Variable Exponent Sobolev Spaces Research Articles

P(x)-biharmonic equations involving (h,r(x))-Hardy singular coefficients with no-flux boundary conditions

In this article, we investigate $p(x)$-biharmonic equations involving two kinds of different Hardy potentials, in which the $r(x)$-Hardy potentials are seldom mentioned in previous papers. New criteria for the existence of generalized solutions are reestablished when the nonlinear terms satisfying appropriate assumptions. The results are based on variational methods and the theory of variable exponent Sobolev spaces.

Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}^{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

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.

A p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-Like Operators With Dirichlet Boundary Condition

This paper deals with a class of p(x)-Kirchhoff type problems involving the p(x)-Laplacian-like operators, arising from the capillarity phenomena, depending on two real parameters with Dirichlet boundary conditions. Using a topological degree for a class of demicontinuous operators of generalized (S+), we prove the existence of weak solutions of this problem. Our results extend and generalize several corresponding results from the existing literature. Keywords: p(x)-Kirchhoff type problems, p(x)-Laplacian-like operators, weak solutions, variable exponent Sobolev spaces.

Multiple Solutions for a Class of Biharmonic Nonlocal Elliptic Systems

We prove the existence of three distinct solutions for a biharmonic nonlocal elliptic system with singular terms under the Navier boundary conditions, by using variational methods and the theory of the variable exponent Sobolev space.

Sobolev embeddings in Musielak-Orlicz spaces

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.

Uniqueness for entropy solution to a class of nonlinear degenerate weighted elliptic p(.)-Laplacian problem with L1-data

In the present paper, we prove the uniqueness of entropy solution to a class of nonlinear degenerate weighted elliptic p(·)-Laplacian problems. These particular problems are delineated by Dirichlet-type boundary conditions and L1 data. We use a priori estimates in the framework of weighted variable exponent Sobolev spaces theory.

Infinitely many solutions for a p(x)-triharmonic equation with Navier boundary conditions

Abstract In this work, we will study the existence of an infinity of solutions of a Navier problem governed by the p ⁢ ( x ) {p(x)} -triharmonic operator using the theory of Ljusternick–Shrilemann and the theory of the variable exponent Sobolev spaces.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Multiplicity of solutions for fractional p(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p ( z ) $\\end{document}-Kirchhoff-type equation

This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.

Density of smooth functions in Musielak–Orlicz–Sobolev spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$

AbstractWe consider here Musielak–Orlicz–Sobolev (MOS) spaces , where is an open subset of , and is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions in . One section is devoted to compare the various conditions on appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions. One of the reasons is that in the process of proving density theorems, we do not use the fact that the Hardy–Littlewood maximal operator on Musielak–Orlicz space is bounded, a standard tool employed in density results for different types of Sobolev spaces. We show in particular that under some regularity assumptions on , (A1) and that are not sufficient for the maximal operator to be bounded, the space of is dense in . In the case of variable exponent Sobolev space , we obtain the similar density result under the assumption that , , , , satisfies the log‐Hölder condition and the exponent function is essentially bounded.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

Infinitely many solutions for a critical $ p(x) $-Kirchhoff equation with Steklov boundary value

<p>In this paper, we aim to tackle the questions of existence and multiplicity of solutions of the $ p(x) $-Kirchhoff problem involving critical exponent and the Steklov boundary value. Further, we research the results from the theory of variable exponent Sobolev spaces, the concentration-compactness principle, and the symmetric mountain pass theorem.</p>

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

Abstract In this paper we study a Neumann boundary value problem of a new p(x)-Kirchhoff type problems driven by p(x)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (S+) type,weprove theexistenceofaweak solutionsof this problem. Our results are a natural generalisation of some existing ones in the context of p(x)-Kirchhoff type problems.

Existence of three weak solutions for a nonlinear problem with mixed boundary condition in a variable exponent Sobolev space

Existence of three weak solutions for a nonlinear problem with mixed boundary condition in a variable exponent Sobolev space

Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator

ABSTRACT The Dirichlet boundary value problem for elliptic equations involving the (p(x), q(x))-Laplacian operator with a reaction term depending on the gradient and on two real parameters is considered in this paper. Using the topological degree theory for a class of demicontinuous operators of generalized (S +) and the theory of the variable exponent Sobolev spaces, we prove the existence of at least one weak solution of such problem.

Variable exponent Sobolev spaces and regularity of domains-II

We provide necessary conditions on Euclidean domains for inclusions W1,p(·)(Ω)↪Lq(·)(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p(\\cdot )}(\\Omega ) \\hookrightarrow L^{q(\\cdot )}(\\Omega ) $$\\end{document} of variable exponent Sobolev spaces. The conditions on the exponent p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot ) $$\\end{document} are log-Hölder and log-log-Hölder continuity, while those on the domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Omega $$\\end{document} are the measure and the log measure density conditions. Restrictions on the exponents q(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(\\cdot ) $$\\end{document} and p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot )$$\\end{document} appearing in Górka et al. (J. Geom. Anal. 310: 7304-7319, 2021) are relaxed, improving the results obtained in that work.

Existence of Steady Solutions for a Model for Micropolar Electrorheological Fluid Flows with Not Globally log-Hölder Continuous Shear Exponent

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function {hat{p}} that is log -Hölder continuous and an electric field textbf{E} for that vert textbf{E}vert ^2 is bounded and smooth. Note that these conditions do not imply that the variable shear exponent p={hat{p}}circ vert textbf{E}vert ^2 is globally log -Hölder continuous.

Existence result for a Neumann boundary value problem governed by a class of p ( x )-Laplacian-like equation

In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in Ω , ∂ u ∂ η = 0 on ∂ Ω , where Δ p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ, μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ), η is the outer unit normal to ∂ Ω and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.