Blow-up solutions for linear perturbations of the Yamabe equation

TL;DR

This paper investigates blow-up solutions for a perturbed Yamabe equation on compact Riemannian manifolds, analyzing how small perturbations affect solution behavior near criticality.

Contribution

It introduces a detailed analysis of blow-up solutions for linear perturbations of the Yamabe equation on compact manifolds, extending understanding of solution behavior under small perturbations.

Findings

Identification of conditions leading to blow-up solutions.

Characterization of the asymptotic behavior of solutions as perturbation parameter tends to zero.

Insights into the stability and structure of solutions near critical points.

Abstract

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$\Delta_g u+(N-2/4(N-1) S_g+\epsilon h)u=u^{N+2/N-2} in M, u>0 in M,$$ where \Delta_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,\alpha}(M)$, and $\epsilon$ is a small parameter.