Some Properties of Musielak-Orlicz-Sobolev spaces with an application in Nonlinear PDE

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).