Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities

Yu-Cheng An; Guai-Qi Tian; Bi-Jun An

doi:10.1515/dema-2025-0130

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Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities

Yu-Cheng An , Guai-Qi Tian and Bi-Jun An

Published/Copyright: July 16, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article is concerned with the existence of nontrivial solutions to critical growth Kohn-Laplace equations with jumping nonlinearities. Or, more specifically, we consider the following Kohn-Laplace problem:

− Δ H u = b u + − a u − + ∣ u ∣ Q ∗ − 2 u , in Ω , u = 0 , on ∂ Ω ,

where Δ H is the Kohn-Laplacian on the Heisenberg group H n with n > 1 , Q ∗ = 2 + 2 n is the critical Sobolev exponent for Δ H , Ω ⊂ H n denotes a smooth bounded domain, both a and b are greater than zero, and u ± = max { ± u , 0 } . When the pair ( a , b ) belongs to some designated regions of R 2 , the existence of nontrivial solutions is proved for the aforementioned Kohn-Laplace equation. The proof is based on two abstract existence results recently obtained by Perera and Sportelli in [Theorems 4.1–4.2].

Keywords: Kohn-Laplace; critical growth; jumping nonlinearities; existence of solutions

MSC 2010: 35J20; 35H20

1 Introduction and main results

At the beginning of this section, we recall some definitions and basic known facts. The Heisenberg group H n , whose points will be denoted by ξ = ( z , t ) = ( x , y , t ) , is the Lie group ( R 2 n + 1 , ∘ ) with composition law defined by ξ ∘ ξ ′ = ( z + z ′ , t + t ′ + 2 ( x ′ ⋅ y − y ′ ⋅ x ) ) , where “ ⋅ ” denotes the inner product in the Euclidean space R n . Let

X j = ∂ x j + 2 y j ∂ t , Y j = ∂ y j − 2 x j ∂ t , j = 1 , 2 , … , n .

It is widely known that the second-order operator

Δ H = ∑ j = 1 n ( X j 2 + Y j 2 )

is called the Kohn-Laplace operator on H n . We shall also denote by

∇ H = ( X 1 , X 2 , … , X n , Y 1 , Y 2 , … , Y n )

the horizontal gradient. For any vector valued function ( ω 1 , ω 2 , … , ω n , ν 1 , ν 2 , … , ν n ) , let

div H ( ω 1 , ω 2 , … , ω n , ν 1 , ν 2 , … , ν n ) = ∑ j = 1 n ( X j ω j + Y j ν j ) ,

and for any ξ = ( z , t ) = ( x , y , t ) ∈ H n ,

A ( ξ ) = I n O 2 y T O I n − 2 x T 2 y − 2 x 4 ∣ z ∣ 2 ,

where ∣ z ∣ 2 is equal to ∑ j = 1 n ( x j 2 + y j 2 ) , O and I n are the n × n zero matrix and identity matrix, respectively. Then, it can be seen that det ( A ( ξ ) ) is equal to zero and

Δ H u = div H ( ∇ H u ) = div ( A ( ξ ) ∇ u ) .

It follows that Δ H is therefore degenerate at any point of H n . But be that as it may, we have the fact that Bony’s maximum principle holds [1], since the operator Δ H satisfies the Hörmander rank condition [2]. Here, we refer to [3] for complete information and details concerning Lie groups and Kohn-Laplace operator.

Now, let us consider the following Kohn-Laplace equation with jumping nonlinearities:

(1.1) − Δ H u = b u + − a u − + ∣ u ∣ Q ∗ − 2 u , in Ω , u = 0 , on ∂ Ω ,

where Ω ⊂ H n be a smooth bounded domain with n > 1 , both a and b are greater than zero, u ± = max { ± u , 0 } , and Q ∗ = 2 + 2 n = 2 Q Q − 2 is the critical Sobolev exponent for Δ H , Q = 2 n + 2 . As is known to us all, when a = b = λ , problem (1.1) reduces to the famous Brézis-Nirenberg problem for the Kohn-Laplacian on the Heisenberg group

(1.2) − Δ H u = λ u + ∣ u ∣ Q ∗ − 2 u , in Ω , u = 0 , on ∂ Ω ,

which is studied by An and Liu [4], Citti [5], and Loiudice [6]. The Dancer-Fučík spectrum Σ of − Δ H in Ω is the set those points ( a , b ) ∈ R 2 such that

(1.3) − Δ H u = b u + − a u − , in Ω , u = 0 , on ∂ Ω ,

has a nontrivial solution. Let { λ k } l = k ∞ denote the sequence of eigenvalues with respect to the Dirichlet problem for − Δ H when a is equal to b . Obviously, the Dancer-Fučík spectrum Σ of − Δ H contains the points ( λ k , λ k ) . Furthermore, by the abstract results for general compact self-adjoint operator in Perera and Schechter [7, Chapter 4], in the square

Q k = ( λ k − 1 , λ k + 1 ) × ( λ k − 1 , λ k + 1 ) ,

the Dancer-Fučík spectrum Σ of − Δ H contains two strictly decreasing curves

C k : b = ν k − 1 ( a ) , C k : b = μ k ( a ) ,

with

ν k − 1 ( a ) ≤ μ k ( a ) , ν k − 1 ( λ k ) = λ k = μ k ( λ k ) .

Here, we refer to [7, Theorem 4.7.9] for details on the Dancer-Fučík spectrum.

