Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
-
and
Abstract
Our primary objective is to study the solvability of two kinds of fractional Kirchhoff double-phase problem involving logarithmic nonlinearity in
1 Introduction and main results
The research on double-phase problems can be traced back to 1986. Zhikov identified a dramatic change in the hardening properties of materials at a specific point, known as Lavrentiev’s phenomenon. For studying the model of strongly anisotropic materials, Zhikov and coworkers [1–4] employed the homogenization theory to study the functional described by:
with
Obviously, the core of double-phase problem is
The working space corresponding to double-phase operators was first given by Colasuonno and Squassina [5] in 2016. They addressed eigenvalue problems involving double-phase variational integrals and established the classical embedding theorem in Musielak-Orlicz-Sobolev space
On the basis of the definitions of double-phase operators and the norm in
When
Some scholars also studied the problem of nonlocal double-phase operators with
They obtained the existence of multiple positive solutions as
Clearly, the operator in (1.3) cannot be addressed under the standard fractional Sobolev spaces.
Hence, it is essential to introduce the definition of fractional Musielak-Sobolev spaces. In 2020, Azroul et al. [18] defined the fractional Musielak-Sobolev space
where
The correlational research on the Kirchhoff equation can be traced back to 1883, which was treated by German physicist Kirchhoff [22] and embraced by Lions [23]. The initial Kirchhoff equations were first proposed as D’Alembert wave equation, which employed to investigate the transverse oscillations of taut strings, with a specific focus on accounting for the alterations in string length resulting from these oscillations. Lions [23] introduced an abstract functional analysis framework and converted the Kirchhoff equation into the following form:
Since then, Kirchhoff problem has been studied extensively [14–17,24–27]. Kirchhoff double-phase problems also have been studied extensively recently [14–17], which extend the Kirchhoff operators to the Musielak-Orlicz-Sobolev spaces. In [15,16], Fiscella et al. considered the following Kirchhoff problems in the bounded domain,
and
where
As far as current literature indicates, most fractional equations only involve power function terms, because logarithmic terms such as
where
By utilizing the concentration-compactness principle and the critical point theory, the existence of ground state solutions in both critical and subcritical conditions was studied.
As mentioned earlier a natural idea arises, which is whether the fractional Musielak-Sobolev space can be applied to study the solvability for the fractional Kirchhoff double-phase problems with the logarithmic terms. In this article, we first deal with following problem:
where
Let’s start with the following assumptions (for short
there exists
Specifically, some hypotheses (for short,
for each
there are
for a.e.
Theorem 1.1
Assume that
For the fourth section of this work, the following Kirchhoff double-phase problem is studied,
where
Theorem 1.2
Suppose that
The features of this work are as follows.
First, we introduce fractional double-phase operators in fractional Musielak-Sobolev space, establishing some corresponding embedding results and properties of the modular function. To address the lack of compactness in the embedding, we revisit the conclusions from [21], providing the equivalent norm forms of fractional double-phase operators (Lemmas 2.4 and 2.5). In addition, when examining fractional Musielak-Sobolev spaces in
Second, unlike [15,16], our consideration is in
Third, the coexistence of logarithmic terms
The structure of this article is as follows. In Sections 2 and 3, we establish some properties of
2 Fractional Musielak-Sobolev spaces framework
In this section, we introduce the fractional Musielak-Sobolev space corresponding to the fractional double-phase operators (1.3), establishing the embedding results and some basic properties.
