Variable Exponent Research Articles

Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections

Abstract In this study, we deal with a multivalued elliptic variational inequality involving a logarithmic perturbed variable exponents double-phase operator. Additionally, it features a multivalued convection term alongside two multivalued terms, one defined within the domain and the other on its boundary. Under the noncoercive framework, we establish the existence results of weak solutions for the multivalued inequality by employing a surjective theorem for multivalued pseudomonotone operators along with the penalty technique. On the other hand, we prove the compactness of solution set by employing the S + {S}_{+} -property of the associated perturbed variable exponent double-phase operator. Finally, we focus on special cases to the multivalued inequality, where K K is a bilateral constraint set, and the two multivalued terms are Clarke’s generalized gradients with respect to two locally Lipschitz functions.

Stein–Weiss inequalities on Herz spaces with variable exponent

This paper extends the celebrated Stein–Weiss inequalities to the Herz spaces with variable exponents. It is an extension of the classical Stein–Weiss inequalities and the Stein–Weiss inequalities on Lebesgue spaces with variable exponents. The Stein–Weiss inequalities to the Herz spaces with variable exponents are presented as the boundedness of the fractional integral operators on the power weighted Herz spaces with variable exponents.

Bounded variation spaces with generalized Orlicz growth related to image denoising

Motivated by the image denoising problem and the undesirable stair-casing effect of the total variation method, we introduce bounded variation spaces with generalized Orlicz growth. Our setup covers earlier variable exponent and double phase models. We study the norm and modular of the new space and derive a formula for the modular in terms of the Lebesgue decomposition of the derivative measure and a location dependent recession function. We also show that the modular can be obtained as the Γ-limit of uniformly convex approximating energies.

Weak Solutions to Leray–Lions-Type Degenerate Quasilinear Elliptic Equations with Nonlocal Effects, Double Hardy Terms, and Variable Exponents

This study investigates the existence and multiplicity of weak solutions for a class of degenerate weighted quasilinear elliptic equations that incorporate nonlocal nonlinearities, a double Hardy term, and variable exponents. The problem encompasses a degenerate nonlinear operator characterized by variable exponent growth, along with a nonlocal interaction term and specific constraints on the nonlinearity. By employing critical point theory, we establish the existence of at least three weak solutions under sufficiently general assumptions.

Decay results for a quasilinear wave equation with a nonlinear time-varying frictional damping of variable exponent

Decay results for a quasilinear wave equation with a nonlinear time-varying frictional damping of variable exponent

On Multiple Degenerate Parabolic Equation with Variable Exponent

On Multiple Degenerate Parabolic Equation with Variable Exponent

Variable Exponent Kirchhoff‐Type Problems Involving Singular and Nonlocal Terms on Riemannian Manifolds

ABSTRACTIn this paper, we investigate a binonlocal elliptic problem of the ‐Kirchhoff type, incorporating a singular term, within the framework of complete noncompact Riemannian manifolds. Our primary goal is to establish the existence of at least two nontrivial weak solutions, provided that a certain parameter remains sufficiently small. To achieve this, we employ the Nehari manifold method combined with the concept of fibering maps. Specifically, we define the associated Nehari manifold by considering the first derivative of the fibering mapping. Subsequently, we partition the Nehari manifold into three disjoint subsets based on the second derivative of the fibering function. It is shown that the global minimizers of the energy functional, when restricted to two of these subsets, correspond to the desired solutions, while the third subset is empty.

