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

Abstract

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 $L^\infty_\text{\scriptsize{loc}}(\rn\setminus\{0\})$.