1. Introduction
We study the nonlocal Schrödinger–Poisson–Slater (SPS)-type equation
where
is the complex-valued wave function,
are the exponents of the local and nonlocal nonlinearities, respectively,
is the coupling constant that measures the strength of the long-range interaction, and
is the Riesz potential of order
, defined for
as
The normalization constant
is chosen so that the kernel
satisfies the semigroup property:
For a detailed discussion, see [
1]. The Riesz potential
in our nonlocal term is intimately connected to fractional Laplacian operators through their shared scaling properties and Fourier multipliers. In fact, just as the fractional Laplacian is a nonlocal operator defined via the Fourier transform with symbol
, the convolution
embodies a similar nonlocal behavior. This connection motivates the use of fractional or Coulomb–Sobolev spaces to capture the appropriate function space setting, as detailed in the works of Moroz and van Schaftingen [
2] and Di Nezza et al. [
3]. The nonlocal convolution term in our equation represents the Coulombic repulsion between electrons. The local term
was introduced by Slater [
4] with
as a local approximation of the exchange potential in the Hartree–Fock model [
5,
6].
In particular, we focus on solitary wave solutions, i.e., standing waves of the form
where we work in atomic units and shift the reference energy so that the coefficient of the linear term is normalised to 1; this convention fixes
, which reduces the time-dependent equation to the stationary equation with
(see also [
2])
where
,
. Equation (
2) thus represents a fractional Schrödinger–Poisson–Slater model in which the local power nonlinearity competes with the nonlocal convolution term, a setting widely used to analyze solitary-wave dynamics.
The natural working space for understanding Equation (
2) is the space
The associated functional for the problem (
2) is
and Equation (
2) is the Euler–Lagrange equation of the functional
on the space
.
To guarantee the well-definedness of the functional, we define the space
with the norm
which makes
a Banach space. The norm ensures that the functional
is finite for all
. Equivalently,
has the product topology induced by the norm
, making both projections continuous. A sequence
in
if and only if
in both
and
.
For more regularity in the solutions, we consider the space
. This space is the intersection of the Coulomb–Sobolev space
and the Sobolev space
, which is defined as
This space ensures that the solutions have the required regularity, with square-integrable gradients (i.e.,
), while also satisfying the nonlocal integrability condition. It is particularly useful for analyzing the stability and behavior of solutions, especially in the presence of nonradial symmetry.
The stationary Equation (
2) can be derived as the Euler–Lagrange equation of the Thomas–Fermi–Dirac–von Weizsöcker (TFDW) energy, which itself originates from the Hartree–Fock self-consistent field method by replacing the nonlocal exchange integral with a local power-law term and adding a gradient correction to the Thomas–Fermi kinetic energy and a Dirac exchange correction, yielding the Coulomb–Dirichlet functional
in the absence of external potentials [
7,
8]. Density functional theory (DFT) expresses the total energy of a quantum system purely in terms of its electron density. In the local-density approximation, this framework recovers the energy functional whose Euler–Lagrange equation becomes the Schrödinger Poisson Slater (SPS) model [
9,
10,
11], thereby unifying long-range Coulomb interactions with a computationally tractable local nonlinear exchange term. Rigorous variational analysis of (
2) hinges on two key points: the Hardy–Littlewood–Sobolev (HLS) inequality [
12,
13], which bounds the nonlocal convolution term, and concentration-compactness methods, which address the lack of compactness in the Coulomb–Sobolev space to establish critical embeddings. Moreover, when
and
, (
2) reduces to the classical Choquard equation whose ground-state theory was pioneered by [
2,
14,
15]. Also, when
,
and
, Jeanjean and Le [
16] studied Equation (
2) does not admit positive solutions.
