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Hessian equations of Krylov type on compact Hermitian manifolds

  • Jundong Zhou EMAIL logo and Yawei Chu
Published/Copyright: October 11, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Keywords: Hessian equations; Hermitian manifolds; subsolution condition
MSC 2010: 35J60; 35B45; 53C55

1 Introduction

Let ( M , ω ) be a compact Kähler manifold of complex dimension n . In 1978, Yau [1] proved the famous Calabi-Yau conjecture by solving the following complex Monge-Ampère equation on M

( ω + − 1 ∂ ∂ ¯ u ) n = f ω n ,

with positive function f . There have been many generalizations of Yau’s work. One extension of Yau’s Theorem is to the case of Hermitian manifolds, which is initiated by Cherrier [2] in 1987. The Monge-Ampère equation on compact Hermitian manifolds was solved by Tosatti and Weinkove [3], building on several earlier works. See [2,4,5, 6,7] and the references therein.

The complex Hessian equation can be expressed as follows:

( ω + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = f ω n , 2 ≤ k ≤ n − 1 .

On compact Kähler manifolds ( M , ω ) , Hou [8] proved the existence of a smooth admissible solution of the complex Hessian equation by assuming the nonnegativity of the holomorphic bisectional curvature. Later, Hou et al. [9] obtained the second-order estimate without any curvature assumption. Using Hou et al.’s estimate, Dinew and Kolodziej [10] applied a blow-up argument to prove the gradient estimate and solved the complex Hessian equation on compact Kähler manifolds. The corresponding problem on Hermitian manifolds was solved by Zhang [11] and Székelyhidi [12] independently.

The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let χ be a real (1, 1) form, the complex Hessian quotient equations can be expressed as follows:

( χ + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = f ( z ) ( χ + − 1 ∂ ∂ ¯ u ) l ∧ ω n − l , 1 ≤ l < k ≤ n , z ∈ M .

When f ( z ) is constant, one special case is the so-called Donaldson equation [13]:

( χ + − 1 ∂ ∂ ¯ u ) n = c ( χ + − 1 ∂ ∂ ¯ u ) n − 1 ∧ ω k .

After some progresses made in [14,15, 16,17], Song and Weinokove [18] solved the Donaldson equation on closed Kähler manifolds via J -flow. Fang et al. [19] extended the Donaldson equation to

( χ + − 1 ∂ ∂ ¯ u ) n = c k ( χ + − 1 ∂ ∂ ¯ u ) n − k ∧ ω k

and solved this equation on closed Kähler manifolds by assuming a cone condition. When f ( z ) is not constant, analogous results were obtained by Sun [20,21] on compact Hermitian manifolds.

In this article, we are concerned with Hessian equations of Krylov type in the form of the linear combinations of the Hessian, which can be written as follows:

(1.1) ( χ + − 1 ∂ ∂ ¯ u ) k ∧ ω n − k = ∑ l = 0 k − 1 α l ( χ + − 1 ∂ ∂ ¯ u ) l ∧ ω n − l , 2 ≤ k ≤ n .

The Dirichlet problem of (1.1) on ( k − 1 ) -convex domain Ω in R n was first studied by Krylov [22] about 20 years ago. He observed that if α l ( x ) ≥ 0 for 0 ≤ l ≤ k − 1 , the natural admissible cone to make (1.1) elliptic is also the Γ k -cone, which is the same as the k -Hessian equation case, where

Γ k = { λ ∈ R n ∣ σ 1 ( λ ) > 0 , … , σ k ( λ ) > 0 } .

Guan and Zhang [23] solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali [24] proved a priori estimates to the following equation in Kähler case:

( χ + − 1 ∂ ∂ ¯ u ) n = ∑ l = 0 n − 1 C n l α l ( χ + − 1 ∂ ∂ ¯ u ) n − k ∧ ω l ,

where α l ≥ 0 are smooth real functions such that either α l = 0 or α l > 0 , and ∑ l = 0 n − 2 α l > 0 . Recently, Phong and Tô [25] solved Hessian equations of Krylov type on compact Kähler manifolds, where α l are non-negative constants for 0 ≤ l ≤ k − 1 . When α l are non-negative smooth functions for 0 ≤ l ≤ k − 1 , analogous results on compact Kähler manifolds are obtained by Chen [26] and Zhou [27] independently.

Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou [27] believed that the condition on α k − 1 ( x ) > 0 is not necessary. In fact, Guan and Zhang [23] considered equation (1.1) without the sign requirement for the coefficient function α k − 1 ( x ) .

In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for α k − 1 ( x ) . Let Γ k − 1 g be the set of all the real (1, 1) forms, eigenvalues of which belong to Γ k − 1 . To ensure the ellipticity and non degeneracy of the equation in Γ k − 1 , we require smooth real functions α l to satisfy the conditions: for 0 ≤ l ≤ k − 2 , either α l > 0 or α l ≡ 0 , and ∑ l = 0 k − 2 α l > 0 . Let χ u = χ + − 1 ∂ ∂ ¯ u and χ u ̲ = χ + − 1 ∂ ∂ ¯ u ̲ . To state our main results, we need also the following condition of C -subsolution, which is similar to C -subsolution introduced by Székelyhidi [12].

Definition 1.1

A smooth real function u ̲ is a C -subsolution to (1.1), if χ u ̲ ∈ Γ k − 1 g , and at each point x ∈ M , the set

λ ( χ ˜ ) ∈ Γ k − 1 ∣ χ ˜ k ∧ ω n − k = ∑ l = 0 k − 1 α l ( x ) χ ˜ l ∧ ω n − l and χ ˜ − χ u ̲ ≥ 0

is bounded.

Theorem 1.2

Let ( M , g ) be a compact Hermitian manifold, χ a real (1, 1) form on M . Suppose that u ̲ is a C -subsolution of equation (1.1) and at each point x ∈ M ,

(1.2) χ u ̲ ( x ) k ∧ ω n − k ≤ ∑ l = 0 k − 1 α l ( x ) χ u ̲ ( x ) l ∧ ω n − l .

Then there exists a smooth real function u on M and a unique constant b solving

(1.3) χ u k ∧ ω n − k = ∑ l = 0 k − 2 α l ( x ) χ u l ∧ ω n − l + ( α k − 1 + b ) χ u k − 1 ∧ ω n − k + 1 ,

with sup M ( u − u ̲ ) = 0 and χ u ∈ Γ k − 1 g .

Corollary 1.3

Let ( M , g ) be a compact Kähler manifold, χ a closed ( 1 , 1 ) -form. Suppose that u ̲ is a C -subsolution of equation (1.1) and

(1.4) ∫ M χ k ∧ ω n − k ≤ ∑ l = 0 k − 1 c l ∫ M χ ∧ ω n − l ,

where c l = inf M α l , 0 ≤ l ≤ k − 1 . Then there exists a smooth real function u on M and a unique constant b solving

(1.5) χ u k ∧ ω n − k = ∑ l = 0 k − 2 α l ( x ) χ u l ∧ ω n − l + ( α k − 1 + b ) χ u k − 1 ∧ ω n − k + 1 ,

with sup M ( u − u ̲ ) = 0 and χ u ∈ Γ k − 1 g .

Lately, Pingali [28] proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle θ ˆ ∈ 1 2 π , 3 2 π on compact Kähler threefold. Let Ω be a closed ( 1 , 1 ) form, Ω u = Ω + − 1 ∂ ∂ ¯ u . From [24], the deformed Hermitian Yang-Mills equation on compact Kähler threefold can be written as follows:

(1.6) Ω u 3 = 3 sec 2 ( θ ˆ ) Ω u ∧ ω 2 + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ω 3 .

