On the least-energy solutions of the pure Neumann Lane-Emden equation

TL;DR

This paper investigates the existence, convergence, and qualitative properties of least-energy solutions to the Neumann Lane-Emden equation across different regimes of the exponent p, revealing domain-dependent behaviors and stability of solutions.

Contribution

It provides new variational characterizations and analyzes the asymptotic behavior of least-energy solutions in subcritical, critical, and supercritical regimes, including symmetry and monotonicity properties.

Findings

Least-energy solutions converge to solutions of related problems as p approaches 0, 1, and 2^*.

No blowup phenomena occur at the critical Sobolev exponent 2^* for Neumann problems.

Domain geometry influences the limit behavior of solutions as p approaches 1.

Abstract

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as $p\searrow 0$; to a l.e.n.s. of the critical problem as $p\nearrow 2^*$ (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent $2^*$); and we show that the limit as $p\to 1$ depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.