On a critical Hamiltonian system with Neumann boundary conditions

Abstract

We consider the Hamiltonian system with Neumann boundary conditions: \[ \begin{cases} -\Delta u + \mu u=v^{q } & \text{in $\Omega$}, \\ -\Delta v+ \mu v=u^{p} & \text{in $\Omega$}, \\ u, v > 0 & \text{in $\Omega$,} \\ \partial_\nu u= \partial_\nu v=0 & \text{on $\partial \Omega$,} \end{cases} \] where $\mu > 0$ is a parameter and $\Omega$ is a smooth bounded domain in $\mathbb R^N .$ When $(p, q)$ approaches from below the critical hyperbola $N/(p+1) + N/$ $(q+1)=N-2$, we build a solution which blows-up at a boundary point where the mean curvature achieves its minimum and negative value.