Existence and nonexistence of solutions to pure supercritical <Emphasis Type="Italic">p</Emphasis>-Laplacian problems

We establish multiplicity and nonexistence of solutions to the quasilinear problem

$$\begin{aligned} -\Delta _{p}v=\left| v\right| ^{q-2}v\,\,\text {in}\,\,\Omega ,\qquad v=0\text { on }{\partial {\Omega }}, \end{aligned}$$

in some bounded smooth domains \(\Omega \) in \(\mathbb {R}^{N}\), for \(1<p<N\) and some supercritical exponents \(q>p^{*}:=\frac{Np}{N-p}\). Multiplicity is established in domains arising from the Hopf maps. We show that, after a suitable change of metric, these maps become p-harmonic morphisms, i.e., they preserve the p-Laplace operator up to a factor. We use them to reduce the supercritical problem to an anisotropic quasilinear critical problem in a domain of lower dimension.