# Weighted Stochastic Field Exponent Sobolev Spaces and Nonlinear Degenerated Elliptic Problem

## TL;DR

This paper introduces weighted stochastic field exponent Sobolev spaces, explores their properties, and applies them to analyze stochastic PDEs with stochastic field growth, advancing the mathematical framework for such problems.

## Contribution

The paper develops new weighted stochastic field exponent Sobolev spaces and demonstrates their application to stochastic partial differential equations with stochastic growth conditions.

## Key findings

- Defined new weighted stochastic field exponent spaces
- Established properties and embeddings of these spaces
- Applied the framework to stochastic PDEs with growth conditions

## Abstract

In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times \Omega \right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times \Omega \right) $. 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.