as → 0. From Theorem 2.1, we can obtain the existence and multiplicity of solutions of the approximation problem (2.2). To study the convergence of solutions of problem (2.2) as → 0 + , we first provide a boundary derivative a priori bound estimate as follows.
The following convergence conclusion is our main result of this section.
Theorem 3.1. For any one-sign solution (λ, v ) of problem (2.2) with any fixed λ, up to a subsequence, the limit of v exists as → 0 + which is denoted by v, and (λ, v) is the solution of problem (1.6).
Proof. By Lemma 2.1, we can easily verify that v ∞ < δ and v ∞ < δ. By the Arzelà-Ascoli Theorem, up to a subsequence, there Obviously, G is uniformly bounded on [κ, 1] with respect to and for any r ∈ [κ, 1]. Then, by the Lebesgue Dominated Convergence Theorem, we obtain that In view of the arbitrariness of κ, it follows that Since v (1) = 0, we have that v(1) = 0. So, it suffices to verify the existence of lim r→0 + v (r) := v (0) and v (0) = 0. From problem (2.2), we can see that It follows that It is easy to verify that Define ψ : (−1, 1) → R by Clearly, ψ is an increasing diffeomorphism. So, we obtain that Integrating the above equation between 0 and 1, we have that By Proposition 3.1, the right term of the above equation is uniformly bounded with respect to . Hence, there exists a positive constant C which is independent on such that Without loss of generality, from now on, we assume that v is positive on [0, 1). In view of Lemma 2.2, we have that By virtue of Fatou Lemma, we infer that which shows that −f v r is an integrable function on (0, 1]. It implies that is also an integrable function on (0, 1]. For any r ∈ (0, 1], integrating equation (3.2) from r to 1, we get that Now, we can see that the limit of the right term exists as r → 0 + . Therefore, we verify the existence of lim r→0 + v (r), which is denoted by v (0). Then, by integrability of − (f v ) /r and f (0) > 0, we derive that v (0) = 0, which is just our desired conclusion.
4. Proofs of Theorems 1.1 and 1.2. From Theorem 3.1, for any solution v ε of problem (2.2), we have that v = lim →0 + v ε is a solution of problem (1.6). It follows that C ν := lim sup ε→0 + C ν ε is the solution set of problem (1.6), where C ν ε is obtained in Theorem 2.1. To study the structure of C ν , we first present the following eigenvalue result.
for v ∈ B. Then, by some elementary calculations, we obtain that By the Implicit function theorem, λ must be an eigenvalue of problem (1.7). So, λ = λ j for some j > 1. Then, by an argument similar to that of Theorem 2.1, we can get a contradiction. So, we have that Clearly, one has that (0, 0) ∈ C ν . Applying Theorem 2.1 of [11], we have that C ν is connected. By Theorem 2.1, we know that C ν joins (0, 0) to (+∞, 1). Reasoning as that of (a), we see that (λ, 0) is a bifurcation of problem (1.6) if and only if λH 0 is an eigenvalue of problem (1.7), which combining H 0 = +∞ implies that C ν ∩ ((0, +∞) × {0}) = ∅. Furthermore, by Lemmas 2.1-2.2 and an argument similar to that of (a), we can obtain the desired conclusions.
Finally, we present the proof of Theorem 1.2.
Proof of Theorem 1.2. Assume that v be any one-sign solution of problem (1.6) with some λ > 0. Multiplying the first equation of problem (1.6) by v, we obtain after integrations by parts that where µ 1 is the first eigenvalue of It follows that λ ≥ µ 1 /( N ).

5.
Entire radially spacelike graphs. In this section, we study the existence of entire radially spacelike graphs of equation (1.4) in Schwarzschild spacetime with a parameter, i.e., the following equation v (0) = 0, (5.1) where λ is a nonnegative parameter.
Theorem 5.1. Besides the assumptions of Theorem 1.1, assume that a : R N → R is a smooth positive radially symmetric function, H : R N × R → R is a continuous function, radially symmetric with respect to x.
In addition, the spacelike slice t = 0 intersects these graphs in the unit ball.
Proof. We only give the proof of (a) because the proofs of other cases are completely analogous. Theorem 1.1 provides at least two solutions v + and v − of problem (1.6) such that νv ν > 0 in (0, 1). It suffices to show that v ν can be extended to (0, +∞) such that they are also solutions of equation (5.1). We consider the following system where F : R + × R × (−δ, δ) → R 2 is continuous.
In addition, the spacelike slice t = 0 intersects these graphs in a ball of radius R 2 .
Consider a(r) = 1 − 2m r and E(r) = 1 where m is the mass of a star or black hole in certain unit system. Taking R 1 > 2m and Ω = R N \ B R1 (0), M is the Schwarzschild exterior spacetime which models the exterior region of a spacetime where there is only a spherically symmetric non-rotating star without charge. The value of the radius r = 2m is known as Schwarzschild radius. When r < 2m, we are in presence of a Schwarzschild black hole. It is not difficult to verify that a and E satisfy the assumptions of Corollary 1.1 and Corollary 5.1. So, the conclusions of Corollary 5.1 can be used on the Schwarzschild exterior spacetime. Another example is the Reissner-Nordström exterior spacetime in which the mass has non-zero electric charge [7,26]. In such case, we have a(r) = 1 − 2m r + r 2 Q r 2 and E(r) = where r Q > 0 is a characteristic length relative to the charge Q of the mass. Taking R 1 > m + m 2 − r 2 Q and Ω = R N \ B R1 (0), M is the Reissner-Nordström exterior spacetime, which can be seen a generalization of the Schwarzschild exterior spacetime. The value of r = m + m 2 − r 2 Q is the exterior event horizon. Clearly, a and E still satisfy the assumptions of Corollary 1.1 and Corollary 5.1. Therefore, the conclusions of Corollary 5.1 can also be applied to the Reissner-Nordström exterior spacetime.