Now, we begin to prepare main results of this article. First, as far as we know, many scholars have studied the jumping problems and proved the existence of solutions for some classical elliptic equations in recent decades (see, e.g., [8–11]). However, the jumping problems for the Kohn-Laplace equations on the Heisenberg have not been studied so far. In fact, since the asymptotic estimates of extremal functions on the Heisenberg group have not been obtained so far and there are many differences between the properties of − Δ H and − Δ [12], e.g., the extreme value function U ( z , t ) is not a Heisenberg radial function (see Section 2) and the critical exponent Q ∗ is strictly less than 2 ∗ when they have the same topological dimension, which have created us some obstacles in proving the existence of solutions to problem (1.1). That is to say, the jumping results of Euclidean case cannot directly generalize to the Heisenberg group case. In order to overcome the aforementioned obstacles, we first establish some crucial estimates in Section 3 and then prove the following existence Theorems 1.1 and 1.2, utilizing two recently obtained abstract existence results by Perera and Sportelli in [13, Theorems 4.1–4.2]. The main results of this article are stated as follows:

Theorem 1.1

Let Ω be a smooth bounded domain of H n with n > 1 . If ( a , b ) belongs to Q k and b is less than ν k − 1 ( a ) for some k ≥ 2 , then problem (1.1) has at least one nontrivial solution.

Theorem 1.2

Let Ω be a smooth bounded domain of H n with n > 1 . If ( a , b ) belongs to Q k and b is greater than or equal to μ k ( a ) for some k ≥ 2 , then problem (1.1) has at least one nontrivial solution.

Remark 1.3

As we mentioned earlier, the jumping problem (1.1) has not been studied so far, and because of the fact that there are many differences between the properties of − Δ H and − Δ [12], the jumping results of Euclidean case cannot directly generalize to the Heisenberg group case, the key to overcoming the obstacle is to establish some asymptotic estimations of the extremal function. Second, the article is written assuming n > 1 . In fact, when n is equal to 1, we think that the conclusion is same as above, the only thing may be that we need to apply the other linking theorem [8] and the asymptotic estimations when n = 1 [4]. Besides, we can also obtain similar results by using the method with few changes in [9].

The structure of the rest of this article is as follows: in Section 2, we collect some properties about the Folland-Stein space S 0 1 ( Ω ) and the Kohn-Laplace operator − Δ H . Furthermore, we also give a crucial Lemma 2.1 according to the two abstract existence results recently obtained by Perera and Sportelli in [13, Theorems 4.1–4.2]. In Section 3, we first prove that the corresponding functional of problem (1.1) satisfies a compactness condition, and then, we divide two Section to complete the proofs of Theorems 1.1 and 1.2. More precisely, we prove some crucial estimates (Lemmas 3.1–3.3) in Section 3.1, and then, we use Lemma 2.1 given in Section 2 to complete the proofs of Theorems 1.1 and 1.2 in Section 3.2.

2 Preliminaries

The Folland-Stein space S 0 1 ( Ω ) is the closure of C 0 ∞ ( Ω ) with respect to the norm

‖ u ‖ S 0 1 ( Ω ) 2 = ∫ Ω ∣ ∇ H u ∣ 2 d ξ .

From now on, we use the notations ‖ ⋅ ‖ and ‖ ⋅ ‖ p to represent the norm ‖ u ‖ S 0 1 ( Ω ) and the L p -norm, respectively, i.e., ‖ u ‖ = ‖ u ‖ S 0 1 ( Ω ) and

‖ u ‖ p p = ∫ Ω ∣ u ∣ p d ξ , u ∈ L p ( Ω ) .

Thanks to the celebrated work in [14], the Folland-Stein space ( S 0 1 ( Ω ) , ‖ ⋅ ‖ ) is a Hilbert space. Moreover, if 1 ≤ p < Q ∗ , then S 0 1 ( Ω ) ↪ L p ( Ω ) is compact; if p = Q ∗ , then S 0 1 ( Ω ) ↪ L p ( Ω ) is only continuous. Let E l denote by the eigenspace of the eigenvalue λ k ,

N k = ⊕ j = 1 k E j , M l = N k ⊥ ⋂ S 0 1 ( Ω ) .

Then, the Folland-Stein space S 0 1 ( Ω ) is the direct sum of vector spaces N k and M k . Moreover, we have the varational inequalities

(2.1) ‖ u ‖ 2 ≤ λ k ∫ Ω ∣ u ∣ 2 d ξ , ∀ u ∈ N k ,

and

(2.2) ‖ u ‖ 2 ≥ λ k + 1 ∫ Ω ∣ u ∣ 2 d ξ , ∀ u ∈ M k .

In addition, Jerison and Lee [15] proved that the best Sobolev constant

(2.3) S = inf u ∈ S 0 1 ( H n ) ∫ H n ∣ ∇ H u ∣ 2 d ξ ∫ H n ∣ u ∣ Q ∗ d ξ 2 Q ∗

is obtained by the smooth function

(2.4) U ( z , t ) = c 0 ( ( 1 + ∣ z ∣ 2 ) 2 + t 2 ) 2 − Q 4 , ξ = ( z , t ) ∈ H n ,

where c 0 is a positive constant (the extremal function U ( z , t ) on the Carnot group can also be seen in [6, Theorem 3.1]). That is to say, U is a solution to the critical equation

(2.5) − Δ H u = u Q + 2 Q − 2 , u ∈ S 0 1 ( H n ) ,

which follows that

(2.6) ∫ H n ∣ ∇ H U ∣ 2 d ξ = ∫ H n ∣ U ∣ 2 Q Q − 2 d ξ = S 2 .

Here, as we mentioned in Section 1, one point worth emphasizing is that the extremal function U ( z , t ) is not a Heisenberg radial function with respect to the Heisenberg distance ∣ ξ ∣ H = ∣ z ∣ 2 + t 2 4 between ξ and the origin, which will bring some difficulties to the asymptotic estimations of the extreme value function (Lemma 3.1). Let δ λ be the natural group of dilations on H n given by

δ λ ( ξ ) = ( λ z , λ 2 t ) , ξ ∈ H n .

