1. Introduction
Double-phase problems with nonlocal nonlinearities arise in numerous domains of mathematical physics, as they are capable of modeling systems characterized by heterogeneous properties and long-range interactions. These problems effectively describe materials with spatially varying stiffness, such as composites, porous media, or biological tissues—by combining distinct growth conditions within a unified framework. This dual-phase nature proves instrumental in analyzing complex phenomena such as anomalous diffusion, nonlinear elasticity, and phase transitions. Applications extend to electrorheological fluids [
1], elasticity theory [
2], and Lavrentiev’s phenomenon [
3].
The incorporation of nonlocal terms enables the models to capture influences that extend beyond immediate neighborhoods, reflecting real-world behaviors more accurately. Such terms are particularly relevant in fields like electromagnetism, quantum mechanics, and population dynamics, where distant interactions play a significant role. Furthermore, double-phase models have found increasing application in image processing, particularly in image restoration [
4,
5,
6] and super-resolution techniques [
7]. Altogether, these models offer a robust approach to studying systems where both local and nonlocal effects significantly impact the dynamics.
The central result of this paper, which establishes the existence of multiple weak solutions, highlights the presence of various equilibrium states or configurations that a physical system can attain under different external influences, such as boundary forces, temperature variations, or external fields. This multiplicity suggests that the system can exhibit multiple stable or metastable states, a key aspect in understanding phase transitions, pattern formation, and bifurcation phenomena across different areas of physics. Consequently, the mathematical analysis of such problems plays a fundamental role in modeling, interpreting, and predicting the behavior of complex physical systems governed by nonuniform and nonlinear interactions.
In [
8], Motivated by recent advances in the study of nonlinear double-phase elliptic problems, the authors considered the following parametric Dirichlet problem involving the
-Laplacian:
where
is a bounded domain with a
-boundary, and
are real parameters. Here,
denotes the
r-Laplacian operator for
with
. The authors proved that the problem exhibits a continuous spectrum, given by the half-line
, where
represents the principal eigenvalue of the
q-Laplacian in
. As a consequence, for every
, problem
admits at least one nontrivial solution. Their findings contribute to the spectral analysis of nonhomogeneous differential operators and reveal the impact of mixed growth conditions on the structure of solutions.
Later, in [
9], the authors obtained infinitely many distinct positive solutions for the following double-phase problem:
where
is a smooth bounded domain in
with
, and the exponents satisfy
. Moreover,
and the function
satisfies the Carathéodory condition. Additionally, there exists
such that
In [
10], the authors explore the existence of weak solutions for a double-phase Dirichlet problem of the following form:
where
is a bounded domain in
with a Lipschitz boundary and dimension
. The parameters
p and
q satisfy the conditions
Here,
is a non-negative coefficient,
represents a real parameter, and
belongs to the class
The function
is continuous and satisfies the condition
This study establishes a critical threshold for in relation to the existence of weak solutions. Specifically, when , only the trivial solution exists, with explicitly defined in terms of the problem’s parameters. On the other hand, for , at least two distinct non-negative weak solutions are found, satisfying an energy-related condition. The thresholds and are determined separately, leaving an open problem regarding the existence of solutions within the intermediate range . These results contribute to a deeper understanding of solution multiplicity in double-phase equations, offering explicit conditions that mark the transition from trivial to multiple solutions.
Furthermore, in [
11], the authors studied a boundary value problem involving a double-phase operator with variable exponents and Dirichlet boundary conditions:
Here,
is a bounded domain with a Lipschitz boundary
, where
. The exponents
satisfy the conditions
The function
is a non-negative element of
, and
is a real parameter. The function
satisfies the Carathéodory condition, exhibits subcritical growth, and has a particular asymptotic behavior as
. Their work establishes the existence of weak solutions under general conditions of subcritical growth and superlinear behavior of the nonlinear term. By employing an abstract critical point theorem, the authors demonstrate the existence of two bounded weak solutions with opposite energy signs. Moreover, in specific cases, these solutions are shown to be non-negative. Further recent developments on double-phase problems can be found in [
12,
13,
14].
Building on previous research, this study examines the multiplicity of weak solutions for a double-phase elliptic problem involving a nonlocal interaction:
Here,
is a bounded domain with a Lipschitz boundary
, and
. The parameter
is a positive constant, and
is a non-negative weight. The function
satisfies the Carathéodory condition and adheres to the following growth restrictions:
where
are positive constants, and the exponents satisfy
. The function
is defined as
The exponents
p and
q satisfy the following conditions:
The double-phase operator
where
is a Musielak–Orlicz–Sobolev space, is closely associated with a two-phase integral functional.