As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Corollary 1.4

Let ( M , g ) be a compact Kähler threefold, constant phase angle θ ˆ ∈ 1 2 π , 3 2 π , and Ω a positive definite closed ( 1 , 1 ) form, satisfying the following conditions:

(1.7) 3 Ω 2 − 3 sec 2 ( θ ˆ ) ω 2 > 0 ,

(1.8) ∫ M Ω 3 = 3 sec 2 ( θ ˆ ) ∫ M Ω ∧ ω 2 + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ∫ M ω 3 ,

(1.9) Ω + sec ( θ ˆ ) ω ∈ Γ 2 g .

Then there exists a smooth solution to equation (1.6) with sup M u = 0 and Ω u ∈ Γ 3 g .

The rest of this article is organized as follows. In Section 2, we set up some notations and provide some preliminary results. In Section 3, we give the C 0 estimate by the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, we establish the C 2 estimate for equation (1.1) by the method of Hou et al. [9] and the C -subsolution condition. In Section 5, we give the gradient estimate. In Section 6, we give the proof of Theorem 1.2, Corollaries 1.3, and 1.4 by the method of continuity. Although the method is very standard in the study of elliptic PDEs, it is not easy to carry out this method on a compact Hermitian manifold. Since the essential C -subsolution condition depends on α 0 , … , α k − 1 , we have to find a uniform C -subsolution condition for the solution flow of the continuity method.

2 Preliminaries

In this section, we set up the notation and establish some lemmas. Let σ k ( λ ) denote the k th elementary symmetric function

σ k ( λ ) = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n λ i 1 ⋯ λ i k , for λ = ( λ 1 , … , λ n ) ∈ R n , 1 ≤ k ≤ n .

For completeness, we define σ 0 ( λ ) = 1 and σ − 1 ( λ ) = 0 . Let σ k ( λ ∣ i ) denote the symmetric function with λ i = 0 and σ k ( λ ∣ i j ) the symmetric function with λ i = λ j = 0 . Also denote by σ k ( A ∣ i ) the symmetric function with A deleting the i th row and i th column, and σ k ( A ∣ i j ) the symmetric function with A deleting the i th, j th rows and i th, j th columns, for all 1 ≤ i , j ≤ n . In local coordinates,

X i j ¯ = X ∂ ∂ z i , ∂ ∂ z ¯ i = χ i j ¯ + u i j ¯ , X ̲ i j ¯ = χ i j ¯ + u ̲ i j ¯ .

Define λ ( χ u ) as the eigenvalue set of { X i j ¯ } with respect to { g i j ¯ } . In local coordinates, equation (1.1) can be written in the following form:

(2.1) σ k ( λ ( χ u ) ) = ∑ l = 0 k − 1 β l ( x ) σ l ( λ ( χ u ) ) ,

where

σ l ( λ ( χ u ) ) C n l = χ u l ∧ ω n − l ω n , β l ( x ) = C n k C n l α l ( x ) .

Equivalently, we can rewrite equation (2.1) as follows:

(2.2) σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = β k − 1 ( x ) .

Lemma 2.1

[29,30] For λ ∈ Γ m and m > l ≥ 0 , r > s ≥ 0 , m ≥ r , l ≥ s , we have

σ m ( λ ) / C n m σ l ( λ ) / C n l 1 m − l ≤ σ r ( λ ) / C n r σ s ( λ ) / C n s 1 r − s .

The following lemma is similar to Lemma 2.3 in [27], but we need to discuss it more widely, that is, λ ( χ u ) ∈ Γ k − 1 instead of λ ( χ u ) ∈ Γ k .

Lemma 2.2

If u ∈ C 2 ( M ) is a solution of (2.2), λ ( χ u ) ∈ Γ k − 1 and β l ( x ) > 0 , 0 ≤ l ≤ k − 2 , then

(2.3) σ l ( λ ) σ k − 1 ( λ ) ≤ C ( n , k , inf 0 ≤ l ≤ k − 2 β l , sup ∣ β k − 1 ∣ ) for 0 ≤ l ≤ k − 2 .

Proof

If σ k σ k − 1 ≤ 1 , then we obtain from equation (2.2)

β l ( x ) σ l σ k − 1 ≤ σ k σ k − 1 − β ( x ) ≤ 1 − β ( x ) ≤ C ( sup M ∣ β k − 1 ∣ ) , for 0 ≤ l ≤ k − 2 .

If σ k σ k − 1 > 1 , i.e., σ k − 1 σ k < 1 , we see from Lemma 2.1 that

σ l σ k − 1 ≤ ( C n k ) k − 1 − l C n l ( C n k − 1 ) k − l σ k − 1 σ k k − 1 − l ≤ ( C n k ) k − 1 − l C n l ( C n k − 1 ) k − l ≤ C ( n , k )

for 0 ≤ l ≤ k − 2 , which completes the proof of Lemma 2.2.□

For any point x 0 ∈ M , choose a local frame such that X i j ¯ = δ i j X i i ¯ . For the convenience of notations, we will write equation (2.2) as follows:

(2.4) F ( X ) = F k ( X ) + ∑ l = 0 k − 2 β l F l ( X ) = β k − 1 ( x ) ,

where F k ( X ) = σ k ( λ ( X ) ) σ k − 1 ( λ ( X ) ) and F l ( X ) = − σ l ( λ ( X ) ) σ k − 1 ( λ ( X ) ) . Let

F i j ¯ ≔ ∂ F ∂ X i j ¯ = ∂ F ∂ λ k ∂ λ k ∂ X i j ¯ ,

then at x 0 , we have

F i j ¯ = F i i ¯ δ i j .

Let

ℱ ≔ ∑ i F i i ¯ .

Lemma 2.3

[23] If λ ∈ Γ k − 1 and α l ( x ) > 0 , 0 ≤ l ≤ k − 2 , then the operator F is elliptic and concave in Γ k − 1 .

From Lemma 2.4 in [27] and Lemma 2.2, we have the following lemma.

Lemma 2.4

If u ∈ C 2 ( M ) is a solution of (2.4), λ ( χ u ) ∈ Γ k − 1 , then at x 0 ,

(2.5) n − k + 1 k ≤ ℱ ≤ C ( n , k , inf 0 ≤ l ≤ k − 2 α l , sup ∣ α k − 1 ∣ ) .

Lemma 2.5

Under assumptions of Theorem 1.2, there is a constant θ > 0 such that

(2.6) F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) ≤ − θ ( 1 + ℱ ) ,

or

(2.7) F 1 1 ¯ ≥ θ .

Proof

Without loss of generality, we may assume that X 1 1 ¯ ≥ ⋯ ≥ X n n ¯ . Thus,

F n n ¯ ≥ ⋯ ≥ F 1 1 ¯ .

Since u ̲ is a C -subsolution, if ε > 0 is small enough, χ u ̲ − ε ω still satisfies Definition 1.1. Since M is compact, there are uniform constants N > 0 and δ > 0 such that

(2.8) F ( X ˜ ) > β k − 1 + δ ,

where

X ˜ = X ̲ − ε g + N 0 ⋯ 0 0 0 ⋯ 0 0 0 ⋯ 0 n × n .

Direct calculation yields

(2.9) F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) = F i i ¯ ( X i i ¯ − X ̲ i i ¯ ) = F i i ¯ ( X i i ¯ − X ˜ i i ¯ ) + F 1 1 ¯ N − ε ℱ .