Then, for any ε > 0 , the scaling function

(2.7) U ε ( ξ ) = ε 2 − Q 2 U ( δ ε − 1 ( ξ ) )

is also a solution of (2.5) and (2.6). Next, we recall the two abstract existence results recently obtained by Perera and Sportelli in [13], and we use it to prove Theorems 1.1 and 1.2. Therefore, we need to consider the following subelliptic equation

(2.8) − Δ H u = b u + − a u − + f ( u ) , in Ω , u = 0 , on ∂ Ω ,

where a , b > 0 and f ∈ C ( S 0 1 ( Ω ) , R ) is a potential operator. For brevity, let

(2.9) I ( u , a , b ) = ∫ Ω ( a ∣ u − ∣ 2 + b ∣ u + ∣ 2 ) d ξ , Φ f ( u ) = ∫ 0 1 f ( s u ) u d s .

Then, if u is a solution to problem (2.8), then u is a critical point of the C 1 -functional J f : S 0 1 ( Ω ) → R defined as follows:

(2.10) J f ( u ) = 1 2 ∫ Ω ∣ ∇ H u ∣ 2 d ξ − 1 2 I ( u , a , b ) − Φ f ( u ) ,

and vice versa. Assume that

( f 1 ) Φ f ( u ) = o ( ‖ u ‖ 2 ) as the norm of u goes to zero;

( f 2 ) Φ f ( u ) is greater than or equal to zero for all u ∈ S 0 1 ( Ω ) ;

( f 3 ) there exists c ∗ is greater than 0 such that for each c ∈ ( 0 , c ∗ ) , every ( P S ) c sequence of J f has a subsequence that converges weakly to a nontrivial critical point of J f .

Now, since − Δ H is a self-adjoint operator on S 0 1 ( Ω ) and ( − Δ H ) − 1 is compact, we may replace the general operator A in [13, Section 2] with − Δ H , and it follows from [13, Theorems 4.1 and 4.2] that the following crucial lemma, i.e.,

Lemma 2.1

Let ( a , b ) ∈ Q k and B = { v + τ ( v , a , b ) : v ∈ N k } , where τ is a positive homogeneous map τ ( ⋅ , a , b ) ∈ C ( N k , M k ) and w = τ ( v , a , b ) is the unique solution of

‖ v + w ‖ 2 − I ( v + w , a , b ) = inf w ′ ∈ M k { ‖ v + w ′ ‖ 2 − I ( v + w ′ , a , b ) } , v ∈ N k .

Assume that ( f 1 ) , ( f 2 ) , and ( f 3 ) , then problem (2.8), has at least one nontrivial solution in each of the following cases:

Assume that b < μ k − 1 ( a ) and there exists e ∈ S 0 1 ( Ω ) \ N k − 1 such that the infimum value of J f is less than c ∗ when u is in the set { v + s e : v ∈ N k − 1 , s ≥ 0 } ;
Assume that b ≥ μ k ( a ) and there exists e ∈ S 0 1 ( Ω ) \ B with − e ∉ B such that the supremum value of J f is less than c ∗ when u is in the set { u + s e : u ∈ B , s ≥ 0 } .

3 Proof of main results

In this section, we prove Theorems 1.1 and 1.2 by applying Lemma 2.1. First, let B r ( ξ ) = B H ( ξ , r ) denote by the ∣ ⋅ ∣ H -ball with center at ξ and radius r , and C , C 1 , C 2 , … denote various positive constants, which may vary from line to line. Second, since u ± = ( − u ) ∓ , u solves (1.1) (resp. (1.3)) if and only if − u solves (1.3) (resp. (1.1)) with a and b interchanged, and hence, the Dancer-Fučík spectrum Σ is symmetric with respect to the line a = b and we may assume that a is less than or equal to b .

Let E ( u ) = J ∣ u ∣ Q ∗ − 2 u ( u ) , that is to say,

(3.1) E ( u ) = 1 2 ∫ Ω ∣ ∇ H u ∣ 2 d ξ − I ( u , a , b ) − Φ ∣ u ∣ Q ∗ − 2 u ( u ) = 1 2 ∫ Ω ∣ ∇ H u ∣ 2 d ξ − 1 2 ∫ Ω ( a ∣ u − ∣ 2 + b ∣ u + ∣ 2 ) d ξ − 1 Q ∗ ∫ Ω ∣ u ∣ Q ∗ d ξ .

Then, it is well known that every critical point of E is a weak solution to problem (1.1).

In the following, we check that the conditions ( f 1 ) , ( f 2 ) , and ( f 3 ) with c ∗ = 1 Q S Q ⁄ 2 for Φ f and E , respectively. First, it is obvious that Φ f satisfies ( f 1 ) and ( f 2 ) by the definition of Φ f . In addition, by the standard arguments [16,17], the function E satisfies the condition ( f 3 ) with c ∗ = 1 Q S Q ⁄ 2 . In fact, let { u n } be a ( P.S. ) c sequence for c < 1 Q S Q ⁄ 2 , i.e.,

(3.2) E ( u n ) → c < 1 Q S Q ⁄ 2 and E ′ ( u n ) → 0 ,

as n → ∞ . It follows from (3.1) and the second formula of (3.2) that

(3.3) E ′ ( u n ) u n = ∫ Ω ∣ ∇ H u n ∣ 2 d ξ − I ( u n , a , b ) − ∫ Ω ∣ u n ∣ Q ∗ d ξ = o ( ‖ u n ‖ ) .

Subtracting (3.3) divided by 2 from the first formula of (3.2) and a simple calculation obtains

(3.4) 1 2 − 1 Q ∗ ∫ Ω ∣ u n ∣ Q ∗ d ξ = c + o ( ‖ u n ‖ ) + o ( 1 ) .

Noting that c + o ( ‖ u n ‖ ) + o ( 1 ) = o ( ‖ u n ‖ ) , then we plug (3.4) back into (3.3), and we obtain that

(3.5) ∫ Ω ∣ ∇ H u n ∣ 2 d ξ − I ( u n , a , b ) = o ( ‖ u n ‖ ) .

Since a is less than or equal to b , from (3.5), we have

(3.6) a ‖ u n ‖ 2 2 − o ( ‖ u n ‖ ) ≤ ‖ u n ‖ 2 ≤ b ‖ u n ‖ 2 2 + o ( ‖ u n ‖ ) .

