Multiplicity of Solutions on a Nehari Set in an Invariant Cone

For $1<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondecreasing 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 minimizer on the Nehari set. This marks a strong difference with the case $p\ge 2$.