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Multiple solutions of semilinear elliptic systems on the Heisenberg group

Gao Jia*, Long-jie Zhang and Jie Chen

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College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

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Boundary Value Problems 2013, 2013:157  doi:10.1186/1687-2770-2013-157


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/157


Received:17 November 2012
Accepted:10 June 2013
Published:1 July 2013

© 2013 Jia et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, a class of semilinear elliptic systems which have a strong resonance at the first eigenvalue on the Heisenberg group is considered. Under certain assumptions, by virtue of the variational methods, the multiple weak solutions of the systems are obtained.

MSC: 35J20, 35J25, 65J67.

Keywords:
semilinear elliptic system; strong resonance; variational method; Heisenberg group

1 Introduction

Let H N be the space R N × R N × R equipped with the following group operation:

η η = ( x , y , t ) ( x , y , t ) = ( x + x , y + y , t + t + 2 ( x y x y ) ) ,

where ‘⋅’ denotes the usual inner-product in R N . This operation endows H N with the structure of a Lie group. The vector fields X 1 , , X N , Y 1 , , Y N , T, given by

X j = x j + 2 y j t , Y j = y j 2 x j t , T = t ,

form a basis for the tangent space at η = ( x , y , t ) .

Definition 1.1 The Heisenberg Laplacian is by definition

Δ H = j = 1 N ( X j 2 + Y j 2 ) ,

and let H u denote the 2N-vector ( X 1 u , , X N u , Y 1 u , , Y N u ) .

Definition 1.2 The space S 0 1 , 2 ( Ω ) is defined as the completion of C 0 ( Ω ) in the norm

u S 0 1 , 2 2 = Ω j = 1 N ( | X j u | 2 + | Y j u | 2 ) = Ω | H u | 2 .

Some existence and nonexistence for the semilinear equations or systems on the Heisenberg group have been studied by Garofalo, Lanconelli and Niu, see [1,2], etc.

In this paper, we study the problems on the existence and multiplicity of solutions for the system

{ Δ H ( u v ) = λ 1 ( a ( x ) b ( x ) b ( x ) d ( x ) ) ( u v ) ( f ( x , u , v ) g ( x , u , v ) ) , x Ω , u = v = 0 , x Ω , (1.1)

where Ω H N is a bounded smooth domain, a , b , d C 0 ( Ω ¯ , R ) and f , g C 1 ( Ω ¯ × R 2 , R ) . Moreover, we assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) such that F = ( f g ) . Here ∇F denotes the gradient in the variable u and v, i.e., F u = f , F v = g .

In fact, the condition in R N was studied by da Silva; we can see [3]. In this paper we study the problem on the Heisenberg group H N . The elliptic problems at resonance have been studied by many authors; see [4-7].

We use the variation methods to solve problem (1.1). Finding weak solutions of (1.1) in E = S 0 1 , 2 ( Ω ) × S 0 1 , 2 ( Ω ) is equivalent to finding critical points of the C 2 functional given by

I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) , (1.2)

where

h E , h = ( h ( 1 ) h ( 2 ) ) , h 2 = Ω | H h ( 1 ) | 2 + | H h ( 2 ) | 2 ,

and , denotes the usual inner product in R 2 .

We introduce the eigenvalue problem with weights. Let us denote by A the set of all continuous, cooperative and symmetric matrices A of order 2, given by

A ( x ) = ( a ( x ) b ( x ) b ( x ) d ( x ) ) ,

where the functions a , b , d C ( Ω ¯ , R ) satisfy the following conditions:

(A1) A ( x ) is cooperative, that is, b ( x ) 0 .

(A2) There is an x 0 Ω such that a ( x 0 ) > 0 or d ( x 0 ) > 0 .

Given A A ( Ω ) , consider the weighted eigenvalue problem

{ Δ H ( h ( 1 ) h ( 2 ) ) = λ A ( x ) ( h ( 1 ) h ( 2 ) ) , in  Ω , h ( 1 ) = h ( 2 ) = 0 , on  Ω ,

if A A ( Ω ) . By virtue of the spectral theory for compact operators, we obtain the sequence of eigenvalues

0 < λ 1 < λ 2 λ 3

such that λ k + as k ; see [6,8,9]. Here, each eigenvalue λ k , k 1 has finite multiplicity, and we have

1 λ k = sup { Ω A h , h , h = 1 , h V k 1 } ,

where V k = span { Φ 1 , , Φ k } with k 1 .

