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

Abstract

We consider the following Lane-Emden system with Neumann boundary conditions \[ -Δu= |v|^{q-1}v \text{ in } Ω,\qquad -Δv= |u|^{p-1}u \text{ in } Ω,\qquad \partial_νu=\partial_νv=0 \text{ on } \partial Ω, \] 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 \[ Δ(|Δu|^{\frac 1 q -1} Δu) = λ|u|^{p-1} u, \quad \partial_νu=\partial_ν(|Δu|^{\frac 1 q -1} Δu)=0 \text{ on } \partial Ω. \] 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.