If it is true that $\int e^u<\infty$ whenever $u\in W^{1,3}$, for $d=3$, and $W^{1,3}$ is not a subset of $L^\infty$, then, since $W_0^{1,3}\subset H_0^1$, there certainly exist essentially unbounded functions $v$ for which both $e^v$ and $e^{-v}$ are integrable ... Or did I miss something?

If it is true that $\int e^u<\infty$ whenever $u\in W^{1,3}$, for $d=3$, and $W^{1,3}$ is not a subset of $L^\infty$, then, since $W_0^{1,3}\subset H_0^1$, there certainly exist essentially unbounded functions $v$ for which both $e^v$ and $e^{-v}$ are integrable ... Or did I miss something?

Example : $v=\log\log\frac1r$ on $B(0,\frac1e)$