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

We consider the following Lane-Emden system with Neumann boundary conditions $$-\mathrm{\Delta }u=\mathrm{\mid }v{\mathrm{\mid }}^{q-1}v\text{ in }\mathrm{\Omega },-\mathrm{\Delta }v=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u\text{ in }\mathrm{\Omega },{\mathrm{\partial }}_{\nu }u={\mathrm{\partial }}_{\nu }v=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ 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 $1\mathrm{/}(p+1)+1\mathrm{/}(q+1)>(N-2)\mathrm{/}N$, or in the critical case $1\mathrm{/}(p+1)+1\mathrm{/}(q+1)=(N-2)\mathrm{/}N$ 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 $$\mathrm{\Delta }(\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^{\frac{1}{q}-1}\mathrm{\Delta }u)=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,{\mathrm{\partial }}_{\nu }u={\mathrm{\partial }}_{\nu }(\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^{\frac{1}{q}-1}\mathrm{\Delta }u)=0\text{ on }\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ 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.