1. Introduction and Main Results
In this article, we investigate the following fractional Schrödinger-Poisson system with logarithmic nonlinearity:
where
,
,
, and
with
denotes the fractional Laplacian operator defined as
where
represents the principal value sense,
represents an appropriate normalization constant. It is worth pointing out that the application background of fractional equations is rooted in areas such as fractional quantum mechanics, physics, finance, conformal geometry, among others; see [
1] for more details. In particular, when
and
system (
1) gains significant interest in physics as it comes from the semi-relativistic theory in the repulsive (plasma Coulomb case).
In recent years, the study of small semiconductor devices has been stimulated increasingly interest, in particular, in the use of quantum-mechanical and numerical methods to explain quantum phenomena like quantum interference, size quantization and tunneling. Since the early 1980s, the Schrödinger-Poisson system, which is the coupling of a Maxwell equations with Schrödinger equation, has been widely adopted as a mathematical framework to explore and evaluate mathematical elements that are crucial for modeling semiconductor heterostructures. For a comprehensive overview of the Schrödinger-Poisson system and related models, for example we refer to [
2].
The single particle system, named the Schrödinger-Poisson system, regulates the temporal evolution of the wave function
, which depicts the condition of a non-relativistic quantum particle in space under the influence of a self-consistent potential
V generated by its own charge. When related to a single particle system in a vacuum, the Schrödinger equation in
is formulated as
where
represents Planck’s constant and
m signifies the mass of the particle. To find
U, we combine this equation with the Poisson equation:
where
represents the anticipated particle density for a pure quantum state in the spatial domain
at time
t. The value of
is +1 when the Coulomb force is repulsive and −1 when it is attractive. Our primary focus in this paper is the repulsive case, and the Poisson equation represents the repulsive character of the Coulomb force.
Over the past three decades, the Schrödinger-Poisson system
has been the subject of extensive research because of its wide physical applications. The model like (
2) proposed by Benci [
3] has been used to describe the relationship between the nonlinear steady-state Schrödinger equation and the electrostatic field, and it is widely used in quantum mechanical models and semiconductor theory. Under the specific hypothesis of
U and
M, Liu and Guo [
4] proved that, by utilizing variational methods, system (
2) has a minimum of one positive ground state solution. In [
5], Zhang et al. demonstrated the existence of high-energy solutions for system (
2) by employing the linking theorem with
. In [
6], Zhong and Tang explored system (
2) where
,
, and established the problem has at least one ground state sign-changing solution by employing the constraint variational method.
In the framework of fractional Laplacian systems, there are numerous results related to the fractional Schrödinger-Poisson system. Here we list some results related to our paper. Zhang et al. [
7] investigated the fractional Schrödinger-Poisson system with subcritical and critical nonlinear terms:
Through a perturbation method, they obtained the existence of positive solutions and detailed the asymptotic of solutions. Employing Pohozaev-Nehari manifold, the monotonicity trick and global compactness Lemma, Teng in [
8] obtained the existence of ground state solutions for (
3) with
. With the help of the Ljusternik-Schnirelmann theory and penalization techniques, Ambrosio in [
9] proved the concentration and multiplicity of positive solutions for system (
3) with
.
Lately, the logarithmic Schrödinger equation expressed as
with
, has garnered significant attention because of its profound impact in various fields, including effective quantum, quantum mechanics, and Bose-Einstein condensation. Finding the standing waves of (
4), which are represented by
where
, is essential. This substitution transforms the equation into
The associated energy functional can be expressed as
Nonetheless,
might not be well-defined in
as there is a
such that
. More precisely, we consider the case where
and
u is a smooth function defined as
for
and
for
. In this scenario,
u belongs to
and
, assuming
V grows slowly enough, such as
when
. To resolve this problem, various techniques have been developed by researchers. Next we review some established results about logarithmic Schrödinger equations. In [
10], the authors applied genus theory and the minimax principles for lower semicontinuous functionals as detailed in [
11] to find multiple solutions for the problem (
5) with periodic potential. Later, inspired by the ideas presented in [
10], Ji and Szulkin in [
12] proved the existence of multiple solutions for the Equation (
5) where
V meets
and .
