Let $p \in [1,d)$, and $p^*$ be the Sobolev conjugate of $p$. I want to prove that there exists a constant $C$ s.t. all $f \in L^1_{loc}(\mathbb{R}^{d})$ vanishing at infinity with $|Df|\in L^p(\mathbb{R}^{d})$ satisfy $$ ||f||_{L^{p^{*}}(\mathbb{R}^{d})} \leq C ||Df|| _{L^p(\mathbb{R}^{d})}$$
Where we say that $f$ vanishes at infnity if for all $\epsilon >0$ the set $\{x \in \mathbb{R}^d : |f(x)|>\epsilon\}$ has finite Lebesgue measure.
Actually I don't know how to proceed. If anyone could help I would be very thankful.
You are talking about "other hypotheses" but I am not sure to understand what are the "classical" hypotheses" you are thinking of.
But anyway, you can find the proof of this version of the Sobolev inequalities in the book $\textit{Analysis}$ by Lieb and Loss (Theorem 8.3 for the $p=2$ case, and section 8.2 for the definitions of the spaces and the explanation on how to do it for $p≠ 2$). The proof is done by using the Hardy-Littlewood-Sobolev inequalities and a duality argument.