2.1 Musielak-Orlicz space
For any
with the seminorm
Let
equipped with the norm
and the seminorm notes as
Now, we introduce the definition of Musielak-Orlicz space. With regard to double-phase problems, we define
where
Then we collect some properties of
Lemma 2.1
Proof
For (i), because
according to Lemma 3.2.2 in [30], and
For (ii),
by Proposition 2.3 in [31], and
For (iii), by Definition 2.3 in [20], we can find
The proof is completed.□
On the basis of Lemma 2.1, we define the Musielak-Orlicz space
with Luxemburg norm
where
The weighted Musielak-Orlicz space
with Luxemburg norm
Proposition 2.2
[32, Theorem 8.5] Let
Here,
Lemma 2.3
If
Proof
For (i), since
For (ii), set
Therefore, if
It means that
For (iii), for each
By utilizing Proposition 2.2, we can obtain
2.2 Fractional Musielak-Sobolev spaces
Then we introduce the fractional Musielak-Sobolev space. By the definition of (1.3), fractional Musielak-Sobolev space associated with
where
with norm
The so-called
The weighted fractional Musielak-Sobolev space is defined by
endowed with norm
We also define the norm
where
Lemma 2.4
Let
For
Proof
(i) It is clear that the mapping
Then (i) and (ii) hold.
To prove (iii), for any constant
We only prove the case that
(iii) can be proved. (iv) follows from (iii) is clear.□
Next, we will prove the equivalence of
Lemma 2.5
Proof
We first verify that
According to the definition of
For each
To prove the triangle inequality, let
Next, we verify the inequality (2.1) holds. By combining with the definition of
According to the definition of
Therefore, we have
and
Thus,
Then, we will give some properties of
Lemma 2.6
If
Proof
We first consider the weighted Sobolev space
endowed with the norm
Since this norm is equivalent to the norm
Theorem 2.7
Proof
We first prove
Because
By (2.2) with
thus
According to Lemma 2.4 (iv),
Now, we prove
Therefore, we can obtain
Case 1:
we can arrive that
It can imply that
Case 2:
where
By the following inequality [33]
(2.3) implies that,
For
By using (2.4), we have
Let
Case 3:
Thus,
Remark 2.8
In [21], completeness, reflexivity, and uniform convexity with respect to
3 Functional setting
For convenience, denote
3.1 Fractional double-phase operator
In this subsection, we give the continuity, boundedness, and
Theorem 3.1
Let T be a mapping of type
then
Proof
Let
By (
In consequence,
From this and (3.2), one yields that
Therefore, by ([34], Proposition 3.131), we can find subsequences satisfying
For each
and
According to (3.3), we have
Combining with (3.1), one has
Applying Young’s inequality, it can obtain
From this, (3.4) and Fatou’s lemma, one leads to
Since
Thus,
and
are uniformly integrable. By applying Vitali theorem again, it follows that
which gives
Theorem 3.2
Suppose that
Proof
We first prove that
Considering the continuity of Kirchhoff operator
Because of
and
Hence, by using Brezis-Lieb lemma (see [35]), we derive that
Since
It can be obtained that
Combining with (3.6), it implies that
Similarly, one yields that
Due to Lemma 2.3, there are positive constants
by Lemma 2.6,
Based on the aforementioned estimates, we have
Therefore,
Finally, we consider the boundedness of
Therefore,
Remark 3.3
In [21], the
3.2 Logarithmic nonlinearity
In this subsection, we establish the relationship between the logarithmic term and the space norm by proving the correlation inequality.
Lemma 3.4
Let
Proof
Because
According to the properties of logarithmic functions, for every
It follows that
where
Thus, by Lemma 2.6, there is
By applying Lebesgue dominated convergence theorem, one has
The proof is completed.□
Lemma 3.5
Let
where
Proof
According to the logarithm algorithm, there is
where
and
For
To estimate
Because of
which completes the proof.□
4 The existence results of Problem (1.5)
In this section, we consider equation (1.5).
Definition 4.1
If
The energy functional
Lemma 4.2
Assumptions
there are two constants
there is
Proof
For (i), due to
By (
By Lemma 2.4 (iii), Lemma 2.6, for
Define
by (
Let
there is
Obviously,
For (ii), by (
By (
Take
Since
The proof is completed.□
Next, we prove the
Lemma 4.3
If
Proof
We first claim the boundedness of
Since
Next, we claim that
Then we have
Applying Lemma 3.4, it follows that
By (
By (
Proof of Theorem 1.1
Lemmas 4.2 and 4.3 mean that
To obtain ground state solution, we define
For any
which means
which implies
Next, let
By Lemma 4.3, we can see that
Hence,
5 The existence results of equation (1.6)
In this section, we study the existence results of equation (1.6).
