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10 pages, 238 KB

Open AccessArticle

Smoothness of the Solution of a Boundary Value Problem for Degenerate Elliptic Equations

by Perizat Beisebay, Yerbulat Akzhigitov, Talgat Akhazhanov, Gulmira Kenzhebekova and Dauren Matin

Symmetry 2025, 17(9), 1562; https://doi.org/10.3390/sym17091562 - 18 Sep 2025

Viewed by 207

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of [...] Read more.

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of degeneracy. The analysis is grounded in several fundamental aspects of symmetry. Structural symmetry is reflected in the formulation of the differential operators; functional symmetry emerges in the properties of the associated weighted Sobolev spaces; and spectral symmetry plays a critical role in the behavior of the eigenvalues and eigenfunctions used to characterize solutions. By employing localization techniques, a priori estimates, and spectral theory, we establish new coefficient conditions ensuring smoothness in both semi-periodic and Dirichlet boundary settings. Moreover, we prove the boundedness and compactness of certain weighted operators, whose definitions and properties are tightly linked to underlying symmetries in the problem’s formulation. These results are not only of theoretical importance but also bear practical implications for numerical methods and models where symmetry principles influence solution regularity and operator behavior. Full article

(This article belongs to the Section Mathematics)

20 pages, 323 KB

Open AccessFeature PaperArticle

Three Solutions for a Double-Phase Variable-Exponent Kirchhoff Problem

by Mustafa Avci

Mathematics 2025, 13(15), 2462; https://doi.org/10.3390/math13152462 - 30 Jul 2025

Viewed by 432

Abstract

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and [...] Read more.

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an n-dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as

C^{1}

-smoothness, the

(S_{+})

-condition, and sequential weak lower semicontinuity. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

13 pages, 1922 KB

Open AccessFeature PaperArticle

On an Ambrosetti-Prodi Type Problem with Applications in Modeling Real Phenomena

by Irina Meghea

Mathematics 2025, 13(14), 2308; https://doi.org/10.3390/math13142308 - 19 Jul 2025

Viewed by 268

Abstract

This work presents a solving method for problems of Ambrosetti-Prodi type involving p-Laplacian and p-pseudo-Laplacian operators by using adequate variational methods. A variant of the mountain pass theorem, together with a kind of Palais-Smale condition, is involved in order to obtain [...] Read more.

This work presents a solving method for problems of Ambrosetti-Prodi type involving p-Laplacian and p-pseudo-Laplacian operators by using adequate variational methods. A variant of the mountain pass theorem, together with a kind of Palais-Smale condition, is involved in order to obtain and characterize solutions for some mathematical physics issues. Applications of these results for solving some physical chemical problems evolved from the need to model real phenomena are displayed. Solutions for Dirichlet problems containing these two operators applied for modeling critical micellar concentration, as well as the volume fraction of liquid mixtures, have been drawn. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)

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11 pages, 288 KB

Open AccessArticle

Uniform Analyticity and Time Decay of Solutions to the 3D Fractional Rotating Magnetohydrodynamics System in Critical Sobolev Spaces

by Muhammad Zainul Abidin and Abid Khan

Fractal Fract. 2025, 9(6), 360; https://doi.org/10.3390/fractalfract9060360 - 29 May 2025

Cited by 1 | Viewed by 488

Abstract

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild [...] Read more.

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework. Full article

(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)

12 pages, 269 KB

Open AccessArticle

A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential

by Khaled Kefi and Mohamad M. Al-Shomrani

Mathematics 2025, 13(9), 1443; https://doi.org/10.3390/math13091443 - 28 Apr 2025

Viewed by 375

Abstract

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain

D \subset R^{N}

(

N \geq 3

) [...] Read more.

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain

D \subset R^{N}

(

N \geq 3

) with homogeneous Dirichlet boundary conditions. A key feature of the system is the presence of a Hardy-type singular term with a variable exponent, where

δ (x)

represents the distance from x to the boundary

\partial D

. By employing a critical point theorem in the framework of variable exponent Sobolev spaces, we establish the existence of a weak solution whose norm vanishes at zero. Full article

14 pages, 292 KB

Open AccessArticle

Positive Normalized Solutions to a Kind of Fractional Kirchhoff Equation with Critical Growth

by Shiyong Zhang and Qiongfen Zhang

Fractal Fract. 2025, 9(3), 193; https://doi.org/10.3390/fractalfract9030193 - 20 Mar 2025

Viewed by 412

Abstract

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies

L^{2}

-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange [...] Read more.

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies

L^{2}

-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given

m > 0,

\int_{R^{N}} {| u |}^{2} d x = m^{2}

). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

22 pages, 353 KB

Open AccessArticle

Ground State Solutions for a Non-Local Type Problem in Fractional Orlicz Sobolev Spaces

by Liben Wang, Xingyong Zhang and Cuiling Liu

Axioms 2024, 13(5), 294; https://doi.org/10.3390/axioms13050294 - 27 Apr 2024

Viewed by 1084

Abstract

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: [...] Read more.

