We study the doubly critical quasilinear problem \( -\Delta _p u - \frac{\lambda }{|x|^p}|u|^{p-2}u = |u|^{p^*-2}u \) in a bounded non-contractible domain \(\Omega \subset \mathbb {R}^N\) containing the origin with zero Dirichlet boundary condition, where \(1<p<N\) , \(0<\lambda <\left( \frac{N-p}{p}\right) ^p\) and \(p^*\) is the critical Sobolev exponent. We show that the equation admits at least \(\textrm{cat}(\Omega ) - 1\) positive solutions provided that \(\lambda \) is small enough, where \(\textrm{cat}(\Omega )\) denotes the Lusternik–Schnirelmann category of \(\Omega \) in itself. Our result is new even in the case \(p=2\) , \(N=3\) .
