Ground State Solutions for a Kind of Schrödinger-Poisson System with Upper Critical Exponential Convolution Term ()
1. Introduction
Recently, the following Schrödinger-Poisson system has been studied widely by researchers
(1.1)
where
,
, the external potential function
and the nonlinearity
. (1.1) is also called Schrödinger-Maxwell system, which appears in an amusing physical background. In fact, based on a classical physical model, coupled nonlinear Schrödinger-Poisson equation can be used to describe the interaction between charge particles and electromagnetic field. For more physical contexts of the Schrödinger-Poisson system, we refer the readers to the papers [1] [2] and the references therein.
There are lots of extended research on (1.1) in
. When
is a constant and
, Khoutir [3] proved that (1.1) possesses a least energy sign-changing solution and a ground state solution by variational methods under some relaxed assumptions on f. When
, (1.1) reduces to the following class of Schrödinger-Possion system
(1.2)
when
, by introducing some new variational and analytic techniques, Chen, Shi and Tang [4] showed that (1.2) has a nontrivial solution of mountain pass type and a ground state solution of Neheri-Pohožaev type in
. By variational methods and Miranda’s theorem, Alves et al. [5] proved that (1.2) admits a least energy sign-changing solution in
when f satisfies some special assumptions. Similarly, combining constraint variational method and quantitative deformation lemma, Shuai and Wang [6] proved that (1.1) possesses a sign-changing solution
. Moreover, they showed that any sign-changing solution of (1.1) has energy exceeding more than twice the least energy. There are a lot of works about (1.2) and we refer to the literature [6] [7] and references therein.
Without the internal potential
, (1.1) reduces to the following Schrödinger equation:
(1.3)
Using Berestycki-Lions conditions on f, Chen and Tang [8] studied generalized nonlinear Schrödinger equation with variable potential. By introducing skillful ideas and relaxed assumptions on
, they obtain a ground state solution of Pohožaev type and a least energy solution. Besides, there are many results of sign-changing ground state solutions of (1.3). We refer to [9] - [15] and references therein.
Let
and
, the Schrödinger-Poisson system (1.1) becomes the following equation with convolution nonlinearity.
(1.4)
where
,
and
. Under mild assumptions on nonlinear perturbation g and V, Chen and Tang [11] proved that (1.4) has a ground state solution in two cases by using new inequalities. In their work, when
, they established the Nehari-Pohožaev manifold and proved that (1.4) has a solution. Next, they defined the Nehari manifold to obtain the existence of the solution when
. For more details about assumptions and techniques of (1.4), we refer to [6] [11] [16] [17] [18].
In this paper, we mainly focus on the following equations:
(1.5)
where V satisfies the following assumptions:
(V1)
,
for all
and
;
(V2)
, the set
has finite Lebesgue measure for every
, and the function
is increasing on
for every
.
In three-dimensional space, the Riesz potential
is defined as a function of
:
where
is the Gamma function. It is widely known that for any
, there exists a unique
such that
by using the Lax-Milgram theorem, moreover,
(1.6)
Inserting (1.6) into (1.5), we get the following equation
(1.7)
The following inequality, which is a special case of Hardy-Littlewood-Sobolev inequality, plays a significant role in resolving the difficulty of relatively compact. There exists sharp constant S, independent of u, such that
(1.8)
whose external function is
(1.9)
is invariant under dilations
[19]. Next, we define the energy functional:
(1.10)
Then for any
,
(1.11)
To state our result, we define the Nehari-Pohožaev manifold as follows:
(1.12)
where
(1.13)
Our main result is as follows.
Theorem 1.1. Assume
, V satisfies (V1), (V2). Then problem (1.5) has a ground state solution
such that
(1.14)
Notations.
●
denotes the usual Sobolev space equipped with the inner product and norm
●
denotes the Lebesgue space with the norm
.
● For any
and
,
.
● For any
,
for
.
●
denote positive constants possibly different in different places.
2. Preliminaries
As usual, we assume
. By (1.11) and (1.13), we have
(2.1)
First, we give some key inequalities.
(2.2)
(2.3)
Inspired by Tang and Chen [11], we establish a key functional inequality as follows.
Lemma 2.1. Assunme that (V1) and (V2) hold. Then
(2.4)
Proof. Note that
(2.5)
Thus, by (1.10), (1.11), (2.2), (2.3) and (2.5), one has
(2.6)
The proof of Lemma 2.1 is complete. □
Assume that
, from (2.4), we have
(2.7)
To solve the trouble caused by the lack of compactness of Sobolev space embedding in
, we define the following energy functional when
(2.8)
According to (1.12) and (2.1), we define
(2.9)
and
(2.10)
From Lemma 2.1, we can deduce the following corollaries.
Corollary 2.2. Assume that (V1) holds. Then
(2.11)
Corollary 2.3. Assume that (V1) and (V2) hold. Then for
(2.12)
Lemma 2.4. ( [11]: Lemma 2.7) Assume that (V1) and (V2) hold. Then there exist
such that
(2.13)
(2.14)
Lemma 2.5. For any
, there exists a unique
such that
.
Proof. Let
be fixed and define a function
on
. Clearly, by (2.1) and (2.5), we have
(2.15)
By (V1), one has
and
for
small and
for t large. Therefore,
has a critical point which corresponds to its maximum, namely, there is a
so that
and
. Then, we claim that
is unique. Similar to the proof of ( [20]: Lemma 3.3), for any
which is given, if there are two positive constants
such that
. Then
. Together with (2.4), we have
(2.16)
and
(2.17)
(2.2), (2.16) and (2.17) imply
. Hence,
is unique for any
. □
Corollary 2.6. For any
, there exists a unique
such that
.
