Existence of Weak Solutions for $p(\mathrm{.})$-Laplacian Equation via Compact Embeddings of the Double Weighted Variable Exponent Sobolev Spaces

In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(\mathrm{.}),p(\mathrm{.})}(\mathrm{\Omega },{\vartheta }_0,\vartheta )$ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(\mathrm{.})$-Laplacian equation \begin{equation*} \left{ \begin{array}{cc} -$\text{div}$\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in \Omega \ f=0 & x\in \partial \Omega \end{array} \right. \end{equation*} under some conditions of compact embedding involving the double weighted variable exponent Sobolev spaces.