On the other hand, by (3.4), the Hölder inequality and the boundedness of Ω , one has

(3.7) ‖ u n ‖ 2 2 ≤ ∣ Ω ∣ 2 ⁄ Q ∫ Ω ∣ u n ∣ Q ∗ d ξ 2 ⁄ Q ∗ = o ( ‖ u n ‖ ) .

It follows from (3.6) and (3.7) that { u n } is bounded in S 0 1 ( Ω ) . Without loss of generality, we do not distinguish the differences between { u n } and its subsequence { u n k } , and therefore, we may assume that

u u ⇀ u in S 0 1 ( Ω ) , u n → u , in L p ( Ω ) for 1 ≤ p < Q ∗ , and u n ( x ) → u ( x ) a.e. x ∈ Ω .

Let u ˜ n = u n − u . We claim that ‖ u ˜ n ‖ → 0 as n → ∞ . If not, there exists ε 0 > 0 such that ‖ u ˜ n ‖ ≥ ε 0 for all large enough n . By the strong convergence and (3.2), we have

(3.8) 1 2 ∫ Ω ∣ ∇ H u n ∣ 2 d ξ − I ( u , a , b ) − 1 Q ∗ ∫ Ω ∣ u n ∣ Q ∗ d ξ = c + o ( 1 )

and

(3.9) ∫ Ω ∣ ∇ H u n ∣ 2 d ξ − I ( u , a , b ) − ∫ Ω ∣ u n ∣ Q ∗ d ξ = o ( 1 ) .

Meanwhile, by the second formula of (3.2) and the weak convergence, we have

(3.10) ∫ Ω ∣ ∇ H u ∣ 2 d ξ − I ( u , a , b ) − ∫ Ω ∣ u ∣ Q ∗ d ξ = 0 .

Therefore, subtracting (3.10) from (3.9) obtains

(3.11) ∫ Ω ( ∣ ∇ H u n ∣ 2 − ∣ ∇ H u ∣ 2 ) d ξ = ∫ Ω ( ∣ u n ∣ Q ∗ − ∣ u ∣ Q ∗ ) d ξ + o ( 1 ) .

It follows from the Brézis-Lieb lemma (see [18, Theroem 1]) and (2.3) that

(3.12) ∫ Ω ∣ ∇ H u ˜ n ∣ 2 d ξ = ∫ Ω ∣ u ˜ n ∣ Q ∗ d ξ + o ( 1 ) .

Noting that ‖ u ˜ n ‖ ≥ ε 0 , it follows from (3.12) and (2.3) that

(3.13) ∫ Ω ∣ ∇ H u ˜ n ∣ 2 d ξ ≥ S Q ⁄ 2 + o ( 1 ) .

On the other hand, subtracting (3.9) divided by Q ∗ from (3.8) obtains

(3.14) 1 Q ∫ Ω ∣ ∇ H u n ∣ 2 d ξ − I ( u , a , b ) = c + o ( 1 ) .

Subtracting (3.10) divided by Q from (3.14) obtains

(3.15) 1 Q ∫ Ω ( ∣ ∇ H u n ∣ 2 − ∣ ∇ H u ∣ 2 ) d ξ + ∫ Ω ∣ u ∣ Q ∗ d ξ = c + o ( 1 ) .

Once again, it follows from the Brézis-Lieb lemma (see [18, Theroem 1]) and (3.15) that

(3.16) 1 Q ∫ Ω ∣ ∇ H u ˜ n ∣ 2 d ξ + ∫ Ω ∣ u ∣ Q ∗ d ξ = c + o ( 1 ) .

By substituting (3.13) into (3.16), we obtain c ≥ 1 Q S Q ⁄ 2 for all large enough n . This is a contradiction, and hence, the claim is true, which means that the function E satisfies the condition ( f 3 ) with c ∗ = 1 Q S Q ⁄ 2 .

In the following, we divide two subsections to complete the proofs of Theorems 1.1 and 1.2.

3.1 Some crucial estimates

In this section, we establish some estimates that play the most important role in the proving process of Theorems 1.1 and 1.2. First, we may assume that ξ 0 ∈ Ω and r 0 is a fixed positive constant such that B r 0 ( ξ 0 ) ⋐ Ω . Let φ : [ 0 , + ∞ ) → [ 0 , 1 ] be a smooth mapping such that φ ( s ) = 0 for s ≥ 1 ⁄ 2 and φ ( s ) = 1 for s ≤ 1 ⁄ 4 .

Now, let us define function u ε , μ : H n → R ,

(3.17) u ε , μ ( ξ ) = φ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) U ε ( ξ 0 − 1 ∘ ξ ) ,

for any ε > 0 and μ > max { 2 ⁄ r 0 , 1 } , where ξ 0 − 1 is the inverse of ξ 0 with respect to the group action ∘ , i.e., ξ 0 − 1 = − ξ 0 . In the following, we give some asymptotic estimations of u ε , μ with ε > 0 small enough.

Lemma 3.1

Let u ε , μ be defined by (3.17). If n > 1 , then we have the following estimates:

(3.18) ∫ Ω u ε , μ 2 d ξ ≥ C 1 ε 2 − C 2 μ Q − 4 ε Q − 2 ,

(3.19) ∫ Ω u ε , μ Q ∗ d ξ ≥ S Q ⁄ 2 − C 3 ( μ ε ) Q ,

(3.20) ∫ Ω u ε , μ d ξ ≤ C 4 μ − 2 ε ( Q − 2 ) ⁄ 2 ,

(3.21) ∫ Ω u ε , μ Q ∗ − 1 d ξ ≤ C 5 ε ( Q − 2 ) ⁄ 2 ,

(3.22) ∫ Ω ∣ ∇ H u ε , μ ∣ 2 d ξ ≤ S Q ⁄ 2 + C 6 ( μ ε ) Q − 2 .

