On a critical Hamiltonian system with Neumann boundary conditions - Pistoia, Angela; Schiera, Delia (Topological Methods in Nonlinear Analysis, 2025)
We consider the Hamiltonian system with Neumann boundary conditions:\[\begin{cases}-\Delta u + \mu u=v^{q } & \text{in $\Omega$}, \\-\Delta v+ \mu v=u^{p} & \text{in $\Omega$}, \\ u, v > 0 & \text{in $\Omega$,} \\\partial_ u u= \partial_ u v=0 & \text{on $\partial \Omega$,}\end{cases}\]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 solutionwhich blows-up at a boundary point where the mean curvature achieves its minimum and negative value.