Nonoscillation theory of nonlinear differential equations of Emden–Fowler type with variable exponents

Nonoscillation theory of nonlinear differential equations of Emden–Fowler type with variable exponents

A Contemporary Review and Data-Driven Evaluation of Archard-Type Wear Laws

Abstract Archard's Wear Law and its variants have remained fundamental to wear prediction for over 70 years, despite frequent criticism regarding their simplicity and the variability of the wear coefficient. Numerous recent sliding wear experiments were evaluated to determine whether these models still hold and to what degree, and to identify areas for improvement. A total of 75 papers with detailed wear data were reviewed, and 39 of them were selected for a regression analysis. Modified Archard models, incorporating variable exponents for load, sliding speed, and material hardness, were optimized for each independent study. A Gaussian mixture model was then used to cluster the optimized exponents into two groups: one centered around the original Archard model, and the other reflecting alternative optimized exponents. The review found that 81% of the papers referencing a wear model employed a variant of the Archard model. Models using material hardness as the primary factor influencing wear volume struggled to make physically sound or accurate predictions. Although the Archard-type model maintains its relevance in relating wear volume to applied load and sliding speed, its dependence on a constant wear coefficient and material hardness falls short in describing wear phenomena comprehensively. Further improvement of the model using mechanics is necessary to enhance the accuracy of wear predictions.

Kirchhoff-type problems with the non-local fractional d(z,.)-Laplacian operator

In this work, we examine whether nonnegative solutions exist for a Kirchhoff-type issue that is driven by a fractional d(z,.)− Laplacian operator, where d is a continuous positive function. We establish the existence of their nonnegative solutions combined with the theory of fractional Sobolev spaces with variable exponents by using approximation methods.

Stability Properties of Distributional Solutions for Nonlinear Viscoelastic Wave Equations with Variable Exponents

A system of nonlinear wave equations in viscoelasticity with variable exponents is considered. It is assumed that the kernel included in the integral term of the equations depends on both the time and the spatial variables. Using the Faedo–Galerkin method and the contraction mapping principle, a theorem of unique solvability of the problem is proved. In addition, under appropriate variable assumptions, an estimate of the stability of the solution to the problem of determining the kernel is obtained. The study is based on Komornik’s inequality. We expand the class of nonlinear boundary value problems that can be investigated by well-known methods.

Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space ℓq(·)logβLp(·), which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when β=0 and to classical Lebesgue spaces when q=∞,β=0. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.

Boundedness of sublinear operators on two-weighted grand Herz spaces with variable exponent

In this article, we introduce the concept of two-weighted grand variable Herz spaces as a natural generalization of variable Herz spaces. We establish the boundedness of sublinear operators within this framework, offering significant insights into their functional behavior. Our findings, incorporating additional weight factors, broaden the scope and applicability of our results.

Blow up, Growth and Decay of Solutions for Class of a Coupled Nonlinear Viscoelastic Kirchhoff Equations with Variable Exponents and Fractional Boundary Conditions

Blow up, Growth and Decay of Solutions for Class of a Coupled Nonlinear Viscoelastic Kirchhoff Equations with Variable Exponents and Fractional Boundary Conditions

Variable Hardy spaces on spaces of homogeneous type and applications to variable Hardy spaces associated with elliptic operators having Robin boundary conditions on Lipschitz domains

AbstractLet be a space of homogeneous type with and a variable exponent function satisfying , where . Assume that is a nonnegative self‐adjoint operator on whose heat kernels satisfy the Gaussian upper bound estimates. In this paper, the authors first establish the atomic, nontangential, and radial maximal function characterizations for the variable Hardy space associated with . Second, as applications, when , is a bounded Lipschitz domain, is a second‐order divergence form elliptic operator with the Robin boundary condition, the authors establish a new atomic characterization of the associated with , and then the authors further show that, for , where and , respectively, denote the variable Hardy spaces on associated with the corresponding elliptic operators having the Dirichlet and the Neumann boundary conditions, and ( and ), respectively, denote “supported type” and “restricted type” Hardy spaces on (its local version), and is a critical index depending on the operators , , and . Third, the authors prove the boundedness of the Riesz transform on the variable Lebesgue space , and from to , or to . Finally, the authors further establish the global regularity estimates in both and for the inhomogeneous Robin problem of on .

