1. Introduction
In this paper, we investigate the existence and regularity of normalized ground state solutions for a class of mixed fractional Schrödinger equations with combined nonlinearities. Specifically, we consider the problem
where
,
,
,
, and
.
is the fractional Laplacian operator defined as
where
is a suitable normalization constant and
is a commonly used abbreviation for the principal value sense.
Equation (
1) has emerged as a fundamental mathematical model with widespread applications in various scientific fields. Its significance stems from its ability to describe complex phenomena driven by the interplay of multiple nonlocal diffusion processes. These mixed fractional operators arise naturally in systems that combine different Lévy processes, from classical Brownian motion to long-range stochastic interactions, and have been widely used to model populations with heterogeneous diffusion mechanisms.
Due to these important applications, research on the elliptic problem of mixed fractional Laplacians is receiving increasing attention. Chergui–Gou–Hajaiej [
1] studied the existence and dynamics of normalized solutions to the following equation with mixed fractional Laplacians
with
. The threshold exponent
is the
-critical exponent or
-critical exponent. Chergui [
2] studied the existence of normalized solutions for equation with Hartree type nonlinearity. Additional advances, including the analysis of ground state solutions under the prescribed
-norm constraints, have been developed in [
3,
4], among others.
The study of normalized solutions (i.e., solutions with prescribed -norm) for nonlinear Schrödinger-type equations has seen significant advances in the past decade, driven by both theoretical questions and applications to Bose–Einstein condensation and nonlinear optics. A critical challenge in this field lies in handling nonhomogeneous nonlinearities or competing interactions, where the interplay between different terms can lead to rich solution structures. Below, we highlight key contributions relevant to our work.
The seminal work of Bellazzini, Jeanjean, and Luo [
5] investigated the existence and instability of standing waves for Schrödinger–Poisson equations with prescribed
-norm constraints.
This direction was further developed by Jeanjean, Luo, and Wang [
6], who established a framework for proving the existence of multiple normalized solutions in quasi-linear Schrödinger equations. By combining mountain pass techniques with Pohozaev constraints, they demonstrated that certain energy functionals admit two critical points under
-constraints. Their methods have inspired subsequent studies on systems with nonlocal terms, including the Schrödinger–Poisson case. For more results on the ground state solutions for the nonlinear fractional Schrödinger equation with prescribed mass, we refer to [
7,
8,
9,
10,
11,
12,
13,
14] and the references therein.
The analysis of equations with combined nonlinearities was advanced by Soave [
15], who systematically studied normalized ground states for the nonlinear Schrödinger equation with mixed power type terms:
where
. By introducing a two-parameter variational approach, Soave characterized the existence regimes for ground states and uncovered threshold phenomena related to the
-critical exponent
. Notably, for
(
-critical) and
, he proved the existence of a second solution with higher energy, complementing earlier results on purely subcritical or supercritical cases. In [
16], Sovae extended (
2) to the Sobolev critical case. The research was further extended to planar systems by Cingolani and Jeanjean [
17], who addressed special challenges in two dimensions and developed refined compactness methods.
In [
18], Yang considered the following equation:
where
,
, and
. By applying a refined version of the minmax principle, he successfully established the existence of a critical point solution to Equation (
3) when the relevant parameters satisfied certain structural conditions.
Most existing results (e.g., [
6,
15,
16,
18]) address classical Laplacians or single-order fractional operators. The case of
(
) is largely unexplored. The combined effects of Choquard terms and power-type nonlinearities under
constraints require new analytical tools, particularly when
p approaches critical exponents.
Our first main result, Theorem 1, establishes key regularity properties of solutions, including boundedness, higher Sobolev regularity, and Pohozaev-type identities. These identities play a crucial role in analyzing the behavior of solutions and deriving necessary conditions for their existence.
Theorem 1. Let , , , , , and be a couple of solution for Equation (1). Then, we have the following results: - (i)
.
- (ii)
.
- (iii)
The following Pohozaev identities hold:
It is well known that the normalized solutions for Equation (
1) are critical points of the energy functional
restricted to the (prescribed
-norm) constraint
We define
In Theorem 2, we prove the existence of a normalized ground state solution , which is radially symmetric and decreasing in . Moreover, we provide an explicit upper bound for the associated Lagrange multiplier , demonstrating its negativity.
Theorem 2. Let , , , and . Then,and is attained at a function with the following properties: - (i)
is radially symmetric and decreasing in .
- (ii)
is the solution of (1) and the corresponding Lagrange multiplier
Furthermore, is a normalized ground state solution of (1). Remark 1. In the process of proving Theorem 2, we must face two fundamental difficulties as follows:
- (1)
The competing effects between the local and nonlocal nonlinearities create new obstacles in the energy estimates and require a delicate analysis of the interaction terms.