, and spectrum .
When the potential meets
, they acquire the existence of infinitely many solutions for (
5) and there exists a ground state solution for (
5) when the potential meets
. By employing variational methods, Alves and Ji in [
13] established the existence of multi-bump positive solutions for the equation similar to (
5). Another subject that has gained growing attention lately is the logarithmic Schrödinger-Poisson system. Recently, Peng [
14] considered existence and concentration of positive solutions for the logarithmic Schrödinger-Poisson system
via variational method and penalization scheme under local assumption that potential meets.
Inspired by the above studies, this paper explores the existence of multiple solutions for the logarithmic fractional Schrödinger-Poisson system. To the best of our knowledge, in the fractional scenario, literature on the Schrödinger-Poisson system with logarithmic features is relatively scarce. In the following, we present the main results.
Theorem 1. Assume that V satisfies with , problem (1) possesses infinitely many solutions such that . Theorem 2. Assume that V satisfies with , problem (1) possesses a ground state solution. Let us outline the main challenges we faced in this paper: Because of the logarithmic terms,
may exist such that
, which can result in the corresponding functional attaining
. Therefore, the functional is not well-defined in
H, which makes traditional variational methods inapplicable in this situation. To find solutions for (
1), similar to [
10], we decompose the functional into the sum of a
functional and a lower semicontinuous convex functional. As far as we know, there are few available results about multiplicity of solutions for fractional Schrödinger-Poisson system, even in the Laplacian setting.
Remark 1. Our results outline two key differences compared to those of [10]: Our equation includes not only logarithmic term but also the nonlocal term ; We extend the Equation (5) to the fractional Laplacian setting. This paper is divided into the following sections. The second section provides a review of several lemmas that are utilized throughout the paper. In the third section, we give the proof of Theorem 1. The fourth section is dedicated to demonstrating Theorem 2.
Throughout this article, we note the following:
C and are different positive constants.
The norm is defined as .
Define as an open ball with radius centered at u, and let .
For a functional I on H, denote by A the critical point set of I, , , and .
2. Preliminaries
Let us first define the homogeneous fractional Sobolev space
as
which represents the closure of
in relation to the norm
for
; see [
1] for more details.
Through the Fourier transform [
1], the fractional Sobolev space
is defined as
equipped with the norm
Based on the Plancherel theorem, it follows that
and
.
Therefore,
Alternatively, the Sobolev space
is described by
This space is endowed with a norm determined by
Based on Propositions 3.4 and 3.6 in [
1], it can be established that
A widely recognized fact is that
is continuously embedded into
for every
and compactly embedded into
for every
, where
; see [
1] for more details.
In Theorem 1, let
with
. Evidently,
H is a Hilbert space endowed with the inner product:
It is standard to prove that the space
H can be continuously embedded into
for all
, and locally compact embedded into
for any
, we refer to [
15] for more details.
In Theorem 2, our working space will be
. When
V meets the conditions stated in Theorem 2,
is endowed with the inner product:
Note that we do not assume the global positivity of
. In fact, we require that
, which implies that the quadratic form
is positive definite on
.
It is well known that if
, there is a unique
for every
, which is guaranteed through the Lax-Milgram theorem, see for example [
16]. This unique function satisfies the equation
which indicates that
is a weak solution to
Additionally, the expression for
is given by
with
This function is referred to as the t-Riesz potential.
By substituting
into system (
1), we see that system (
1) can be reformulated as a single equation
It is standard to show that the energy functional
I related to problem (
7) is
Definition 1. A solution to Equation (7) means a function such that and First of all, we outline several properties of .
Lemma 1. If and , then the following properties hold:
- (i)
in and ,
- (ii)
If with , it follows that and
- (iii)
If , it follows that and in
- (iv)
If , are bounded in H with , then - (v)
If in H, then it follows that for any and the compact support K of v.
