Nonexistence results of positive solutions of Heisenberg Hessian equations and inequalities on the Heisenberg group

Chuanqiang Chen; Yan Ma

doi:10.1515/agms-2025-0029

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Nonexistence results of positive solutions of Heisenberg Hessian equations and inequalities on the Heisenberg group

Chuanqiang Chen and Yan Ma

Published/Copyright: November 21, 2025

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From the journal Analysis and Geometry in Metric Spaces Volume 13 Issue 1

Abstract

In this article, we study Liouville type theorems of fully nonlinear elliptic partial differential equations on the Heisenberg group and obtain some nonexistence results of positive solutions of Heisenberg Hessian equations (and inequalities), including Heisenberg Hessian quotient equations (and inequalities) and mixed Heisenberg Hessian quotient equations (and inequalities). The proofs are based on integration by parts and the Newton-Maclaurin inequality.

Keywords: Heisenberg group; Liouville theorem; integration by parts; Newton-Maclaurin inequality

MSC 2020: Primary 35J60; Secondary 35B53

1 Introduction

In this article, we study Liouville type theorems of fully nonlinear elliptic partial differential equations on the Heisenberg group:

(1.1) F ( − D H 2 u ) = u α , in H n ,

where the Heisenberg group H n is the set C n × R with the coordinates ( z , t ) and the group law

( z , t ) ( w , s ) = z + s , t + s + 2 Im ∑ i = 1 n z i w ¯ i , ( z , t ) , ( w , s ) ∈ C n × R .

Moreover, the solution u is a real-valued function defined in H n , and the Heisenberg Hessian matrix of u is

D H 2 u ≕ ( u i j ¯ + u i ¯ j ) 1 ≤ i , j ≤ n ,

which is a real-valued symmetric matrix (see the proof in Section 2). Hence, the Heisenberg Laplacian of u is

Δ H u ≕ ∑ i = 1 n ( u i i ¯ + u i ¯ i ) = Tr ( D H 2 u ) .

Furthermore, we denote the homogeneous dimension of H n by Q ≕ 2 n + 2 .

It is well known that there are many important studies of Liouville type theorems on R n . The most famous results concern the Lane-Emden equation:

(1.2) − Δ u = u α , in R n .

For α ≤ n n − 2 , the only nonnegative solution of (1.2) is u = 0 [12]. For 1 < α < n + 2 n − 2 , Gidas and Spruck [15] proved the nonexistence of positive solutions by integration by parts and the suitable choice of a vector field. For the critical case α = n + 2 n − 2 , equation (1.2) is related to Yamabe problem, and the classification result gives a complete classification of metrics on R n , which are conformal to the standard one. By utilizing the Alexandrov reflection technique as developed by Gidas et al. [13,14], Caffarelli et al. [5] proved that all the positive solutions of (1.2) are of the form

u ( x ) = λ n ( n − 2 ) λ 2 + ∣ x − x 0 ∣ 2 n − 2 2 , with λ > 0 , x 0 ∈ R n .

Later, Chen and Li [10] gave a new proof by the moving plane method. Moreover, Liouville type theorems (nonexistence results of positive solutions) and classifications of positive solutions are studied for semilinear and quasilinear elliptic equations on R n . See [2,19–21,27] for some results about semilinear equations, [1,6,11,18,27,28] for some results about quasilinear equations and [8,23–26] for fully nonlinear elliptic equations.

On the Heisenberg group H n , there are also some important Liouville type results for semilinear equations. Birindelli et al. [3] proved that for 1 < α ≤ Q Q − 2 , the only nonnegative solutions of

(1.3) − Δ H u = u α , in H n

are the trivial ones, where they also showed that α = Q Q − 2 is sharp. Birindelli and Prajapat [4] proved that, for 0 < α < Q + 2 Q − 2 , equation (1.3) has no any positive cylindrical solution in H n . Xu [30] obtained a Liouville type theorem for the nonnegative solution of equation (1.3), in the range of n > 1 and 1 < α ≤ O + 2 Q − 2 − 1 ( Q − 2 ) ( Q − 1 ) 2 , which means the unique solution is trivial solution. Later, we mention a recent advance made by Ma and Ou [22], in which the authors established the Liouville theorem for the classical solution of (1.3) for the subcritical case 1 < α < Q + 2 Q − 2 . The method used by Ma and Ou [22] is the integral estimate, which deduced from a generalized formula of that found by Jerison and Lee [16].

