Sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \ &in\ \Omega,\\ \partial_\nu u_1=\partial_\nu u_2=0,\ &on\ \partial\Omega \end{cases} \end{equation*} where $\Omega=B_1(0)$ is the unit ball in $\mathbb{R}^n$ ($n\geq4$) centered at the origin, $p_\epsilon=p+\alpha\epsilon, q_\epsilon=q+\beta\epsilon$ with $\alpha,\beta>0$ and $\frac1{p+1}+\frac1{q+1}=\frac{n-2}n$. We show the existence and multiplicity of concentrated solutions based on the Lyapunov-Schmidt reduction argument incorporating the zero-average condition by certain symmetries. It is worth noting that we simultaneously consider two cases: $p>\frac n{n-2}$ and $p<\frac n{n-2}$. The coupling mechanisms of the system are completely different in these different cases, leading to significant changes in the behavior of the solutions. The research challenges also vary. Currently, there are very few papers that take both ranges into account when considering solution construction. Therefore, this is also the main feature and new ingredient of our work.