Variable Exponent Research Articles

Different aspects of blow-up property for a nonlinear wave equation

In this work, we are concerned with a nonlinear wave equation with variable exponents. In the presence of the logarithmic nonlinear source, we established a global nonexistence result with negative initial data and without imposing the Sobolev Logarithmic Inequality. The blow-up time is established with upper bound and lower bound. In addition, under some conditions on the initial data and for a specific class of relaxation functions, we established an infinite time blow-up result.

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Existence result for a Steklov problem involving a singular nonlinearity and variable exponents

Abstract In this paper, we use the variational method to study some singular Steklov-type problem with variable exponents. More precisely, we use the min-max method in order to prove the existence of a solution to such a problem.

Marcinkiewicz spaces with variable exponents

Abstract Marcinkiewicz spaces with variable exponents are defined and some basic properties are given.

Triple weak solution for p(x)-Laplacian like problem involving no flux boundary condition

Abstract In this paper, we consider a class of p ⁢ ( x ) {p(x)} -Laplacian like problems with indefinite weight involving no flux boundary condition. Using variational techniques and the critical point theorem of Bonanno and Marano [4], we prove the existence of at least three weak solutions to the problem in Sobolev variable exponent spaces.

The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t−s)−α(t) with 0≤α(t)<1. A Nyström method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, superconvergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nyström method.

Positive solution for a nonlinear elliptic equation on symmetric domains

We show the existence of a positive solution for the Schrodinger quasilinear equation with variable exponents above the critical regime. For that matter, we show an embedding into an Orlicz space of functions modeled over radially symmetric domains. Then we use a Galerkin method combined with a fixed-point argument to obtain a solution. For more information see https://ejde.math.txstate.edu/Volumes/2024/41/abstr.html

Nonexistence Results for the Elliptic Equations with Fractional Laplacian and Variable Exponents Nonlinearities

Nonexistence Results for the Elliptic Equations with Fractional Laplacian and Variable Exponents Nonlinearities

On a class of nonhomogeneous anisotropic elliptic problem with variable exponents

On a class of nonhomogeneous anisotropic elliptic problem with variable exponents

Continuous imbedding theorems in Musielak spaces

We provide continuous Sobolev-type imbedding results for Musielak spaces. In the first result the target space is a Musielak space bigger than the considered one, while the second is that of bounded and continuous functions. Continuous imbeddings into a particular Musielak space built upon Sobolev conjugate functions generate serious difficulties and does not holds in general for arbitrary Musielak functions, as it is well known in variable exponent Sobolev spaces. We overcome this problem by imposing only a regularity assumption on the Musielak functions without resorting to other restrictions such as the Δ2/∇2 conditions. Nevertheless, the continuous embedding in the space of bounded continuous functions is obtained by using the convex envelope, which then allows us to use a result already obtained in Orlicz spaces. The results we obtain extend and unify those obtained in classical Sobolev spaces, variable exponent Sobolev spaces as well as those obtained by Donaldson-Trudinger in the setting of Orlicz spaces.

Existence of weak solutions of the fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problem by topological degree

In this paper, we prove the existence result of weak solutions to a class fractional ‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type . The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results.

Stability analysis for wave equations with variable exponents and acoustic boundary conditions

In this paper, we consider a wave equation with nonlinearities of variable exponents and acoustic boundary conditions. Using the multiplier method, we establish general stability results to the equation when the damping term has a time‐dependent coefficient and the exponent has extended range.

Variable exponents anisotropic nonlinear elliptic systems with L p ′ → ( ⋅ ) -data

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary W ˚ 1 , p → ( ⋅ ) ( Ω , R m ) for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with L p ′ → ( ⋅ ) data for the datum f (i.e. ∈ L p ′ → ( ⋅ ) ( Ω , R m ) ), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives ∂ i u , i = 1 , … , N .

Nonlinear weighted elliptic problem with variable exponents and $$L^1$$ data

Nonlinear weighted elliptic problem with variable exponents and $$L^1$$ data

On the superlinear Kirchhoff problem involving the double phase operator with variable exponents

On the superlinear Kirchhoff problem involving the double phase operator with variable exponents

Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent

The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.