Since F is concave in Γ k − 1 , from (2.8), we obtain

(2.10) ∑ i = 1 n F i i ¯ ( X i i ¯ − X ˜ i i ¯ ) ≤ F ( X ) − F ( X ˜ ) ≤ − δ .

By substituting (2.10) into (2.9), we obtain

F i i ¯ ( u i i ¯ − u ̲ i i ¯ ) ≤ − δ + F 1 1 ¯ N − ε ℱ .

Set θ = min δ 2 , ε , δ 2 N . If F 1 1 ¯ N ≤ δ 2 , we have (2.6); otherwise, (2.7) must be true.□

3 C 0 estimate

In this section, we obtain the C 0 estimate by using the Alexandroff-Bakelman-Pucci maximum principle and prove the following Proposition 3.1, which is similar to the approach of Székelyhidi [12].

Proposition 3.1

Let α l ( x ) > 0 for 0 ≤ l ≤ k − 2 and χ be a smooth real ( 1 , 1 ) form on ( M , g ) . Assume that u and u ̲ are solution and C -subsolution to (1.1) with λ ( χ u ) ∈ Γ k − 1 , λ ( χ u ̲ ) ∈ Γ k − 1 , respectively. We normalize u such that sup M ( u − u ̲ ) = 0 . There is a constant C depending on the given data, such that

(3.1) sup M ∣ u ∣ < C .

Proof

To simplify notation, we can assume u ̲ = 0 , otherwise we modify the background form χ . Therefore, sup M u = 0 . The following goal is to prove that L = inf M u has a uniform lower bound. Notice that λ ( χ u ) ∈ Γ k − 1 , so λ ( χ u ) ∈ Γ 1 , that is,

Δ u = g i p ¯ u i p ¯ > − g i p ¯ χ i p ¯ ≥ − C ^ .

Let G : M × M → R be the Green’s function of a Gauduchon metric conformal to g . From [1], there is a uniform constant K such that

G ( x , y ) + K ≥ 0 , ∀ ( x , y ) ∈ M × M , and ∫ y ∈ M G ( x , y ) ω n ( y ) = 0 .

Since sup M u = 0 , there is a point x 0 ∈ M such that u ( x 0 ) = 0 . Hence,

u ( x 0 ) = ∫ M u d μ − ∫ y ∈ M G ( x 0 , y ) Δ u ( y ) ω n ( y ) = ∫ M u d μ − ∫ y ∈ M ( G ( x 0 , y ) + K ) Δ u ( y ) ω n ( y ) ≤ ∫ M u d μ + C ^ K ,

which yields

∫ M ∣ u ∣ d μ ≤ C ^ K .

Next, we choose local coordinates at the minimum point of u and L = inf M u = u ( 0 ) . Let B ( 1 ) = { z : ∣ z ∣ < 1 } and v = u + ε ∣ z ∣ 2 for a small ε > 0 . From the Alexandroff-Bakelman-Pucci maximum principle, we obtain

(3.2) c 0 ε 2 n ≤ ∫ Ω det ( D 2 v ) ,

where

Ω = x ∈ B ( 1 ) : ∣ D v ( x ) ∣ < ε 2 , v ( y ) ≥ v ( x ) + D v ( x ) ⋅ ( y − x ) , ∀ y ∈ B ( 1 ) .

Let λ ˜ − λ ( χ u ̲ ) ≥ 0 and

σ k ( λ ˜ ) σ k − 1 ( λ ˜ ) − ∑ l = 0 k − 2 β l σ l ( λ ˜ ) σ k − 1 ( λ ˜ ) = β k − 1 ( x ) .

u ̲ is a C subsolution, which means that ∣ λ ˜ ∣ is bounded. Since M is compact, there is uniform constant η > 0 such that λ ( χ u ̲ ) − η 1 satisfies Definition 1.1. Since Ω is a contact set, we have D 2 v ( x ) ≥ 0 , for x ∈ Ω , which implies u i j ¯ ( x ) + ε δ i j ≥ 0 . Choosing ε such that 0 < ε ≤ η , on Ω , we have

λ ( χ u ) − ( λ ( χ u ̲ ) − η 1 ) ≥ λ ( χ u ) − ( λ ( χ u ̲ ) − ε 1 ) = λ ( u i j ¯ ) + ε 1 ≥ 0 .

Since

σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = β k − 1 ( x ) ,

we obtain that ∣ λ ( χ u ) ∣ is bounded, which yields ∣ u i j ¯ ∣ ≤ C . Then

det ( D 2 v ( x ) ) ≤ 2 2 n det ( v i j ¯ ) 2 ≤ C .

From this and (3.2), we obtain

(3.3) c 0 ε 2 n ≤ ∫ Ω det ( D 2 v ) ≤ C ⋅ vol ( Ω ) .

On the other hand, we have for x ∈ Ω

v ( 0 ) ≥ v ( x ) − D v ( x ) ⋅ x > v ( x ) − ε 2 ,

so

∣ v ( x ) ∣ > ∣ L + ε 2 ∣ .

Then,

∫ M ∣ v ( x ) ∣ ≥ ∫ Ω ∣ v ( x ) ∣ ≥ ∣ L + ε 2 ∣ ⋅ vol ( Ω ) .

Since ∫ M ∣ v ( x ) ∣ is uniformly bounded, this inequality contradicts (3.3) if L is very large.□

4 C 2 estimate

In this section, we establish the C 2 estimate to equation (1.1). Our calculation is similar to that in [27], but on Hermitian manifolds, equation (1.1) are much more difficult to treat due to the torsion terms.

4.1 Notations and lemma

In local coordinates z = ( z 1 , … , z n ) , the Chern connection ∇ and torsion are given, respectively, by

∇ ∂ ∂ z i ∂ ∂ z j = Γ i j k ∂ ∂ z k , Γ i j k = g k l ¯ ∂ g j l ¯ ∂ z i , T i j k = Γ i j k − Γ j i k ,

while the curvature tensor R i j ¯ k l ¯ by

R i j ¯ k l ¯ = g p l ¯ ∂ Γ i k p ∂ z ¯ j .

For u ∈ C 4 ( M ) , we denote

u i j = ∇ j ∇ i u , u i j ¯ = ∇ j ¯ ∇ i u .

We have (see [4,11,31])

(4.1) u i j ¯ l = u l j ¯ i + T i l p u p j ¯ , u i j ¯ k = u i k j ¯ − g l m ¯ R k j ¯ i m ¯ u l , u i j ¯ k ¯ = u i k ¯ j ¯ + T j k l ¯ u i l ¯ , u i j ¯ k = u k i j ¯ − g l m ¯ R i j ¯ k m ¯ u l + T i k l u l j ¯ ,

and

(4.2) u i j ¯ k l ¯ = u k l ¯ i j ¯ + g p q ¯ ( R k l ¯ i q ¯ u p j ¯ − R i j ¯ k q ¯ u p l ¯ ) + T i k p u p j ¯ l ¯ + T j l q ¯ u i q ¯ k − T i k p T j l q ¯ u p q ¯ .

Let A i j ¯ = g j p ¯ X i p ¯ , λ ( A ) = ( λ 1 , … , λ n ) and λ 1 ≥ ⋯ ≥ λ n . For a fixed point x 0 ∈ M , choose a local coordinates such that A i j ¯ = A i i ¯ δ i j . Since λ 1 , … , λ n need not be distinct at x 0 , we will perturb χ u slightly such that λ 1 , … , λ n become smooth functions near x 0 . Let D be a diagonal matrix such that D 11 = 0 and 0 < D 22 < ⋯ < D n n are small, satisfying D n n < 2 D 22 . Define the matrix A ˜ = A − D . At x 0 , A ˜ has eigenvalues

λ ˜ 1 = λ 1 , λ ˜ i = λ i − D i i , i ≥ 2 .

