Abstract
In this article, we study the elliptic system:
where
1 Introduction
In this article, we study the following elliptic system:
where
System (1.1) is the nonautonomous case of the following elliptic system:
where
Gidas et al. [5] studied the elliptic equation:
They proved that if
where
Following these ideas, we prove the following.
Theorem 1.1
Let
In [8,10,11], Kajikiya obtained the multiple nonradial solutions of equation (1.4) by the group critical theory. By the unique and nondegenerate results, they proved, if
Theorem 1.2
For all
This article is organized as follows. In Section 2, we present several preliminary results, and in Sections 3 and 4, we present the proof of the main results.
2 Some preliminaries and lemmas
Let
We assume
The energy functionals of (1.1) and (1.2) are
and
The Nehari manifolds are
and
Definition 2.1
(
If the infimum is obtained from the radial functions, we denote it as
By the moving plane method, one can prove that the solutions of system (1.2), denoted
has only trivial solutions, in this case
3 The Proof of Theorem 1.1
Lemma 3.1
For any
Proof
For any
and thus,
We have completed the proof.□
Lemma 3.2
where
Proof
By the variational method, we can prove that
Let
When
where
and
By combining these results, we have that
Let
On the other hand, we have
and similarly, we have that
Combining the previous inequalities (3.2)–(3.4), we have that
where
Lemma 3.3
Let
where
Proof
For convenience, we replace
where
then we have
and
and
and
By combining the previous equalities, we obtain the following:
Since
Thus,
By divergence theorem, we obtain that , i.e.,
Again since
Since
and
Since
and
and then we obtain
By the divergence theorem, we have
and
and
Thus, we obtain that
and
We also obtain that
and by the previous estimations, one can obtain that
From (3.10), we obtain
and then we have
and
If
where
Now we list the proof of Theorem 1.1.
The proof of Theorem 1.1
By Lemmas 3.2 and 3.3, we can prove that if
then
4 The Proof of Theorem 1.2
By using the idea of Wei and Yao [20], we obtain the following results.
Lemma 4.1
Let
Proof
Following the idea of Gidas et al. [5], Damascelli and Pacella [14], or Troy [15], we prove that the positive solutions are radially symmetric.
Let
where
Let
By the compact theorem, one can prove that
Next we present the proof of Theorem 1.2.
By contradiction, the ground state solutions of (1.1) are nonradial. By the variational method, we prove that (1.1) has a nontrivial radial solution, one can see [22], denoting
where
By Hölder’s and Young’s inequality, we have that
where
Then we obtain that
Again by Hölder and Sobolev embedding inequality, we obtain
and
Thus, we obtain that
Now let
Notice that
and thus,
Let
Following the idea of [8], by contradiction,
then
then
By the
where
Now let
It is a contradiction, then we complete the proof.
Acknowledgement
The authors thank the referees for valuable comments and suggestions, which improved the presentation of the manuscript.

Funding information: Lou is supported by NSFC 11571339 and 12101192 and Key Scientific Research Projects of Higher Education Institutions in Henan Province 20B110004.

Conflict of interest: The authors state no conflict of interest.

Data availability statement: No data, models, or code are generated or used during the study.
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