- (2)
The interaction between different fractional orders creates competing regularity requirements that complicate the analysis of critical points, particularly when combined with the nonlocal Hartree nonlinearity.
Remark 2. For radially symmetric solutions, we can refer to [19]. For
and
, let
for a.e.
. This yields that
.
We introduce the fibering map
Firstly, we consider the case
. For every
, by Lemma 2, we obtain that
If
we derive that
for all
. This implies that
is strictly increasing, and we present the following non existence result.
Theorem 3. Let , , and (6) hold. Then, the functional has no critical point on . In what follows, we focus on the case
. We use the notation
where
It is obvious that all critical points of
stay in
according to the Pohozaev identity. From a similar discussion to ([
20], Lemmas 2.12 and 2.13), we deduce that
is a natural constraint.
Proposition 1. Let , . Then, is a smooth manifold with codimension 2 in and 1 in . Furthermore, if is a critical point of , then u is a critical point of .
We will show that
is bounded from below. The structure of
is strongly influenced by the monotonicity and convexity properties of
. Through simple calculations, we see that
which yields that
is a critical point of
if and only if
. Moreover,
. Let us consider decomposing
into disjoint union sets
where
Therefore, for
, we derive that
According to
, we have
and
Using the Pohozaev set , based on the above discussion, we can obtain the next result.
Theorem 4. Let , , , and and assume that satisfyThen, there exists a constant such that for any , (1) admits a radial ground state solution and the corresponding Lagrange multiplier . In this paper, we use the following notations:
denotes a Lebesgue space; the norm in is denoted by .
denote (possibly different) any positive constant.
The rest of this paper is organized as follows. In
Section 2, we present some preliminary results. We obtain the regularity of solutions to Equation (
1) in
Section 3 and prove Theorem 2 in
Section 4.
Section 5 is devoted to the proof of Theorem 4.
2. Preliminaries
In this section, we begin by summarizing key established results on fractional Sobolev spaces. For
, the space
is defined to be the completion of
with the Gagliaardo seminorm
and the fractional Sobolev space is defined as
endowed with the natural norm
For the convenience of readers, we introduce some preliminary results to prove our main theorems.
Firstly, we review the following compactness result, which can be found in [
21].
Lemma 1 (See [
21])
. Let and . Then, there exists a constant such thatFurthermore, is continuously embedded into for all and compactly embedded into for all . Before describing more details, let us introduce the following fractional Gagliardo–Nirenberg–Sobolev inequality in [
19].
Lemma 2. For and . Then, there exists a constant such that Proof. To interpolate between
and
, express
as
where
. Then, we derive that
which yields that
Then,
follows from
. Hölder’s inequality for
gives
which implies that
Applying Lemma 1, we deduce that
□
We also require the following Hardy–Littlewood–Sobolev inequality.
Lemma 3 ([
22])
. Let with For any and , one has Lemma 4 ([
22])
. Let and . Then, there exists a constant such thatwhere It follows from Lemma 4 that for any
,
, and
From Lemma 3, for any
, if
with
, thus
is well defined. Together with Lemma 4, we obtain that
are the Hardy–Littlewood–Sobolev lower and upper critical exponent, respectively. Particularly, for any
,
where
It follows from [
22] (Theorem 4.3) that
is attained by
for some
and
.
The next two lemmas are useful in proving the splitting property of the energy functional.
Lemma 5 ([
23], Lemma 2.4)
. Let , , , and be a bounded sequence in . If a.e. on as , then The next result is a splitting property of the nonlocal energy functional for fractional Choquard equation in with purely power.
Lemma 6 ([
24], Lemma 2.7)
. Let and , and such that in . Then, for all ,andas . 5. Proof of Theorem 4
In this section, we shall prove Theorem 4.
Lemma 16. Let , . For each , admits a unique critical point such thatParticularly, the map is of class . Proof. For
, we see that
Thanks to
and
, we can derive that
as
and
as
. Furthermore, from (
8), we conclude that
has a unique zero point
, which is the unique maximum point of
. Together with (
5) and (
7), (
38) holds.
We denote by the function . Applying the implicit functon theorem to the function , we can complete the proof. □
Setting
we see that
, according to (
9).
Lemma 17. Let , and (9) hold. Then, is coercive on and Proof. For
, by
and (
10), we conclude that
which yields that
Thus, for each
, from (
9), (
12), and (
40), we observe that
□
Proof. From
, we see that
Therefore, it is only necessary to prove that
For this purpose, let
denote the symmetric decreasing rearrangement of
. According to (
35)–(
37), we derive that
and
It follows from (
35), (
36) and (
39) that
. Together with Lemma 16, we conclude that
According to
, we obtain that
and then
Since
, we infer that
which yields that (
41) holds. □
Lemma 19. For and , the mapis a linear isomorphism with inverse , where is the tangent space to in u. Proof. The proof is standard, see ([
14], Lemma 5.5). □
Next, we consider the functional
defined by
It follows from Lemma 16 that
is of class
. Similar to ([
17], Lemma 3.15), we derive the following result.
