Weighted stochastic field exponent Sobolev spaces and nonlinear degenerated elliptic problem with nonstandard growth

Abstract

In this study, we consider weighted stochastic field exponent function spaces L p(.,.) ϑ (D × Ω) and W k,p(.,.) ϑ (D × Ω). Also, we investigate some basic properties and embeddings of these spaces. Finally, we present an application of these spaces to the stochastic partial differential equations with stochastic field growth.

Dirichlet Type Problem With P(.)-Triharmonic Operator

“…In recent years, variational mathematical problems with ( )-growth have been studied in several topics, such as electrorheological fluids, image processing, elastic mechanics, fluid dynamics and calculus of variations [1][2][3][4][5]. Moreover, using compact embedding theorems and equivalent norms in variable exponent Sobolev spaces (weighted or unweighted) give good results to find weak solutions for elliptic and parabolic problems involving ( )-Laplacian operator [3,[6][7][8][9][10][11].…”

Three solutions to a Steklov problem involving the weighted p(.)-Laplacian

“…Because several physical problems, such as electrorheological fluids, elastic mechanics, stationary thermo-rheological viscous flows of non-Newtonian fluids, exponential growth and image processing, can be modeled by such kind of equations, see [13], [21], [23]. In recent years p(x)-Laplacian equations have attracted great attention, see [6], [10], [11], [12], [18], [22]. Moreover, Steklov problems involving p(x)-Laplacian operator have been studied by many authors, see [2], [3], [4], [7], [8], [9], [14].…”

On some general multiplying solutions results of a Robin problem

“…Assume that ω 1 and ω 2 are weight functions. The aim of this study is to discuss the three solutions for the following Robin problem (1.1) − div ω 1 (x) |∇u| p(x)−2 ∇u = λω 2 (x)f (x, u), x ∈ Ω ω 1 (x) |∇u| p(x)−2 ∂u ∂υ + β(x) |u| p(x)−2 u = 0, x ∈ ∂Ω, where ∂u ∂υ is the outer unit normal derivative of u with respect to ∂Ω, λ > 0, p, q ∈ C Ω with inf In recent years, the investigating of the existence of weak solutions of partial differential equations involving weighted p(x)-Laplacian in variable exponent (weighted or unweighted) Sobolev spaces has been very popular (see [4], [6], [9], [11], [12], [17], [23]). Because some such type of equations can explain several physical problems such as electrorheological fluids, image processing, elastic mechanics, fluid dynamics and calculus of variations, see [14], [18], [20], [24].…”