Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 48, pp. 1–10. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.48 LOCAL BIFURCATION STRUCTURE AND STABILITY OF THE MEAN CURVATURE EQUATION IN THE STATIC SPACETIME SIYU GAO, QINGBO LIU, YINGXIN SUN Abstract. We consider the curvature equation in the static spacetime, div ( f(x)∇u√ 1− f2(x)|∇u|2 ) + ∇u∇f(x)√ 1− f2(x)|∇u|2 = λNH in Ω, where Ω is a bounded domain in RN , N ≥ 1; the function H gives the mean curvature. We investigate the local bifurcation structure and stability of the solutions to this equation. 1. Introduction and main results We consider a domain Ω ⊆ RN , where N is greater than or equal to 1. Let f be a smooth positive function on Ω. Consider the N+1-dimensional product manifold M = I × Ω equipped with the Lorentzian metric g = −f2(x) dt2 + dx2. In [22, Lemma 12.37], it was established that M is static with respect to ∂t/f . For each u ∈ C2(Ω), let M = {(x, u) : x ∈ Ω, u ∈ C2(Ω)}. A spacetime M is termed static in relation to an observer field Q if Q is irrotational and if there exists a smooth positive function such that fQ is a Killing vector field. Then, (M, g) = U represents an N -dimensional hypersurface in M at time t, which can be depicted by the graph of t = u. U is referred to as spacelike if |∇u| < 1/f in Ω (see [20]). We define U as being weakly spacelike if |∇u| ≤ 1/f , i.e., if it is in Ω. Given the mean curvature H for a spacelike graph U , Problem (1.1) has implications for classical relativity [4] and cosmology research [5, 19, 21]. For the case in which f is constantly equal to 1, Calabi [9] explored the properties of maximal surfaces and demonstrated that when N ≤ 4, equation (1.2) allows only linear solutions. Cheng and Yau [10] further investigated maximal surfaces, extending Calabi’s findings to all dimensions, and proposed the Bernstein theorem. For cases in which f is constantly equal to 1, Treibergs [24] provided significant results for entire surfaces with a constant mean curvature. For cases in which f equals 1, Bartnik and Simon [4] considered the Dirichlet problem for equation (1.2) with surfaces of bounded mean curvature. 2020 Mathematics Subject Classification. 35B32, 35J93, 35B35. Key words and phrases. Bifurcation; mean curvature operator; stability. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 10, 2024. Published August 26, 2024. 1 2 S. GAO, Q. LIU, Y. SUN EJDE-2024/48 The authors of [6, 11] used critical point theory and topological degree arguments to explore the nonexistence, existence, and multiplicity of positive solutions for f ≡ 1 in bounded domains. In [13], the authors investigated the nonexistence, existence, and multiplicity of positive radial solutions of equation (1.2) with NH = −λf(x, s) on the unit ball via the bifurcation method. This work was later extended to general domains in [14, 16]. The author in [18] studied the existence and uniqueness of classical solutions, the multiplicity of strong solutions, and the symmetry of positive solutions. The global structure of the positive solutions for this problem was also delineated. For more research results on the mean curvature equation, see references [7, 8, 17, 15] and their cited literature. In [1], the stability of hypersurfaces with a constant mean curvature was studied through the calculus of variations. The relationship between