Spike layered solutions for elliptic systems on Riemannian Manifolds

TL;DR

This paper analyzes spike layered solutions for a Hamiltonian elliptic system on Riemannian manifolds, showing solutions concentrate at points of maximum scalar curvature as a parameter tends to zero.

Contribution

It establishes the existence and concentration behavior of solutions to a coupled elliptic system on manifolds, linking concentration points to scalar curvature maxima.

Findings

Solutions concentrate at points where scalar curvature is maximal.

Concentration occurs as the parameter epsilon approaches zero.

Solutions are positive and of least energy.

Abstract

In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}\Delta_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}\Delta_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v >0 && \text{ in } \mathcal{M}, \end{aligned} \end{cases} \end{equation*} where $\mathcal{M}$ is a smooth, compact and connected Riemannian manifold of dimension $N\geq 3$ without boundary. The exponents $p,q>1$ are assumed to lie below the critical hyperbola, ensuring subcritical growth conditions. We investigate a sequence of least energy critical points of the associated dual functional and analyze their concentration behavior as $\varepsilon \to 0$. Our main result shows that the sequence of solutions exhibits point concentration, with the concentration occurring at a point where the scalar curvature of $\mathcal{M}$ attains its maximum.