Lemma 20. It holds thatfor each and . Once again, analogue to ([
17], Lemma 3.16), we obtain the existence of Palais–Smale sequences to a general homotopy-stable family, according to Lemmas 19 and 20.
Lemma 21. Let be a homotopy-stable family of compact subsets of with a closed boundary and defineAssume that is contained in a connected component of andThen, there exists a Palais–Smale sequence of restricted to at level . Applying Lemma 21, we shall present the existence of a Palais–Smale sequence of restricted to at level .
Lemma 22. Let , and and (9) hold. Then, there exists a Palais–Smale sequence for at level . Proof. Let
be a family of all singletons belonging to
. Clearly, the boundary
is empty. Thus, it is a homotopy-stable family of compact subset of
without a boundary, due to ([
28], Definition 3.1). Taking into account of Lemma 18, we derive that
Therefore, using Lemma 21, we end the proof. □
Next, we discuss the convergence of special Palais–Smale sequences that satisfy appropriate additional conditions, following the idea first proposed by Jeanjean in [
29].
Lemma 23. Let , and be a Palais–Smale sequence for at level . If is bounded in , then there eixsts such that for each , up to a subsequence, strongly in .
Proof. The proof is divided into five main steps.
Step 1. Since
is bounded and the embedding
is compact for
, there exists
such that
Moreover, there exists a sequence
such that for any
,
Taking
in (
43), we observe that
which yields that
is bounded. Then, up to a subsequence, there exists
such that
as
.
Step 2.
. From
and
, we infer that
which leads to
. We will show that
; if not, due to (
44) and
, we can see that
, which contradicts Lemma 17. Thus,
Step 3.
. Assume by contradiction that
. Hence,
. Together with (
44) and
, we deduce that
On the other hand,
which yields a contradiction. Therefore,
.
Step 4. The upper bound of
. By (
13) and (
14), we obtain that
Set
for a.e.
. Clearly,
and
. From Lemma 16, there exists a unique
such that
Lemma 18 yields that
Furthermore, by direct calculations, it can be concluded that
Therefore, taking
we observe that for any
,
Therefore, applying (
45), we conclude that
Now, we define a function
Obviously, there exists a unique critical point
and
is the maximum of
g. Hence, it holds that
Step 5.
in
. Since
in
, from (
43) and Lemma 6, we conclude that
is a weak solution of
Then, we obtain
Set
, then
in
. Hence,
From (
42) and Lemma 5, we observe that
and
Therefore, applying
and
, we obtain that
On the other hand, from (
47), we obtain that for any
,
Taking
in (
43) and (
51), we conclude that
Applying (
49) and (
50), we induce that
Recalling (
12), we see that
If
, then
in
, and we end the proof. If
, by (
48)–(
50), we can observe that
Together with (
48)–(
50) and (
52), recalling that
, we have that
which contradicts (
46). Then, we complete the proof. □
Proof of Theorem 4. By Lemma 22,
is a Palais–Smale sequence for
at level
. Due to Lemma 17, we obtain that
is bounded in
. Then, from Lemma 23, there eixsts
such that for each
, up to a subsequence,
strongly in
. Lemma 18 indicates that
is a radial minimizer of
on
, and it is a solution of (
1) with
. Taking into account Lemma 16, we derive that
is a ground state solution of
on
. The proof is completed. □
6. Conclusions
This paper has studied the existence and regularity of normalized ground state solutions for a mixed-order fractional Schrödinger equation involving combined local and nonlocal nonlinearities. The main results establish the key regularity properties of the solution, derive essential Pohozaev identities, and determine precise parameter regimes under which normalized solutions exist.
For -subcritical nonlinear interactions, we obtain the attainment of energy minimizers and characterize their geometric properties, showing radial symmetry and monotonicity while also obtaining sharp bounds on the associated Lagrange multiplier. The analysis reveals how competing effects between different fractional orders and nonlinear terms introduce delicate analytical challenges, particularly in maintaining coercivity and compactness under critical scaling conditions.
Furthermore, we identify a -critical exponent threshold beyond which no constrained critical points exist, demonstrating the structural limitations imposed by the interplay of nonlocal diffusion and Hartree-type interactions. In the -supercritical case, a refined small--constraint guarantees the existence of ground states, provided the local nonlinearity dominates in a controlled manner.
These results extend the understanding of constrained variational problems with mixed nonlinearities, offering new insights into the role of mixed-order fractional operators and nonlocal nonlinearities in the existence of solitary waves. Future directions may include studying multi-peak solutions, or systems with competing fractional orders in more general domains.