Proof

Since n > 1 , Q > 4 . By the definition of u ε , μ , φ and B r 0 ( ξ 0 ) ⋐ Ω , it is easy to obtain

(3.23) ∫ Ω u ε , μ 2 d ξ = ∫ Ω ( φ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) U ε ( ξ 0 − 1 ∘ ξ ) ) 2 d ξ = ∫ B r 0 μ ( ξ 0 ) ( φ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) U ε ( ξ 0 − 1 ∘ ξ ) ) 2 d ξ ≥ ∫ B r 0 4 μ ( ξ 0 ) ε 2 − Q U 2 ( δ ε − 1 ( ξ 0 − 1 ∘ ξ ) ) d ξ = ε 2 ∫ B r 0 4 μ ε ( 0 ) U 2 ( η ) d η

(3.23) = ε 2 ∫ B 1 ( 0 ) U 2 ( η ) d η + ∫ B r 0 4 μ ε ( 0 ) \ B 1 ( 0 ) U 2 ( η ) d η ≥ C 1 ε 2 1 + ∫ 1 r 0 4 μ ε ρ 3 − Q d ρ ≥ C 1 ε 2 − C 2 μ Q − 4 ε Q − 2 .

This completes the proof of estimate (3.18). Now, we prove estimate (3.19).

(3.24) ∫ Ω u ε , μ Q ∗ d ξ = ∫ Ω U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ + ∫ Ω ( φ Q ∗ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) − 1 ) U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ = S Q ⁄ 2 + I ( φ , ε ) ,

where

I ( φ , ε ) = ∫ Ω ( φ Q ∗ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) − 1 ) U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ − ∫ H n \ Ω U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ .

If − I ( φ , ε ) ≤ C ( μ ε ) Q , then estimate (3.19) holds. In fact, by μ > max 2 r 0 , 1 , we have

(3.25) − I ( φ , ε ) = ∫ Ω ( 1 − φ Q ∗ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) ) U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ + ∫ H n \ Ω U ε Q ∗ ( ξ 0 − 1 ∘ ξ ) d ξ ≤ ∫ H n \ B r 0 μ ( 0 ) U ε Q ∗ ( η ) d η + ∫ H n \ B r 0 μ ( 0 ) U ε Q ∗ ( η ) d η ≤ 2 ∫ H n \ B r 0 μ ( 0 ) U ε Q ∗ ( η ) d η ≤ C ∫ H n \ B r 0 μ ε ( 0 ) U ( ∣ η ∣ H ) d η ≤ C ∫ r 0 μ ε + ∞ ρ − ( Q + 1 ) d ξ ≤ C ( μ ε ) Q .

It follows from (3.24) and (3.25) that estimate (3.19) is true. Next, let us prove (3.20):

(3.26) ∫ Ω u ε , μ d ξ = ∫ Ω φ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) U ε ( ξ 0 − 1 ∘ ξ ) d ξ = ∫ B r 0 μ ( 0 ) φ ( μ ∣ η ∣ H ) U ε ( η ) d η ≤ ε Q + 2 2 ∫ B r 0 μ ε ( 0 ) U ( η ) d η ≤ C 4 μ − 2 ε ( Q − 2 ) ⁄ 2 .

Then, estimate (3.20) is proved. The proof of (3.21) is similar to that of (3.20). In fact,

(3.27) ∫ Ω u ε , μ Q ∗ − 1 d ξ = ∫ Ω ( φ ( μ ∣ ξ 0 − 1 ∘ ξ ∣ H ) U ε ( ξ 0 − 1 ∘ ξ ) ) Q ∗ − 1 d ξ = ∫ B r 0 μ ( 0 ) ( φ ( μ ∣ η ∣ H ) U ε ( η ) ) Q ∗ − 1 d η

(3.27) ≤ ∫ B r 0 μ ( 0 ) ε 2 − Q 2 U ( δ ε − 1 ( η ) ) Q ∗ − 1 d η = ε Q − 2 2 ∫ B r 0 μ ε ( 0 ) U Q ∗ − 1 ( η ) d η ≤ C 5 ε ( Q − 2 ) ⁄ 2 .

It remains to show estimate (3.22). In fact,

(3.28) ∫ Ω ∣ ∇ H u ε , μ ∣ 2 d ξ = ∫ Ω ∇ H U ε ( ξ 0 ∘ ξ ) ⋅ ∇ H ( φ 2 ( μ ∣ ξ 0 ∘ ξ ∣ H ) U ε ( ξ 0 ∘ ξ ) ) d ξ + ∫ Ω ∣ ∇ H φ ( μ ∣ ξ 0 ∘ ξ ∣ H ) ∣ 2 U ε 2 ( ξ 0 ∘ ξ ) d ξ = ∫ Ω φ 2 ( μ ∣ ξ 0 ∘ ξ ∣ H ) U ε Q ∗ ( ξ 0 ∘ ξ ) d ξ + ∫ Ω ∣ ∇ H φ ( μ ∣ ξ 0 ∘ ξ ∣ H ) ∣ 2 U ε 2 ( ξ 0 ∘ ξ ) d ξ = ∫ Ω ∣ ∇ H φ ( μ ∣ ξ 0 ∘ ξ ∣ H ) ∣ 2 U ε 2 ( ξ 0 ∘ ξ ) d ξ + ∫ H n U ε Q ∗ ( η ) d η + I ( φ , ε ) ,

where I ( φ , ε ) is the same as in (3.24). Now, let us evaluate the first integral in the right-hand side of (3.28). By the definition of φ , (3.24), and the fact that I ( φ , ε ) is less than or equal to zero, we have

(3.29) ∫ Ω ∣ ∇ H φ ( μ ∣ ξ 0 ∘ ξ ∣ H ) ∣ 2 U ε 2 ( ξ 0 ∘ ξ ) d ξ ≤ ∫ Ω ∣ ∇ H φ ( μ ∣ ξ 0 ∘ ξ ∣ H ) ∣ 2 U ε 2 ( ξ 0 ∘ ξ ) d ξ − I ( φ , ε ) ≤ C ∫ B r 0 μ ( 0 ) \ B r 0 4 μ ( 0 ) U ε 2 ( η ) d ξ + C ( μ ε ) Q = C ε 2 ∫ B r 0 μ ε ( 0 ) \ B r 0 4 μ ε ( 0 ) U 2 ( η ) d η + C ( μ ε ) Q ≤ C ε 2 ∫ r 0 4 μ ε r 0 μ ε ρ 3 − Q d ρ + C ( μ ε ) Q ≤ C 6 ( μ ε ) Q − 2 .