Remark 1.1

(1) E = V k V k for k 1 .

(2) The following variational inequalities hold:

h 2 λ k Ω A h , h , h V k , k 1 , (1.3)

h 2 λ k + 1 Ω A h , h , h V k , k 0 . (1.4)

The variational inequalities will be used in the next section. We would like to mention that the Φ 1 is positive in Ω. In the paper, without loss of generality, we assume that λ 1 = 1 .

We now state the assumptions and the main results in this paper. Firstly, we define the following functions:

{ T + = lim inf ( u , v ) ( , ) F ( x , u , v ) , S + = lim sup ( u , v ) ( , ) F ( x , u , v ) , T = lim inf ( u , v ) ( , ) F ( x , u , v ) , S = lim sup ( u , v ) ( , ) F ( x , u , v ) . (1.5)

The above functions belong to L 1 ( Ω ) and the limits are taken a.e. and uniformly in x Ω .

Now we make the following basic hypotheses:

(E0) There exists k C ( Ω ¯ ) such that

lim | h | F ( x , h ) = 0 , | F ( x , h ) | k ( x ) , a.e.  x Ω , h R 2 .

(E1) F ( x , h ) 1 2 ( 1 λ 2 ) A h , h + b 1 | Ω | 1 , b 1 0 , ( x , h ) Ω × R 2 .

(E2) A h , h 0 , ( x , h ) Ω × R 2 .

(E3) There exist α ( 0 , 1 ) and δ > 0 such that

F ( x , h ) 1 α 2 A h , h , x Ω  and  | z | < δ .

(E4) Ω S + 0 and Ω S 0 .

(E5) There exists t 0 R such that

Ω F ( x , t 0 Φ 1 ) < min { Ω T + , Ω T } .

(E6) There are t 1 < 0 and t 1 + > 0 such that

Ω F ( x , t 1 ± Φ 1 ) < min { Ω T + , Ω T } .

We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.

Theorem 1.1Let Ω H N be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) . Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) such that F u = f , F v = g . Furthermore, if the conditions (E0), (E1), (E2) are satisfied, problem (1.1) has at least one solution z 1 E .

Remark 1.2 For the hypotheses F ( x , 0 , 0 ) 0 and F ( x , 0 , 0 ) 0 , problem (1.1) admits the trivial solution ( u , v ) = 0 . In this case, the main point is to assure the existence of nontrivial solutions.

Theorem 1.2Let Ω H N be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) . Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) such that F u = f , F v = g . Furthermore, if the conditions (E0), (E2), (E3), (E4) and (E5) are satisfied, then problem (1.1) has at least two nontrivial solutions.

Theorem 1.3Let Ω H N be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) . Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) such that F u = f , F v = g . Furthermore, if the conditions (E0), (E1), (E2), (E3), (E4) and (E6) are satisfied, then problem (1.1) has at least three nontrivial solutions.

2 Preliminaries and fundamental lemmas

In this section, we prove some lemmas needed in the proof of our main theorems.

We first introduce the Folland-Stein embedding theorem (see [10]) as follows.

Lemma 2.1Let Ω H N be a bounded domain and let Q = 2 N + 2 . Then S 0 1 , 2 ( Ω ) compactly embedding in L p ( Ω ) , where 2 p < 2 Q Q 2 .

To establish Lemmas 2.7 and 2.8, we introduce the following corollary of the Ekeland variation principle (see [11]).

Lemma 2.2Xis a metric space, I C 1 ( X , R ) is bounded from below, which satisfies the ( P S ) c condition, then c = inf x X E ( x ) is a critical value ofE.

Next, we describe some results under the geometry for the functional I.

Lemma 2.3Under hypotheses (E0) and (E1), the functionalIhas the following saddle geometry:

(L3-1) I ( h ) if h with h V 1 .

(L3-2) There is α R such that I ( h ) α , z V 1 .

