Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

We study the following Lane-Emden system $$-\mathrm{\Delta }u=\mathrm{\mid }v{\mathrm{\mid }}^{q-1}v\text{ in }\mathrm{\Omega },-\mathrm{\Delta }v=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u\text{ in }\mathrm{\Omega },u_{\nu }=v_{\nu }=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ with $\mathrm{\Omega }$ a bounded regular domain of ${\mathbb{R}}^N$, $N\ge 4$, and exponents $p,q$ belonging to the so-called critical hyperbola $1\mathrm{/}(p+1)+1\mathrm{/}(q+1)=(N-2)\mathrm{/}N$. We show that, under suitable conditions on $p,q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N\ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $\mathrm{\Omega }$ is a ball or an annulus.