  For $1<p<2$ and $q$ large, we prove the existence of two positive,
nonconstant, radial and radially nondreacreasing solutions of the supercritical
equation \[-\Delta_p u+u^{p-1}=u^{q-1}\] under Neumann boundary conditions, in
the unit ball of $\mathbb R^N$. We use a variational approach in an invariant
cone. We distinguish the two solutions upon their energy: one is a ground state
inside a Nehari-type subset of the cone, the other is obtained via a mountain
pass argument inside the Nehari set. As a byproduct of our proofs, we detect
the limit profile of the low energy solution as $q\to\infty$ and show that the
constant solution 1 is a local minimum on the Nehari set.

Analysis of PDEs