Lemma 4.1

(4.3) λ ˜ 1 , i i ¯ ≥ X i i ¯ 1 1 ¯ + 2 Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 λ 1 − C 0 .

Proof

Commuting derivative of λ 1 ˜ gives

λ ˜ 1 , i = ∂ λ ˜ 1 ∂ A ˜ p q ¯ ∂ A ˜ p q ¯ ∂ z i = X 1 1 ¯ i − ( D 11 ) i ,

(4.4) λ ˜ 1 , i i ¯ = ∂ 2 λ ˜ 1 ∂ A ˜ r s ¯ ∂ A ˜ p q ¯ ∂ A ˜ p q ¯ ∂ z i ∂ A ˜ r s ¯ ∂ z ¯ i + ∂ λ ˜ 1 ∂ A ˜ p q ¯ ∂ 2 A ˜ p q ¯ ∂ z ˜ i ∂ z i = ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 λ 1 − λ ˜ p − 2 ∑ p ≥ 2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) + Re ( ( D p 1 ) i ¯ X 1 p ¯ i ) λ 1 − λ ˜ p + ∑ p ≥ 2 ( D 1 p ) i ( D p 1 ) i ¯ + ( D p 1 ) i ( D 1 p ) i ¯ λ 1 − λ ˜ p + X 1 1 ¯ i i ¯ + ( D 11 ) i i ¯ .

λ ( A ) ∈ Γ 1 implies that ∣ λ p ∣ < ( n − 1 ) λ 1 , p ≥ 2 . If the matrix D is sufficiently small, then ∣ λ ˜ p ∣ < ( n − 1 ) λ 1 , p ≥ 2 , which means that

1 n λ 1 ≤ 1 λ 1 − λ ˜ p ≤ 1 D p p .

We are trying to bound λ 1 from mentioned earlier, so we can assume λ 1 > 1 . Hence,

(4.5) ∑ p ≥ 2 ( D 1 p ) i ( D p 1 ) i ¯ + ( D p 1 ) i ( D 1 p ) i ¯ λ 1 − λ ˜ p + ( D 11 ) i i ¯ ≥ − C 0 .

From here on, C 0 will always denote such a constant, which depends on the given data and may vary from line to line. Using

2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) ≤ 1 2 ∣ X p 1 ¯ i ∣ 2 + C 0 ,

we have

(4.6) ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 λ 1 − λ ˜ p − 2 ∑ p ≥ 2 Re ( ( D 1 p ) i ¯ X p 1 ¯ i ) + Re ( ( D p 1 ) i ¯ X 1 p ¯ i ) λ 1 − λ ˜ p ≥ 1 2 n λ 1 ∑ p ≥ 2 ( ∣ X 1 p ¯ i ∣ 2 + ∣ X p 1 ¯ i ∣ 2 ) − C 0 .

From (4.2), we obtain

(4.7) u 1 1 ¯ i i ¯ = u i i ¯ 1 1 ¯ + R i i ¯ 1 p ¯ u p 1 ¯ − R 1 1 ¯ i p ¯ u p i ¯ + T 1 i p u p 1 ¯ i ¯ + T 1 i p ¯ u 1 p ¯ i − T 1 i p T 1 i q ¯ u p q ¯ .

From this, we have

(4.8) X 1 1 ¯ i i ¯ = X i i ¯ 1 1 ¯ + χ 1 1 ¯ i i ¯ − χ i i ¯ 1 1 ¯ + R i i ¯ 1 p ¯ u p 1 ¯ − R 1 1 ¯ i p ¯ u p i ¯ + T 1 i p u p 1 ¯ i ¯ + T 1 i p ¯ u 1 p ¯ i − T 1 i p T 1 i q ¯ u p q ¯ ≥ X i i ¯ 1 1 ¯ + λ 1 R i i ¯ 1 1 ¯ − λ i R 1 1 ¯ i i ¯ + 2 Re ( X 1 p ¯ i T 1 i p ¯ ) − λ p ∣ T 1 i p ∣ 2 − C 0 ≥ X i i ¯ 1 1 ¯ + 2 Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − 1 2 n λ 1 ∑ p ≥ 2 ∣ X 1 p ¯ i ∣ 2 − C 0 λ 1 − C 0 .

Substituting (4.5), (4.6), and (4.8) into (4.4) gives (4.3).□

4.2 C 2 estimate

Proposition 4.2

Let α l ( x ) > 0 for 0 ≤ l ≤ k − 2 and χ be a smooth real ( 1 , 1 ) form on ( M , g ) . Assume that u and u ̲ are solution and C -subsolution to (1.1) with λ ( χ u ) ∈ Γ k − 1 , λ ( χ u ̲ ) ∈ Γ k − 1 , respectively. Then there is an estimate as follows:

sup M ∣ ∂ ∂ ¯ u ∣ ≤ C ( sup M ∣ ∇ u ∣ 2 + 1 ) ,

where C is a uniform constant.

Proof

We assume that the C subsolution u ̲ = 0 , since otherwise we modify the background form χ . We normalize u so that sup M u = 0 . Consider the function

(4.9) W = log λ ˜ 1 + φ ( ∣ ∇ u ∣ 2 ) + ψ ( u ) .

Here,

φ ( t ) = − 1 2 log 1 − t 2 K , 0 ≤ t ≤ K − 1 , ψ ( t ) = − E log 1 + t 2 L , − L + 1 ≤ t ≤ 0 ,

where

K = sup M ∣ ∇ u ∣ 2 + 1 , L = sup M ∣ u ∣ + 1 , E = 2 L ( C 1 + 1 ) ,

and C 1 is to be determined later. Direct calculation gives

(4.10) 0 < 1 4 K ≤ φ ′ ≤ 1 2 K , φ ″ = 2 ( φ ′ ) 2 ,

and

(4.11) C 1 + 1 ≤ − ψ ′ ≤ 2 ( C 1 + 1 ) , ψ ″ ≥ 4 ε 1 − ε ( ψ ′ ) 2 , for all ε ≤ 1 4 E + 1 .

Since M is compact, W attains its maximum at some point x 0 ∈ M . From now on, all the calculations will be carried out at the point x 0 and the Einstein summation convention will be used. Calculating covariant derivatives, we obtain

(4.12) 0 = W i = X 1 1 ¯ i λ 1 + φ ′ ( ∣ ∇ u ∣ 2 ) i + ψ ′ u i − ( D 11 ) i λ 1 , 1 ≤ i ≤ n ,

(4.13) 0 ≥ W i i ¯ = λ ˜ 1 , i i ¯ λ 1 − λ ˜ 1 , i λ ˜ 1 , i ¯ λ 1 2 + ψ ′ u i i ¯ + ψ ″ ∣ u i ∣ 2 + φ ′ ( ∣ ∇ u ∣ 2 ) i i ¯ + φ ″ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 .

Multiplying (4.13) by F i i ¯ and summing it over index i yield

(4.14) 0 ≥ F i i ¯ λ ˜ 1 , i i ¯ λ 1 − F i i ¯ ∣ λ ˜ 1 , i ∣ 2 λ 1 2 + ψ ′ F i i ¯ u i i ¯ + ψ ″ F i i ¯ ∣ u i ∣ 2 + φ ′ F i i ¯ ( ∣ ∇ u ∣ 2 ) i i ¯ + φ ″ F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 .

