Multiplicity of solutions on a Nehari set in an invariant cone

For $1<p<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondreacreasing solutions of the supercritical equation $$-{\mathrm{\Delta }}_{p}u+u^{p-1}=u^{q-1}$$ 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 $q\to \mathrm{\infty }$ and show that the constant solution 1 is a local minimum on the Nehari set.