With the change of variable
,
, we convert the problem (
2) to
The parameter
controls the scaling of the solution and spatial variables, enabling asymptotic analysis by homogenizing different scales in the system. Motivated by this equation, we study the limit problem first
Throughout this paper, we assume that
; hence, we omit its explicit mention in subsequent sections. According to [
17], we can also say that it is a “zero-mass” problem, since the zero linearization operator only involves the Laplacian. The associated energy functional is
,
It is well-established that
is well-defined, and
and its critical point are associated with the solution of (
4).
Our goal in this paper is to establish the existence of ground-state solutions and to prove their orbital stability for the time-dependent problem (
1), and to investigate the qualitative properties of solutions to the associated stationary problem (
2).
For a classical case in (
2), see the Schrödinger–Poisson–Slater problem:
where
,
and
. Ruiz [
18] demonstrates a lower bound for the Coulomb energy and proposes an inequality that is versatile across different frameworks. Importantly, the derived inequality is nearly optimal, underlining its close proximity to optimality within the specified context.
Ruiz’s approach for controlling the Riesz potential fails when because it relies on the specific structure of the Coulomb-type convolution, which is valid only for . In this case, the nonlocal term corresponds to the Coulomb kernel, enabling the use of symmetry-based compactness arguments. However, for , the convolution becomes more complex and loses this structure, complicating the analysis due to the altered interaction between nonlocal and nonlinear terms. Consequently, it is crucial to carefully examine how p and q interact in order to properly balance the effects of nonlocality and nonlinearity. This requires developing new mathematical techniques that go beyond the traditional Coulomb framework.
In [
19], Mercuri et al. studied the nonlocal Schrödinger–Poisson–Slater equation
with
,
,
. They introduced the Coulomb–Sobolev space and established optimal interpolation inequalities that yield existence results in specific parameter regimes. Their embedding results have been particularly influential in our study.
We allow arbitrary dimension
, Riesz potential order
, and convolution exponent
, thereby generalizing the problem beyond the constraints considered by Mercuri et al. [
19]. By incorporating the linear mass term, we unify the analysis of zero-frequency and nonzero-frequency cases, capturing the frequency’s impact on ground state existence and stability, a factor not addressed in Ruiz’s bounded domain analysis [
18]. Also, in contrast to Bellazzini et al. [
20], who mainly focused on the existence of standing waves under an
mass constraint, we eliminate this constraint using the linear term, making the energy functional coercive. We prove the orbital stability of radial ground states when
p lies below the critical threshold
, and establish nonexistence when
p exceeds the Sobolev critical exponent
using the Pohozaev identity. This precise delineation of the supercritical regime, which previous studies either partially addressed or overlooked [
19,
20].
Moreover, the framework developed by Moroz and Van Schaftingen [
2] for fractional Choquard equations, along with insights from Servadei and Valdinoci [
21], underscores the importance of nonlocal variational methods. Recent contributions by Tang et al. [
22] and Liu et al. [
23] further demonstrate existence, nonexistence, and symmetry-breaking phenomena in related models, albeit within restricted parameter ranges and dimensions. These developments collectively motivate our broader investigation into fractional nonlocal Schrödinger–Poisson–Slater equations. Notably, Zhang and Hou [
24] and Wang et al. [
25] have provided crucial insights into the existence and radial symmetry of ground states, while Zhang et al. [
26] and Zhang et al. [
27] advanced the analysis of fractional
p-Laplacian systems. Furthermore, recent work by Zhang and Nie [
28] and by Wang et al. [
29] has introduced novel tools for fractional operators that are instrumental in addressing symmetry-breaking in the nonradial setting.
It is noteworthy that our results have broad implications; indeed, several existing works (e.g., [
18,
23,
30,
31,
32]) appear as particular cases when the parameters
, and
are suitably specialized. By refining the Coulomb–Sobolev embedding of Mercuri et al. [
19], we prove convergence and existence across a substantially wider parameter domain-recovering Ruiz’s coercivity for
,
[
18]-and further extending their framework to supercritical
p and nonradial minimisers via a nonlocal Brezis–Lieb lemma (see Theorem 1 and
Section 4).
Throughout this paper, we fix
and set
And
represents the subspace of radial functions in the space
.