Definition 5.1
If
The energy functional
Lemma 5.2
Assumptions
there are two constants
there is
Proof
For (i), due to
By
The rest proof can refer to Lemma 4.2 (i). We can obtain (i) hold when
For (ii), by (
Take
Since
The proof is completed.□
Lemma 5.3
If assumptions
Proof
Let
We first show the boundedness of
Case 1:
By Lemma 2.4, (
Since
Case 2:
One of
Similar to Case 1, we can obtain
By Lemma 2.4, we can obtain
Thus,
Next, we verify that
Then
Next, we will divide two cases to consider the behavior of
Case 1:
Because of
Subcase 1.1:
When
Subcase 1.2:
When
By (5.6) and
By (5.5) and
Thus,
If
If
Thus, by (5.8), we can obtain
Denote
On account of
which is a contraction with
Subcase 1.3:
When
By (5.5) and
which gives
Similar with subcase 1.2, we obtain that
Because
It can ensure that
where
Thus, according to (5.10), we have
Hence, by applying Lemma 2.4, we obtain
Subcase 1.4:
When
According to the convexity, we can obtain
Using (
since
Case 2:
If
Proof of Theorem 1.2
The functional
To obtain ground state solution, we denote by
For any
which means
Thus,
Next, let
By Lemma 4.3, we can see that
Hence,
Acknowledgments
The authors express their sincere gratitude to the anonymous referees for their careful reading of the manuscript and valuable comments.
-
Funding information: This work was supported by the National Natural Science Foundation of China (12371173), the Shandong Natural Science Foundation of China (ZR2023QA025 and ZR2021MA064), and the Taishan Scholar project of China.
-
Author contributions: All authors contributed equally to this work.
-
Conflict of interest: The authors declare that there is no conflict of interest.
-
Ethical approval: Not applicable.
-
Data availability statement: No data was used for the research described in the article.
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- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
- A new perspective on generalized Laguerre polynomials
- Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
- Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
- ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
- New developments for the Jacobi polynomials
- Generalization of Sheffer-λ polynomials
- Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
- A new type of soft multi rough sets
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
- Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
- Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
- Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
- Weak solutions to an asymptotic equation of the variational sine-Gordon equation
- Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
Articles in the same Issue
- Research Articles
- On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
- Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
- Grand Triebel-Lizorkin-Morrey spaces
- Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
- Proofs of two conjectures involving sums of normalized Narayana numbers
- On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
- New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
- Some inequalities for rational function with prescribed poles and restricted zeros
- Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
- Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
- Fixed point results for contractions of polynomial type
- Estimation for spatial semi-functional partial linear regression model with missing response at random
- Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
- New forms of bilateral inequalities for K-g-frames
- Rate of pole detection using Padé approximants to polynomial expansions
- Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
- Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
- Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
- New topologies derived from the old one via operators
- Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
- Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
- Nonparametric expectile shortfall regression for functional data
- Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
- A generalized p-Laplacian problem with parameters
- A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
- Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
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- Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
- Solutions of several general quadratic partial differential-difference equations in ℂ2
- Inequalities for the generalized trigonometric functions with respect to weighted power mean
- Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
- Hankel determinants for q-starlike functions connected with q-sine function
- System of partial differential hemivariational inequalities involving nonlocal boundary conditions
- A new family of multivalent functions defined by certain forms of the quantum integral operator
- A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
- Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
- Behavior of spatial curves under different transformations in Euclidean 4-space
- Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
- A new kind of Durrmeyer-Stancu-type operators
- A study of generalized Mittag-Leffler-type function of arbitrary order
- On the approximation of Kantorovich-type Szàsz-Charlier operators
- Split quaternion Fourier transforms for two-dimensional real invariant field
- Quantum injectivity of G-frames in Hilbert spaces
- Some results on disjointly weakly compact sets
- On Motzkin sequence spaces via q-analog and compact operators
- Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ3
- Stability analysis of linear time-invariant difference-differential system with constant and distributed delays