An interesting question is whether Liouville type theorems (nonexistence results of positive solutions) hold for fully nonlinear elliptic partial differential equations (1.1) on the Heisenberg group.

First, we deduce a Liouville type theorem for the Heisenberg Laplacian equation and inequality as follows.

Theorem 1.1

For n ≥ 1 and α ∈ ( − ∞ , Q Q − 2 ] , the Heisenberg Laplacian equation and inequality

(1.4) − Δ H u ≥ u α , i n H n

have no positive solution u ∈ C 2 ( H n ) .

Second, we deduce a Liouville type theorem for the Heisenberg Hessian quotient equation (and inequality) as follows.

Theorem 1.2

For 0 ≤ l < k ≤ n , and α ∈ ( − ∞ , Q ( k − l ) Q − 2 ] , the Heisenberg Hessian quotient equation (and inequality)

(1.5) σ k ( − D H 2 u ) σ l ( − D H 2 u ) ≥ u α , i n H n

has no positive Γ k -admissible solution u ∈ C 2 ( H n ) , where the Γ k -admissible solution means − D H 2 u ∈ Γ k for any point x ∈ H n , and Γ k is defined in (2.2).

Moreover, we deduce a Liouville type theorem for the mixed Heisenberg Hessian quotient equation (and inequality) as follows.

Theorem 1.3

For 0 ≤ l < k ≤ n and α ∈ ( − ∞ , Q ( k − l ) Q − 2 ] , the mixed Heisenberg Hessian quotient equation (and inequality)

(1.6) σ k ( − Δ H u I n + D H 2 u ) σ l ( − Δ H u I n + D H 2 u ) ≥ u α , i n H n

has no positive Γ k ˜ -admissible solution u ∈ C 2 ( H n ) , where I n is the n × n identity matrix, the Γ k ˜ -admissible solution means − D H 2 u ∈ Γ k ˜ for any point x ∈ H n , and Γ k ˜ is defined in (2.4).

Finally, we can establish the nonexistence result for general Heisenberg Hessian type equation (and inequality) as follows.

Theorem 1.4

For n ≥ 1 , the Heisenberg Hessian type equation (and inequality)

(1.7) F ( − D H 2 u ) ≥ u α , in H n

satisfies the following property

(1.8) F ( A ) ≤ C 0 ⋅ σ 1 ( A ) , for A ∈ { A : F ( A ) > 0 } ,

where C 0 is a positive constant. Then (1.7) has no positive solution u ∈ C 2 ( H n ) in { A : F ( A ) > 0 } for α ∈ ( − ∞ , Q Q − 2 ] .

The rest of the article is organized as follows. In Section 2, we introduce some covariant derivatives rules on the Heisenberg group, and the k -Hessian operators, mixed Hessian operators, and some properties. In Section 3, we prove Theorem 1.1 by integration by parts, and in Section 4, we prove Theorems 1.2 and 1.3 by Newton-MacLaurin inequality. Finally, we prove Theorem 1.4 and give some discussions.

2 Preliminaries

In this section, we introduce some covariant derivatives rules on the Heisenberg group, and the k -Hessian operators, mixed Hessian operators, and some properties. These properties are well known and can be similarly found in [7,9,17].

2.1 Heisenberg group

We shall first give a brief introduction to the Heisenberg group H n and some notations. We consider H n as the set C n × R with the coordinates ( z , t ) and the group law

( z , t ) ( w , s ) = ( z + w , t + s + 2 Im ∑ i = 1 n z i w ¯ i ) , ( z , t ) , ( w , s ) ∈ C n × R ,

where in the sequel, the repeated indices are sum from 1 to n . For ( z , t ) ∈ C n × R as an element of H n , the norm is defined by ∣ ( z , t ) ∣ ≕ ( ∣ z ∣ 4 + ∣ t ∣ 2 ) 1 4 . Define B R ( ( z , t ) ) as the metric ball centered at ( z , t ) with radius R > 0 , and the volume ∣ B R ( ( z , t ) ) ∣ ≈ R 2 n + 2 . The associated homogeneous dimension of the Heisenberg group is Q = 2 n + 2 , and the Heisenberg group is a dilation group.