(L3-3) I ( h ) b 1 , z V 1 .

Proof (L3-1). From (1.2), (1.4) we have

I ( h ) 1 2 ( 1 1 λ 2 ) h 2 + Ω F ( x , h ) , h V 1 .

Using (E0), we have J ( h ) , as h .

(L3-2). By simple calculation, we get

I ( h ) = Ω F ( x , h ) , h V 1 .

By using (E0), we have

I ( h ) = Ω F ( x , h ) Ω k ( x ) .

So, we choose α = Ω k ( x ) .

(L3-3). By (E1) and the variational inequality (1.4), we have

I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) 1 2 h 2 λ 2 2 Ω A h , h + b 1 b 1 , z V 1 ,

the proof of this lemma is completed. □

Next, we prove the Palais-Smale conditions at some levels for the functional I. We recall that I: E R is said to satisfy the Palais-Smale conditions at the level c R ( ( P S ) c in short) if any sequence { h n } E such that

I ( h n ) c , I ( h n ) 0 ,

as n , possesses a convergent subsequence in E. Moreover, we say that I satisfies the (PS) conditions when we have ( P S ) c for all c R .

Lemma 2.4Assume that the condition (E0) holds. Then the functionalIhas the ( P S ) c conditions whenever c < min { Ω T + , Ω T } or c > max { Ω S + , Ω S } .

Proof We only prove the condition for all c < min { Ω T + , Ω T } . For the case c > max { Ω S + , Ω S } , we can use similar methods.

1. Boundedness of the (PS) sequence.

The proof is by contradiction. Suppose that there exists a ( P S ) c unbounded sequence { h n } E such that c < min { Ω T + , Ω T } . For the ease of notation and without loss of generality, we assume that

h n ,

I ( h n ) c ,

I ( h n ) 0 , n .

We define h ¯ n = h n h n , hence there is an h ¯ E with the following properties:

h ¯ n h ¯ in E,

h ¯ n h ¯ in L p ( Ω ) × L p ( Ω ) , where 2 p < 2 and 2 = 2 N + 2 N ,

h ¯ n h ¯ a.e. in Ω.

For any Φ E , obviously I ( h n ) Φ h n 0 . By simple calculation, it is easy to obtain

I ( h n ) Φ = Ω H h n , H Φ Ω A h n , Φ + Ω F ( x , h n ) , Φ ,

where h n = ( h n ( 1 ) h n ( 2 ) ) , Φ = ( Φ ( 1 ) Φ ( 2 ) ) . We have

I ( h n ) Φ h n = Ω H h ¯ n , H Φ Ω A h ¯ n , Φ + Ω F ( x , h n ) , Φ h n 0 as  n .

From the convergence of { h ¯ n } , we have

Ω H h ¯ , H Φ = Ω A h ¯ , Φ .

We see that λ 1 = 1 , and by the definition of λ 1 , we obtain that h ¯ = ± Φ 1 . So, we suppose initially that h ¯ = Φ 1 . Because Φ 1 is positive, i.e., Φ 1 ( 1 ) > 0 , Φ 1 ( 2 ) > 0 , it is obvious that h n ( 1 ) , h n ( 2 ) , x Ω as n .

Hence, we can take h n = t n Φ 1 + ω n , where { t n } R , { ω n } V 1 , and we have

I ( h n ) = 1 2 t n Φ 1 + ω n 2 1 2 Ω A ( t n Φ 1 + ω n ) , t n Φ 1 + ω n + Ω F ( x , h n ) = 1 2 ω n 2 Ω A ω n , ω n + Ω F ( x , h n ) .

Using (1.4), we obtain

I ( h n ) 1 2 ( 1 1 λ 2 ) ω n 2 + Ω F ( x , h n ) . (2.1)

Since I ( h n ) c , it is easy to obtain that the sequence { ω n } is bounded. On the other hand, because of h n , on a subsequence | t n | , without loss of generality, we assume t n .

Now, using Hölder’s inequality and (E0), we have

| Ω F ( x , h n ) ω n | C ( Ω | F ( x , h n ) | 2 ) 1 2 .

