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 group1 Introduction
Let
where ‘⋅’ denotes the usual innerproduct in
form a basis for the tangent space at
Definition 1.1 The Heisenberg Laplacian is by definition
and let
Definition 1.2 The space
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
where
In fact, the condition in
We use the variation methods to solve problem (1.1). Finding weak solutions of (1.1)
in
where
and
We introduce the eigenvalue problem with weights. Let us denote by
where the functions
(A_{1})
(A_{2}) There is an
Given
if
such that
where
Remark 1.1
(1)
(2) The following variational inequalities hold:
The variational inequalities will be used in the next section. We would like to mention
that the
We now state the assumptions and the main results in this paper. Firstly, we define the following functions:
The above functions belong to
Now we make the following basic hypotheses:
(E_{0}) There exists
(E_{1})
(E_{2})
(E_{3}) There exist
(E_{4})
(E_{5}) There exists
(E_{6}) There are
We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.
Theorem 1.1Let
Remark 1.2 For the hypotheses
Theorem 1.2Let
Theorem 1.3Let
2 Preliminaries and fundamental lemmas
In this section, we prove some lemmas needed in the proof of our main theorems.
We first introduce the FollandStein embedding theorem (see [10]) as follows.
Lemma 2.1Let
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,
Next, we describe some results under the geometry for the functional I.
Lemma 2.3Under hypotheses (E_{0}) and (E_{1}), the functionalIhas the following saddle geometry:
(L31)
(L32) There is
(L33)
Proof (L31). From (1.2), (1.4) we have
Using (E_{0}), we have
(L32). By simple calculation, we get
By using (E_{0}), we have
So, we choose
(L33). By (E_{1}) and the variational inequality (1.4), we have
the proof of this lemma is completed. □
Next, we prove the PalaisSmale conditions at some levels for the functional I. We recall that I:
as
Lemma 2.4Assume that the condition (E_{0}) holds. Then the functionalIhas the
Proof We only prove the condition for all
1. Boundedness of the (PS) sequence.
The proof is by contradiction. Suppose that there exists a
We define
For any
where
From the convergence of
We see that
Hence, we can take
Using (1.4), we obtain
Since
Now, using Hölder’s inequality and (E_{0}), we have
Thus, applying the dominated convergence theorem, we conclude that
On the other hand,
Using (2.2), (1.4), we obtain
as
Consequently, by virtue of Fatou’s lemma and (E_{0}), we have
which contradicts the condition
2. Various convergence of
Since
3.
From the definition of
By Fatou’s lemma and the above convergence of
as
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 (E_{0}) and (E_{3}) are satisfied. Then the origin is a local minimum for the functional I.
Proof Using (E_{3}), we can choose
Consequently, we have
where ρ is small enough and
To complete the mountain pass geometry, we prove the following result.
Lemma 2.6Let the hypotheses (E_{0}), (E_{4}) and (E_{5}) hold. Then there exist
Proof Using (E_{2}) and (E_{5}), we take
and
Lemma 2.7Under hypotheses (E_{0}), (E_{4}) and (E_{5}), problem (1.1) has at least one nontrivial solution
Proof By (E_{0}) and (1.4), we obtain
Therefore, the functional I is bounded below. In this case, we would like to mention that I has the
Consequently, applying Lemma 2.2, we have one critical point
To prove Theorem 1.3, we establish the following lemma.
Lemma 2.8Assume that the conditions (E_{0}), (E_{1}), (E_{4}) and (E_{6}) hold. Then problem (1.1) has at least two nontrivial solutions with negative energy.
Proof Define
We have
Firstly, we consider the functionals
In this way, by using Lemma 2.2 for the functional
Moreover, we affirm that
and I restricted to
Next, we prove that
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
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
On the other hand, by Lemma 2.7, we get another solution
Proof of Theorem 1.3 Since the conditions (E_{0}), (E_{3}), (E_{4}) and (E_{5}) imply that Lemma 2.5 and Lemma 2.6 hold. Thus, we have one solution
On the other hand, using Lemma 2.8, we obtain two distinct critical points
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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