The Cauchy-Riemann structure of H n is given by the bundle ℋ spanned by the left-invariant vector field

Z i = ∂ ∂ z i + − 1 z ¯ i ∂ ∂ t , i = 1 , … , n .

The standard (left-invariant) contact form on H n is Θ = d t + − 1 ( z i d z ¯ i − z ¯ i d z i ) . With respect to the standard holomorphic frame { Z i } and dual admissible coframe d z i for i = 1 , … , n . Covariant derivatives on H n are given by u j 1 … j r = Z j r … Z j 1 u , for j s = i , i ¯ , or 0 , with the conventions that Z 0 = ∂ ∂ t and Z i ¯ = Z ¯ i . For the first covariant derivatives, we know u i ¯ = u i ¯ , and denote

∂ u = ( u 1 , … , u n ) , ∣ ∂ u ∣ 2 ≕ ∑ i = 1 n u i u i ¯ .

For the second covariant derivatives of a real-valued function u , we have the following commutation relations:

u i j − u j i = 0 ,

u i j ¯ − u j ¯ i = 2 − 1 δ i j u 0 ,

u 0 i = u i 0 .

In fact, we have

(2.1) u i j ¯ = u j ¯ i ¯ , i , j = 1, 2 , … , n .

Precisely, for i , j = 1, 2 , … , n , we have

u i j ¯ = Z j ¯ Z i u = ∂ ∂ z j ¯ − − 1 z j ∂ ∂ t ∂ u ∂ z i + − 1 z ¯ i ∂ u ∂ t = ∂ 2 u ∂ z i ∂ z ¯ j − − 1 z j ∂ 2 u ∂ z i ∂ t + − 1 δ i j ∂ u ∂ t + − 1 z ¯ i ∂ 2 u ∂ z ¯ j ∂ t + z j z ¯ i ∂ 2 u ∂ t ∂ t ,

and

u i ¯ j = Z j Z i ¯ u = ∂ ∂ z j + − 1 z ¯ j ∂ ∂ t ∂ u ∂ z ¯ i − − 1 z i ∂ u ∂ t = ∂ 2 u ∂ z ¯ i ∂ z j + − 1 z ¯ j ∂ 2 u ∂ z ¯ i ∂ t − − 1 δ i j ∂ u ∂ t − − 1 z i ∂ 2 u ∂ z j ∂ t + z ¯ j z i ∂ 2 u ∂ t ∂ t .

Hence,

u i ¯ j ¯ = ∂ 2 u ∂ z i ∂ z ¯ j − − 1 z j ∂ 2 u ∂ z i ∂ t + − 1 δ i j ∂ u ∂ t + − 1 z ¯ i ∂ 2 u ∂ z ¯ j ∂ t + z j z ¯ i ∂ 2 u ∂ t ∂ t = u i j ¯ .

Hence, (2.1) holds. Therefore, we know

u i j ¯ + u i ¯ j = u i j ¯ + u i j ¯ ¯ = 2 Re ( u i j ¯ ) .

Hence, the Heisenberg Hessian matrix of u is

D H 2 u ≕ ( u i j ¯ + u i ¯ j ) 1 ≤ i , j ≤ n ,

which is a real-valued symmetric matrix. Moreover, the Heisenberg Laplacian of u is

Δ H u ≕ ∑ i = 1 n ( u i i ¯ + u i ¯ i ) = Tr ( D H 2 u ) .

