Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-\Delta_p u-\mu|x|^{ -p}{ u^{ p-1}}=|x|^{ -s}{ u^{ \crits-1}}+u^{ \crit-1}$ admits a positive weak solution in $\rn$ of class $\dunp\cap C^1(\rn\setminus\{ 0\})$, whenever $\mu<\mu_1$, and $\mu_1=[(n-p)/p]^p$. The technique is based on the existence of extremals of some Hardy--Sobolev type embeddings of independent interest. We also show that if $u\in\dunp$ is a weak solution in $\rn$ of $-\Delta_p u-\mu|x|^{ -p}{ |u|^{ p-2}u}=|x|^{ -s}{ |u|^{ \crits-2}u}+|u|^{ q-2}u$, then $u\equiv0$ when either $1<q<\crit$, or $q>\crit$ and $u$ is also of class $L^\infty_\text{ \scriptsize{ loc}}(\rn\setminus\{ 0\})$.
Categories: math.AP