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: $$\begin{array}{r}-\mathrm{\Delta }u=|v{|}^{q-1}v\text{in}\mathrm{\Omega },-\mathrm{\Delta }v=|u{|}^{p-1}u\text{in}\mathrm{\Omega },{\mathrm{\partial }}_{\nu }u={\mathrm{\partial }}_{\nu }v=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ 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 $1/(p+1)+1/(q+1)>(N-2)/N$ , 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 the least energy levels in terms of the nonlinear eigenvalue problem $$\begin{array}{r}\mathrm{\Delta }(|\mathrm{\Delta }u{|}^{(1/q)-1}\mathrm{\Delta }u)=\lambda |u{|}^{p-1}u\text{in}\mathrm{\Omega },{\mathrm{\partial }}_{\nu }u={\mathrm{\partial }}_{\nu }(|\mathrm{\Delta }u{|}^{(1/q)-1}\mathrm{\Delta }u)=0\text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$$ 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.