stability and constant mean curvature was presented under the condition that the hypersurface is compact. In [2], the stability of hypersurfaces with constant mean curvature in Riemannian manifolds was studied. Barros, Brasil, and Caminha [3] investigated stability issues concerning the generalized Robertson-Walker spacetime. In this work, we investi- gate the local bifurcation structure and stability of the mean curvature equation in static the spacetime. We consider the following 0-Dirichlet problem involving the mean curvature op- erator in Minkowski space: −div ( f2(x)∇u√ 1− f2(x)|∇u|2 ) = −λNf(x)H(x, u) in Ω, u = 0 on ∂Ω. (1.1) Here, λ is a nonnegative parameter representing the strength of the mean curvature function, the real-valued function H gives the mean curvature, Ω is a C2,α bounded domain in RN with N ≥ 1 for some α > 0, and f ∈ C0,α(Ω × [−d, d]), where d is the diameter of Ω. Using the equation div ( f(x)∇u√ 1− f2(x)|∇u|2 ) + ∇u∇f(x)√ 1− f2(x)|∇u|2 = NH, (1.2) we can derive that − div ( f2(x)∇u√ 1− f2(x)|∇u|2 ) = −div ( f(x) · f(x)∇u√ 1− f2(x)|∇u|2 ) = −f(x) div ( f(x)∇u√ 1− f2(x)|∇u|2 ) − f(x) ∇u∇f(x)√ 1− f2(x)|∇u|2 = −Nf(x)H. From [18], we have div ( f2(x)∇u√ 1− f2(x)|∇u|2 ) = Nf(x)H. This equation is equivalent to (1.2). Next, we present the main theorem. Theorem 1.1. Suppose that H is C3 with respect to its second argument and that H0 ∈ (0,+∞). Then, all the solutions of problem (1.1) near (λ1/H0, 0) can be EJDE-2024/48 LOCAL BIFURCATION STRUCTURE AND STABILITY 3 expressed as (λ(s), sφ1+ sz(s)) for s in an open interval (−δ, δ), where δ > 0, such that λ(0) = λ1/H0 and λ′(0) = − − λ1 H0 N ∫ Ω f(x)Huu(x, 0)φ 3 1 dx 2H0 ∫ Ω f(x)φ2 1 dx . Here, z : (−δ, δ) → Z is a C2 function that satisfies z(0) = 0. Additionally, if Huu(x, 0) is exactly zero on Ω, then we obtain λ′′(0) = − − λ1 H0 N ∫ Ω f(x)Huuu(x, 0)φ 4 1dx 3H0 ∫ Ω f(x)φ2 1dx , where Huuu(x, 0) denotes the third derivative of H with respect to its second variable at 0. By Theorem 1.1, we can deduce the following stability result. Theorem 1.2. Let λ1/H0 be a bifurcation point for the equation F (λ, u) = 0 in a Banach space X , and assume that 0 is a simple eigenvalue of the linearized operator Fu(λ1/H0, 0). Suppose further that Huu(x, 0) ≡ 0 in Ω and that Huuu(x, 0) ̸= 0 in Ω. Then, the stability of the solutions u(s) near the bifurcation point (λ1/H0, 0) is determined as follows: (1) If Huuu(x, 0) > 0 in Ω, then the solutions are asymptotically linearly stable, i.e., λ′′(0) > 0. (2) If Huuu(x, 0) < 0 in Ω, then the solutions are asymptotically linearly un- stable, i.e., λ′′(0) < 0. This article is organized as follows. Section 2 discusses the local bifurcation structure of the solution set of equation (1.1). Section 3 presents the stability results near the bifurcation point. 2. Local bifurcation structure In this section, we provide the proof of Theorem 1.1. Consider the set X defined as X = {u ∈ C1(Ω) : u = 0 on ∂Ω} with the norm ∥u∥ := ∥f(x)∇u∥∞. Let φ1 be a positive eigenfunction correspond- ing to λ1 with ∥φ1∥ = 1. Let X0 be a closed subspace of X such that X = X1 ⊕X0, where X1 = span{φ1}. By applying the Hahn–Banach theorem, we can find a linear continuous functional l ∈ X∗ satisfying l(φ1) = 1 and X0 = {u ∈ X : l(u) = 0}. Proof of Theorem 1.1. Define X = {u ∈ C2(Ω) : u = 0, on ∂Ω}, Y = C(Ω). Consider the function defined by F (λ, u) = div ( f2(x)∇u√ 1− f2(x)|∇u|2 ) − λNf(x)H(x, u). Since H0 ∈ (0,+∞) and H(x, 0) = 0 holds for any x ∈ Ω, we have F (λ, 0) = div ( f2(x)∇0√ 1− f2(x)|∇0|2 ) − λNf(x)H(x, 0) = 0 4 S. GAO, Q. LIU, Y. SUN EJDE-2024/48 (a) Transcritical bifurcation (b) Supercritical pitchfork (c) Subcritical pitchfork Figure 1. Bifurcation diagrams of Theorem 1.1 for any λ. The partial derivative of F (λ, u) with respect to λ is Fλ(λ, u) = −Nf(x)H(x, u). According to [18], we have lim t→0+ NH(x, t) t = −H0. This holds because H is a function with third-order continuous derivatives with respect to its second variable, and F is C3 with respect to u in some small neigh- borhood V ⊂ X of 0. By calculation, we obtain Fu(λ, 0)[φ1] = div(f2(x)∇φ1) + λf(x)H0φ1, Fu( λ1 H0 , 0)[φ] = div(f2(x)∇φ) + λ1 H0 f(x)H0φ, where φ ∈ X . The function φ1 is a positive eigenfunction corresponding to the principal eigenvalue λ1 of the linearized problem associated with equation (1.1). Specifically, φ1 is a solution of Fu(λ1/H0, 0)[φ] = 0. Then, we have div(f2(x)∇φ1) + λ1 H0 f(x)H0φ1 = 0. EJDE-2024/48 LOCAL BIFURCATION STRUCTURE AND STABILITY 5 Since φ1 is a nontrivial solution, it follows that φ1 ̸= 0 and φ2 1 ≥ 0. Hence, the integral ∫ Ω φ2 1 dx > 0. Thus, the kernel space is N ( Fu( λ1 H0 , 0) ) = span{φ1}. The codimension of the image space is R ( Fu( λ1 H0 , 0) ) = { v ∈ Y : ∫ Ω vφ1 dx = 0 } . Therefore, dimN ( Fu( λ1 H0 , 0) ) = codimR ( Fu( λ1 H0 , 0) ) = 1. (2.1) Clearly, F is C1 with respect to λ, and Fλu exists and remains continuous in a small neighborhood of (λ1/H0, 0). From the calculations, we obtain Fλu( λ1 H0 , 0)[φ1] = fH0φ1, (2.2) Fuu( λ1 H0 , 0)[φ1] 2 = − λ1 H0 NfHuu(x, 0)φ 2 1 and Fuuu( λ1 H0 , 0)[φ1] 3 = − λ1 H0 NfHuuu(x, 0)φ 3 1. Combining this with H0 > 0, we obtain H0 ∫ Ω φ2 1 dx ̸= 0. Thus, ∫ Ω fH0φ 2 1 dx ̸= 0. This leads to the conclusion that Fλu(λ1/H0, 0)[φ1] ̸∈ R(Fu(λ1/H0, 0)). (2.3) By applying [16], we deduce that all the solutions near (λ1/H0, 0) for problem (1.1) can be expressed as (λ(s), sφ1+sz(s)), where s belongs to the interval (−δ, δ) for some positive value of δ, and that they satisfy the conditions λ(0) = λ1/H0 and z(0) = 0. We rescale φ1 so that ∫ Ω φ2 1 dx = 1. (2.4) Then, we define the linear functional l(u) = ∫ Ω uφ1 dx (2.5) and N (l) = R(Fu( λ1 H0 , 0)). Furthermore, by employing formula (4.5) derived in [23], we can deduce that λ′(0) = − ⟨l, Fuu( λ1 H0 , 0)[φ1] 2⟩ 2⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ . 6 S. GAO, Q. LIU, Y. SUN EJDE-2024/48 Subsequently, ⟨l, Fuu( λ1 H0 , 0)[φ1] 2⟩ = ∫ Ω Fuu( λ1 H0 , 0)[φ1] 2φ1 dx = − λ1 H0 N ∫ Ω f(x)Huu(x, 0)φ 3 1 dx (2.6) and 2⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ = 2 ∫ Ω Fλu( λ1 H0 , 0)[φ1]φ1 dx = 2H0 ∫ Ω f(x)φ2 1 dx. (2.7) From (2.6) and (2.7), we obtain λ′(0) = − ⟨l, Fuu( λ1 H0 , 0)[φ1] 2⟩ 2⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ = − − λ1 H0 N ∫ Ω f(x)Huu(x, 0)φ 3 1 dx 2H0 ∫ Ω f(x)φ2 1 dx . If Huu(x, 0) ≡ 0 in Ω, using [23, (4.6)], we deduce that λ′′(0) = − ⟨l, Fuuu( λ1 H0 , 0)[φ1] 3⟩ 3⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ , where ⟨l, Fuuu( λ1 H0 , 0)[φ1] 3⟩ = ∫ Ω Fuuu( λ1 H0 , 0)[φ1] 3φ1 dx = − λ1 H0 N ∫ Ω f(x)Huuu(x, 0)φ 4 1 dx and 3⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ = 3 ∫ Ω Fλu( λ1 H0 , 0)[φ1]φ1 dx = 3H0 ∫ Ω f(x)φ2 1 dx. Thus, λ′′(0) = − ⟨l, Fuuu( λ1 H0 , 0)[φ1] 3⟩ 3⟨l, Fλu( λ1 H0 , 0)[φ1]⟩ = − − λ1 