It follows from (3.28) and (3.29) that (3.22) and the proof of Lemma 3.1 is complete.□

Remark 3.2

In the process of the aforementioned proof, we first used the fact that u ( ξ ) ≃ C ∣ ξ ∣ H 2 − Q as ∣ ξ ∣ H goes to infinity (see [6, Lemma 3.2] or [19, Theorem 3.4]). Second, we also used the left translation invariance and the Heisenberg polar coordinates (see [20]), i.e., let f : R + → R and r > 0 . Then,

∫ B r ( ξ 0 ) f ( ∣ ξ 0 − ∘ ξ ∣ H ) d ξ = ∫ B r ( 0 ) f ( ∣ η ∣ H ) d η = ∫ 0 r f ( s ) s Q − 1 d s ,

whenever one of the previous integral exists. In addition, we remark also that the asymptotic estimates (3.20) and (3.21) have not been proved in [6, Lemma 3.3] and the proofs of (3.18), (3.19), and (3.22) are slightly different from that contained in [6, Lemma 3.3], since the existence of the parameter μ .

Lemma 3.3

Let Ω ⊂ H n with n > 1 be a smooth bounded domain. Assume Q + 2 Q < β < Q − 2 2 and

(3.30) sup u ∈ K ‖ u ‖ C 2 ( B 1 ⁄ μ 0 ( ξ 0 ) ¯ ) < + ∞ ,

where K is a subset of the intersection of C 2 ( Ω ) and M = { u ∈ S 0 1 ( Ω ) : ‖ u ‖ = 1 } . Let

ϕ ( u ) = ∫ Ω ∣ ∇ H u ∣ 2 d ξ , ψ p ( u ) = ∫ Ω ∣ u ∣ p d ξ .

Then, there exist C 7 , C 8 , … , C 14 > 0 such that for all ε > 0 , μ ≥ μ 0 , u ∈ K , and s , t ≥ 0 ,

(3.31) ϕ ( t u + s u ε , μ ) ≤ ( 1 + C 7 μ − ( Q + 2 ) ) t 2 + ( S Q ⁄ 2 + C 8 ( μ ε ) Q − 2 ) s 2 ,

(3.32) ψ Q ∗ ( t u + s u ε , μ ) ≥ ψ Q ∗ ( t u ) + S Q 2 − C 9 ( μ ε ) Q − C 10 ε 2 Q β Q + 2 s Q ∗ − C 11 ( μ − Q + ε Q ( Q − 2 β − 2 ) ⁄ ( Q − 2 ) ) t Q ∗ ,

(3.33) I ( t u + s u ε , μ , a , b ) ≥ ( I ( u , a , b ) − C 12 μ − 4 ) t 2 + ( C 13 ε 2 − C 14 μ Q − 4 ε Q − 2 ) s 2 .

Proof

First of all, from the definition of φ and Cauchy’s inequality, we have to admit the fact that u ε , μ = 0 on ∂ Ω and

2 s t μ − 2 ε Q − 2 ≤ μ − ( Q + 2 ) t 2 + ( μ ε ) Q − 2 s 2 .

Meanwhile, it follows from u ∈ K ⊂ M ∩ C 2 ( Ω ) that u and Δ H u are bounded. Then, by (3.20) and (3.22), we have

(3.34) ϕ ( t u + s u ε , μ ) = t 2 + 2 s t ∫ Ω ∇ H u ⋅ ∇ H u ε , μ d ξ + s 2 ∫ Ω ∣ ∇ H u ε , μ ∣ 2 d ξ = t 2 − 2 s t ∫ Ω u ε , μ Δ H u d ξ + s 2 ∫ Ω ∣ ∇ H u ε , μ ∣ 2 d ξ ≤ t 2 − C s t μ − 2 ε ( Q − 2 ) ⁄ 2 + ( S Q ⁄ 2 + C 1 ( μ ε ) Q − 2 ) s 2 ≤ ( 1 + C 7 μ − ( Q + 2 ) ) t 2 + ( S Q ⁄ 2 + C 8 ( μ ε ) Q − 2 ) s 2 .

This completes the proof of estimate (3.31). Now, we proceed to prove estimate (3.32). In fact, for any x 1 , x 2 ∈ R and p > 2 , there exists a positive constant C such that

(3.35) ∣ x 1 + x 2 ∣ p ≥ ∣ x 1 ∣ p + ∣ x 2 ∣ p − C ( ∣ x 1 ∣ p − 1 ∣ x 2 ∣ + ∣ x 2 ∣ p − 1 ∣ x 1 ∣ ) .

Hence, it follows from (3.19), (3.20), (3.21), and (3.35) that

(3.36) ψ Q ∗ ( t u + s u ε , μ ) ≥ ψ Q ∗ ( t u ) + ψ Q ∗ ( s u ε , μ ) − C s t Q ∗ − 1 ∫ Ω ∣ u ∣ Q ∗ − 1 u ε , μ d ξ + t s Q ∗ − 1 ∫ Ω ∣ u ε , μ ∣ Q ∗ − 1 u d ξ ≥ ψ Q ∗ ( t u ) + s Q ∗ ( S Q ⁄ 2 − C 3 ( μ ε ) Q ) − C s t Q ∗ − 1 ∫ Ω ∣ u ε , μ ∣ d ξ + t s Q ∗ − 1 ∫ Ω ∣ u ε , μ ∣ Q ∗ − 1 d ξ ≥ ψ Q ∗ ( t u ) + s Q ∗ ( S Q ⁄ 2 − C 3 ( μ ε ) Q ) − C ( s t Q ∗ − 1 μ − 2 ε ( Q − 2 ) ⁄ 2 + t s Q ∗ − 1 ε ( Q − 2 ) ⁄ 2 ) .