2.2 k -Hessain operators

Definition 2.1

For any k = 1, 2 , … , n , the k -elementary symmetric polynomial is defined by

σ k ( λ ) = ∑ 1 ≤ i 1 < i 2 < … < i k ≤ n λ i 1 λ i 2 … λ i k ,

where λ = ( λ 1 , … , λ n ) ∈ R n . For convenience, let σ 0 ( λ ) = 1 and σ k ( λ ) = 0 for k > n .

For real symmetric matrix A = ( A i j ) , denote its eigenvalues by λ ( A ) = ( λ 1 , … , λ n ) and define the k-Hessian operators by

σ k ( A ) = σ k ( λ ( A ) ) = ∑ 1 ≤ i 1 < i 2 < … < i k ≤ n λ i 1 λ i 2 … λ i k ,

we also set σ 0 ( A ) = 1 and σ k ( A ) = 0 for any k > n or k < 0 . Also, we can define the Gårding’s cone

(2.2) Γ k = { A : σ s ( A ) > 0, 1 ≤ s ≤ k } .

In fact, ∂ σ k ( A ) ∂ A i j > 0 for A ∈ Γ k .

The following Newton-Maclaurin inequality for Hessian quotient operators is well known, and its proof can be found in [29].

Proposition 2.1

For A ∈ Γ k and 0 ≤ l < k ≤ n , 0 ≤ s < r ≤ k , s ≤ l , we have

σ k ( A ) ⁄ C n k σ l ( A ) ⁄ C n l 1 k − l ≤ σ r ( A ) ⁄ C n r σ s ( A ) ⁄ C n s 1 r − s ,

and the equality holds if and only if λ 1 = λ 2 = … = λ n > 0 . In particular, we have

(2.3) σ k ( A ) σ l ( A ) ≤ C n k C n l 1 n k − l [ σ 1 ( A ) ] k − l .

2.3 Mixed k -Hessian operators

For real symmetric matrix A , denote its eigenvalues by λ ( A ) = ( λ 1 , … , λ n ) . Consider the matrix Tr ( A ) I n − A , and denote its eigenvalues by η ( A ) = ( η 1 , η 2 , … , η n ) . Then, we have

η i = σ 1 ( λ ) − λ i

for i = 1, 2 , … , n . As in [7,9], we define the mixed k -Hessian operators

σ k ( Tr ( A ) I n − A ) = σ k ( η ( A ) ) = ∑ 1 ≤ i 1 < i 2 < … < i k ≤ n η i 1 η i 2 … η i k

and define the corresponding cones

(2.4) Γ k ˜ = { A : σ i ( Tr ( A ) I n − A ) > 0, 1 ≤ i ≤ k } .

In particular, we have

σ 1 ( Tr ( A ) I n − A ) = ( n − 1 ) σ 1 ( A ) .

In fact, if A ∈ Γ k ˜ , we have ∂ σ k ( Tr ( A ) I n − A ) ∂ A i j > 0 .

When A ∈ Γ k ˜ , we have Tr ( A ) I n − A ∈ Γ k , and then from Proposition 2.1, we have the following Newton-Maclaurin inequality for mixed Hessian quotient operators.

Proposition 2.2

For A ∈ Γ k ˜ and 0 ≤ l < k ≤ n , 0 ≤ s < r ≤ k , s ≤ l , we have

σ k ( Tr ( A ) I n − A ) ⁄ C n k σ l ( Tr ( A ) I n − A ) ⁄ C n l 1 k − l ≤ σ r ( Tr ( A ) I n − A ) ⁄ C n r σ s ( Tr ( A ) I n − A ) ⁄ C n s 1 r − s ,

and the equality holds if and only if λ 1 = λ 2 = … = λ n > 0 . In particular, we have

(2.5) σ k ( Tr ( A ) I n − A ) σ l ( Tr ( A ) I n − A ) ≤ C n k C n l ( n − 1 ) k − l n k − l [ σ 1 ( A ) ] k − l .

3 The proof of theorem 1.1

In this section, we prove Theorem 1.1 by using integration by parts.