- The discriminant of quasi m-boundary singularities
- Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics
- Fuzzy stability of multi-additive mappings
- On inequalities involving n-polynomial exponential-type convex functions
- Singularities of multiplicative spherical Darboux image and multiplicative rectifying developable surface
- A golden ratio technique for equilibrium problem in reflexive Banach spaces
- A parallel inertial three-step iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces endowed with graphs applicable to signal recovery problems
- Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term
- New soliton solutions for a nonlinear complex hyperbolic Schrödinger dynamical equation with a truncated M-fractional derivative
- On a generalization of derangement polynomials and numbers
- The description of entire solutions of complex PDEs and PDDEs
- A modified RMIL conjugate gradient-based projection algorithm for constrained nonlinear equations: application to image denoising
- Fast solution strategies for time-space fractional linear complementarity problems governing American options pricing
- Existence results for Robin problems involving p(x)-Laplacian-like operators with convection term
- On asymptotic behaviors of a specific cubic functional equation and its hyperstability
-
The description of entire solutions for some class of complex nonlinear partial differential equation (systems) in
- Variations in the geometry of the basins of escape in a modified Hénon–Heiles potential
- A Rothe method for a viscoelastic contact problem involving time-fractional derivatives in locking materials
- Upper and lower solution method for a higher order ϕ-Laplacian BVPs on an infinite interval
- Weyl almost periodic functions on time scales and their Fourier series
- Integrable system of null curve and Betchov-Da Rios equation
- Fekete–Szegö problems for (β, Φ)-spirllike mapping of complex order γ in Banach space
- Modulated convergence: a deferred approach
- Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions
- Existence and nonexistence of normalized solutions for the Biharmonic equation with combined nonlinearities
- Ekeland’s variational principle for interval-valued functions with an α-level set in Kaleva-Seikkala’s type fuzzy metric spaces
- On Kurzweil integral of fuzzy number valued functions with two variables
- On split common null point and common fixed point problems for multivalued demicontractive mappings
- Approximation by weighted Durrmeyer-type max-product neural network operators
- A new predictor-corrector interior-point algorithm for semidefinite optimization
- Densities of measures: fine properties and examples
- Review Articles
- Characterization generalized derivations of tensor products of nonassociative algebras
- On the performance of the new minimax shrinkage estimators for a normal mean vector
- Special Issue on Differential Equations and Numerical Analysis - Part II
- Existence and optimal control of Hilfer fractional evolution equations
- Persistence of a unique periodic wave train in convecting shallow water fluid
- Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
- Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
- Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
- Stability and bifurcation analysis of a modified chemostat model
- Some new quantum derivatives and integrals with their applications in integral error bounds
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
- Analytic solutions of a generalized complex multi-dimensional system with fractional order
- Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
- Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
- On the sum form functional equation related to diversity index
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
- Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
- Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
- Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
- Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
- New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
- Special Issue on Approximation Theory and Special Functions 2024 conference
- Ulam-type stability for Caputo-type fractional delay differential equations
- Faster approximation to multivariate functions by combined Bernstein-Taylor operators
- (λ, ψ)-Bernstein-Kantorovich operators
- Some special functions and cylindrical diffusion equation on α-time scale
- (q, p)-Mixing Bloch maps
- Orthogonalizing q-Bernoulli polynomials
- On better approximation order for the max-product Meyer-König and Zeller operator
- Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
- A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
- A new perspective on generalized Laguerre polynomials
- Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
- Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
- ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
- New developments for the Jacobi polynomials
- Generalization of Sheffer-λ polynomials
- Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function
- A new type of soft multi rough sets
- Special Issue on Variational Methods and Nonlinear PDEs
- A note on mean field type equations
- Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
- Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
- Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
- Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
- Ground state solutions to singularly perturbed Chern-Simons-Schrödinger systems with a neutral scalar field
- Weak solutions to an asymptotic equation of the variational sine-Gordon equation
- Multiplicity of positive solutions for a concave-convex fractional elliptic system with critical growth
