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We consider the following slightly supercritical problem for the Lane-Emdensystem with Neumann boundary conditions: \begin{equation*} \begin{cases}-\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Deltau_2=|u_1|^{q_\epsilon-1}u_1, \ &in\ \Omega,\\ \partial_\nu u_1=\partial_\nuu_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 andmultiplicity of concentrated solutions based on the Lyapunov-Schmidt reductionargument incorporating the zero-average condition by certain symmetries.  It is worth noting that we simultaneously consider two cases: $p>\fracn{n-2}$ and $p<\frac n{n-2}$. The coupling mechanisms of the system arecompletely different in these different cases, leading to significant changesin the behavior of the solutions. The research challenges also vary. Currently,there are very few papers that take both ranges into account when consideringsolution construction. Therefore, this is also the main feature and newingredient of our work.

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