Thus, applying the dominated convergence theorem, we conclude that

lim n Ω F ( x , h n ) ω n = 0 . (2.2)

On the other hand,

I ( h n ) ω n = Ω H h n , H ω n Ω A h n , ω n + Ω F ( x , h n ) , ω n = ω n 2 Ω A h n , ω n + Ω F ( x , h n ) , ω n .

Using (2.2), (1.4), we obtain

( 1 1 λ 2 ) ω n 2 | I ( h n ) ω n | + | Ω F ( x , h n ) , ω n | 0

as n . Therefore, by variational inequalities (1.3) and (1.4), we obtain that

ω n 2 Ω A h n , ω n 0 as  n .

Consequently, by virtue of Fatou’s lemma and (E0), we have

c = lim n ( 1 2 ω n 2 Ω A ω n , ω n + Ω F ( x , h n ) ) = lim inf n Ω F ( x , t n Φ 1 + ω n ) Ω lim inf n F ( x , t n Φ 1 + ω n ) = Ω T ,

which contradicts the condition c < min { Ω T + , Ω T } . Hence, the ( P S ) c sequence is bounded.

2. Various convergence of { h n } .

Since { h n } is a bounded sequence, there is an h E with the following properties:

h n h in E,

h n h in L p ( Ω ) × L p ( Ω ) , where 2 p < 2 and 2 = 2 N + 2 N ,

h n h a.e. in Ω.

3. { h n } convergence to h in E.

From the definition of ( P S ) c sequence, we have, as n ,

I ( h n ) h = Ω H h n , H h Ω A h n , h + Ω F ( x , h n ) , h 0 , I ( h n ) h n = Ω | H h n | 2 Ω A h n , h n + Ω F ( x , h n ) , h n 0 .

By Fatou’s lemma and the above convergence of { h n } , it is easy to show that

Ω A h n , h Ω A h , h , Ω F ( x , h n ) , h Ω F ( x , h ) , h , Ω A h n , h n Ω A h , h , Ω F ( x , h n ) , h n Ω F ( x , h ) , h

as n . Hence, we have

Ω H h n , H h Ω A h , h Ω F ( x , h ) , h as  n , (2.3)

Ω | H h n | 2 Ω A h , h Ω F ( x , h ) , h as  n . (2.4)

By weak convergence, we have

Ω H h n , H h Ω H h , H h as  n . (2.5)

Using (2.3), (2.4) and (2.5), by simple calculation, we obtain

Ω | H h n H h | 2 0 as  n .

The proof is completed. □

Lemma 2.5Suppose that (E0) and (E3) are satisfied. Then the origin is a local minimum for the functional I.

Proof Using (E3), we can choose p ( 2 , 2 ) and a constant C > 0 such that

F ( x , h ) 1 α 2 A h , h C | h | p , ( x , h ) Ω × R 2 .

Consequently, we have

I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) 1 2 ( 1 α ) h 2 C Ω | h | p 1 2 ( 1 α ) h 2 C h p 1 4 ( 1 α ) h 2 , h < ρ ,

where ρ is small enough and 0 < ρ < t 0 , t 0 is provided by (E5). Therefore the proof has been completed. □

To complete the mountain pass geometry, we prove the following result.

Lemma 2.6Let the hypotheses (E0), (E4) and (E5) hold. Then there exist h 0 E and ρ > 0 such that I ( h 0 ) < 0 and h 0 > ρ .

Proof Using (E2) and (E5), we take h 0 = t 0 Φ 1 , where t 0 is provided by (E5). Thus, we obtain

I ( t 0 Φ 1 ) = 1 2 t 0 Φ 1 2 1 2 Ω A ( t 0 Φ 1 ) , t 0 Φ 1 + Ω F ( x , t 0 Φ 1 ) , = Ω F ( x , t 0 Φ 1 ) < min { Ω T + , Ω T } < max { Ω S + , Ω S } 0 ,

and t 0 Φ 1 = t 0 . If we take 0 < ρ < t 0 , then the conclusion follows. □

Lemma 2.7Under hypotheses (E0), (E4) and (E5), problem (1.1) has at least one nontrivial solution h 0 E . Moreover, h 0 has negative energy, i.e., J ( h 0 ) < 0 .

