Variable Exponent Sobolev Spaces Research Articles

On compact and bounded embedding in variable exponent Sobolev spaces and its applications

For a weighted variable exponent Sobolev space, the compact and bounded embedding results are proved. For that, new boundedness and compact action properties are established for Hardy’s operator and its conjugate in weighted variable exponent Lebesgue spaces. Furthermore, the obtained results are applied to the existence of positive eigenfunctions for a concrete class of nonlinear ode with nonstandard growth condition.

Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space

In this paper, we are concerned with the equation $$\begin{aligned} u_t= \sum _{i=1}^N\frac{\partial }{\partial x_i} \left( a_i(x)\left| u_{x_i} \right| ^{p_i(x) - 2} u_{x_i} \right) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ where $$a_i(x)|_{x\in \partial \Omega }=0$$ and $$a_i(x)|_{x\in \Omega }>0$$ . By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since $$a_i(x)$$ is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.

Existence results of infinitely many solutions for p(x)-Kirchhoff type triharmonic operator with Navier boundary conditions

This paper is concerned with the existence of weak solutions to a class of nonlinear elliptic Navier boundary value problem involving the p(x)-Kirchhoff type triharmonic operator. By means of a variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence of weak solutions for the problem.

On Lewy–Stampacchia Inequalities for a Pseudomonotone Elliptic Bilateral Problem in Variable Exponent Sobolev Spaces

The aim of this work is to consider Lewy–Stampacchia inequalities for pseudomonotone elliptic operators in very general situations. This generalizes the results, and simplifies the proofs, proposed in the unilateral obstacle case, as well as the one in the bilateral case. By an ad hoc perturbation of the operator and a penalization of the constraint, one is able to reduce significantly the usual assumptions on the data and to consider a pseudomonotone elliptic operator defined on variable exponent Sobolev spaces.

Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces with variable exponent

‎‎In this paper‎, ‎we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces‎, ‎by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces‎.

New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

We establish a new characterization of the Musielak-Orlicz-Sobolev space on ℝn, which includes the classical Orlicz-Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hasto and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f ∈ L1(ℝn) into f ∈ L1loc(ℝn), which was conjectured to be true by Hasto and Ribeiro in the aforementioned same article.

Correction to: A $${\texttt {p}}({\cdot })$$p(·)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

Correction to: A $${\texttt {p}}({\cdot })$$p(·)-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

Existence results for Schrödinger [formula omitted]-Laplace equations involving critical growth in [formula omitted

We establish some existence results for Schrödinger p(⋅)-Laplace equations in RN with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space RN. The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings.

Global gradient estimates for asymptotically regular problems of p(x)-Laplacian type

We study an asymptotically regular problem of [Formula: see text]-Laplacian type with discontinuous nonlinearity in a nonsmooth bounded domain. A global Calderón–Zygmund estimate is established for such a nonlinear elliptic problem with nonstandard growth under the assumption that the associated nonlinearity has a more general kind of the asymptotic behavior near the infinity with respect to the gradient variable. We also address an optimal regularity requirement on the nonlinearity as well as a minimal geometric assumption on the boundary of the domain for the nonlinear Calderón–Zygmund theory in the setting of variable exponent Sobolev spaces.

A $$\texttt {p}(\cdot )$$-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

We prove a \(\texttt {p}(\cdot )\)-Poincare-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\)-capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.

Existence of entropy solutions to nonlinear parabolic problems with variable exponent and $$L^1$$ L 1 -data

In the present paper we prove existence results of entropy solutions to a class of nonlinear parabolic p(.)-Laplacian problem with Neumann-type boundary conditions and \(L^1\) data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

Existence of solutions for a nonlocal (P_1(x), P_2(x))-Laplace equation with dependence on the gradient

The object of this work is to study the existence of solutions for a nonlocal (p1(x), p2(x)) Laplace equation with dependence on the gradient. We establish our results by using the degree theory for operators of (S+) type in the framework of variable exponent Sobolev spaces.