Proof of Theorem 1.1

Assume u > 0 be a real-valued solution of (1.4) in C 2 ( H n ) . Choose the cut-off function η ∈ C 0 ∞ ( H n ) , 0 ≤ η ≤ 1 , and

η = 1 , for ∣ ( z , t ) ∣ ≤ 1 2 R ,

0 ≤ η ≤ 1 , for 1 2 R < ∣ ( z , t ) ∣ ≤ R ,

∣ ∂ η ∣ ≤ C R , for 1 2 R ≤ ∣ ( z , t ) ∣ ≤ R .

Multiplying both sides of (1.4) by u − δ η θ (where δ > 0 and θ > Q to be determined later), and integrating over H n , we can obtain

(3.1) ∫ u α − δ η θ ≤ − ∫ Δ H u u − δ η θ = − ∫ ( u i i ¯ + u i ¯ i ) u − δ η θ = − δ ∫ u i u i ¯ u − δ − 1 η θ + θ ∫ u i η i ¯ u − δ η θ − 1 − δ ∫ u i ¯ u i u − δ − 1 η θ + θ ∫ u i ¯ η i u − δ η θ − 1 = − 2 δ ∫ ∣ ∂ u ∣ 2 u − δ − 1 η θ + θ ∫ ( u i η i ¯ + u i ¯ η i ) u − δ η θ − 1 = − 2 δ M 1 + θ E 1 ,

where M 1 ≕ ∫ ∣ ∂ u ∣ 2 u − δ − 1 η θ and E 1 ≕ ∫ ( u i η i ¯ + u i ¯ η i ) u − δ η θ − 1 .

By Young’s inequality, we have

(3.2) ∣ E 1 ∣ ≤ ε ∫ ∣ ∂ u ∣ 2 u − δ − 1 η θ + C ( ε ) ∫ u − δ + 1 ∣ ∂ η ∣ 2 η θ − 2 ≤ ε ∫ ∣ ∂ u ∣ 2 u − δ − 1 η θ + C ( ε ) R 2 ∫ u − δ + 1 η θ − 2 ,

where ε > 0 is small. By substituting (3.2) into (3.1), we have

(3.3) ∫ u α − δ η θ + ( 2 δ − ε ) M 1 ≤ C ( ε ) R 2 ∫ u − δ + 1 η θ − 2 .

In the following, we prove Theorem 1.1 for α ∈ ( − ∞ , Q Q − 2 ] by dividing into four cases.

Case 1: α ∈ ( − ∞ , 1 ) .

For this case, we choose δ > ( n + 1 ) − n α , and θ > 2 α − δ α − 1 , then it is easy to check that α − δ < − δ + 1 < 0 , and 2 n + 2 − 2 α − δ α − 1 < 0 .

From (3.3), we can obtain by Young’s inequality with exponent pair α − δ 1 − δ , α − δ α − 1

∫ u α − δ η θ + ( 2 δ − ε ) M 1 ≤ C ( ε ) R 2 ∫ u − δ + 1 η θ − 2 ≤ ε ∫ u α − δ η θ + C ( ε ) R 2 α − δ α − 1 ∫ η θ − 2 α − δ α − 1 ≤ ε ∫ u α − δ η θ + C ( ε ) R 2 n + 2 − 2 α − δ α − 1 .

Hence,

(3.4) ( 1 − ε ) ∫ u α − δ η θ + ( 2 δ − ε ) M 1 ≤ C ( ε ) R 2 n + 2 − 2 α − δ α − 1 .

We reach a contradiction if R → + ∞ in (3.4).

Case 2: α = 1 .

For this case, we choose δ = α = 1 and θ > 2 n + 2 , then from (3.3), we can obtain

∫ η θ + ( 2 − ε ) M 1 ≤ C ( ε ) R 2 ∫ η θ − 2 ≤ ε ∫ η θ + C ( ε ) R 2 n + 2 − θ .

Hence, we have

(3.5) ( 1 − ε ) ∫ η θ + ( 2 − ε ) M 1 ≤ C ( ε ) R 2 n + 2 − θ .

Letting R → + ∞ , we obtain a contradiction in (3.5).

Case 3: α ∈ ( 1 , Q Q − 2 ) .

