Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions

In this paper we study the necessary and sufficient conditions on domain for Musielak-Orlicz-Sobolev embedding of the space $W^{1,\mathrm{\Phi }(\cdot ,\cdot )}(\mathrm{\Omega })$ where $\mathrm{\Phi }(x,t):=t^{p(x)}{(\log (e+t))}^{q(x)}\mathrm{.}$