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 nondecreasing solutions of the supercritical equation −∆pu + u^p−1 = u^q−1 under Neumann boundary conditions, in the unit ball of 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 → ∞ and show that the constant solution 1 is a local minimizer on the Nehari set. This marks a strong difference with the case p ≥ 2.