Some Properties of Musielak-Orlicz-Sobolev spaces with an application in Nonlinear PDE
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Date
2025-06-15
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M'sila
Abstract
This work investigates some properties of Musielak-Orlicz-Sobolev spaces W1,θ(Ω) and their
application to the nonlinear double-phase problem with variable exponents:
−div |∇u|
p(x)−2∇u + µ(x)|∇u|
q(x)−2∇u
= f(x, u,∇u), in Ω,
u = 0, on ∂Ω,
(4.39)
Where Ω ⊂ RN (N ≥ 2) is a bounded domain with Lipschitz boundary ∂Ω, p, q ∈ C(Ω) satisfy
1 < p(x) < N, p(x) < q(x) for all x ∈ Ω, µ ∈ L
∞(Ω), and f is a Carathéodory function.
We study the existence and uniqueness results for solutions to problem (4.39) in the Musielak Orlicz-Sobolev space framework. The variational methods cannot be applied here due to
problem (4.39) does not have variational structure. This why our approach employs non variational method following [20], building on the analytical techniques developed in [12]
and [14] for studying partial differential equations with non-standard growth conditions.
These methods provide a robust framework for analyzing sequences of approximate solu tions and their convergence properties.
The analysis highlights the fundamental role of Musielak-Orlicz-Sobolev spaces in functional
analysis, extending the classical variable exponent Lebesgue space theory [3]. We examine
several key properties of these spaces that are essential for handling the nonlinearities and
variable growth conditions in problem (4.39).
Description
Keywords
Musielak-Orlicz Spaces, Musielak-Orlicz-Sobolev Spaces, Double Phase Operator, Variable Exponent, Nonlinear PDEs, Existence Results, Uniqueness