Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 18, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu k-HESSIAN CURVATURE TYPE EQUATIONS IN SPACE FORMS JUNDONG ZHOU Abstract. In this article, we study closed star-shaped (η, k)-convex hyper- surfaces in space forms satisfying a class of k-Hessian curvature type equations. Firstly, using the maximum principle, we obtain a priori estimates for the class of Hessian curvature type equations. Secondly, we obtain an existence result by using standard degree theory based on a priori estimates. 1. Introduction Suppose that M is an immersed hypersurface in Euclidean space Rn+1. Define a (0, 2)-tensor η on M by ηij = Hgij − hij , where gij , hij and H are the first, second fundamental forms and mean curvature of M respectively. In fact, η is the first Newton transformation of h with respect to g, see [18]. Let κ = (κ1, . . . , κn) be the vector whose components κi are the principal curvatures of M . Using λ(η) to denote the vector whose components are the eigenvalues of η, we have that λ(η) = (H − κ1, . . . ,H − κn). Then k-Hessian equation of λ(η) can be written as σk(λ(η)) = f(X, ν(X)), 1 ≤ k ≤ n, X ∈M, (1.1) where ν is the normal vector field along M and σk is the k-th elementary symmetric function σk(λ) = ∑ i1<··· 0,∀1 ≤ i ≤ k}. For example, when k = n, it becomes det(η(X)) = f(X, ν), for X ∈M. (1.3) The (η, n)-convex hypersurface has been studied intensively by Sha [21, 22], Wu [24], Harvey and Lawson [14]. (η, n)-convexity is called (n− 1)-convexity in [14, 21, 22]. In complex geometry, when k = n, Equation (1.1) is called the (n − 1) Monge- Ampère equation, which is related to the Gauduchon conjecture (see [8]). Compared to (1.2), it is interesting that the curvature estimate of (1.1) can be established for 1 ≤ k ≤ n. Chu and Jiao [5] established curvature estimates for (η, k)-convex hypersurface and proved the existence for (1.1). Chen, Tu and Xiang [4] extended it to a class of Hessian quotient equations. In this article, we give a simpler proof of the result of Chu and Jiao [5], and extend it to space forms. Let Nn+1(K) be a space form of sectional curvature K = −1, 0, or 1. It is known that the space forms can be viewed as Euclidean space Rn+1 equipped with a metric tensor gN , that is, Nn+1(K) = (Rn+1, gN ), gN = dρ2 + φ2(ρ)dz2, where φ(ρ) =  sin(ρ), ρ ∈ [0, π2 ), if K = 1, ρ, ρ ∈ [0,+∞), if K = 0, sinh(ρ), ρ ∈ [0,+∞), if K = −1, where dz2 denotes the standard