For this case, we choose δ > ( n + 1 ) − n α , and θ > 2 α − δ α − 1 , then it is easy to check that α − δ > − δ + 1 > 0 , and 2 n + 2 − 2 α − δ α − 1 < 0 .

From (3.3), we can obtain by Young’s inequality with exponent pair α − δ 1 − δ , α − δ α − 1

Hence,

(3.6) ( 1 − ε ) ∫ u α − δ η θ + ( 2 δ − ε ) M 1 ≤ C ( ε ) R 2 n + 2 − 2 α − δ α − 1 .

We reach a contradiction if R → + ∞ in (3.6).

Case 4: α = Q Q − 2 .

For this case, we choose θ > 0 large enough, and we have from (1.4)

(3.7) ∫ u α η θ ≤ − ∫ Δ H u η θ = − ∫ ( u i i ¯ + u i ¯ i ) η θ = θ ∫ ( u i η i ¯ + u i ¯ η i ) η θ − 1 .

Denote E 1 ≕ ∫ ( u i η i ¯ + u i ¯ η i ) η θ − 1 , and we estimate E 1 in the following.

First, we have

∣ E 1 ∣ ≤ C R ∫ ∣ ∂ u ∣ η θ − 1 ≤ ε ∫ ∣ ∂ u ∣ 2 u − δ − 1 η θ + C ε R 2 ∫ u δ + 1 η θ − 2 ,

where 0 < δ < 1 n is fixed. Choosing ε = R − 2 n + 2 α δ , we have

∣ E 1 ∣ ≤ C R 2 n + 2 α δ − 2 ∫ u δ + 1 η θ − 2 + C R − 2 n + 2 α δ − 2 ∫ u − δ + 1 η θ − 2 .

Next, by Hölder inequality, we have

R 2 n + 2 α δ − 2 ∫ u δ + 1 η θ − 2 ≤ R 2 n + 2 α δ − 2 ∫ u α η θ 1 + δ α ∫ η θ − 2 α α − δ − 1 α − δ − 1 α ≤ C R 2 n + 2 α δ − 2 ∫ u α η θ 1 + δ α ⋅ R ( 2 n + 2 ) α − δ − 1 α = C R α − 1 α 2 n + 2 − 2 α α − 1 ∫ u α η θ 1 + δ α ,

and

R − 2 n + 2 α δ − 2 ∫ u − δ + 1 η θ − 2 ≤ R − 2 n + 2 α δ − 2 ∫ u α η θ 1 − δ α ∫ η θ − 2 α α + δ − 1 α + δ − 1 α ≤ C R − 2 n + 2 α δ − 2 ∫ u α η θ 1 − δ α ⋅ R ( 2 n + 2 ) α + δ − 1 α = C R α − 1 α 2 n + 2 − 2 α α − 1 ∫ u α η θ 1 − δ α .

Since α = Q Q − 2 = n + 1 n , i.e. 2 n + 2 − 2 α α − 1 = 0 , we can see

(3.8) ∣ E 1 ∣ ≤ C ∫ u α η θ 1 + δ α + C ∫ u α η θ 1 − δ α .

Recall the definition of “ E 1 ” , all the integrations in (3.8) are taken over the domain Ω = { x ∈ H n ⊂ R 2 n + 1 : ∣ x ∣ ≤ R } . Hence, combining (3.7) with (3.8), we can obtain

(3.9) ∫ Ω u α η θ ≤ C ∫ Ω u α η θ 1 + δ α + C ∫ Ω u α η θ 1 − δ α .

Since 0 < 1 + δ α , 1 − δ α < 1 , (3.9) shows that

∫ Ω u α η θ → 0 , as R → + ∞ .

This is a contradiction, and hence, the proof of Theorem 1.1 goes to end.□

4 The proof of Theorems 1.2 and 1.3

In this section, we prove Theorems 1.2 and 1.3 by the Newton-Maclaurin inequality.

4.1 Proof of Theorem 1.2

Assume (1.5) has a positive Γ k -admissible solution u , then we prove Theorem 1.2 by contradiction in the following.