This paper establishes the existence of a single solution as well as the existence of three distinct solutions, without imposing additional constraints on the exponents
p and
q beyond the condition (1.2) imposed in [
9]. The primary analytical tools employed are critical point theorems as developed in [
15,
16].
The structure of this paper is as follows: the next section introduces the variational framework and relevant preliminaries, while the final section presents the main results.
2. Variational Framework and Preliminaries
Let be a bounded domain in with and a Lipschitz boundary . For any , we denote by the usual Lebesgue space, equipped with the norm . When , we consider the Sobolev spaces and , endowed with the norms and , respectively.
Define the function
by
which corresponds to the modular function
The associated Musielak–Orlicz space
is defined as
which is equipped with the Luxemburg norm
The fundamental properties of Musielak–Orlicz spaces play a crucial role in the analysis of functionals with nonstandard growth conditions. For a comprehensive exposition of these properties, we refer the reader to the work of Diening et al. [
17].
Proposition 1. Let p, q satisfy the condition , let , and let . Then, the following properties hold:
- (1)
- (2)
(respectively , ) if and only if (respectively , ).
- (3)
- (4)
- (5)
if and only if .
- (6)
if and only if .
- (7)
if and only if .
- (8)
If in , then
The Musielak–Orlicz–Sobolev space
is defined as
equipped with the corresponding norm:
where
. We denote by
the completion of
in
.
It can be shown that both the Musielak–Orlicz and Musielak–Orlicz–Sobolev spaces are uniformly convex (and therefore reflexive) Banach spaces; see Crespo-Blanco et al. [
18].
Furthermore, the following embedding results hold.
Proposition 2 (see [
18] (Propositions 2.16 and 2.18)).
Suppose that holds. Then the following are true:- (1)
is compact.
- (2)
is compact for all .
- (3)
is continuous for all .
- (4)
is continuous for all .
Furthermore, a Poincaré-type inequality holds, which allows us to consider in
the equivalent norm
Let
be such that the embedding
is continuous. We denote by
the best constant for which the following inequality holds:
In other words,
is the operator norm of the embedding
.
The differential operator in
is the so-called double-phase operator:
Definition 1. Let be fixed, and by a weak solution to , we mean a function , such that Let us define the functional
as
where
and
It is easy to see that, under condition
and
,
and
are well defined and continuously Gâteaux differentiable with
and
Lemma 1 (see [
19], (Proposition 3.1)).
The functional is strictly monotone in
The functional is a mapping of type, i.e., if in and then in
The functional is a homeomorphism.
Lemma 2. The functional is compact.
Proof. Condition
, together with the compact embeddings
, where
(see
, in Proposition 2), imply that
is compact. In fact, let us consider a sequence
such that
. The embedding
, where
, is compact; therefore, there exists a subsequence of
, which we still denote by
, such that
strongly in
and
for almost every
. The continuity of
with respect to
w ensures that
Now, since
, then, from condition
, one has
By using the dominated Convergence theorem, we can write
Then, from the condition
, we affirm the continuity of the Nemytskii operator
, as
is a Carathéodory function that satisfies condition
, and then
in
. Next, by utilizing the Hölder inequality, for any
, we have the estimate
where
is the constant associated with the embedding
with
. By combining the results from Equations (
8) and (
9), we deduce that
in
, which implies that
is completely continuous. Therefore,
is compact. □
The forthcoming definition and critical point theorems serve as fundamental tools in establishing our main results. To formulate our existence theorem, we first introduce the necessary preliminary definitions and key theorems.
Definition 2. Let and be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix . The functional is said to verify the Palais–Smale condition cut of upper at d (in short, ) if any sequence such that the following apply:
If , the functional satisfies the Palais–Smale condition.
Our main existence result is due to the following Theorem.
Theorem 1 (Theorem 3.2 [
15]).
Let X be a real Banach space, and let be two continuously Gâteaux differentiable functionals such thatAssume that there exists the positive constant and with such thatandThen, for every , there is such that for all and . Theorem 2 ([
16] Theorem 3.6).
Let X be a reflexive real Banach space and assume the following: is a coercive functional that is continuously Gateaux differentiable and weakly lower semicontinuous in the sequential sense.
The Gateaux derivative of has a continuous inverse on the dual space .
is a continuously Gateaux differentiable functional with a compact Gateaux derivative.
Furthermore, suppose that There exists a positive constant d and a point such that , and the following conditions are satisfied: Then, for any , has at least three distinct critical points in X.