Multiplicity of solutions to critical p-Laplace equations involving a Hardy potential

Abstract

In this paper, we prove the existence of at least $N$ pairs of nontrivial solutions to the doubly critical quasilinear elliptic problem \[ -\Delta_p u - \frac{\lambda}{|x|^p}|u|^{p-2}u =a(x)|u|^{p-2}u + |u|^{p^*-2}u \] in $\mathbb{R}^N$, as well as in smooth bounded domains, where $1< p< N$, $0< \lambda< (({N-p})/{p})^p$ and $a$ is strictly positive in a small ball. Our results hold under the assumption that $N\ge p^2$ and $\lambda$ and $\|a^+\|_{L^{N/p}}$ are small enough. To circumvent difficulties due to the lack of compactness of the problem, we combine Krasnosel'ski{\u\i}'s genus with a recent classification result by Oliva, Sciunzi, Vaira (J. Math. Pures Appl. {\bf 140} (2020), 89-109) and global compactness results by Li, Guo, Niu (Nonlinear Anal. {\bf 74} (2011), no.\ 4, 1445-1464).