Solutions of evolutionary $${\varvec{p(x)}}$$-Laplacian equation based on the weighted variable exponent space

The paper studies the equation $$\begin{aligned} u_t= {\text {div}} (a(x)\left| \nabla u \right| ^{p(x) - 2}\nabla u) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ with the boundary degeneracy due to $$a(x)\mid _{x\in \partial \Omega }=0$$ , and $$\Omega \subset \mathbb {R}^{N}$$ , where N is a positive integer. By the theory of the weighted variable exponent Sobolev spaces, the well posedness of weak solutions of this equation is discussed. The novelty of our results lies in the fact that under certain conditions, if a(x) satisfies $$\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x<\infty $$ , the global stability of weak solutions can be established without any boundary value condition. While $$\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x=\infty $$ , the local stability of weak solutions can be obtained without any boundary value condition.

The stability of the anisotropic parabolic equation with the variable exponent

Consider the equation \t\t\tut=div(a(x)|∇u|p(x)−2∇u→)=∑i=1N(ai(x)|uxi|pi(x)−2uxi)xi,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$${u_{t}}= \\operatorname{div} \\bigl(\\overrightarrow{a(x){ \\vert{ \\nabla u} \\vert^{p(x) - 2}}\\nabla u} \\bigr)=\\sum_{i=1}^{N} \\bigl(a_{i}(x) \\vert u_{x_{i}} \\vert^{p_{i}(x)-2}u_{x_{i}} \\bigr)_{x_{i}}, $$\\end{document} with a_{i}(x), p_{i}(x)in C^{1}(overline{Omega}), p_{i}(x)>1, a_{i}(x)geq 0. Basing on the weighted variable exponent Sobolev space, a new kind of weak solutions of the equation is introduced. Whether the usual Dirichlet homogeneous boundary value condition can be imposed depends on whether a_{i}(x) is degenerate on the boundary or not. If some of { a_{i}(x)} are degenerate on the boundary, a partial boundary value condition is imposed. If every a_{i}(x) is degenerate on the boundary, by the new definition of a weak solution, the stability of weak solutions can be proved without any boundary value condition.

Existence results for a class of p(x)-Kirchhoff problems

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhofftype equations with Robin boundary data as follows: in Ω, on ∂Ω, where β ∈ L∞ (∂Ω) and f : Ω × → satisfies the Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

Infinitely Many Solutions of the Neumann Problem for Elliptic systems in Anisotropic Variable Exponent Sobolev Spaces

Abstract In this paper, we prove the existence of in finitely many solutions for the following system by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces 2010 Mathematics Subject Classification. 35K05 - 35K55.

RETRACTED ARTICLE: Existence of weak solutions for two point boundary value problems of Schr\xf6dingerean predator-prey system and their applications

By means of a variational analysis and the theory of variable exponent Sobolev spaces, the existence of weak solutions for two point boundary value problems of Schrödingerean predator-prey system with latent period is investigated either analytically or numerically. More precisely, the local stability of the Schrödingerean equilibrium and endemic equilibrium of the model are discussed in detail. And we specially analyzed the existence and stability of the Schrödingerean Hopf bifurcation by using the center manifold theorem and the bifurcation theory. As applications, theoretic analysis and numerical simulation show that the Schrödingerean predator-prey system with latent period has very rich dynamic characteristics.

Existence and multiplicity results for a class of Kirchho type problems involving the p(x)-biharmonic operator

The aim of this paper is to establish the existence and multiplicity of solutions for a class of nonlocal problem involving the p(x)-biharmonic operator. Our technical approach is based on direct variational method and the theory of variable exponent Sobolev spaces.

Characterization of the variable exponent Sobolev norm without derivatives

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.

Existence results for a class of nonlocal problems involving the -Laplace operator

This paper is concerned with the existence of solutions to a class of -Kirchhoff-type equations with Dirichlet boundary condition of the form:By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions.