Multiple solutions of semilinear elliptic systems on the Heisenberg group
Boundary Value Problemsvolume 2013, Article number: 157 (2013)
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.
Let be the space equipped with the following group operation:
where ‘⋅’ denotes the usual inner-product in . This operation endows with the structure of a Lie group. The vector fields , , T, given by
form a basis for the tangent space at .
Definition 1.1 The Heisenberg Laplacian is by definition
and let denote the 2N-vector .
Definition 1.2 The space is defined as the completion of in the norm
In this paper, we study the problems on the existence and multiplicity of solutions for the system
where is a bounded smooth domain, and . Moreover, we assume that there is some function such that . Here ∇F denotes the gradient in the variable u and v, i.e., , .
In fact, the condition in was studied by da Silva; we can see . In this paper we study the problem on the Heisenberg group . 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 is equivalent to finding critical points of the functional given by
and denotes the usual inner product in .
We introduce the eigenvalue problem with weights. Let us denote by the set of all continuous, cooperative and symmetric matrices A of order 2, given by
where the functions satisfy the following conditions:
(A1) is cooperative, that is, .
(A2) There is an such that or .
Given , consider the weighted eigenvalue problem
if . By virtue of the spectral theory for compact operators, we obtain the sequence of eigenvalues
where with .
The following variational inequalities hold:(1.3)(1.4)
The variational inequalities will be used in the next section. We would like to mention that the is positive in Ω. In the paper, without loss of generality, we assume that .
We now state the assumptions and the main results in this paper. Firstly, we define the following functions:
The above functions belong to and the limits are taken a.e. and uniformly in .
Now we make the following basic hypotheses:
(E0) There exists such that
(E1) , , .
(E2) , .
(E3) There exist and such that
(E4) and .
(E5) There exists such that
(E6) There are and such that
We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.
Theorem 1.1 Let be a bounded smooth domain, and . Assume that there is some function such that , . Furthermore, if the conditions (E0), (E1), (E2) are satisfied, problem (1.1) has at least one solution .
Remark 1.2 For the hypotheses and , problem (1.1) admits the trivial solution . In this case, the main point is to assure the existence of nontrivial solutions.
Theorem 1.2 Let be a bounded smooth domain, and . Assume that there is some function such that , . Furthermore, if the conditions (E0), (E2), (E3), (E4) and (E5) are satisfied, then problem (1.1) has at least two nontrivial solutions.
Theorem 1.3 Let be a bounded smooth domain, and . Assume that there is some function such that , . 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 ) as follows.
Lemma 2.1 Let be a bounded domain and let . Then compactly embedding in , where .
To establish Lemmas 2.7 and 2.8, we introduce the following corollary of the Ekeland variation principle (see ).
Lemma 2.2 X is a metric space, is bounded from below, which satisfies the condition, then is a critical value of E.
Next, we describe some results under the geometry for the functional I.
Lemma 2.3 Under hypotheses (E0) and (E1), the functional I has the following saddle geometry:
(L3-1) if with .
(L3-2) There is such that , .
(L3-3) , .
Proof (L3-1). From (1.2), (1.4) we have
Using (E0), we have , as .
(L3-2). By simple calculation, we get
By using (E0), we have
So, we choose .
(L3-3). By (E1) and the variational inequality (1.4), we have
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: is said to satisfy the Palais-Smale conditions at the level ( in short) if any sequence such that
as , possesses a convergent subsequence in E. Moreover, we say that I satisfies the (PS) conditions when we have for all .
Lemma 2.4 Assume that the condition (E0) holds. Then the functional I has the conditions whenever or .
Proof We only prove the condition for all . For the case , we can use similar methods.
Boundedness of the (PS) sequence.
The proof is by contradiction. Suppose that there exists a unbounded sequence such that . For the ease of notation and without loss of generality, we assume that
We define , hence there is an with the following properties:
in , where and ,
a.e. in Ω.
For any , obviously . By simple calculation, it is easy to obtain
where , . We have
From the convergence of , we have
We see that , and by the definition of , we obtain that . So, we suppose initially that . Because is positive, i.e., , , it is obvious that , , as .
Hence, we can take , where , , and we have
Using (1.4), we obtain
Since , it is easy to obtain that the sequence is bounded. On the other hand, because of , on a subsequence , without loss of generality, we assume .
Now, using Hölder’s inequality and (E0), we have
Thus, applying the dominated convergence theorem, we conclude that
On the other hand,
Using (2.2), (1.4), we obtain
as . Therefore, by variational inequalities (1.3) and (1.4), we obtain that
Consequently, by virtue of Fatou’s lemma and (E0), we have
which contradicts the condition . Hence, the sequence is bounded.
Various convergence of .
Since is a bounded sequence, there is an with the following properties:
in , where and ,
a.e. in Ω.
convergence to h in E.
From the definition of sequence, we have, as ,
By Fatou’s lemma and the above convergence of , it is easy to show that
as . Hence, we have
By weak convergence, we have
Using (2.3), (2.4) and (2.5), by simple calculation, we obtain
The proof is completed. □
Lemma 2.5 Suppose that (E0) and (E3) are satisfied. Then the origin is a local minimum for the functional I.
Proof Using (E3), we can choose and a constant such that
Consequently, we have
where ρ is small enough and , is provided by (E5). Therefore the proof has been completed. □
To complete the mountain pass geometry, we prove the following result.
Lemma 2.6 Let the hypotheses (E0), (E4) and (E5) hold. Then there exist and such that and .
Proof Using (E2) and (E5), we take , where is provided by (E5). Thus, we obtain
and . If we take , then the conclusion follows. □
Lemma 2.7 Under hypotheses (E0), (E4) and (E5), problem (1.1) has at least one nontrivial solution . Moreover, has negative energy, i.e., .
Proof By (E0) and (1.4), we obtain
Therefore, the functional I is bounded below. In this case, we would like to mention that I has the conditions with . For seeing this, by Lemma 2.4, we take provided by (E5) we can obtain
Consequently, applying Lemma 2.2, we have one critical point such that . The proof of this lemma is completed. □
To prove Theorem 1.3, we establish the following lemma.
Lemma 2.8 Assume that the conditions (E0), (E1), (E4) and (E6) hold. Then problem (1.1) has at least two nontrivial solutions with negative energy.
We have . Hence, we minimize the functional I restricted to and .
Firstly, we consider the functionals . Using Lemma 2.4, possesses the conditions whenever . Therefore, we obtain that satisfies the conditions with .
In this way, by using Lemma 2.2 for the functional , we obtain two critical points which we denote by and , respectively. Thus, we have and .
Moreover, we affirm that and are nonzero critical points. To see this, from (E4) and (E6), we obtain that
and I restricted to is nonnegative. More specifically, given , using (L3-3) in Lemma 2.3, we have
Next, we prove that and are distinct. The proof of this affirmation is by contradiction. If , then . Using (2.6), we obtain . Therefore, we have a contradiction. Consequently, we get . 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 conditions for some levels . Set , where . Using Lemma 2.3, we get that the functional I satisfies the saddle point geometry (see , Theorem 1.11). This implies that I has one critical point . 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 conditions for all . Thus, we have a solution given by the mountain pass theorem. Obviously, the solution satisfies .
On the other hand, by Lemma 2.7, we get another solution and . 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 which satisfies .
On the other hand, using Lemma 2.8, we obtain two distinct critical points such that . Therefore, we obtain that problem (1.1) has at least three nontrivial solutions. The proof is completed. □
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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).
The authors declare that they have no competing interests.
We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.