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
Definition 1.1 The Heisenberg Laplacian is by definition
In this paper, we study the problems on the existence and multiplicity of solutions for the system
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].
(2) The following variational inequalities hold:
We now state the assumptions and the main results in this paper. Firstly, we define the following functions:
Now we make the following basic hypotheses:
We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.
Theorem 1.2Letbe a bounded smooth domain, and. Assume that there is some functionsuch that, . Furthermore, if the conditions (E0), (E2), (E3), (E4) and (E5) are satisfied, then problem (1.1) has at least two nontrivial solutions.
Theorem 1.3Letbe a bounded smooth domain, and. Assume that there is some functionsuch 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.
To establish Lemmas 2.7 and 2.8, we introduce the following corollary of the Ekeland variation principle (see ).
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:
Proof (L3-1). From (1.2), (1.4) we have
(L3-2). By simple calculation, we get
By using (E0), we have
(L3-3). By (E1) and the variational inequality (1.4), we have
the proof of this lemma is completed. □
1. Boundedness of the (PS) sequence.
Using (1.4), we obtain
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
Consequently, by virtue of Fatou’s lemma and (E0), 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 2.5Suppose that (E0) and (E3) are satisfied. Then the origin is a local minimum for the functional I.
Consequently, we have
To complete the mountain pass geometry, we prove the following result.
Proof By (E0) and (1.4), we obtain
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.
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 .
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.
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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