Proof. We just need to verify (
) and (v) since the verifications for (i), (
) and (
) are available in Lemma 2.1 of [
14]. Following the ideas from Lemma 2.2 in [
16], we can prove (
) and (v).
Verification of (
): Applying Hölder’s inequality along with the condition
which implies
we obtain
for any
.
Verification of (v): Using Hölder’s inequality,
and (
) which implies
Then we can conclude that
As desired.
□
As in [
10], we define
and
Consequently,
By choosing a sufficiently small
, we know that
is convex,
and
, where
.
Hence,
,
,
. Obviously,
G is a convex function. Besides, by Fatou’s lemma, we may conclude that
G is lower semicontinuous (see [
17], Lemma 2.9). Therefore, the critical point theory described in [
13] is applicable to the functional
I.
Definition 2 (see [
11]).
Let H be a Banach space and , where and . Moreover, G is lower semicontinuous, convex and .- (i)
is named the effective domain of I.
- (ii)
For any . We define as the subdifferential of I at u, with .
- (iii)
For all , supposing that and , i.e. then is a critical point of I.
- (iv)
Assume is bounded and there exists such that then is a Palais-Smale sequence for I.
- (v)
I fulfills the Palais-Smale condition if Palais-Smale sequence has a convergent subsequence.
Lemma 2 (see Proposition 2.3 of [
12]).
If , then there is a unique such that , i.e.,This unique ξ is defined as .
Lemma 3. - (i)
If , then if and only if is a solution of (
1).
- (ii)
If is bounded, then is a Palais-Smale sequence if and only if .
- (iii)
If is bounded above, and , it follows that u is a critical point.
Proof. The proof of (i) and (
) is similar to Lemma 2.4 of [
12]. Now we prove (
). In fact, we can deduce that
G is weakly lower semicontinuous due to the lower semicontinuity and convexity of
G. Therefore,
and
. By
of Lemma 1 and
in
for each
,
for all
. □
For all , we define by A the critical point set of I, for which . The subsequent pseudo-gradient vector field will be significant in the upcoming sections:
Proposition 1 (see Lemma 2.7 of [
10]).
If there is a set of points , a locally finite countable covering of and a locally Lipschitz continuous vector field , then the following conclusions hold:- (i)
and , where for all i such that .
- (ii)
F is odd in u.
- (iii)
F possesses locally compact support. That is, for each there is a neighbourhood of in and such that for any .
Corollary 1. For any , we can construct , and F on , where , and F satisfy all properties in Proposition 1. (i.e., can be substituted with all the time).
In addition, we will require a logarithmic Sobolev inequality in [
18] applicable to all
, stated as follows:
for any
.
3. Proof of Theorem 1
This section introduces several lemmas that will be utilized later. Firstly, we will demonstrate that the functional I fulfills the Palais-Smale condition.
Lemma 4. The functional I fulfills the Palais-Smale condition.
Proof. First, let us demonstrate the boundedness of the sequence
. Select
such as
for all
n. As
, we have
Using (
9), we conclude that
by choosing a sufficiently small
. Consequently, by employing (
10) and (
11), we have
where we take
. Therefore, the sequence
is bounded and for some
u,
in
H, after passing to a subsequence. Due to the compactness of the embedding
for
, as shown in [
19],
in
. Substituting
into (
8), we deduce that
where
as
. Thus,
Since
and
, the above inequalities lead to
. Consequently,
in
H. □
Lemma 5 (see Lemma 3.3 of [
12]).
Suppose , there is such that no Palais-Smale sequences exists in . Assume
and consider
as defined in Lemma 5. Define
as an even function is locally Lipschitz continuous, with
on
and
elsewhere. Let the flow
be denoted by
The vector field
F which is defined on
is described according to Corollary 1. It should be noted that
if
. In [
10], it has been proved
is differentiable and
Therefore, according to (i) of Propositon 1, we see that
is non-increasing. Taking into account
and
for any
, there exists
.
