On a $p$--Laplace equation with multiple critical nonlinearities

Using the Mountain–Pass Theorem of Ambrosetti and Rabinowitz we prove that $-{\mathrm{\Delta }}_{p}u-\mu \mathrm{\mid }x{\mathrm{\mid }}^{-p}{u}^{p-1}=\mathrm{\mid }x{\mathrm{\mid }}^{-s}{u}^{\text{crits}-1}+u^{\text{crit}-1}$ admits a positive weak solution in $\text{rn}$ of class $\text{dunp}\cap C^1(\text{rn}\setminus \{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1=[(n-p)\mathrm{/}p{]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in \text{dunp}$ is a weak solution in $\text{rn}$ of $-{\mathrm{\Delta }}_{p}u-\mu \mathrm{\mid }x{\mathrm{\mid }}^{-p}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\mathrm{\mid }x{\mathrm{\mid }}^{-s}\mathrm{\mid }u{\mathrm{\mid }}^{\text{crits}-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$, then $u\equiv 0$ when either $1<q<\text{crit}$, or $q>\text{crit}$ and $u$ is also of class $L_{\text{loc}}^{\mathrm{\infty }}(\text{rn}\setminus \{0\})$.