On the other hand, by Cauchy’s inequality, we have

(3.37) s t Q ∗ − 1 μ − 2 ε ( Q − 2 ) ⁄ 2 ≤ 1 Q ∗ μ − Q t Q ∗ + 1 − 1 Q ∗ ( μ ε ) Q s Q ∗

and

(3.38) t s Q ∗ − 1 ε ( Q − 2 ) ⁄ 2 ≤ 1 Q ∗ ε Q [ 1 − 2 β ⁄ ( Q − 2 ) ] t Q ∗ + 1 − 1 Q ∗ ε 2 Q β ⁄ ( Q + 2 ) s Q ∗ .

By substituting (3.36) and (3.37) into (3.38), we obtain (3.32).

The rest of the proof is estimate (3.33). In fact, u ε , μ is equal to zero outside B 1 ⁄ μ ( ξ 0 ) and a is less than or equal to b , and we have

(3.39) I ( t u + s u ε , μ , a , b ) = t 2 ∫ Ω \ B 1 μ ( ξ 0 ) [ a ( u − ) 2 + b ( u + ) 2 ] d ξ + ∫ B 1 μ ( ξ 0 ) [ a ( ( t u + s u ε , μ ) − ) 2 + b ( ( t u + s u ε , μ ) + ) 2 ] d ξ ≥ t 2 I ( u , a , b ) + g ( u μ , ε , a , b ) ,

where

g ( u μ , ε , a , b ) = ∫ B 1 μ ( ξ 0 ) [ a ( t u + s u ε , μ ) 2 − b ( t u ) 2 ] d ξ .

In the following, let us evaluate the asymptotic behavior of g ( u μ , ε , a , b ) . In fact, it follows from (3.18), (3.20), and (3.30) that

(3.40) g ( u μ , ε , a , b ) = a s 2 ∫ B 1 μ ( ξ 0 ) u ε , μ 2 d ξ + 2 t s ∫ B 1 μ ( ξ 0 ) u u ε , μ d ξ + ( a − b ) t 2 ∫ B 1 μ ( ξ 0 ) u 2 d ξ ≥ ( C 1 ε 2 − C 2 μ Q − 4 ε Q − 2 ) s 2 − C μ − Q t 2 − 2 C s t ( μ − 2 ε ( Q − 2 ) ⁄ 2 ) .

Note that the inequality

(3.41) 2 t s μ − 2 ε ( Q − 2 ) ⁄ 2 ≤ μ − 4 t 2 + ε Q − 2 s 2 .

It follows from (3.39), (3.40), and (3.41) that estimate (3.33) holds. So, the proof of Lemma 3.3 is complete.□

Lemma 3.4

Let Ω ⊂ H n with n > 1 be a smooth bounded domain and (3.30) holds. Assume that

(3.42) ‖ u ‖ 2 − I ( u , a , b ) ≤ 0 , ∀ u ∈ K ,

where K is the same as in (3.30). Then, for any small positive number ε ,

(3.43) sup u ∈ K , s , t ≥ 0 E ( t u + s u ε , ε − 1 ) < 1 Q S Q ⁄ 2 .

Proof

It follows from u ∈ K , a ≤ b , (3.42) and Hölder’s inequality that

(3.44) 1 = ‖ u ‖ 2 ≤ b ∫ Ω u 2 d ξ ≤ b ∣ Ω ∣ 2 ⁄ Q Q ∗ ∫ Ω ∣ u ∣ Q ∗ d ξ 2 ,

which means that

(3.45) inf u ∈ K ∫ Ω ∣ u ∣ Q ∗ d ξ > 0 .

On the other hand, by n > 1 and Q = 2 n + 2 , we have 1 ⁄ Q < γ < 1 − 2 ⁄ ( Q − 2 ) , and hence, we may fix 1 ⁄ Q < γ < 1 − 2 ⁄ ( Q − 2 ) . Let μ = ε − γ , then one has

( μ ε ) Q − 2 = ε ( Q − 2 ) ( 1 − γ ) , ( Q − 2 ) ( 1 − γ ) > 2 , ( Q − 2 ) ( 1 − γ ) + 2 γ > 2 .

So, for any arbitrarily small ε > 0 , u ∈ K , and s , t ≥ 0 , we consider Lemma 3.3 taking μ = ε − γ , it follows from (3.31), (3.32), and (3.33) that

(3.46) E ( t u + s u ε , ε − γ ) = 1 2 ϕ ( t u + s u ε , ε − γ ) − 1 2 I ( t u + s u ε , ε − γ ) − 1 Q ∗ ψ Q ∗ ( t u + s u ε , ε − γ ) ≤ 1 2 ( ‖ u ‖ 2 − I ( u , a , b ) + C 15 ε 4 γ ) t 2 + 1 2 ( S Q ⁄ 2 − C 16 ε 2 ) s 2 − 1 Q ∗ ( ‖ u ‖ Q ∗ Q ∗ − C 11 ε Q γ − C 11 ε Q [ 1 − 2 β ⁄ ( Q − 2 ) ] ) t Q ∗ − 1 Q ∗ ( S Q ⁄ 2 − C 9 ε Q ( 1 − γ ) − C 10 ε 2 Q β ⁄ ( Q + 2 ) ) s Q ∗ .

It follows from (3.42), (3.45), and (3.46) that

(3.47) E ( t u + s u ε , ε − γ ) ≤ Σ ( ε , s ) S Q ⁄ 2 + τ ( ε , t ) ,

where

τ ( ε , t ) = C 17 ε 4 γ t 2 − C 18 t Q ∗ , Σ ( ε , s ) = 1 2 ( 1 − C 19 ε 2 ) s 2 − 1 Q ∗ ( 1 − C 20 ε Q ( 1 − γ ) − C 21 ε 2 Q β ⁄ ( Q + 2 ) ) s Q ∗ .

