On a critical Hamiltonian system with Neumann boundary conditions

TL;DR

This paper constructs boundary blow-up solutions for a coupled Hamiltonian system with Neumann boundary conditions, especially near the critical hyperbola, revealing the influence of domain curvature on solution behavior.

Contribution

It introduces a method to build boundary blow-up solutions for a critical Hamiltonian system with Neumann conditions, focusing on the role of domain curvature.

Findings

Solutions blow up at boundary points with minimal negative mean curvature.

Behavior near the critical hyperbola is characterized by boundary concentration.

The approach links geometric properties of the domain to solution singularities.

Abstract

We consider the Hamiltonian system with Neumann boundary conditions: \[ -\Delta u + \mu u=v^{q }, \quad -\Delta v+ \mu v=u^{p} \quad \text{ in $\Omega$}, \qquad u, v >0 \quad \text{ in $\Omega$,} \qquad \partial_\nu u= \partial_\nu v=0 \quad \text{ on $\partial \Omega$, } \] 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.