H0 N ∫ Ω f(x)Huuu(x, 0)φ 4 1 dx 3H0 ∫ Ω f(x)φ2 1 dx . (2.8) We observe that λ′(0) ̸= 0 if Huu(x, 0) ̸= 0 in Ω. This indicates the occurrence of a transcritical bifurcation, characterized by λ′(0) ̸= 0 (see Figure 1). If Huu(x, 0) ≡ 0 in Ω but Huuu(x, 0) ̸= 0 in Ω, it follows that λ′(0) = 0 and λ′′(0) ̸= 0, implying a pitchfork bifurcation, characterized by λ′′(0) ̸= 0. Specifically, if λ′′(0) > 0, a supercritical pitchfork bifurcation occurs. If λ′′(0) < 0, a subcritical pitchfork bifurcation occurs (see Figure 1). Hence, the desired conclusions are obtained. □ EJDE-2024/48 LOCAL BIFURCATION STRUCTURE AND STABILITY 7 3. Stability properties In this section, we provide the formal stability results near the bifurcation point. The stability properties are obtained via the exchange of stability theorem presented in [12], which is our fundamental tool. Theorem 3.1 (Crandall-Rabinowitz). Let X and Y be real Banach spaces, and let K : X → Y be two bounded linear operators. Assume that F : R × X → Y is C2 near (λ∗, 0) ∈ R ×X with F (λ, 0) = 0 for a sufficiently small |λ∗ − λ|. Let T = Fu(λ∗, 0). If β = 0 is a Fλu(λ∗, 0)-simple eigenvalue of operator T and a K-simple eigenvalue of T , then there locally exists a curve (λ(s), u(s)) ∈ R × X such that (λ(0), u(0)) = (λ∗, 0) and F (λ(s), u(s)) = 0. Moreover, if F (λ, u) = 0 with u ̸= 0 and (λ, u) near (λ∗, 0), then (λ, u) = (λ(s), u(s)) for some s ̸= 0. Furthermore, there are eigenvalues β(s) and βtriv(λ) ∈ R with eigenvectors φ(s) and φtriv(λ) ∈ X such that Fu(λ(s), u(s))φ(s) = β(s)Kφ(s), Fu(λ, 0)φtriv(λ) = βtriv(λ)Kφtriv(λ), with β(0) = βtriv(λ∗) = 0, φ(0) = φtriv(λ∗) = φ∗. Each curve is C1 if F is C2. Then, dβtriv(λ) dλ |λ=λ∗ ̸= 0, lim s→0,β(s)̸=0 sλ′(s) β(s) = − 1 β′ triv(λ∗) . By Theorem 3.1, we obtain the following formula, which is convenient to be used. Proposition 3.2. Under the assumption of Theorem 3.1, we have that lim s→0,β(s)̸=0 sλ′(s) β(s) l(Fλu( λ1 H0 , 0)φ1) l(Kφ1) = −1, where l ∈ X∗ satisfies N (l) = R(Fu(λ1/H0, 0)), with X∗ being the dual space of X. In particular, if K = Fλu(λ1/H0, 0), then lim s→0,β(s)̸=0 sλ′(s) β(s) = −1, and β′ triv(λ1/H0) = 1. Proof. By differentiating Fu(λ, 0)φtriv(λ) = βtriv(λ)Kφtriv(λ), we have Fλu(λ, 0)φtriv(λ) + Fu(λ, 0)φ ′ triv(λ) = βtriv(λ)Kφ′ triv(λ) + β′ triv(λ)Kφtriv(λ). Taking λ = λ1/H0, we can obtain Fλu ( λ1 H0 , 0 ) φ1 + Fu ( λ1 H0 , 0 ) φ′ triv ( λ1 H0 ) = β′ triv ( λ1 H0 ) Kφ1. (3.1) Since β = 0 is an Fλu(λ1/H0, 0)-simple eigenvalue of operator Fu(λ1/H0, 0), we have Fλu ( λ1 H0 , 0 ) φ1 ̸∈ R ( Fu ( λ1 H0 , 0 )) . 8 S. GAO, Q. LIU, Y. SUN EJDE-2024/48 By taking l on both sides of equation (3.1) and using the fact that N (l) = R(Fu( λ1 H0 , 0)), we obtain that l ( Fλu ( λ1 H0 , 0 ) φ1 ) = β′ triv ( λ1 H0 ) l(Kφ1). It can be deduced that l(Kφ1) ̸= 0; consequently, β′ triv ( λ1 H0 ) = l ( Fλu ( λ1 H0 , 0 ) φ1 ) l(Kφ1) , which yields the desired formula. □ Before providing the stability result (in the linearized sense) for (1.1) near the bifurcation point, we review the concept of stability. The operator equation F (λ, x) = 0 represents the equilibrium form of the evolution equation dx dt = F (λ, x). (3.2) Suppose that F (λ0, x0) = 0. If all the eigenvalues of Fx(λ0, x0) are negative, then x0 is called an asymptotically linearly stable solution of (3.2). On the other hand, if a positive eigenvalue of Fx(λ0, x0) exists, then