We will control some terms in (4.14). Covariant differentiating equation (2.4) twice in the ∂ ∂ z 1 direction and the ∂ ∂ z ¯ 1 direction, we have

(4.15) F i i ¯ X i i ¯ 1 + ∑ l = 0 k − 2 ( β l ) 1 F l = ( β k − 1 ) 1

and

(4.16) F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ + F i i X i i ¯ 1 1 ¯ + 2 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 + ∑ l = 0 k − 2 ( β l ) 1 1 ¯ F l = ( β k − 1 ) 1 1 ¯ .

Direct calculation deduces that

(4.17) F i i ¯ X i i ¯ = F i i ¯ λ i = F k i i ¯ λ i + ∑ l = 0 k − 2 β l F l i i ¯ λ i = β k − 1 − ∑ l = 0 k − 2 ( k − l ) β l F l .

From Lemmas 4.1 and (4.16), we can estimate the first term in (4.14)

(4.18) F i i ¯ λ ˜ 1 , i i ¯ λ 1 ≥ 1 λ 1 F i i ¯ X i i ¯ 1 1 ¯ + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ = − 1 λ 1 F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 − 1 λ 1 ∑ l = 0 k − 2 ( β l ) 1 1 ¯ F l + ( β k − 1 ) 1 1 ¯ λ 1 + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ .

It is shown by Krylov in [22] that the σ k − 1 σ l 1 k − l − 1 is concave in Γ k − 1 for 0 ≤ l ≤ k − 2 , which means that

( − F l ) − 1 k − l − 1 i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ ≤ 0 .

Direct computation gives

− F l i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ≥ k − l k − l − 1 ( − F l ) − 1 ∣ F l i i ¯ X i i ¯ 1 ∣ 2 ,

which yields

− F i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ λ 1 − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 ≥ ∑ l = 0 k − 2 k − l k − l − 1 β l λ 1 ( − F l ) − 1 F l i i ¯ X i i ¯ 1 + k − l − 1 k − l ( β l ) 1 ¯ β l F l 2 + ∑ l = 0 k − 2 k − l − 1 k − l ( β l ) 1 2 β l λ 1 F l

≥ ∑ l = 0 k − 2 k − l − 1 k − l ( β l ) 1 2 β l λ 1 F l ≥ − C 0 ,

where the last inequality is given by Lemma 2.2. Noting that

− F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ ≥ − F i i ¯ , j j ¯ X i i ¯ 1 X j j ¯ 1 ¯ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 ,

we have

(4.19) − 1 λ 1 F i j ¯ , p q ¯ X i j ¯ 1 X p q ¯ 1 ¯ − 2 λ 1 Re ∑ l = 0 k − 2 ( β l ) 1 ¯ F l i i ¯ X i i ¯ 1 ≥ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 − C 0 .

Substituting (4.19) into (4.18) and by Lemma 2.2,

(4.20) F i i ¯ λ ˜ 1 , i i ¯ λ 1 ≥ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 + 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 ℱ − C 0 .

Since

(4.21) X 1 1 ¯ i = χ 1 1 ¯ i + u 1 1 ¯ i = ( χ 11 i − χ i 11 + T i 1 p χ p 1 ¯ ) + X i 1 ¯ 1 − T i 1 1 λ 1 ,

we have

(4.22) ∣ X 1 1 ¯ i ∣ 2 ≤ ∣ X i 1 ¯ 1 ∣ 2 − 2 λ 1 Re ( X i 1 ¯ 1 T i 1 1 ¯ ) + C 0 ( λ 1 2 + ∣ X 1 1 ¯ i ∣ ) .

From

λ ˜ 1 , i = X 1 1 ¯ i − ( D 11 ) i ,

we estimate the second term in (4.14)

(4.23) − F i i ¯ ∣ λ ˜ 1 , i ∣ 2 λ 1 2 = − F i i ¯ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 + 2 λ 1 2 F i i ¯ Re ( X 1 1 ¯ i ( D 11 ) i ¯ ) − F i i ¯ ∣ ( D 11 ) i ∣ 2 λ i 2 ≥ − F i i ¯ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 − C 0 λ 1 2 F i i ¯ ∣ X 1 1 ¯ i ∣ − C 0 ℱ ≥ − F i i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 2 − 2 λ 1 F i i ¯ Re ( X 1 1 ¯ i T 1 i 1 ¯ ) − C 0 λ 1 2 F i i ¯ ∣ X 1 1 ¯ i ∣ − C 0 ℱ ,

where the last inequality is given by (4.21) and (4.22). By (4.1), we have the identities

(4.24) u p i i ¯ = u i i ¯ p − T i p i λ i + T i p q χ q i ¯ + R i i ¯ p q ¯ u q ,

(4.25) u p ¯ i i ¯ = u i i ¯ p ¯ − T i p i ¯ λ i + T i p q ¯ χ i q ¯ .

It follows from (4.24) and (4.25) that

(4.26) F i i ¯ u p i i ¯ u p ¯ = F i i ¯ u i i ¯ p u p ¯ − F i i ¯ T i p i λ i u p ¯ + F i i ¯ T i p q χ q i ¯ u p ¯ + F i i ¯ R i i ¯ p q ¯ u q u p ¯ = − F i i ¯ χ i i ¯ p u p ¯ − ∑ l = 0 k − 2 ( β l ) p F l u p ¯ + ( β k − 1 ) p u p ¯ − F i i ¯ T i p i λ i u p ¯ + F i i ¯ T i p q χ q i ¯ u p ¯ + F i i ¯ R i i ¯ p q ¯ u q u p ¯ ≥ − C 0 K 1 2 ℱ + K 1 2 + K 1 2 + K 1 2 ℱ + K ℱ − C 0 K 1 2 F i i ¯ λ i ,

where the second equality is given by (4.15) and the last inequality given by Lemma 2.2. From (4.16) and K 1 2 ≤ 1 4 + K , we obtain

(4.27) − C 0 K 1 2 F i i ¯ λ i = − C 0 K 1 2 β k − 1 − ∑ l = 0 k − 2 ( k − l ) β l F l ≥ − C 0 − C 0 K .

Noting φ ′ ≥ 1 4 K , we substitute (4.27) into (4.26),

(4.28) φ ′ F i i ¯ u p i i ¯ u p ¯ ≥ − C 0 ℱ − C 0 K ℱ − C 0 − C 0 K .

The same estimate also holds for φ ′ F i i ¯ u p ¯ i i ¯ u p . From (4.28), we can estimate the fifth term in (4.14)

(4.29) φ ′ F i i ¯ ( ∣ ∇ u ∣ 2 ) i i ¯ = φ ′ F i i ¯ ( u p i i ¯ u p ¯ + u p ¯ i i ¯ u p ) + φ ′ F i i ¯ ∑ p ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) ≥ − C 0 ℱ − C 0 K ℱ − C 0 − C 0 K + 1 4 K F i i ¯ ∑ p ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) .

Substituting (4.20), (4.23), and (4.29) into (4.14)

(4.30) 0 ≥ − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 − F i i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 2 − C 0 λ 1 F i i ¯ ∣ X 1 1 ¯ i ∣ λ 1 + 1 4 K F i i ¯ ∑ p ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) + ψ ′ F i i ¯ u i i ¯ + ψ ″ F i i ¯ ∣ u i ∣ 2 + φ ″ F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − C 0 ℱ − C 0 − C 0 K ℱ − C 0 K .

We set

(4.31) δ = min 1 1 + 4 E , 1 2 ,

where

1 1 + 4 E = 1 1 + 8 L ( C 1 + 1 ) , C 1 = 1 + 1 K C 0 θ ,

with θ in Lemma 2.6. Then we have two cases to consider.