We have the following result:
Theorem 1. Let and . assume eitherThen, the functionalis coercive and weakly lower semicontinuous. In particular, if , J attains a negative minimum, so that Equation (4) admits a positive solution in . Our approach extends Theorem 1.3 in [
18], which proved coercivity and minimizers for the radial Schrödinger–Poisson–Slater functional
J with
in
, by generalizing to multiparameter dependencies, regime bifurcations involving both cases
and
, and higher-dimensional settings. This requires overcoming new challenges in scaling, critical exponents, and nonlocal decay.
Remark 1. In the special case , and , Theorem 1.3 of [18] recovers a similar result. Our work broadens this framework, allowing exponents p and q to exceed the critical threshold . Notably, our proof leverages the compact immersion from [20], enabling this extension to supercritical regimes. Theorem 2. Let . Assume thatThen, the functional associated with Equation (2), is coercive, weakly lower semicontinuous, attains a minimum, and satisfies the Palais–Smale condition. After establishing the existence of ground states, we now derive the Pohozaev identity to pinpoint the parameter regimes that rule out nontrivial solutions.
Theorem 3 (Nonexistence of Ground States)
. Let and . Assume thatThen, any solution of Equation (2) must be trivial, i.e., . In nonlocal Schrödinger–Poisson–Slater equations, symmetry-breaking can occur when the nonlocal interactions, represented by the Riesz potential, dominate over local terms. Mercuri et al. [
19] analyze this competition and show how the nonlocal term affects the symmetry of solutions, particularly in regimes where nonlocality becomes dominant. Using Coulomb–Sobolev spaces and Brezis–Lieb-type inequalities, they explore the existence and properties of ground states, highlighting how the symmetry of solutions can break depending on the relative strengths of the local and nonlocal terms. Liu et al. [
23] further explored this, identifying both radial and nonradial minimizers, while Ruiz [
18] highlighted the conditions that lead to symmetry-breaking in the SPS system.
In the following, we demonstrate that the interplay between the nonlocal and nonlinear terms governs the existence, with the choice and interplay of parameter regimes playing a decisive role.
Theorem 4. Let and . Assume thatThen, there exists a ground-state solution u to Equation (2) that is orbitally stable. We will write some common symbols as follows:
To provide a clear overview of the main results discussed in this paper, we present a summary of the parameter ranges and corresponding conclusions of the four primary theorems in
Table 1.
The paper is organized as follows:
Section 2 presents the variational framework, functional spaces (including
), and preliminary tools.
Section 3 proves the coercivity, existence and nonexistence of minimizers for
J and
E (Theorems 1–3).
Section 4 establishes the existence and orbital stability of ground states (Theorem 4).
2. Preliminaries
In this section, we establish some notation that will be used throughout the article. We also study some basic properties of the space .
Definition 1. The space is defined aswith the normand define the space with the norm of As shown in [
19] (see also [
20,
33]), the space
is a normed space, and it is also complete. That is to say, the space
is a Banach space (see Proposition 1 below). For convenience, we state it below for our later use.
Proposition 1. For every and , the normed space is a Banach space, i.e., is a norm of , and is a uniformly convex Banach space. Moreover, it is a complete normed space.
We shall use the following result, which is also from [
19,
20]. The Coulomb–Sobolev space
can be naturally approximated in the
norm by the set of testing functions
, which is proved in [
19]. For the sake of readers, we recall its statement here.
Proposition 2 (Density of test functions). Let , and . Then, is dense in .
We adapt the following compact embedding result from [
19,
20], adjusted to our parameter framework:
Lemma 1. Let , , and . Assume p satisfies:where is the critical Sobolev exponent and . Then, the embeddingis compact. We employ a scaling-invariant inequality from [
20], adjusted to our parameter regime. Specifically, for our choice of
, and
p, the following holds:
Lemma 2 (Scaling–Invariant Sobolev–Coulomb inequality)
. Let , , and satisfy . There exists a constant such that the scaling-invariant inequalityholds for every if and only if Proposition 3 (Sobolev embedding)
. Let . Sobolev space continuously embeds into . Specifically, there exists a constant such that: Lemma 3. For , the nonlocal term modifies the mountain-pass geometry.