x0 is called an unstable solution of (3.2). Proof of Theorem 1.2. Let X be a Banach space. From equations (2.1) and (2.3), it can be inferred that 0 is a simple eigenvalue of Fu(λ1/H0, 0) := T . Let φ1 represent the eigenfunction corresponding to the eigenvalue 0 with ∥φ1∥ = 1. We denote T = Fu(λ1/H0, 0) and K = Fλu(λ1/H0, 0). According to Theorem 3.1, we have K[φ1] = Fλ(λ, u) ∣∣ (λ,u)=(λ1/H0,0) [φ1]. Thus, we obtain K[φ1] = −Nf(x)H(x, 0)φ1. Next, we need to verify whether 0 is a simple eigenvalue of K. Consider the eigen- value problem K[φ1] = −Nf(x)H(x, 0)φ1 = 0. This implies that φ1 satisfies f(x)H(x, 0)φ1 = 0. For nonzero φ1, this can only hold if f(x)H(x, 0) = 0. However, since f(x) ̸= 0 and H(x, 0) ̸= 0, φ1 must be zero, which contradicts our assumption. This indicates that K[φ1] /∈ R(T ). Therefore, 0 is a K-simple eigenvalue of Fu(λ1/H0, 0). According to Theorem 3.1, there exist eigenvalues β(s) and βtriv(λ) ∈ R and eigenvectors ϕ1(s) and ϕtriv(λ) in the vector space X such that Fu(λ(s), u(s))ϕ1(s) = β(s)Fλu(λ1/H0, 0)ϕ1(s), Fu(λ, 0)ϕtriv(λ) = βtriv(λ)Fλu(λ1/H0, 0)ϕtriv(λ) with β(0) = βtriv(λ1/H0) = 0, ϕ1(0) = ϕtriv(λ1/H0) = φ1. Each curve is C1 if F belongs to C2; then, dβtriv(λ) dλ ∣∣ λ=λ1/H0 ̸= 0, lim s→0,β(s) ̸=0 sλ′(s) β(s) = − 1 β′ triv(λ1/H0) . EJDE-2024/48 LOCAL BIFURCATION STRUCTURE AND STABILITY 9 By (2.2) and (2.5), we obtain l ( Fλu ( λ1 H0 , 0 ) φ1 ) = ∫ Ω Fλu ( λ1 H0 , 0 ) φ1 · φ1 dx. From (2.4), we have l ( Fλu ( λ1 H0 , 0 ) φ1 ) = fH0 ∫ Ω φ2 1 dx = fH0 · 1 = fH0 > 0. This suggests that if β(s) > 0 (β(s) < 0), u(s) is (formally) unstable (stable). By Proposition 3.2, we can deduce that lim s→0 sλ′(s) β(s) = −1. If Huu(x, 0) ≡ 0 in Ω but Huuu(x, 0) ̸= 0 in Ω, it has been demonstrated that λ′(0) = 0. We can write λ′(s) = sλ′′(0) +O(s2). Thus, we see that lim s→0 s2λ′′(0) +O(s3) β(s) = −1. Furthermore, it can be observed that lim s→0 λ′′(0) β(s) s2 = −1. We therefore have lim s→0 β(s) s2 = −λ′′(0). (3.3) If λ′′(0) > 0, we can conclude from (3.3) that for small values of |s|, β(s) is negative. On the other hand, if λ′′(0) < 0, we can conclude from (3.3) that for small values of |s|, β(s) is positive. Therefore, when Huuu(x, 0) > 0 in Ω, we have λ′′(0) > 0 along the nontrivial bifurcating curve passing through (λ1/H0, 0), indicating asymptotic linear stability of the nontrivial solutions. Similarly, when Huuu(x, 0) < 0 in Ω, we have λ′′(0) < 0 along the nontrivial bifurcating curve passing through (λ1/H0, 0), indicating the asymptotic linear instability of the nontrivial solutions. □ Acknwledgments. We would like to express our deepest gratitude to our doc- toral adviser, Professor Guowei Dai, for his invaluable guidance and support in the selection of our research topic, as well as through the writing and research process. His contributions have been instrumental in the successful completion of this work. This research was supported by the NNSF of China (No. 12371110). References [1] J. L. Barbosa, M. D. Carmo; Stability of hypersurfaces with constant mean curvature, Math. Z., 185 (1984), 339–353. [2] J. L. Barbosa, M. D. Carmo, J. Eschenburg; Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z., 197 (1988), 123–138. [3] A. Barros, A. Brasil, A. Caminha; Stability of spacelike hypersurfaces in foliated spacetimes, Differential Geom. Appl., 26 (2008), 357-365. [4] R. Bartnik, L. Simon; Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982-1983), 131–152. 