By Newton-Maclaurin inequality (2.3) for A = − D H 2 u ∈ Γ k , we have

σ k ( − D H 2 u ) σ l ( − D H 2 u ) ≤ C n k C n l 1 n k − l [ σ 1 ( − D H 2 u ) ] k − l = C n k C n l 1 n k − l [ − Δ H u ] k − l .

Hence, we have from (1.5),

(4.1) C ⋅ [ − Δ H u ] ≥ u 1 k − l α ,

where C is a positive constant depending only on n , k , l . Since α ∈ ( − ∞ , Q ( k − l ) Q − 2 ] , then 1 k − l α ∈ ( − ∞ , Q Q − 2 ] . From Theorem 1.1, we know (4.1) has no such positive solution u ∈ C 2 ( H n ) . The proof is finished.

4.2 Proof of Theorem 1.3

We can obtain Theorem 1.3 by similar process as in the proof of Theorem 1.2.

Assume (1.6) has a positive Γ k ˜ -admissible solution u , then we prove Theorem 1.3 by contradiction in the following.

By Newton-Maclaurin inequality (2.5) for A = − D H 2 u ∈ Γ k ˜ , we have

σ k ( − Δ H u I n + D H 2 u ) σ l ( − Δ H u I n + D H 2 u ) ≤ C n k C n l ( n − 1 ) k − l n k − l [ σ 1 ( − D H 2 u ) ] k − l = C n k C n l ( n − 1 ) k − l n k − l [ − Δ H u ] k − l .

Hence, we have from (1.6),

(4.2) C ⋅ [ − Δ H u ] ≥ u 1 k − l α ,

where C is a positive constant depending only on n , k , l . Since α ∈ ( − ∞ , Q ( k − l ) Q − 2 ] , then 1 k − l α ∈ ( − ∞ , Q Q − 2 ] . From Theorem 1.1, we know (4.2) has no such positive solution u ∈ C 2 ( H n ) . The proof is finished.

5 Discussions of Theorem 1.4

In this section, we prove Theorem 1.4 and give some discussions.

5.1 The Proof of Theorem 1.4

Assume (1.7) has a positive solution u , then we prove Theorem 1.4 by contradiction in the following.

From (1.7) and (1.8), we have

(5.1) C 0 ⋅ [ − Δ H u ] ≥ u α , in H n .

Since α ∈ ( − ∞ , Q Q − 2 ] , we know (5.1) has no such positive solution u ∈ C 2 ( H n ) from Theorem 1.1. The proof is finished.

5.2 Discussion

In Theorem 1.4, we do not require the operator F is divergence free, that is,

∑ i = 1 n ∂ ∂ z i ∂ F ( A ) ∂ A i j = 0 , for j = 1, 2 , … , n ,

∑ j = 1 n ∂ ∂ z ¯ j ∂ F ( A ) ∂ A i j = 0 , for i = 1, 2 , … , n

cannot vanish. Even in Theorem 1.4, we do not require F is elliptic.

Hence, it is easy to establish nonexistence results of positive admissible solutions for many Hessian type inequalities. For example,

∑ i = 0 k − 1 C i ⋅ [ σ k − i ( − D H 2 u ) ] k k − i ≥ u α , in H n ,

∑ i = 0 l C i ⋅ σ k − i ( − D H 2 u ) σ l − i ( − D H 2 u ) ≥ u α , in H n ,

∑ i = 0 k − 1 C i ⋅ ln [ σ k − i ( − D H 2 u ) ] ≥ u α , in H n ,

arctan [ σ k ( − D H 2 u ) ] ≥ u α , in H n .

Acknowledgments

The authors would like to thank Prof. Lu Xu for the helpful discussions in this subject.

Funding information: This work was partially supported by NSFC No. 12171260 and ZJNSF No. LRG25A010002.
Author contributions: All authors contributed equally to the writing of this article. All authors read the final manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data was used for the research described in the article.

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Received: 2025-04-16

Accepted: 2025-08-18

Published Online: 2025-11-21

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

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https://doi.org/10.1515/agms-2025-0029

Keywords for this article

Heisenberg group; Liouville theorem; integration by parts; Newton-Maclaurin inequality

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