metric on Sn induced from Rn+1. We define the vector field V = φ(ρ) ∂∂ρ . In fact, V is a conformal Killing field in Nn+1(K) and V is just the position vector field in Rn+1. We consider the k-Hessian equation of λ(η) in Nn+1(K), σk(λ(η)) = f(V, ν), 2 ≤ k ≤ n, (1.4) and obtain the main result as follows. Theorem 1.1. Let f(V, ν) ∈ C2(Γ) be a positive function and Γ be an open neigh- borhood of the unit normal bundle of M in Nn+1×Sn. Assume that there exist two positive constants r1, r2 and r1 < 1 < r2, such that f(V, V |V | ) ≤ Ckn(n− 1)k (φ′(r2) φ(r2) )k , for ρ = r2, (1.5) f(V, V |V | ) ≥ Ckn(n− 1)k (φ′(r1) φ(r1) )k , for ρ = r1, (1.6) ∂ ∂ρ [ φkf(V, ν) ] ≤ 0, for r1 ≤ ρ ≤ r2. (1.7) Then there exists a C4,δ closed star-shaped (η, k)-convex hypersurface satisfying (1.4) for any δ ∈ (0, 1). EJDE-2022/18 k-HESSIAN CURVATURE TYPE EQUATIONS 3 The rest of this article is organized as follows. In Section 2, we give some definitions and important formulas. In Section 3, we prove C0, C1 and C2 estimates of (1.4). In Section 4, we give the proof for the existence, that is Theorem 1.1. 2. Preliminaries In this section, we recall some geometric objects and related formulas on hy- persurfaces in space forms. Let M be an immersed star-shaped hypersurface in Nn+1(K), which is expressed as M = {(z, ρ(z)) : z ∈ Sn}. Let∇′ and∇ denote the covariant derivatives with respect to the standard spherical metric and the covariant derivatives with respect to the induced metric on M , respectively. Following the notations in [1], the induced metric, its inverse, unit normal vector and second fundamental form on M are respectively by gij = φ2eij +∇′iρ∇′jρ, gij = 1 φ2 ( eij − ρiρj φ2 + |∇′ρ|2 ) , (2.1) ν = −∇′ρ+ φ2 ∂ ∂ρ√ φ4 + φ2|∇′ρ|2 , (2.2) hij = φ√ φ2 + |∇′ρ|2 ( −∇′ijρ+ 2φ′ φ ∇′iρ∇′jρ+ φφ′eij ) . (2.3) where eij is the standard spherical metric and eij is inverse of it. We define Φ(ρ) =∫ ρ 0 φ(r)dr and u = 〈V, ν〉. Let {e1, . . . , en} be a local orthonormal frame on M . By direct calculations, we have the following formulas (see [13, 23]): ∇iΦ = 〈V, ei〉, ∇ijΦ = φ′gij − uhij , (2.4) ∇iu = gklhik∇lΦ, (2.5) ∇iju = gkl∇khij∇lΦ + φ′hij − ugklhikhjl, (2.6) ∇iν = gklhikel, (2.7) ∇ijhkl = ∇klhij − hml(himhkj − hijhmk)− hmj(hmihkl − hilhmk) +Khml(δijδkm − δimδkj) +Khmj(δilδkm − δimδkl). (2.8) For simplicity, we denote G(η) := σ 1/k k (λ(η)), Gij(η) := ∂G ∂ηij , Gij,rs(η) := ∂2G ∂ηijηrs , F ii = ∑ k 6=i Gkk. If (hij) is diagonal and h11 ≥ · · · ≥ hnn, then η11 ≤ · · · ≤ ηnn, G11 ≥ · · · ≥ Gnn, F 11 ≤ · · · ≤ Fnn. 3. A priori estimates In this section, we obtain C0, C1 and C2 estimates for (1.4). Let us consider a family of functions, for t ∈ [0, 1], f t(V, ν) = tf(V, ν)+(1− t)Ckn(n−1)k [(φ′(ρ) φ(ρ) )k +ε ((φ′(ρ) φ(ρ) )k−(φ′(1) φ(1) )k)] , (3.1) 4 J. ZHOU EJDE-2022/18 where the constant ε is small sufficiently such that min r1≤ρ≤r2 [(φ′(ρ) φ(ρ) )k + ε ((φ′(ρ) φ(ρ) )k − (φ′(1) φ(1) )k)] ≥ c0 > 0. It is easy to see that f t(V, ν) satisfies (1.5), (1.6) and (1.7) with strict inequalities for 0 < t < 1. To prove Theorem 1.1, we consider the family of equations σk(λ(η)) = f t(V, ν), 0 ≤ t ≤ 1. (3.2) 3.1. C0 estimates. Now, we prove the following proposition which asserts that the solutions of (3.2) have uniform C0 bounds. Proposition 3.1. Let f t(V, ν) ∈ C2(Nn+1 × Sn) is a positive function. Under assumptions (1.5) and (1.6), if Mt = {(z, ρ(z)) : z ∈ Sn} ⊂ Nn+1(K) is a star- shaped (η, k)-convex hypersurface satisfying Equation (3.2) for 0 < t < 1, then r1 < ρt < r2. Proof. Suppose that ρt(z) attains its maximum at z0 ∈ Sn and ρt(z0) ≥ r2. Then ∇′ρ = 0, at z0. Therefore, from (2.1) and (2.3) we obtain gij = φ−2eij , hij = −∇′ijρ+ φφ′eij , which implies that hij = gikhkj = − eik∇′kjρ φ2 + φ′ φ δij ≥ φ′ φ δij . It follows that ηij = Hδij − hij ≥ (n− 1) φ′ φ δij . Noticing that σk is elliptic in Γk, we have σk(λ(η)) ≥ Ckn(n− 1)k (φ′ φ )k . (3.3) On the other hand, the unit outer normal vector ν = V |V | at z0 and f t(V, ν) satisfies (1.5) with strict inequality for 0 < t < 1. If ρt(z0) = r2, then Ckn(n− 1)k (φ′(r2) φ(r2) )k > f t(V, V |V | ) = f t(V, ν) = σk(λ(η)). (3.4) This contradicts (3.3), and shows that . supMt ρt < r2. Similarly, we prove infMt ρt > r1. � Now, we prove the following uniqueness result. Proposition 3.2. For t = 0, there exists unique (η, k)-convex solution of Equation (3.2), namely, M0 is an unit sphere in Nk(K). Proof. Let M0 be a solution of (3.2) for t = 0. Assume the height function ρ(z) of M0 achieves its maximum ρmax at z0 ∈ Sn, then Ckn(n− 1)k [(φ′(ρmax) φ(ρmax) )k + ε ((φ′(ρmax) φ(ρmax) )k − (φ′(1) φ(1) )k)] = σk(λ(η)) ≥ Ckn(n− 1)k (φ′(ρmax) φ(ρmax) )k , EJDE-2022/18 k-HESSIAN CURVATURE TYPE EQUATIONS 5 which implies φ′(ρmax) φ(ρmax) ≥ φ′(1) φ(1) . (3.5) Noting that φ′(ρ) φ(ρ) =  cot(ρ), if K = 1, 1 ρ , if K = 0, coth(ρ), if K = −1, we obtain ρmax ≤ 1. Similarly, ρmin ≥ 1. Thus, ρ = 1 is the unique solution of (3.2) for t = 0. � 3.2. C1 estimates. In this section, we follow the ideas in [3] and [10] to obtain C1 estimates for the height function ρ. Proposition 3.3. Let M be a closed star-shaped (η, k)-convex hypersurface in Nk(K) satisfying (3.2). Under assumption (1.7), if ρ has positive upper and lower bounds, there exists a constant C depending on infMρ, supM ρ, and ‖f‖C1(M) such that |∇ρ| ≤ C. Proof. Since u = 〈V, ν〉 = φ2 φ2 + |∇′ρ|2 , it is sufficient to obtain a positive lower bound of u. We consider a test function ϕ = − log u+ γ(Φ(ρ)), where γ(t) is a function which will be chosen later. Assume that ϕ achieves its maximum value at z0 ∈ Sn, we will show that u(z0) = |V (z0)|, that is, V (z0) = φ(ρ(z0))ν(z0), which implies a uniform lower bound for u on M . If not, we may choose a local orthonormal frame {e1, . . . , en} around (z0, ρ(z0)) ∈ M such that 〈V, e1〉 6= 0 and 〈V, ei〉 = 0, i ≥ 2. Using (2.5), we have at (z0, ρ(z0)) ∈M , 0 = ∇iϕ = −∇iu u + γ′∇iΦ = −hi1〈V, e1〉 u + γ′〈V, ei〉. (3.6) It follows from (3.6) that h11 = uγ′, hi1 = 0, i ≥ 2. (3.7) Rotate {e2, . . . , en} around (z0, ρ(z0)) ∈ M such that hij is diagonal. Covariantly differentiating ϕ twice yields 0 ≥ F ii∇iiϕ = −F ii∇iiu u + F ii |∇iu|2 u2 + γ′′F ii|∇iΦ|2 + γ′F ii∇iiΦ = − 1 u F ii ( hii1∇1Φ + φ′hii − uh2 ii ) + ( (γ′)2 + γ′′ ) F ii|∇iΦ|2 + γ′F ii ( φ′δii − uhii ) , (3.8) where the second equality is given by using (2.4), (2.5) and (2.6). Then ηii = ∑ j 6=i hjj 6 J. ZHOU EJDE-2022/18 implies ∑ i ηii = (n− 1) ∑ i hii, hii = 1 n− 1 ∑ k ηkk − ηii, which results in∑ i F iihii = ∑ i (∑ k Gkk −Gii )( 1 n− 1 ∑ k ηkk − ηii ) = ∑ i Giiηii = f1/k(V, ν), (3.9) ∑ i F iihiij = ∑ i Giiηiij . (3.10) Notice that (1.4) can be written as G(η) = f1/k(V, ν) = f̃(V, ν). (3.11) By (2.7) and covariantly differentiating (3.11) with respect to e1, we have Giiηii1 = dV f̃(∇e1V ) + h11dν f̃(e1). (3.12) Taking (2.4), (3.9), (3.10) and (3.12) in (3.8) yields 0 ≥ − 1 u ( dV f̃(∇e1V )〈V, e1〉+ φ′f̃ + h11dν f̃(e1)〈V, e1〉 ) + ( (γ′)2 + γ′′ ) F 11〈V, e1〉2 + γ′φ′ ∑ i F ii − γ′uf̃ ≥ − 1 u ( dV f̃(∇e1V )〈V, e1〉+ φ′f̃ ) − γ′dν f̃(e1)〈V, e1〉 + ( (γ′)2 + γ′′ ) F 11〈V, e1〉2 + γ′φ′ ∑ i F ii − γ′uf̃ , (3.13) where the second inequality is obtained by (3.7). Since V = 〈V, e1〉e1 + 〈V, ν〉ν at z0, dV f̃(V ) = 〈V, e1〉 ( dV f̃ ) (∇e1V ) + u ( dV f̃ ) (∇νV ). (3.14) From this and the assumption (1.7), we see that 0 ≥ ∂ ∂ρ ( φkf(V, ν) ) = k ( φf̃ )k−1( φ′f̃ + dV f̃(V ) ) = k ( φf̃ )k−1 ( φ′f̃ + 〈V, e1〉 ( dV f̃ ) (∇e1V ) + u ( dV f̃ ) (∇νV ) ) . (3.15) Combining this with (3.13) gives 0 ≥ dV f̃(∇νV ) + ( (γ′)2 + γ′′ ) F 11〈V, e1〉2 + γ′φ′ ∑ i F ii − γ′uf̃ − γ′dν f̃(e1)〈V, e1〉. (3.16) Now we choose γ(t) = α t , (3.17) where α is sufficiently large. Recalling that h11 = γ′u at (z0, ρ(z0)), we have h11 < 0. Since H > 0, there exists k0 with 2 ≤ k0 ≤ n such that hk0k0 > h11. Combining this with the definitions of ηii and Gii yields ηK0k0 < η11, Gk0k0 ≥ G11. EJDE-2022/18 k-HESSIAN CURVATURE TYPE EQUATIONS 7 Thus, F 11 = ∑ j 6=1 Gjj ≥ 1 2 ∑ i Gii = 1 2(n− 1) ∑ i F ii ≥ 1 2 ( Ckn )1/k . (3.18) Putting (3.17) and (3.18) in (3.16), we obtain 0 ≥ 〈V, e1〉2 2(n− 1) ( α2Φ−4 + 4α2Φ−6 )∑ i F ii − αΦ−2φ′ ∑ i F ii − αΦ−2 ∣∣V ∣∣∣∣dν f̃(e1) ∣∣− ∣∣dV f̃(∇νV ) ∣∣, (3.19) which leads to a contradiction when α is large. Therefore u(z0) = |V (z0)|. � 3.3. C2 estimates. To obtain C2 estimates for (3.2), we prove that the principal curvatures have uniform bounds. Proposition 3.4. Let M = {(z, ρ(z)) : z ∈ Sn} be a closed star-shaped (η, k)- convex hypersurface in Nk(K) satisfying (3.2), where f(V, ν) ∈ C2(Γ) is a positive function and Γ is an open neighborhood of the unit normal bundle of M in Nn+1× Sn. If 0 < r1 ≤ ρ(z) ≤ r2, ‖ρ‖C1 ≤ r3, then there exists a constant C depending on n, k, r1, r2, r3, ‖f‖C2(M) and infM f such that max Sn |κi| ≤ C, for 1 ≤ i ≤ n, where (κ1, . . . , κn) is the principal curvatures vector of M . Proof. Since H > 0, it suffices to prove that the largest curvature κmax is uniformly bounded from above. From Propositions 3.1 and 3.3, we know that 1 C ≤ inf M u ≤ u ≤ sup M u ≤ C, where the positive constant C depends on infM ρ and ‖ρ‖C1 . Taking the auxiliary function Q = eβΦκmax u− a , (3.20) where a = 1 2 infM u and β is a large constant to be determined later. Assume that (z0, ρ(z0)) is the maximum point of the function Q, we can choose a local orthonormal frame {e1, . . . , en} around (z0, ρ(z0)) such that hij is diagonal and h11 ≥ · · · ≥ hnn at (z0, ρ(z0)). In the rest of proof, all computations will be carried out at (z0, ρ(z0)). Since h11 = κmax, the function logQ = log h11 − log(u− a) + βΦ has a local maximum at (z0, ρ(z0)). Therefore, 0 = ∇ih11 h11 − ∇iu u− a + β∇iΦ, (3.21) 0 ≥ F ii∇iih11 h11 − F ii ( ∇ih11 )2 h2 11 − F ii∇iiu u− a + F ii ( ∇iu )2 (u− a)2 + βF ii∇iiΦ. (3.22) By (2.4) and (3.9), we have βF ii∇iiΦ = βφ′ ∑ i F ii − βuf̃ . (3.23) 8 J. ZHOU EJDE-2022/18 It follows from (2.6) and (3.12) that −F ii∇iiu u− a = −F iihiij∇jΦ u− a − φ′f̃ u− a + uF iih2 ii u− a ≥ −dV f̃(∇eiV )∇iΦ u− a − hiidν f̃(ei)∇iΦ u− a − φ′f̃ u− a + uF iih2 ii u− a . (3.24) Applying (2.8) and (3.9), we obtain F ii∇iih11 = F ii∇11hii − h11F iih2 ii + F iihiih 2 11 −KF ii(h11δ 2 1i − h11δii + hii − hi1δi1) = F ii∇11hii − h11F iih2 ii + f̃h2 11 +Kh11 ∑ i F ii − f̃K. (3.25) Covariantly differentiating (3.11) twice yields F ii∇11hii = Gii∇11ηii ≥ −Gij,rs∇1ηij∇1ηrs+ ∑ i h11idν f̃(ei)−C1(1+h2 11), (3.26) where the positive constant C1 depends on ‖f‖C2 . The concavity of G and Codazzi formula give Gij,rs∇1ηij∇1ηrs ≥ −2 ∑ i≥2 G1i,i1|∇1η1i|2 = −2 ∑ i≥2 G1i,i1|∇ih11|2. (3.27) Combining (3.25) and (3.26) with (3.27), we obtain F ii∇iih11 h11 ≥ − 2 h11 ∑ i≥2 G1i,i1|∇ih11|2 − F iih2 ii + h11idν f̃(ei) h11 +K ∑ i F ii + h11f̃ − Kf̃ h11 − C1( 1 h11 + h11). (3.28) Putting (3.23), (3.24) and (3.28) in (3.22), 0 ≥ − 2 h11 ∑ i≥2 G1i,i1|∇ih11|2 − F ii|∇ih11|2 h2 11 + a u− a F iih2 ii + F ii|∇iu|2 (u− a)2 + ∑ i (∇ih11 h11 − hii∇iΦ u− a ) dν f̃(ei) + (K + βφ′) ∑ i F ii − C2(1 + h11) ≥ − 2 h11 ∑ i≥2 G1i,i1|∇ih11|2 − F ii|∇ih11|2 h2 11 + a u− a F iih2 ii + F ii|∇iu|2 (u− a)2 + (K + βφ′) ∑ i F ii − C2(β + h11), (3.29) where C2 depends r1, r2, r3, and ‖f‖C2 . The second inequality is obtained by (3.21). We divide the rest of proof into three steps. Step 1. We prove that a 2(u− a) F iih2 ii + 1 2 (K + βφ′) ∑ i F ii ≥ C2h11. (3.30) The proof of step 1 is split into two cases. EJDE-2022/18 k-HESSIAN CURVATURE TYPE EQUATIONS 9 Case 1. |hii| ≤ δh11 for all 2 ≤ i ≤ n, δ is a small constant to be chosen. We obtain |η11| ≤ (n−1)δh11, ( 1−(n−2)δ ) h11 ≤ η22 ≤ · · · ≤ ηnn ≤ ( 1+(n−2)δ ) h11. (3.31) This shows that σk−1(η) = σk−1(η|1) + η11σk−2(η|1) ≥ Ck−1 n−1 ( 1− (n− 2)δ )k−1 hk−1 11 − Ck−2 n−1 ( 1 + (n− 2)δ )( 1− (n− 2)δ )k−2 hk−1 11 . (3.32) Choosing δ sufficiently small and using k ≥ 2, we have σk−1(η) ≥ 1 2 hk−1 11 ≥ 1 2 h11. (3.33) It follows from (3.33) and the definitions of Gii and F ii that∑ i F ii = (n− 1) ∑ i Gii = (n− 1)(n− k + 1) k σ 1 k−1 k (η)σk−1(η) ≥ (n− 1)(n− k + 1) 2k infM f1− 1 k h11. (3.34) Choosing β sufficiently large gives 1 2 (K + βφ′) ∑ i F ii ≥ C2h11. (3.35) Case 2. h22 > δh11 or hnn < −δh11. We obtain a 2(u− a) F iih2 ii ≥ a 2(supM u− a) ( F 22h2 22 + Fnnh2 nn ) ≥ aδ2 2(supM u− a) F 22h2 11. (3.36) Applying Maclaurin’s inequality, we have F 22 = ∑ i 6=2 Gii ≥ 1 2 ∑ i Gii ≥ 1 2 (Ckn)1/k. (3.37) Inserting into (3.36) yields a 2(u− a) F iih2 ii ≥ aδ2 4(supM u− a) (Ckn)1/kh2 11 ≥ C2h11, (3.38) where