Since τ ( ε , 0 ) = Σ ( ε , 0 ) = 0 and

lim t → ∞ τ ( ε , t ) = lim s → ∞ Σ ( ε , s ) = − ∞ ,

both τ ( ε , t ) and Σ ( ε , s ) have maximum values. Moreover, it can easily be shown that

(3.48) τ ( ε , t ) ≤ C 22 ε 2 Q γ

and

(3.49) Σ ( ε , s ) ≤ 1 Q ( 1 − C 23 ε 2 ) Q ⁄ 2 ( 1 − C 24 ε 2 Q β ⁄ ( Q + 2 ) − C 25 ε Q ( 1 − γ ) ) ( Q − 2 ) ⁄ 2 .

Note that the fact that 2 Q β ⁄ ( Q + 2 ) , Q ( 1 − γ ) and 2 Q γ are all greater than 2, and it follows from (3.47), (3.48), and (3.49) that (3.43) follows. This completes the proof of Lemma 3.4.□

3.2 Proof of Theorems 1.1 and 1.2

In this subsection, we prove Theorems 1.1 and 1.2 by using Lemma 2.1 and some crucial estimates (Lemmas 3.1, 3.3, and 3.4) proved in Section 3.1.

Proof of Theorem 1.1

We consider Lemma 2.1 taking e = u ε , ε − γ with ε > 0 sufficiently small. Functions u ε , ε − γ with different centers ξ 0 and sufficiently small ε have disjoint supports, and hence, they are linearly independent. Because of the fact that N k − 1 is finite dimensional, it follows that ξ 0 ∈ Ω can be chosen so that u ε , ε − γ does not belong to N k . Setting K = M ∩ N k − 1 in (3.30) (or (3.42)), it follows from ( a , b ) ∈ Q k that both a and b are greater than λ k − 1 for k ≥ 2 . Hence, for any u ∈ K , we have

(3.50) ‖ u ‖ 2 − I ( u , a , b ) ≤ ∫ Ω ∣ ∇ H u ∣ 2 d ξ − λ k − 1 ( ‖ u + ‖ 2 2 + ‖ u − ‖ 2 2 ) = ∫ Ω ∣ ∇ H u ∣ 2 d ξ − λ k − 1 ∫ Ω ∣ u ∣ 2 d ξ .

Then, it follows from (2.1) and (3.50) that (3.42) holds. In addition, it follows from the sub-elliptic estimation [20,21] that the regularity of eigenfunctions, i.e., set N k is contained in set C 0 2 ( Ω ) . Hence, (3.30) holds in Lemma 3.18. In a word, there exists

e = u ε , ε − γ ∈ S 0 1 ( Ω ) \ N k − 1 ,

such that the infimum value of E is less than c ∗ = 1 ⁄ Q S Q ⁄ 2 when u is in the set

Γ = { v + s u ε , ε − γ : v ∈ N k − 1 , s ≥ 0 } ,

i.e., all the conditions of Lemma 2.1 (i) hold. Hence, it follows Lemma 2.1 that the existence of nontrivial solutions, and therefore, we have completed the proof of Theorem 1.1.□

Proof of Theorem 1.2

As in the proof of Theorem 1.1, we also consider Lemma 2.1 taking e = u ε , ε − γ with ε > 0 sufficiently small, and for simplicity, let

Ψ ( u ε , ε − γ , a , b ) = ‖ u ε , ε − γ ‖ 2 − I ( u ε , ε − γ , a , b ) .

Then, Gradients Ψ ′ ( u ε , ε − γ , a , b ) with different centers ξ 0 and sufficiently small ε have disjoint supports, and hence, they are linearly independent. It follows from [13, Proposition 2.1] that Ψ ′ ( v + τ ( v , a , b ) , a , b ) is perpendicular to M k for all v ∈ N k , and therefore,

{ Ψ ′ ( v + τ ( v , a , b ) , a , b ) : v ∈ N k }

is a subset of N k . Because of the fact that N k is finite dimensional, it follows that ξ 0 ∈ Ω can be chosen so that u ε , ε − γ and − u ε , ε − γ do not belong to the set

B = { v + τ ( v , a , b ) : v ∈ N k } .

Setting K = M ∩ N k − 1 in (3.30) (or (3.42)). We apply [13, Theorem 1.2] taking N = N k , M = M k , and τ = τ ( ⋅ , a , b ) . Then, it follows from b ≥ μ k ( a ) that

sup v ∈ N k ∩ M Ψ ( v + τ ( v , a , b ) , a , b ) ≤ 0 .

Hence, for any v ∈ N k ∩ M , it follows that Ψ ( v + τ ( v , a , b ) , a , b ) is less than or equal to zero, which means that (3.42) holds.

In the following, we claim that (3.30) holds in Lemma 3.18. In fact, by the regularity of subelliptic operator equations [20,21], we also have the fact that set N k is contained in set C 0 2 ( Ω ) , and the proof of the claim is therefore quite similar to that given in [13] and so is omitted. Therefore, all the conditions of Lemma 2.1 (ii) hold. Then, it follows Lemma 2.1 the existence of nontrivial solutions, and we have thus proved Theorem 1.2.□

Acknowledgments

The authors would like to thank the reviewers for careful reading and valuable suggestions.

Funding information: This work was supported by the Education Department Youth Science Fund Project of Guizhou Province (No.QJJ[2022]204), the Science and Technology Project of Bijie (Nos.BKH[2023]26, BKH[2025]13).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors contributed equally to the writing of this article.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
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: 2024-10-04

Revised: 2025-01-29

Accepted: 2025-04-23

Published Online: 2025-07-16

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https://doi.org/10.1515/dema-2025-0130

Keywords for this article

Kohn-Laplace; critical growth; jumping nonlinearities; existence of solutions

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