On a critical Hamiltonian system with Neumann boundary conditions

Abstract

We consider the Hamiltonian system with Neumann boundary conditions: \[ -Δu + μu=v^{q }, \quad -Δv+ μv=u^{p} \quad \text{ in $\Omega$}, \qquad u, v >0 \quad \text{ in $\Omega$,} \qquad \partial_νu= \partial_νv=0 \quad \text{ on $\partial \Omega$, } \] 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.