On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity

TL;DR

This paper investigates the behavior, multiplicity, and symmetry properties of least energy solutions to a Neumann boundary Lane-Emden system, especially as parameters approach zero, revealing convergence to sign-changing nonlinear problems and symmetry breaking phenomena.

Contribution

It introduces the first analysis of least energy solutions in this setting with exponents tending to zero, and characterizes their convergence and symmetry breaking.

Findings

Least energy solutions converge to sign nonlinearity problems as exponents tend to zero.

Symmetry breaking occurs in solutions of higher-order equations with sign nonlinearity.

The approach links solutions to a nonlinear eigenvalue problem involving the bilaplacian.

Abstract

We consider the following Lane-Emden system with Neumann boundary conditions \[ -\Delta u= |v|^{q-1}v \text{ in } \Omega,\qquad -\Delta v= |u|^{p-1}u \text{ in } \Omega,\qquad \partial_\nu u=\partial_\nu v=0 \text{ on } \partial \Omega, \] where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ with $N \ge 1$. We study the multiplicity of solutions and the convergence of least energy (nodal) solutions (l.e.s.) as the exponents $p, q > 0$ vary in the subcritical regime $1/(p+1) + 1/(q+1) > (N-2)/N$, or in the critical case $1/(p+1) + 1/(q+1) =(N-2)/N$ with some additional assumptions. We consider, for the first time in this setting, the cases where one or two exponents tend to zero, proving that l.e.s. converge to a problem with a sign nonlinearity. Our approach is based on an alternative characterization of least energy levels in terms of the nonlinear eigenvalue problem \[ \Delta (|\Delta u|^{\frac 1 q -1} \Delta u) = \lambda |u|^{p-1} u, \quad \partial_\nu u=\partial_\nu(|\Delta u|^{\frac 1 q -1} \Delta u)=0 \text{ on } \partial \Omega. \] As an application, we show a symmetry breaking phenomenon for l.e.s. of a bilaplacian equation with sign nonlinearity and for other equations with nonlinear higher-order operators.