10 S. GAO, Q. LIU, Y. SUN EJDE-2024/48 [5] C. Bereanu, D. de la Fuente, A. Romero, P.J. Torres; Existence and multiplicity of en- tire radial space like graphs with prescribed mean curvature function in certain Friedmann- Lemâıtre-Robertson-Walker space times, Commun. Contemp. Math., 19 (2017), 1–18. [6] C. Bereanu, P. Jebelean, J. Mawhin; The Dirichlet problem with mean curvature operator in Minkowski space-a variational approach, Adv. Nonlinear Stud. 14 (2014), 315–326. [7] C. Bereanu, P. Jebelean, P. J. Torres; Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270–287. [8] C. Bereanu, P. Jebelean, P. J. Torres; Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), 644– 659. [9] E. Calabi; Examples of Berstein problems for some nonlinear equations, Proc. Sym. Global Analysis, Univ. of Calif., Berkeley, 1968. [10] S.-Y. Cheng, S.-T. Yau; Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104 (1976), 407–419. [11] C. Corsato, F. Obersnel, P. Omari, S. Rivetti; Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227–239. [12] M. G. Crandall, P. H. Rabinowitz; Bifurcation, perturbation of simple eigenvalues and lin- earied stability, Arch. Ration. Mech. Anal., 52 (1973), 161–180. [13] G. Dai; Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var., 55 (2016), 1–17. [14] G. Dai; Global bifurcation for problem with mean curvature operator on general domain, Nonlinear Differ. Equ. Appl., 24 (2017), 30. [15] G. Dai; Global structure of one-sign solutions for problem with mean curvature operator. Nonlinearity, 31 (2018), 5309–5328. [16] G. Dai; Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J., 67 (2018), 2103–2121. [17] G. Dai, Z. Zhang; Spectrum, bifurcation and hypersurfaces of prescribed k-th mean curvature in Minkowski space. Asymptot. Anal. 136 (2024), 257–289. [18] G. Dai, S. Gao, H. Luo; Existence and regularity of spacelike hypersurfaces for mean curvature equation in the static spacetime, Rocky Mountain J. Math. 2024. Accepted for publication, available at https://api.semanticscholar.org/CorpusID:270309258 [19] D. de la Fuente, A. Romero, P.J. Torres; Radial solutions of the Dirichlet problem for the prescribed mean curvature equation, Adv. Nonlinear Stud. 15 (2014), 171–181. [20] D. Fuente, A. Romero, P. J. Torres; Entire spherically symmetric spacelike graphs with pre- scribed mean curvature function in Schwarzschild and Reissner-Nordström spacetimes, Class. Quantum Grav., 32 (2015), 035018 (17pp). [21] J. Mawhin, P. J. Torres; Prescribed mean curvature graphs with Neumann boundary condi- tions in some FLRW spacetimes, J. Differential Equations 261 (2016), 7145–7156. [22] B. O’Neill; Semi-Riemannian geometry, Academic Press, 1983. [23] J. Shi; Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (2) (1999), 494–531. [24] A. E. Treibergs; Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39–56. Siyu Gao (corresponding author) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address: gao15898107523@163.com Qingbo Liu School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address: liuqingbo@mail.dlut.edu.cn Yingxin Sun School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address: sunyingxin2023@mail.dlut.edu.cn 1. Introduction and main results 2. Local bifurcation structure 3. Stability properties Acknwledgments References