For \(p \in (1, N)\) and a weight function \(g\), the authors study positive solutions \(u\) to the equation \[ -\Delta_p u - \mu g u^{p-1} = u^{\frac{Np}{N-p}- 1 } \] with a parameter \(\mu > 0\), on a domain \(\Omega \subset \mathbb R^N\). The exponent on the right side being Sobolev-critical, the existence of a positive solution through a variational argument is not straightforward.
The authors give sufficient conditions on \(g\) so that a positive solution exists for some range of \(\mu > 0\). These conditions consist in strict inequalities for certain variational quantities, in the spirit of the classical result by H. Brézis et al. [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)] for \(p=2\). Unbounded domains \(\Omega\) are also considered, and special attention is paid to treating a natural optimal class of functions \(g\), namely \(g\) for which a Hardy-type inequality with exponent \(p\) holds. When the above conditions fail, the authors show that a variant of them, which also leads to existence, can sometimes be restored under additional symmetry assumptions on \(g\) and \(\Omega\). Finally, a Pohozaev-type identity for solutions on \(\Omega = \mathbb R^N\) is shown.