Proof. Recall the energy functional associated with (
2):
Now, fix a nonzero
u. Define
As
, we have
so that
for sufficiently small
t. On the other hand, as
, the local term
dominates provided
. In fact,
for
t sufficiently large. Next, differentiating
with respect to
t, we obtain
Setting
for
and dividing by
t, we obtain
Next, differentiating twice with respect to
t yields
When
and
, and when
and
, we have that for
, the quadratic term dominates so that
; whereas for
, the negative term
eventually dominates, and hence,
. This change in sign implies that
has a unique positive maximum at some
.
Hence, by continuity of , there is at least one positive solution to . We claim the uniqueness of positive solution . Suppose, for contradiction, there exist two positive roots of . By continuity, must change sign at , becoming either strictly positive or negative for . However, we already know that as . Hence, cannot become positive again after , ruling out a second zero at .
Therefore, the energy functional possesses the mountain-pass geometry. □
To establish various key estimates in the proof, we rely on the well-known Hardy–Littlewood–Sobolev inequality, which plays a crucial role in controlling nonlocal terms and ensuring the coercivity of the energy functional [
13].
Lemma 4 ([
13])
. Let and with . Let and . Then, there exists a sharp constant , independent of f and h, such thatThe sharp constant satisfies 4. The Existence of the Solution in the Nonradial Case
In this section, we will prove the existence and orbital stability of ground-state solutions to the nonlocal Schrödinger–Poisson–Slater equation. The proof proceeds in two main steps: (i) establishing coercivity of the energy functional, which ensures boundedness of solutions, and (ii) demonstrating existence through minimization on the Nehari manifold.
Before proceeding with the proof of Theorem 4, we provide a visual representation of the symmetry-breaking phenomenon and the relevant parameter space.
Figure 3 visually highlights the transition from a radially symmetric solution to a nonradially symmetric one, illustrating the core idea of symmetry-breaking that will be formalized in Theorem 4.
Figure 4 reveals the critical regions for the existence and stability of ground-state solutions in the fractional nonlocal Schrödinger–Poisson–Slater equation. Specifically, the green dashed line marks the critical value for
q beyond which solutions become unstable, while the orange and red lines delineate the bounds for the exponent
p, indicating the regions where different types of solutions exist. The area between the orange and red lines (shown in yellow) is where nonradial solutions are stable, according to Theorem 4.
Proof of Theorem 4. First, we assume that the equation conserves both the mass
and the energy
We consider the minimization problem on the set
□
Now, we claim that, under our assumptions on
p and
q (
5), the energy
is coercive. Indeed, on the one hand, by the Hardy–Littlewood–Sobolev inequality, there exists a constant
such that
where
and
. On the other hand, For the local term, by the Gagliardo–Nirenberg inequality, there exists a constant
and an exponent
such that
Hence,
where
and
. Above all, under our assumptions that
, and
, we obtain
for some
and constant
. Thus, the energy
is coercive and, being conserved, it controls the
norm of the solution.
We now prove the existence of a ground-state solution by minimizing the energy functional over the Nehari manifold. The Nehari manifold is defined as
and we claim the following two properties:
Claim 1. The infimum of the energy functional over is strictly positive, i.e., . .
For ,It implies thatThen, we obtainHence Claim 2. There exists a function such that , meaning that u minimizes the energy functional. There exists , such that .