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces:

{(- Δ_{Φ})}^{s} u + V (x) a (| u |) u = f (x, u), x \in R^{N}

, where

{(- Δ_{Φ})}^{s} (s \in (0, 1))

denotes the non-local and maybe non-homogeneous operator, the so-called fractional

Φ

-Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function

V (x)

. The proof is based on a variant version of the mountain pass theorem and a Lions’ type result in fractional Orlicz–Sobolev spaces. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

9 pages, 286 KB

Open AccessArticle

Three Weak Solutions for a Neumann p(x)-Laplacian-like Problem with Two Control Parameters

by Khaled Kefi

Mathematics 2023, 11(23), 4789; https://doi.org/10.3390/math11234789 - 27 Nov 2023

Viewed by 1131

Abstract

We establish the existence of at least three weak solutions for a Neumann

p (x)

-Laplacian-like problem with two control parameters. Our main result is due to the critical theorem of Bonanno and Marano. Full article

25 pages, 359 KB

Open AccessArticle

Well-Posedness of a Class of Radial Inhomogeneous Hartree Equations

by Saleh Almuthaybiri, Radhia Ghanmi and Tarek Saanouni

Mathematics 2023, 11(23), 4713; https://doi.org/10.3390/math11234713 - 21 Nov 2023

Cited by 1 | Viewed by 1017

Abstract

The present paper investigates the following inhomogeneous generalized Hartree equation

i \dot{u} + Δ u = \pm {| u |}^{p - 2} {| x |}^{b} (I_{α} {* | u |}^{p} {| \cdot |}^{b}) u

, [...] Read more.

The present paper investigates the following inhomogeneous generalized Hartree equation

i \dot{u} + Δ u = \pm {| u |}^{p - 2} {| x |}^{b} (I_{α} {* | u |}^{p} {| \cdot |}^{b}) u

, where the wave function is

u : = u (t, x) : R \times R^{N} \to C

, with

N \geq 2

. In addition, the exponent

b > 0

gives an unbounded inhomogeneous term

{| x |}^{b}

and

I_{α} \approx {| \cdot |}^{- (N - α)}

denotes the Riesz-potential for certain

0 < α < N

. In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term

{| x |}^{b}

for certain

b > 0

. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

30 pages, 417 KB

Open AccessArticle

Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales

by Xing Hu and Yongkun Li

Fractal Fract. 2022, 6(2), 121; https://doi.org/10.3390/fractalfract6020121 - 19 Feb 2022

Cited by 4 | Viewed by 2423

Abstract

Using the concept of fractional derivatives of Riemann–Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some embeddings. Finally, [...] Read more.

Using the concept of fractional derivatives of Riemann–Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some embeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational methods and critical point theory, and, by constructing an appropriate variational setting, we obtain two existence results of the problem. Full article

(This article belongs to the Section General Mathematics, Analysis)

17 pages, 347 KB

Open AccessFeature PaperArticle

The Existence of Solutions for Local Dirichlet (r(u),s(u))-Problems

by Calogero Vetro

Mathematics 2022, 10(2), 237; https://doi.org/10.3390/math10020237 - 13 Jan 2022

Cited by 7 | Viewed by 1843

Abstract

In this paper, we consider local Dirichlet problems driven by the

(r (u), s (u))

-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents [...] Read more.

In this paper, we consider local Dirichlet problems driven by the

(r (u), s (u))

-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents

r, s

are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications 2021)

20 pages, 336 KB

Open AccessArticle

Fractional N-Laplacian Problems Defined on the One-Dimensional Subspace

by Q-Heung Choi and Tacksun Jung

Symmetry 2021, 13(10), 1819; https://doi.org/10.3390/sym13101819 - 29 Sep 2021

Viewed by 1767

Abstract

The research of the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators will give the development of nonlinear elasticity theory, electro rheological fluids, non-Newtonian fluid theory in a porous medium as well as Probability and Analysis as they proved to be accurate models [...] Read more.

The research of the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators will give the development of nonlinear elasticity theory, electro rheological fluids, non-Newtonian fluid theory in a porous medium as well as Probability and Analysis as they proved to be accurate models to describe different phenomena in Physics, Finance, Image processing and Ecology. We study the number of weak solutions for one-dimensional fractional N-Laplacian systems in the product of the fractional Orlicz-Sobolev spaces, where the corresponding functionals of one-dimensional fractional N-Laplacian systems are even and symmetric. We obtain two results for these problems. One result is that these problems have at least one nontrivial solution under some conditions. The other result is that these problems also have infinitely many weak solutions on the same conditions. We use the variational approach, critical point theory and homology theory on the product of the fractional Orlicz-Sobolev spaces. Full article

(This article belongs to the Section Mathematics)

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