Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions

In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,\mathrm{\Phi }(\cdot ,\cdot )}(\mathrm{\Omega })$ where $\mathrm{\Phi }(x,t):=t^{p(x)}+a(x)t^{q(x)}{\log }^{r(x)}(e+t)$ with $1\le p(x)\le q(x)\mathrm{.}$ We have established subcritical embedding in bounded John domains under some regularity assumptions on exponents $p,$ $q,$ $r$, and $a$. Conversely, we have proved that if the embedding holds in any domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n,$ then $\mathrm{\Omega }$ must satisfy the log-measure density condition.