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

Abstract

In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,\hat{I}¦(\cdot ,\cdot )}(\hat{I}\text{c◯})$ where $\hat{I}¦(x,t):=t^{p(x)}+a(x)t^{q(x)}{\log }^{r(x)}(e+t)$ with $1\le p(x)\le q(x).$ 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 $\hat{I}\text{c◯}$ in ${\mathbb{R}}^n,$ then $\hat{I}\text{c◯}$ must satisfy the log-measure density condition.