Lemma 6 (see Proposition 3.4 of [
12]).
Assume and let be as defined in Lemma 5. If , then for each compact set , there is such that . Given that
H is separable and
is dense in
H, it is possible to find a sequence of subspaces, denoted as
, each within
and of dimension
k, such that
. Define
as the orthogonal complement of
in
H, denoted by
. Let
where
.
Lemma 7 (see Lemma 3.4 of [
20]).
If is odd and , then . Lemma 8. There exists such that Proof. Set
with
and
. Subsequently,
Considering that all norms in are equivalent, coupled with , we can conclude that both integrals above are uniformly bounded. Therefore, as , uniformly for all . This implies that there exists such that . Additionally, can be selected to be arbitrarily large as needed.
Set
In order to prove
, we refer to Lemma 3.8 in [
20]. Specifically, we give the following proof. The sequence
is both positive and decreasing, leading to the conclusion that
. Additionally, there exists a sequence
with
such that
. Considering that
in
H, it follows
in
. Consequently, we can conclude that
.
Employing (
11) as demonstrated in Lemma 4, one has
where
. Set
and
. Thus, as
, it can be concluded that
which implies that
. Given that
can be selected such that
, this proof is thus completed. □
Proof of Theorem 1. Set
and
According to Lemma 7, we find that
, which leads to the conclusion that
. What remains to be shown is that
for sufficiently large
k. Assuming the opposite, select
and
T according to Lemma 6. Consider
such that
. Set
, with
representing the flow given in (
12). By (
) of Proposition 1,
is odd. Given that
for any
, it follows
, and therefore
. According to Lemma 6, we have
, which contradicts the definition of
.
4. Proof of Theorem 2
In this section, our work is conducted in the space
where the functional is defined as
Lemma 9. If is bounded above and , then is bounded.
Proof. Selecting
such as
for any
n, we derive as
where
. Here, we have used (
10) once more, by choosing
in (
9) to be sufficiently small. Consequently,
in (
11) is replaced by a constant
a, ensuring that
. □
We next consider a limiting problem
The associated energy functional is given by
Consider the Nehari manifold for
I, denoted as
In a similar way, the Nehari manifold for
is denoted by
. Following the ideas from [
10], we can prove problem (
14) exists a nontrivial solution
and
. It is obvious that
is a ground state solution to (
14). We first elaborate on the differences from the Section 2.1 of [
10]. It is worth noting that Lemma 10, Lemma 12, and Lemma 13 in this paper correspond to Lemma 2.10, Lemma 2.13, and Lemma 2.14 in [
10], respectively. For the reader’s convenience, we restate these lemmas below.
Lemma 10 (see Lemma 2.10 of [
10]).
Lemma 11 (see Lemma 2.11 of [
10]).
If are two Palais-Smale sequence, then one of the following holds: or as . Proof. Choose
, where
q is in
. Hence there exists a constant
to ensure
. Let us first assume
as
. By Lemma 9, it follows that
are bounded in
H. By (
8) and (
) of Lemma 1, we conclude that
So
as
.
Suppose now that
. Using Lions’ lemma (see Lemma 1.21 of [
20] or Lemma I.1 of [
21]), it is easy to find a sequence
and
such that, for sufficiently large
n,
By (
) of Lemma 1,
I is invariant under translations by elements of
, the subsequence
can be assumed to be bounded. Thus, after taking a subsequence,
,
and
. By (
) of Lemma 3, we have
. Hence
This finishes the proof. □
Remark 2. As in Remark 2.12 of [10], the conclusions of Lemmas 10–13 remain valid on , we just need to show that within the argument for Lemma 11. By the lower semicontinuity of and () of Lemma 1,Therefore, and similarly, . Now, we turn our focus to the flow
as defined by
Denote the maximal existence time for the trajectory
as
.
Lemma 12 (see Lemma 2.13 of [
10]).
Let . Then there are two possible outcomes: either , or exists and is a critical point of I. Let , select such that .
