Abstract
In this article, we consider the non-linear Choquard equation
where
1 Introduction and main results
In the past two decades, many authors have devoted to the study of existence, multiplicity, and properties of the solutions of the non-linear Choquard equation (1.1),
In a early paper [1], Lieb proved that the ground state
is radial and unique up to translations. While Lions [2] showed the existence of a sequence of radially symmetric solutions via variational methods. In [3,4], the authors proved, if
and can be spanned by
where
for some
The aim of the present article is to consider the following non-linear Choquard equation:
where potential
as
To apply variational methods, we introduce the energy functional associated with equation (1.4) by
The Hardy-Littlewood-Sobolev inequality implies that
The main result of this article is to establish the existence of infinitely many non-radial solution for (1.4) under assumption
Theorem 1.1
Suppose that assumption
To prove the main results, we will adopt the idea introduced by Wei and Yan in [15] to use the unique ground state
and
where
where
To prove Theorem 1.1 we only need to prove the following result:
Theorem 1.2
Suppose that
where
This article is organized as follows. In Section 2, we prove two basic estimates. In Section 3, we carry out the reduction. Then, we study the reduced finite dimensional problem and prove Theorem 1.2 in Section 4.
2 Preliminaries
Throughout this article we write
Let
Then, we have the following basic estimates:
Lemma 2.1
For any
Proof
For any
If
So, for any
Thus,
Lemma 2.2
For any
Proof
In view of the symmetry, we only estimate the function
By Lemma 2.1, we have
Thus for any
Therefore,
and
Consequently, (2.2) follows.□
3 The reduction argument
Let
Applying Lemma 2.2, there exists a bounded linear operator
Thus, we have
Lemma 3.1
There is a constant
Next, we show that
Lemma 3.2
There is a constant
Proof
Suppose to the contrary that there are
Then
We may assume that
By symmetry, we see from (3.1),
In particular,
and
Let
So, we may assume that there is a
and
Since
we obtain
So,
Now, we claim that
Define
For any
By Lemma 2.2, we know
Similarly,
Thus, we have
On the other hand, since
But (3.7) holds for
As a result,
On the other hand, it follows from Lemma 2.1 that for any small
Thus,
This is a contradiction to (3.3).□
Let
Expand
where
and
In order to find a critical point
Lemma 3.3
There is a constant
and
Proof
Similar to the proof of (3.1), we have that for any
and
Lemma 3.4
Moreover, there is a small
Proof
By the symmetry of the problem,
because
By Lemma 2.1, we have
and
Combining these with
we have,
Since
Thus, we have
Proposition 3.5
There is an integer
Moreover, there is a small
Proof
Since
Thus, finding a critical point for
By Lemma 3.2,
Let
So, from Lemma 3.4,
Thus,
By (3.11), we have,
So, we have proved that
4 Proof of Theorem 1.2
Lemma 4.1
There is a small constant
where
and
Proof
Using the symmetry,
It follows from Lemma 2.1 that
Using Lemma 2.1 and the fact
we obtain that if
Thus, we have
On the other hand, we have
and
So,
where
and
We are ready to prove Theorem 1.2. Let
With the same argument in [15,16], we can easily check that for
Proof of Theorem 1.2
It follows from Lemmas 3.1 and 3.3 that
So, Proposition 3.5 and Lemma 4.1 give
where
Consider
where
has a maximum point
which is an interior point of
is a solution of (1.4).□
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Funding information: Fashun Gao was partially supported by NSFC (11901155). Minbo Yang is the corresponding author who was partially supported by NSFC (11971436, 12011530199) and ZJNSF (LZ22A010001, LD19A010001).
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Conflict of interest: Authors state no conflict of interest.
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© 2022 Fashun Gao and Minbo Yang, published by De Gruyter
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