the second inequality is obtained from h11 ≥ 4(supM u− a) aδ2 (Ckn)− 1 kC2, otherwise, the proof is complete. Step 2. We prove that |hii| ≤ βC3, for 2 ≤ i ≤ n, 10 J. ZHOU EJDE-2022/18 where C3 depends r1, r2, r3, and ‖f‖C2 . Combining step 1 and (3.29) gives 0 ≥ − 2 h11 ∑ i≥2 G1i,i1|∇ih11|2 − F ii|∇ih11|2 h2 11 + a 2(u− a) F iih2 ii + F ii|∇iu|2 (u− a)2 + 1 2 (K + βφ′) ∑ i F ii − C2β. (3.39) From (3.21) and Cauchy-Schwarz inequality, we have − F ii|∇ih11|2 h2 11 ≥ − 1 + ε (u− a)2 F ii|∇iu|2 − (1 + 1 ε )β2F ii|∇iΦ|2. (3.40) Note that − 2 h11 ∑ i≥2 G1i,i1|∇ih11|2 ≥ 0. (3.41) Using (3.40) and (3.41) in (3.39) yields 0 ≥ ( a 2(u− a) − ε|∇Φ|2 (u− a)2 ) F iih2 ii − C2β + (1 2 (K + βφ′)− (1 + 1 ε )β2|∇Φ|2 )∑ i F ii, (3.42) where ∇iu = hii∇iΦ. Recalling that F ii ≥ F 22 ≥ 1 2(n− 1) ∑ i F ii ≥ 1 2 (Ckn)1/k and choosing ε sufficiently small such that a 2(u− a) − ε|∇Φ|2 (u− a)2 ≥ c0 > 0, we deduce that 0 ≥ c0 2(n− 1) ∑ j≥2 h2 jj + (1 2 (K + βφ′)− (1 + 1 ε )β2|∇Φ|2 ) − C2β∑ i F ii . (3.43) Therefore, ∑ i≥2 h 2 ii ≤ β2C2 3 . Step 3. We show that there exists a constant C depending r1, r2, r3, ‖f‖C2 , and infM f , such that h11 ≤ C. From (3.21) and Cauchy-Schwarz inequality, we obtain − F ii|∇ih11|2 h2 11 ≥ − 1 + ε (u− a)2 F 11|∇1u|2 − (1 + 1 ε )β2F 11|∇1Φ|2 − ∑ i≥2 F ii|∇ih11|2 h2 11 . (3.44) Choosing ε sufficiently small, we obtain − ε (u− a)2 F 11|∇1u|2 = − ε|∇1Φ|2 (u− a)2 F 11h2 11 ≥ − a 16(u− a) F iih2 ii. (3.45) Without loss of generality, we assume that h2 11 ≥ max {32(supM u− a)β2 aε |∇Φ|2, β 2C2 3 α2 } , EJDE-2022/18 k-HESSIAN CURVATURE TYPE EQUATIONS 11 where α will be determined later (α < 1). This gives − (1 + 1 ε )β2F 11|∇1Φ|2 ≥ −2 ε β2F 11|∇Φ|2 ≥ − a 16(u− a) F iih2 ii. (3.46) By step 2, |hii| ≤ αh11, for i ≥ 2, (3.47) which implies that 1 h11 ≤ 1 + α h11 − hii . (3.48) Noting that −G1i,i1 = G11 −Gii ηii − η11 = F ii − F 11 h11 − hii , we have − ∑ i≥2 F ii|∇ih11|2 h2 11 ≥ − ∑ i≥2 F ii − F 11 h2 11 |∇ih11|2 − ∑ i≥2 F 11|∇ih11|2 h2 11 ≥ −1 + α h11 ∑ i≥2 F ii − F 11 h11 − hii |∇ih11|2 − ∑ i≥2 F 11|∇ih11|2 h2 11 = 1 + α h11 ∑ i≥2 Gi1,1i|∇ih11|2 − ∑ i≥2 F 11|∇ih11|2 h2 11 . (3.49) Using (3.21), (3.47), and Cauchy-Schwarz inequality we have − ∑ i≥2 F 11|∇ih11|2 h2 11 ≥ −2 ∑ i≥2 F 11|∇iu|2 (u− a)2 − 2β2 ∑ i≥2 F 11|∇iΦ|2 ≥ −2(n− 1)α2|∇Φ|2 a2 aF 11h2 11 u− a − ε(u− a) 16(supM u− a) aF 11h2 11 u− a . (3.50) Choosing α sufficiently small gives − ∑ i≥2 F 11|∇ih11|2 h2 11 ≥ −aF 11h2 11 8(u− a) ≥ − aF iih2 ii 8(u− a) . (3.51) Putting (3.44), (3.45), (3.46), (3.49), and (3.51) in (3.39) yields 0 ≥ F ii|∇iu|2 4(u− a)2 + 1 2 (K + βφ′) ∑ i F ii − C2β ≥ C2 2 h11 − C2β. (3.52) Thus h11 ≤ 2β. � 4. Existence In this section, we use the degree theory for nonlinear elliptic equation developed in [16] to prove Theorem 1.1. After establishing the a priori estimates in Proposi- tions 3.1, 3.3 and 3.4, we know that (3.2) is uniformly elliptic. From Evans-Krylov estimates [7, 15], and Schauder estimates, we obtain ‖ρ‖C4,δ ≤ C (4.1) 12 J. ZHOU EJDE-2022/18 for any (η, k)-convex solution M = {(z, ρ(z)) : z ∈ Sn} to (1.4). We consider a family of the mappings for t ∈ [0, 1], F (·; t) : C4,δ 0 (Sn)→ C2,δ(Sn), defined by F (z, ρ(z); t) = σk(λ(η))− f t(V, ν), where f t(V, ν) = tf(V, ν) + (1− t)Ckn(n− 1)k [(φ′(ρ) φ(ρ) )k + ε ((φ′(ρ) φ(ρ) )k − (φ′(1) φ(1) )k)] , where the constant ε is sufficiently small such that min r1≤ρ≤r2 [(φ′(ρ) φ(ρ) )k + ε ((φ′(ρ) φ(ρ) )k − (φ′(1) φ(1) )k)] ≥ c0 > 0, for some positive constant c0. We set OR = {ρ ∈ C4,δ 0 (Sn) : ‖ρ‖C4,δ(Sn) < R}, which is an open set of C4,δ 0 (Sn). If R is sufficiently large, F (z, ρ(z); t) = 0 has no solution on ∂OR by the a priori estimates in (4.1). Therefore, the degree of deg(F (·; t),OR, 0) is well-defined. Using the homotopic invariance of the degree, we have deg(F (·; 1),OR, 0) = deg(F (.; 0),OR, 0). At t = 0, by Proposition 3.2, ρ0 = 1 is the unique solution of (3.2) in OR. Direct calculations yields F (z, ρ; 0) = −εCkn(n− 1)k ((φ′(ρ) φ(ρ) )k − (φ′(1) φ(1) )k) . By the definition of φ(ρ), we obtain δρ0F (z, ρ0; 0) = d ds |s=1F (z, sρ0; 0) = −εkCkn(n− 1)k (φ′(1) φ(1) )k−1φ′′(1)φ(1)− φ′(1)φ′(1)( φ(1) )2 > 0, where δF (z, ρ0; 0) is the linearized operator of F at ρ0. Then δF (z, ρ0; 0) takes the form δϕF (z, ρ0; 0) = −aij∇′ijϕ+bi∇′iϕ−εkCkn(n−1)k (φ′(1) φ(1) )k−1φ′′(1)φ(1)− φ′(1)φ′(1)( φ(1) )2 , where (aij) is a positive definite matrix. Clearly, δρ0F (z, ρ0; 0) is an invertible operator. Therefore, deg(F (.; 1),OR, 0) = deg(F (.; 0),OR, 0) 6= 0. It implies that there is a solution of Equation (3.2) at t = 1. 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Jundong Zhou School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236037, An- hui, China. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China Email address: zhou109@mail.ustc.edu.cn 1. Introduction 2. Preliminaries 3. A priori estimates 3.1. C0 estimates 3.2. C1 estimates 3.3. C2 estimates 4. Existence Acknowledgments References