Take a minimizing sequence , . By this way, we assume that for some uniform constant ,For the nonlocal term, the Hardy–Littlewood–Sobolev inequality yieldsSince when , applying the Sobolev embedding , we obtainand soBy the variational characterization of ,for , we obtainHence, for , the energy is nonnegative. According to the nonlocal Brezis–Lieb lemma [
19], we know that
By Proposition 4.3 and Proposition 3.3 in [
19], equality in the nonlocal Brezis–Lieb inequality holds if
in
for
. Here, by Rellich–Kondrachov theorem, it is satisfied when
. Thus, we conclude that
Using the Sobolev embedding
again, we have
Then, we deduce
By contradiction, if
, and
for
small, then we have
for some
. Then,
and
. But we compute:
which is absurd. Then,
,
, i.e.,
u is a minimizer of
E on
. Following the Lagrange multiplier,
, and taking
,
that is
It implies that
As we know
,
and
,
, we obtain
. Therefore, we obtain
for all
. Hence,
u is a critical point. Then,
is a ground state.
Consider the Cauchy problem
Define the linear operator
Write the equation in Duhamel form, we have
where
Fix
and define
where
and we choose the Schrödinger-admissible pair
It follows from the classical Strichartz estimates, that is
In order to balance the interplay between the nonlocal term and the local term, define
with dual exponents
and
. By the Hardy–Littlewood–Sobolev inequality and Sobolev’s embedding, we obtain
and using Sobolev’s embedding,
hence, the Duhamel operator is bounded. This implies that
is locally Lipschitz from
into the dual space
.
where
. Thus, the overall estimate for the inhomogeneous term becomes
where
is time integration.
Above all, we obtain
here
. For any
,
so
so the mapping
is a contraction on a ball
, where
for some
provided
is sufficiently small. Thus, there exists a unique local solution
. It implies that if the initial data have finite energy Then, the solution cannot blow up in finite time.
By the conservation of energy and mass, and the coercivity inequality (
15),
the local solution extends globally in time, i.e.,
Moreover, the solution depends continuously on the initial data.
In the end, we prove orbital stability of the ground state. By contradiction, assume that orbital stability fails. Then, there exists an
, a sequence of initial data
, and a sequence of positive times
such that
but the corresponding solutions
of Equation (
2) with initial data
satisfy
Denote
. At time
, we have
Next, denote by
the modulated initial data sequence. Since
is a ground state to (
2), by definition, we have
By Sobolev embedding, we still have
so
in
. Then, conservation laws imply that, up to translations and phase shifts,
In order to conclude strong convergence of the sequence, we must rule out vanishing and dichotomy in the concentration-compactness principle.
Rule out Vanishing. If the mass of “spreads out” to infinity, then would not converge to contradicting energy conservation.
Assume vanishing, which means
By Lions’ vanishing lemma [
35], for
,
. The nonlocal term is controlled by the Hardy–Littlewood–Sobolev inequality (
6):
Since
(from
), vanishing implies
, so
Rule out Dichotomy. Let
denote the minimal energy among functions with fixed mass
, i.e.,
where
and the mass is
Assume by contradiction that there exists a minimizing sequence
for
, which exhibits dichotomy. That is, after extracting a subsequence, there exist sequences
and
with asymptotically disjoint supports such that
for some
. In this scenario, the splitting of
into two disjoint parts implies that the strict subadditivity of
is violated. Denote
Define the scaling
where
and
. Through direct computation, we obtain
Under the parameter assumption, we deduce that
By the definition of
and taking the infimum over all
yields
Here, we employ a key lemma in concentration-compactness from [
12]. The lemma asserts that if a function
satisfies
then it is strictly subadditive
The strict subadditivity assumption then reads
Since
is a minimizing sequence with
and
, one expects that if dichotomy occurs Then, the energy asymptotically splits as
By the definition of minimal energy, we have
Passing to the limit, we obtain
which contradicts the dichotomy.
Since neither vanishing nor dichotomy occurs, the concentration-compactness principle ensures that there exists a sequence
such that the translated sequence
Thus,
achieves the minimal energy
and is a ground state. Consequently, for large
n,
Thus, by ruling out vanishing and dichotomy and applying the concentration compactness principle, we conclude that the sequence
converges strongly in
, and the ground-state solution is orbitally stable.