Multiple solutions of semilinear elliptic systems on the Heisenberg group
© Jia et al.; licensee Springer. 2013
Received: 17 November 2012
Accepted: 10 June 2013
Published: 1 July 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.
form a basis for the tangent space at .
and let denote the 2N-vector .
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].
and denotes the usual inner product in .
where the functions satisfy the following conditions:
(A1) is cooperative, that is, .
(A2) There is an such that or .
where with .
- (2)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 .
The above functions belong to and the limits are taken a.e. and uniformly in .
Now we make the following basic hypotheses:
(E1) , , .
(E2) , .
(E4) and .
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) , .
Using (E0), we have , as .
So, we choose .
the proof of this lemma is completed. □
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 .
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.
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 .
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 .
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.
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.
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 .
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., .
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 .
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. □
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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