Existence and asymptotic stability for a semilinear damped wave equation with dynamic boundary conditions involving variable nonlinearity

We investigate the solvability of a class of quasilinear elliptic equations characterized by a (p(x); k(x)) growth structure and nonlinear boundary conditions, speci cally in the context of Kelvin-Voigt damping with arbitrary data. Our approach involves analyzing the problem within appropriate functional spaces, utilizing Lebesgue and Sobolev spaces with variable exponents. In the rst step, we establish the existence and uniqueness of results for solutions to the model, provided the data meet certain regularity conditions. Our methodology primarily relies on xed-point theory and Faedo-Galerkin techniques, incorporating some novel strategies. In the second part, we consider scenarios with su ciently large data sets and show that the system's energy grows exponentially.

Maximal Convergence by Faber Series in Morrey-Smirnov Classes with Variable Exponents

In this paper, we assume that G is a domain bounded by Γ Dini-smooth curve and R > 1 is the largest number such that a function f is analytic inside the level curve ΓR in the exterior of Γ. By taking the function f in the Morrey-Smirnov classes with variable exponents , we obtain a rate of maximal convergence of the nth partial sums of the Faber series of the function f in the uniform norm on the closure of G. Here the rate of maximal convergence depends on the best approximation number .

Multiple solutions for some classes of non-linear elliptic equations with variable exponents

Abstract This article concerns the existence and multiplicity of solutions for the following class of non-linear elliptic equations with variable exponents \begin{equation*} \left\{ \begin{array}{l} -\Delta u+\lambda u=f(x,u), \quad \text{in} \quad \mathbb{R}^N, \\ \,u\in H^{1}(\mathbb{R}^N),\\ \end{array} \right. \end{equation*} where λ > 0, $N\geq 2$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a function of the following types: Type 1: The subcritical case: \begin{equation*}f(x,t)=|t|^{p(\varepsilon x)-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*} Type 2: The critical case: \begin{equation*}f(x,t)=\mu|t|^{p(\varepsilon x)-2}t+|t|^{2^*-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*} Type 3: The exponential subcritical growth case: \begin{equation*} f(x,t)=\mu|t|^{p(\varepsilon x)-2}te^{\alpha|t|^\beta}, \quad \forall (x,t) \in \mathbb{R}^2\times \mathbb{R}, \end{equation*} where parameter ɛ > 0, α > 0, $\beta \in (0,2)$ , $2^*=\frac{2N}{N-2}$ if $N \geq 3$ and $2^*=+\infty$ if N = 2. Related to the function $p:\mathbb{R}^{N}\rightarrow \mathbb{R}$ , we assume that it is a continuous function with $p_{\max}, p_{\min} \in (2,2^*)$ , where $p_{\max}=\displaystyle \max_{x \in \mathbb{R}^N}p(x)$ and $p_{\min}=\displaystyle \min_{x \in \mathbb{R}^N}p(x)$ . We show that for each λ > 0 the number of solutions is associated with the number of global maximum or global minimum points of p when ɛ is small enough. The proof of is based on the variational methods, Ekeland’s variational principle, Trundiger–Moser inequality, and the monotonicity of the ground state energy with respect to p. Our results extend those of Cao and Noussair (Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$ . Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 13 (1996), 567–588) and Ji, Wang and Wu (A monotone property of the ground state energy to the scalar field equation and applications. J. Lond. Math. Soc., II. Ser.100 (2019), 804–824).

A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents

A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents

A general stability result for a nonlinear viscoelastic equation with variable exponents

This work is concerned with a viscoelastic equation with nonlinearities of variable exponents. Firstly, the global existence of solution is obtained. Then, a general decay result of the global solution is derived via the perturbed energy method and some formulas related to convex functions. The kernel function under consideration belongs to a wider class of functions. More precisely, it does not need to satisfy the condition related to differential inequalities involving itself; it just needs to stay below a certain known function. The results in this work expand and generalize those in the literature regarding problems with variable exponents.