Proof By (E0) and (1.4), we obtain

I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) Ω F ( x , h ) Ω k ( x ) .

Therefore, the functional I is bounded below. In this case, we would like to mention that I has the ( P S ) c conditions with c = inf { I ( h ) : h E } . For seeing this, by Lemma 2.4, we take t 0 R provided by (E5) we can obtain

c I ( t Φ 1 ) = Ω F ( x , t Φ 1 ) < min { Ω T + , Ω T } 0 .

Consequently, applying Lemma 2.2, we have one critical point h 0 E such that I ( h 0 ) = inf { I ( h ) : h E } I ( t Φ 1 ) < 0 . The proof of this lemma is completed. □

To prove Theorem 1.3, we establish the following lemma.

Lemma 2.8Assume that the conditions (E0), (E1), (E4) and (E6) hold. Then problem (1.1) has at least two nontrivial solutions with negative energy.

Proof Define

M + = { t Φ 1 + ω , t 0 , ω V 1 } , M = { t Φ 1 + ω , t 0 , ω V 1 } .

We have M + = M = V 1 . Hence, we minimize the functional I restricted to M + and M .

Firstly, we consider the functionals I ± = I | M ± . Using Lemma 2.4, I ± possesses the ( P S ) c conditions whenever c < min { Ω T + , Ω T } . Therefore, we obtain that I ± satisfies the ( P S ) c conditions with c ± = inf { I ± ( h ) : h M ± } .

In this way, by using Lemma 2.2 for the functional I ± , we obtain two critical points which we denote by h 0 + and h 0 , respectively. Thus, we have c + = I + ( h 0 + ) = inf h M + { I ( h ) } and c = I ( h 0 ) = inf h M { I ( h ) } .

Moreover, we affirm that h 0 + and h 0 are nonzero critical points. To see this, from (E4) and (E6), we obtain that

I ± ( h 0 ± ) I ± ( t 1 ± Φ 1 ) = Ω F ( x , t 1 ± Φ 1 ) < min { Ω T + , Ω T } 0 ,

and I restricted to V 1 is nonnegative. More specifically, given ω V 1 , using (L3-3) in Lemma 2.3, we have

I ( ω ) b 1 0 . (2.6)

Next, we prove that h 0 + and h 0 are distinct. The proof of this affirmation is by contradiction. If h 0 + = h 0 , then h 0 + = h 0 V 1 . Using (2.6), we obtain I ( h 0 + ) < 0 I ( h 0 + ) . Therefore, we have a contradiction. Consequently, we get h 0 + h 0 . Thus problem (1.1) has at least two nontrivial solutions. Moreover, these solutions have negative energy. □

3 Proof of main theorems

In this section, we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.

Proof of Theorem 1.1 From Lemma 2.4, the functional I satisfies the ( P S ) c conditions for some levels c R . Set E = V 1 V 1 , where V 1 = span { Φ 1 } . Using Lemma 2.3, we get that the functional I satisfies the saddle point geometry (see [12], Theorem 1.11). This implies that I has one critical point h 1 E . Theorem 1.1 is proved. □

Proof of Theorem 1.2 From Lemma 2.5 and Lemma 2.6, we know that the functional I satisfies the geometric conditions of the mountain pass theorem. Moreover, the functional I satisfies the ( P S ) c conditions for all c 0 . Thus, we have a solution h 2 E given by the mountain pass theorem. Obviously, the solution h 2 satisfies I ( h 2 ) > 0 .

On the other hand, by Lemma 2.7, we get another solution h 0 and I ( h 0 ) < 0 . It follows that problem (1.1) has at least two nontrivial solutions. The proof is completed. □

Proof of Theorem 1.3 Since the conditions (E0), (E3), (E4) and (E5) imply that Lemma 2.5 and Lemma 2.6 hold. Thus, we have one solution h 2 which satisfies I ( h 2 ) > 0 .

On the other hand, using Lemma 2.8, we obtain two distinct critical points h 0 ± such that I ( h 0 ± ) < 0 . Therefore, we obtain that problem (1.1) has at least three nontrivial solutions. The proof is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.

Acknowledgements

The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work was supported by the National Natural Science Foundation of China (11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

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