Case 1 λ n < − δ λ 1 . By using the critical point condition (4.12), we obtain

(4.33) − F i i ¯ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 = − F i i ¯ ∣ φ ′ ( ∣ ∇ u ∣ 2 ) i + ψ ′ u i − ( D 11 ) i λ 1 ∣ 2 ≥ − 2 ( φ ′ ) 2 F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 2 F i i ¯ ∣ ψ ′ u i − ( D 11 ) i λ 1 ∣ 2 ≥ − 2 ( φ ′ ) 2 F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 4 ∣ ψ ′ ∣ 2 K ℱ − C 0 ℱ .

It follows from (4.17) that

(4.34) ψ ′ F i i ¯ u i i = ψ ′ F i i ¯ ( X i i − χ i i ) ≥ ψ ′ β k − 1 − ∑ l = 0 k − 2 ( k − l ) β l F l − C 0 ℱ ≥ ψ ′ ( 1 + ℱ ) C 0 .

By the fact that

∣ X 1 1 ¯ i ∣ λ 1 = − φ ′ ( u p i u p ¯ + u p u p ¯ i ) − ψ ′ u i + ( D 11 ) i λ 1 ,

we have

(4.35) − C 0 λ 1 F i i ¯ ∣ X 1 1 ¯ i ∣ λ 1 ≥ − C 0 λ 1 K − 1 2 F i i ¯ ( ∣ u p i ∣ + ∣ u p ¯ i ∣ ) + C 0 λ 1 ψ ′ K 1 2 ℱ − C 0 ℱ .

Since

− F i 1 ¯ , 1 ¯ i = F i i ¯ − F 1 1 ¯ λ 1 − λ i and λ 1 ≥ ⋯ ≥ λ n ,

we have

(4.36) − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 ≥ 0 .

Since φ ″ = 2 ( φ ′ ) 2 and ψ ″ > 0 , substituting (4.32), (4.33), (4.34), and (4.35) into (4.30), we obtain

(4.36) 0 ≥ 1 4 K F i i ¯ ∑ p ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) − C 0 λ 1 K 1 2 F i i ¯ ∑ p ( ∣ u p i ∣ + ∣ u p ¯ i ∣ ) + C 0 λ 1 K 1 2 ψ ′ ℱ + ψ ′ C 0 ( 1 + ℱ ) − 4 ( ψ ′ ) 2 K ℱ − C 0 ( 1 + ℱ ) 1 + 1 K ≥ 1 8 K F i i ¯ ∑ p ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) + C 0 λ 1 K 1 2 ψ ′ ℱ + ψ ′ C 0 ( 1 + ℱ ) − 4 ( ψ ′ ) 2 K ℱ − C 0 ( 1 + ℱ ) 1 + 1 K ,

where the last inequality is obtained by using the first term absorbing the ∣ u p i ∣ , ∣ u p ¯ i ∣ terms.

From Lemma 2.4, we know that ℱ is controlled by the uniform positive constant, which means that ℱ can be absorbed by C 0 . Noticing that

F i i ¯ ∣ u i i ¯ ∣ 2 = F i i ¯ ( λ i − χ i i ¯ ) 2 ≥ 1 2 F i i ¯ λ i 2 − C 0 ℱ , and F n n ¯ ≥ ℱ n ≥ n − k + 1 n k ,

we have

0 ≥ 1 16 K F i i ¯ λ i 2 − C 0 K 1 2 ( 1 + C 1 ) − C 0 ( 1 + C 1 ) − C 0 ( 1 + C 1 ) 2 K − C 0 1 + 1 K ≥ ( n − k + 1 ) δ 2 16 n k K λ 1 2 − C 0 ( 1 + C 1 ) 2 K − C 0 1 + 1 K .

This inequality implies λ 1 ≤ C K .

Case 2 λ n ≥ − δ λ 1 . Let

I = { i ∈ { 1 , … , n } ∣ F i i ¯ > δ − 1 F 1 1 ¯ } .

For those indices, which are not in I , we have

(4.37) − ∑ i ∉ I F i i ¯ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 = − ∑ i ∉ I F i i ¯ φ ′ ( ∣ ∇ u ∣ 2 ) i + ψ ′ u i − ( D 11 ) i λ 1 2 ≥ − 2 ( φ ′ ) 2 ∑ i ∉ I F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 2 ∑ i ∉ I F i i ¯ ∣ ψ ′ u i − ( D 11 ) i λ 1 ∣ 2 ≥ − 2 ( φ ′ ) 2 ∑ i ∉ I F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 4 K δ ∣ ψ ′ ∣ 2 F 1 1 ¯ − C 0 ℱ .

From (4.12), (4.37), and

∣ X i 1 ¯ 1 ∣ 2 λ 1 2 ≤ ∣ X 1 1 ¯ i ∣ 2 λ 1 2 + C 0 1 + ∣ X 1 1 ¯ i ∣ λ 1 ,

we obtain

(4.38) − ∑ i ∉ I F i i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 2 ≥ − 2 ( φ ′ ) 2 ∑ i ∉ I F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 4 K δ ∣ ψ ′ ∣ 2 F 1 1 ¯ + C 0 φ ′ ∑ i ∉ I F i i ¯ ( ∣ ∇ u ∣ 2 ) i + C 0 ψ ′ ∑ i ∉ I F i i ¯ u i − C 0 ℱ ≥ − 2 ( φ ′ ) 2 ∑ i ∉ I F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 − 4 K δ ∣ ψ ′ ∣ 2 F 1 1 ¯ − C 0 φ ′ K 1 2 F i i ¯ ∑ p ( ∣ u p i ∣ + ∣ u p ¯ i ∣ ) + C 0 K 1 2 ψ ′ δ − 1 F 1 1 ¯ − C 0 ℱ .

Since

− F i 1 ¯ , 1 ¯ i = F i i ¯ − F 1 1 ¯ X 1 1 ¯ − X i i ¯ and λ i ≥ λ n ≥ − δ λ 1 ,

we have

(4.39) − ∑ i ∈ I F i 1 ¯ , 1 ¯ i ≥ 1 − δ 1 + δ 1 λ 1 ∑ i ∈ I F i i ¯ ,

which yields

(4.40) − F i 1 ¯ , 1 i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 ≥ 1 − δ 1 + δ ∑ i ∈ I F i i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 2 .

Recalling that φ ″ = 2 ( φ ′ ) 2 and 0 < δ ≤ 1 2 , we obtain from (4.12)

(4.41) ∑ i ∈ I φ ″ F i i ¯ ∣ ( ∣ ∇ u ∣ 2 ) i ∣ 2 = 2 ∑ i ∈ I F i i ¯ X i 1 ¯ 1 λ 1 + ψ ′ u i − ( D 11 ) i λ 1 − T i 1 1 + χ 11 i − χ i 11 + T i 1 p χ p 1 ¯ λ 1 2 ≥ 2 ∑ i ∈ I F i i ¯ δ X i 1 ¯ 1 λ 1 2 − 2 δ 1 − δ ( ψ ′ ) 2 ∣ u i ∣ 2 − C 0 ≥ 2 δ ∑ i ∈ I F i i ¯ X i 1 ¯ 1 λ 1 2 − 4 δ 1 − δ ( ψ ′ ) 2 F i i ¯ ∣ u i ∣ 2 − C 0 ℱ .

