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 \[ -\Delta u= |v|^{q-1}v \text{ in } \Omega,\qquad -\Delta v= |u|^{p-1}u \text{ in } \Omega,\qquad \partial_\nu u=\partial_\nu v=0 \text{ on } \partial \Omega, \] 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.