Multiplicity of solutions on a Nehari set in an invariant cone

TL;DR

This paper proves the existence of two distinct positive radial solutions to a supercritical p-Laplacian equation in a ball, using variational methods within an invariant cone, and analyzes their asymptotic behavior as the exponent grows.

Contribution

It introduces a variational approach in an invariant cone to find multiple solutions of a supercritical PDE, distinguishing them by energy and analyzing their limits.

Findings

Existence of two positive solutions with different energies.

Identification of the limit profile as q approaches infinity.

Constant solution 1 is a local minimum on the Nehari set.

Abstract

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.