Combing Corollary 2.3 and Lemma 2.5, we get
and the following minimax characterization.
Lemma 2.7. ( [11]: Lemma 2.10) Assume (V1) and (V2) hold. Then
Lemma 2.8. Assume that (V1), (V2) hold. Then
(i) There exists
such that
,
;
(ii)
.
Proof. (i) Since
,
, by (1.8), (2.1) and Sobolev embedding theorem, one has
(2.18)
which implies
(2.19)
(ii) Let
be such that
. Cases: (1)
; (2)
exist.
Case (1).
. By (2.4), one has
(2.20)
Case (2).
. From (2.19), we have
(2.21)
By Hardy-Littlewood-Sobolev inequality, one has
(2.22)
Let
, then (2.21) implies that
is bounded. Since
, it follows from (2.15) and (2.22) that
(2.23)
Cases (1) and (2) show that (ii) holds. □
Lemma 2.9. Assume that (V1) and (V2) hold. Then
.
Proof. In view of Lemma 2.1 and Corollary 2.3, we have
. By contradiction, we assume that
. Let
. Then there exists
such that
(2.24)
In view of Lemma 2.5, there exists
such that
. Hence, joining with (V1), (V2), (1.10), (2.5), (2.11) and (2.24), we have
(2.25)
This is a contradiction. Therefore, the conclusion of Lemma 2.11 is true. □
Lemmma 2.10. ( [11]: Lemma 2.12) Assume that (V1) and (V2) hold. If
in
, then along a subsequence,
(2.26)
(2.27)
(2.28)
Lemma 2.11. Assume that (V1), (V2) hold. Then m is achieved.
Proof. In view of Lemmas 2.5 and 2.8, we have
and
. Let
be such that
. Then it follows from (1.10), (2.1) and (2.27) that
(2.29)
It indicates that
is bounded. Next, we will verify that
is also bounded. From (V1), (1.10), (2.1), (2.13), (2.29) and the Soblev embbeding inequality, we derive
(2.30)
Together with (2.29), (2.30) implies that
is bounded in
. Passing to a subsequence, we can get
in
. Then
in
for
and
a.e. in
. For
, there are two cases: (1)
and (2)
.
Case (1).
, i.e.
in
. Then
in
for
and
a.e. in
. Using (V1) and (V2), it is easy to prove that
(2.31)
From (1.10), (2.1), (2.8), (2.10) and (2.31), one has
(2.32)
By (1.8), (2.1) and Lemma 2.8 (i), we have
(2.33)
According to (2.33) and Lion’s concentration compactness priciple ( [21]: Lemma 1.21), we can prove that there exist
and
such that
. Let
. Then we have
and
(2.34)
Hence, there exists
such that, passing to a subsequence,
(2.35)
Let
. Then (2.35) and Lemma 2.10 yield
(2.36)
We set
(2.37)
From (2.8), (2.10), (2.24), (2.36) and (2.37), one has
(2.38)
If there exists a subsequence
of
such that
, then we have
(2.39)
Next, we consider that
. We claim that
. By contradiction, when
, that is (2.38) implies
for large n. In view of Corollary 2.6, there exists
such that
for large n. From (2.8), (2.10), (2.11), (2.38) and Lemma 2.9, one has
(2.40)
Since
, the above result is impossible, this shows that
. In view of Lemma 2.1, there exists
such that
. From (2.8), (2.10), (2.11), (2.32), (2.34), (2.37) Fatou’s lemma and Lemma 2.9, one has
(2.41)
which implies (2.39) holds also. In view of Lemma 2.3, there exists
such that
, moreover, it follows from (V1), (1.10), (2.8), (2.39) and Corollary 2.3 that
(2.42)
Case (ii).
. In this case, the proof is similar to (2.39), by using E and J instead of
and
, we can deduce that
and
.
Similar to the [22] and [23], we can obtain the following conclusion. □
Lemma 2.12. Assume that (V1), (V2) hold. If
and
, then
is a critical point of E.
Proof. From (2.1) and Lemma 2.5, there exist
and
such that
(2.43)
From (2.2) and (2.4), we have
(2.44)
and (2.44) implies
(2.45)
The next proof steps are routine. Similar to [22], we can verify Lemma 2.12 by using (2.43) and (2.44) instead of ( [22]: (2.34) and (2.35)). □
Proof of Theorem 1.1. In view of Lemmas 2.9 and 2.10, there exists
such that
(2.46)
This shows that
is a ground state solution of (1.4). □
3. Conclusion
Although one can establish a (PS) sequence in a nonstandard way, it is not easy to prove its boundness because of the convolution term
and lack of Ambrosetti-Rabinowitz condition of Choquard type. To overcome this difficulty, we introduce an auxiliary function. Firstly, we proved that there exists a unique
such that
. What’s more, we find out the minima of the energy functional. Next, we get that m is achieved, that is the energy value of the minima of the energy functional is achieved by Mountain Pass Theorem. Finally, we proved that the limit of the (PS) sequence, that is
, is the critical point of E. It is obvious that for the Schrödinger-Poisson system with upper critical exponential convolution term, its ground state solution also exists. We hope the result can be widely used in Schrödinger-Poisson systems.
Acknowledgements
The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.
Funding
This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040).
Authors’ Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.