Noticing the fact that ψ ″ ≥ 4 ε 1 − ε ( ψ ′ ) 2 , for all ε ≤ 1 4 E + 1 = δ , we have

(4.42) ψ ″ F i i ¯ ∣ u i ∣ 2 − 4 δ 1 − δ ( ψ ′ ) 2 F i i ¯ ∣ u i ∣ 2 ≥ 0 .

Inserting (4.38), (4.40), (4.41), and (4.42) into (4.30), we deduce that

(4.43) 0 ≥ 1 4 K ∑ p F i i ¯ ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) + 1 − δ 1 + δ + 2 δ − 1 ∑ i ∈ I F i i ¯ ∣ X i 1 ¯ 1 ∣ 2 λ 1 2 + ψ ′ F i i ¯ u i i ¯ − 4 K δ ( ψ ′ ) 2 F 1 1 ¯ + C 0 K 1 2 ψ ′ δ − 1 F 1 1 ¯ − C 0 λ 1 F i i ¯ ∣ X 1 1 ¯ i ∣ λ 1 − 1 + 1 K C 0 ℱ − C 0 − C 0 K − C 0 K − 1 2 F i i ¯ ∑ p ( ∣ u p i ∣ + ∣ u p ¯ i ∣ ) ≥ 1 8 K ∑ p F i i ¯ ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) + ψ ′ F i i ¯ u i i ¯ − C 0 λ 1 F i i ¯ ∣ X 1 1 ¯ i ∣ λ 1 − 4 K δ ( ψ ′ ) 2 F 1 1 ¯ + C 0 K 1 2 ψ ′ δ − 1 F 1 1 ¯ − 1 + 1 K C 0 ℱ − C 0 − C 0 K ,

where the last inequality is given by using the first term absorbing the ∣ u p i ∣ , ∣ u p ¯ i ∣ terms. Combining (4.34) with (4.43), we obtain

(4.44) 0 ≥ 1 16 K ∑ p F i i ¯ ( ∣ u p i ∣ 2 + ∣ u p ¯ i ∣ 2 ) + ψ ′ F i i ¯ u i i ¯ − 4 K δ ( ψ ′ ) 2 F 1 1 ¯ + C 0 K 1 2 ψ ′ δ − 1 F 1 1 ¯ + C 0 λ 1 ψ ′ K 1 2 ℱ − 1 + 1 K C 0 ℱ − C 0 − C 0 K .

From Lemma 2.4, we know that ℱ is controlled by the uniform positive constant, which means that ℱ can be absorbed by C 0 . Noticing that

F i i ¯ ∣ u i i ¯ ∣ 2 = F i i ¯ ( λ i − χ i i ¯ ) 2 ≥ 1 2 F i i ¯ λ i 2 − C 0 ℱ ,

we obtain

(4.45) 0 ≥ 1 32 K ∑ p F i i ¯ λ i 2 + ψ ′ F i i ¯ u i i ¯ − 4 K δ ( ψ ′ ) 2 F 1 1 ¯ + C 0 K 1 2 ψ ′ δ − 1 F 1 1 ¯ + C 0 λ 1 ψ ′ K 1 2 − 1 + 1 K C 0 .

We may assume λ 1 ≥ 2 ( C 1 + 1 ) K , then

C 0 λ 1 ψ ′ K 1 2 ≥ − C 0 K − 1 2 ≥ − 1 + 1 K C 0 .

There are two cases to consider from Lemma 2.5.

If (2.6) holds, then we obtain ψ ′ F i i ¯ u i i ¯ ≥ ( C 1 + 1 ) θ ( 1 + ℱ ) ≥ ( C 1 + 1 ) θ . Substituting this into (4.45) yields

0 ≥ 1 32 K F 1 1 ¯ λ 1 2 + ( C 1 + 1 ) θ − 16 ( C 1 + 1 ) 2 K δ F 1 1 ¯ − C 0 ( C 1 + 1 ) K 1 2 δ F 1 1 ¯ − 1 + 1 K C 0 .

Recall that

C 1 = 1 + 1 K C 0 θ .

We then obtain

0 ≥ 1 32 K λ 1 2 − 16 ( C 1 + 1 ) 2 K δ − C 0 ( C 1 + 1 ) K 1 2 δ ,

which implies λ 1 ≤ C K .

If (2.7) holds, then, by (4.17),

0 ≥ 1 32 K λ 1 2 − 16 ( C 1 + 1 ) 2 K δ − C 0 ( C 1 + 1 ) K 1 2 δ − θ 1 + 1 K C 0 − θ ( C 1 + 1 ) C 0 .

This inequality again implies λ 1 ≤ C K .□

5 C 1 estimates

In this section, we obtain the following gradient estimate by the blowup method and the Liouville theorem as suggested by Dinew and Kolodziej [10]. The argument follows closely Proposition 5.1 in [27], so we omit the proof.

Proposition 5.1

Let α l ( x ) > 0 for 0 ≤ l ≤ k − 2 and χ be a smooth real ( 1 , 1 ) -form on ( M , g ) . Assume that u and u ̲ are solution and C -subsolution to (1.1) with λ ( χ u ) ∈ Γ k − 1 , λ ( χ u ̲ ) ∈ Γ k − 1 , respectively. We normalize u such that sup M ( u − u ̲ ) = 0 . Then there is an estimate

sup M ∣ ∇ u ∣ ≤ C ,

where C is a uniform constant.

6 Proof of main theorem

From the standard regularity theory of uniformly elliptic partial differential equations, we can obtain the high order regularity. We refer the readers to Tosatti et al. [32]. In this section, we prove Theorem 1.2 and Corollaries 1.3 and 1.4 by the method of continuity. As explicitly shown in the proofs of the estimates up to second-order, we have to find a uniform C -subsolution condition for all the solution flow of the continuity method.

Proof of Theorem 1.2

Proof

Define ϕ ̲ by

σ k ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) = ϕ ̲ ( x ) ,

where λ ( χ u ̲ ) ∈ Γ k − 1 . It is easy to see that

(6.1) lim t → ∞ σ k ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) > ϕ ̲ ( x ) ,

where e i is the i th standard basis vector in R n . Since u ̲ is a C -subsolution of equation (1.1),

(6.2) lim t → ∞ σ k ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) > β k − 1 .

We consider

(6.3) σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = ( 1 − t ) ϕ ̲ ( x ) + t β k − 1 ( x ) + b t ,

where λ ( χ u ) ∈ Γ k − 1 and b t is a constant for each t . Set

T ≔ { t ′ ∈ [ 0 , 1 ] ∣ ∃ u ∈ C 2 , α ( M ) and b t solving ( 6.3 ) for t ∈ [ 0 , t ′ ] } .

As shown in [20], the continuity method works if we can guarantee (1) 0 ∈ T and (2) uniform C ∞ estimates for all u . When t = 0 , b 0 = 0 by the uniqueness, the first requirement is naturally met. For the second requirement, we only need to show a uniform C -subsolution for all the solution flow. The condition (1.2) yields

ϕ ̲ ( x ) ≤ β k − 1 ( x ) .

At the maximum point of u − u ̲ ,

σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) ≤ σ k ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) − ∑ l = 0 k − 2 α l β l ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) = ϕ ̲ ( x ) ,

which means that

( 1 − t ) ϕ ̲ ( x ) + t β k − 1 ( x ) + b t ≤ ϕ ̲ ( x ) ,

so

b t ≤ 0 .

This means that

( 1 − t ) ϕ ̲ ( x ) + t β k − 1 ( x ) + b t ≤ β k − 1 ( x ) .