Lemma 13 (see Lemma 2.14 of [
10]).
For every there is such thatFurthermore, for all , we have .
Lemma 14. There is such that holds for any and for any .
Proof. For the proof we mimic that of Lemma 2.15 in [
10]. In view of (i) of Lemma 1,
and
, we find that
. Therefore, the desired result follows. □
Inspired by the idea of [
10], we can prove that problem (
7) possesses a ground state solution. In fact, for any
, we define
; then, we obtain
Rewrite
, where
It can be easily infered from (
18) that
and
. Obviously, for all
,
, so there exists a unique
such that
. Note that
, we confirm that
from (
17) and
is the only intersection of the
with
. Furthermore,
as
. When
, the mapping
increases for all
(where
is independent of
u) and
increases for all
thanks to its convexity. Therefore,
is bounded away from the origin. Set
and
Based on Lemma 14, we have that
. Obviously,
. Suppose that for some
, there are no nontrivial solution with energy levels below
. By Remark 2, we can apply Lemma 13 with
and a sufficiently small
. We explore the flow denoted by
where
is locally Lipschitz continuous such that
on
,
elsewhere. By Lemma 13, we obtain a contradiction and a sequence of nontrivial solutions
. Hence we deduce that
and thus
. Furthermore, we assume
in
H as
. According to (
) of Lemma 3,
u is a solution of (
14). If
, then
which implies that
as
. This contradicts the assumption that
. Therefore,
and then we can find a sequence
and
such that for large
n,
thanks to Lions’ lemma (see Lemma 1.21 of [
20] or Lemma I.1 of [
21]). Using the method applied in Lemma 11, we can assume that the sequence
remains bounded after necessary translations. Hence, for the (translated) sequence
, it follows
as
. Based on (
) of Lemma 3,
, so
. Following the reasoning in Remark 2, we also conclude that
. Thus,
, indicating that
u is a ground state solution.
Remark 3. It is worth mentioning that if , then the above results remains valid. In this case, there is a nontrivial solution for (
14)
and satisfying . Lemma 15. - (i)
If , then , where
- (ii)
If and then ; after takeing a subsequence, u is a critical point of I and .
Proof. (i) Choose
such that
, where
is a ground state solution of (
14). Considering
in some open set and
,
for all
has a unique maximum at
,
(ii) According to Lemma 9, we have
in
H after passing to a subsequence. Furthermore, as stated in (
) of Lemma 3,
u is a critical point of
I. Following the same argument as in Remark 2, we get
. Now there is only the task of demonstrating that
. Indirectly, let us assume
. Given that
as
and
in
, we find
Thus,
. By applying the Sobolev inequality and the Hölder inequality, and choosing
v such that
, we derive
The expression on the right converges to 0 uniformly when
,
. Therefore,
. If
, according to (
20), we have
in
H. By setting
in (
8), we obtain
This implies that
. Thus,
, which contradicts the assumption that
. Consequently,
. By means of Lions’ lemma, we deduce that there exist sequences
and
such that for large
n,
Set
. By (
) of Lemma 1,
is invariant under translations by elements of
, thus
and
. Furthermore,
and hence
after taking a subsequence. Thus
w is a nontrivial critical point of
satisfying
. Consequently, this leads to a contradiction. □
Proof of Theorem 2. According to Remark 3, if , then is the exact solution we seek for. Therefore, suppose for some x. Let us assume that there is such that there are no Palais-Smale sequences in . Let . Select so that . It can be assumed . Let be such that if , if and . Let . We observe that uniformly in as . Given that , there is such that for any , we have . Additionally, since is convex and , we conclude that . Therefore, and , where . By the definition of compact support, we can conclude that has compact support and since , we obatin . According to Lemma 6, let and we derive , leading to , thereby conflicting with the definition of c. Due to the fact that may be taken arbitrarily small, there is a sequence such that and . Through Lemma 15, we acquire a nontrivial critical point u of I, fulfilling . Therefore, . Consequently, and u is a ground state solution. Thus, the proof is complete.