The Kolmogorov–Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces

Abstract 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.

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Nonlinear elliptic problems in weighted variable exponent Sobolev spaces by topological degree

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-2019-04-0048 ◽

2019 ◽

Vol 38 (4) ◽

pp. 733-751

Author(s):

Mustapha Ait Hammou ◽

El Houssine Azroul

Keyword(s):

Sobolev Spaces ◽

Topological Degree ◽

Variable Exponent ◽

Elliptic Problems ◽

Nonlinear Elliptic Problems ◽

Nonlinear Elliptic ◽

Variable Exponent Sobolev Spaces ◽

Weighted Variable

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Compact embeddings on a subspace of weighted variable exponent Sobolev spaces

Advances in Operator Theory ◽

10.15352/aot.1803-1335 ◽

2019 ◽

Vol 4 (2) ◽

pp. 388-405 ◽

Cited By ~ 2

Author(s):

Cihan Unal ◽

Ismail Aydın

Keyword(s):

Sobolev Spaces ◽

Variable Exponent ◽

Compact Embeddings ◽

Variable Exponent Sobolev Spaces ◽

Weighted Variable

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Weighted Variable Exponent Sobolev spaces on metric measure spaces

Moroccan Journal of Pure and Applied Analysis ◽

10.1515/mjpaa-2018-0007 ◽

2018 ◽

Vol 4 (2) ◽

pp. 62-76

Author(s):

Moulay Cherif Hassib ◽

Youssef Akdim

Keyword(s):

Sobolev Spaces ◽

Variable Exponent ◽

Metric Spaces ◽

Maximal Inequality ◽

Outer Measure ◽

Sobolev Function ◽

Convergence Results ◽

Variable Exponent Sobolev Spaces ◽

Measure Spaces ◽

Weighted Variable

AbstractIn this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

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