Then C -subsolution condition is uniform for all the solution flow. As a result, we have uniform C ∞ estimates of u .□

Proof of Corollary 1.3

Proof

We can find a smooth real function h satisfying that all x ∈ M

h ( x ) ≥ max { ϕ ̲ ( x ) , β k − 1 ( x ) }

and

(6.4) lim t → ∞ σ k ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ̲ ) + t e i ) σ k − 1 ( λ ( χ u ̲ ) + t e i ) > h ( x ) ,

which means that the set

χ ˜ ∈ Γ k − 1 g ∣ χ ˜ k ∧ ω n − k ≤ ∑ l = 0 k − 2 α l ( x ) χ ˜ l ∧ ω n − l + C n k − 1 C n k h ( x ) χ ˜ k − 1 ∧ ω n − k + 1 and χ ˜ − χ u ̲ ≥ 0

is bounded. First, we consider

(6.5) σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = ( 1 − t ) ϕ ̲ ( x ) + t h ( x ) + a t ,

where λ ( χ u ) ∈ Γ k − 1 and a t is a constant for each t . Set

T 1 ≔ { t ′ ∈ [ 0 , 1 ] ∣ ∃ u ∈ C 2 , α ( M ) and a t solving ( 6.5 ) for t ∈ [ 0 , t ′ ] } .

When a 0 = 0 , 0 ∈ T 1 . At the maximum point of u − u ̲ , we obtain

σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) ≤ σ k ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) − ∑ l = 0 k − 2 α l β l ( λ ( χ u ̲ ) ) σ k − 1 ( λ ( χ u ̲ ) ) = ϕ ̲ ( x ) ,

which means that

( 1 − t ) ϕ ̲ ( x ) + t h ( x ) + a t ≤ ϕ ̲ ( x ) ,

so

a t ≤ 0 .

Obviously,

( 1 − t ) ϕ ̲ ( x ) + t h ( x ) + a t ≤ h ( x ) .

Second, we consider the family of equations:

(6.6) σ k ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) − ∑ l = 0 k − 2 β l σ l ( λ ( χ u ) ) σ k − 1 ( λ ( χ u ) ) = ( 1 − t ) h ( x ) + t β k − 1 ( x ) + b t ,

where λ ( χ u ) ∈ Γ k − 1 and b t is a constant for each t . Set

T 2 ≔ { t ′ ∈ [ 0 , 1 ] ∣ ∃ u ∈ C 2 , α ( M ) and b t solving ( 6.6 ) for t ∈ [ 0 , t ′ ] } .

Clearly, 0 ∈ T 2 with b 0 = a 1 . Integrating (6.6) on M , we have

∫ M χ k ∧ ω n − k = ∑ l = 0 k − 2 ∫ M α l χ u l ∧ ω n − l + ∫ M ( 1 − t ) C n k − 1 C n k h + t α k − 1 + C n k − 1 C n k b t χ u k − 1 ∧ ω n − k + 1 ≥ ∑ l = 0 k − 2 ∫ M c l χ u l ∧ ω n − l + ∫ M α k − 1 + C n k − 1 C n k b t χ u k − 1 ∧ ω n − k + 1 ≥ ∑ l = 0 k − 1 c l ∫ M χ l ∧ ω n − l + C n k − 1 C n k b t ∫ M χ k − 1 ∧ ω n − k + 1 ≥ ∫ M χ k ∧ ω n − k + C n k − 1 C n k b t ∫ M χ k − 1 ∧ ω n − k + 1 .

The last inequality is given by the condition (1.4). Hence,

b t ≤ 0 .

This means that

( 1 − t ) h ( x ) + t β k − 1 ( x ) + b t ≤ h ( x ) .

Then C -subsolution condition is uniform for all the solution flow. As a result, we have uniform C ∞ estimates of u .□

Proof of Corollary 1.4

Proof

The cone condition (1.7) is equivalent to C -subsolution of equation (1.6) satisfying u ̲ ≡ 0 . To solve equation (1.6), we consider two cases.

Case 1 tan ( θ ˆ ) ≥ 0 .

Since 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ≥ 0 , equation (1.6) is a special case of equation (1.5) and all conditions in Corollary 1.3 are satisfied. Then there exists a smooth function to solve equation (1.6) and Ω u ∈ Γ 2 g . Denote the eigenvalue of g i k ( Ω k j ¯ + u k j ¯ ) as λ = ( λ 1 , λ 2 , λ 3 ) , and λ 1 ≥ λ 2 ≥ λ 3 . Obviously, λ 1 ≥ λ 2 > 0 . From equation (1.6), we have

λ 1 λ 2 λ 3 = sec 2 ( θ ˆ ) ( λ 1 + λ 2 + λ 3 ) + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) > 0 .

Hence, λ 3 > 0 , which implies Ω u ∈ Γ 3 g .

Case 2 tan ( θ ˆ ) < 0 .

In this case, θ ˆ ∈ ( π 2 , π ) . The sign of 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) does not satisfy our requirement. Let Ω u = Ω ˜ u − sec ( θ ˆ ) ω . Substituting Ω u into equation (1.6), we obtain

(6.7) Ω ˜ u 3 = 3 sec ( θ ˆ ) Ω ˜ u 2 ∧ ω + 2 sec 2 ( θ ˆ ) ( sec ( θ ˆ ) − tan ( θ ˆ ) ) ω 3 ,

where 2 sec 2 ( θ ˆ ) ( sec ( θ ˆ ) − tan ( θ ˆ ) ) > 0 . From the cone condition (1.7), we have

3 ( Ω + sec ( θ ˆ ) ω ) 2 − 6 sec ( θ ˆ ) ( Ω + sec ( θ ˆ ) ω ) ∧ ω > 0 ,

which means that equation (6.7) also satisfies the cone condition. Hence, equation (6.7) satisfies all the conditions in Corollary 1.3. Then there exists a smooth function to solve equation (6.7) and Ω ˜ u ∈ Γ 2 g . Denote the eigenvalue of g i k ( Ω k j ¯ + sec ( θ ˆ ) g k j ¯ + u k j ¯ ) as λ ˜ = ( λ ˜ 1 , λ ˜ 2 , λ ˜ 3 ) , and λ ˜ 1 ≥ λ ˜ 2 ≥ λ ˜ 3 . Obviously, λ ˜ i = λ i + sec ( θ ˆ ) , for any 1 ≤ i ≤ 3 . Since λ ˜ ∈ Γ 2 , we obtain

λ 1 + λ 2 + λ 3 > − 3 sec ( θ ˆ ) .

Therefore,

λ 1 λ 2 λ 3 = sec 2 ( θ ˆ ) ( λ 1 + λ 2 + λ 3 ) + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) ≥ − 3 sec 3 ( θ ˆ ) + 2 tan ( θ ˆ ) sec 2 ( θ ˆ ) > 0 ,

which implies Ω u ∈ Γ 3 g .□

Acknowledgements

The authors are grateful to the anonymous referees for many constructive feedback and useful suggestions.

  1. Funding information: Research of the authors were supported by the Natural Science Foundation of Anhui Province Education Department (No. KJ2021A0659); University Excellent Young Talents Research Project of Anhui Province (No. gxyq2022039); Quality Enginering Project of Anhui Province Education Department (Nos. 2018jyxm0491, 2019mooc205, and 2020szsfkc0686); and Doctoral Scientific Research Initiation Project of Fuyang Normal University (No. 2021KYQD0011).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-10-14
Revised: 2022-04-11
Accepted: 2022-09-09
Published Online: 2022-10-11

© 2022 Jundong Zhou and Yawei Chu, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0504
Keywords for this article
Hessian equations; Hermitian manifolds; subsolution condition
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  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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