Stable and singular solutions of the equation $\Delta u = 1/u$

TL;DR

This paper investigates the existence and stability of singular solutions to the elliptic equation Δu=1/u, revealing nonexistence in low dimensions under stability constraints and connecting these solutions to minimal hypersurface singularities.

Contribution

It establishes nonexistence results for stable singular solutions in low dimensions and links the problem to minimal hypersurface singularities.

Findings

No stable singular solutions in low dimensions

Connection between singular solutions and minimal hypersurface singularities

Insights into the structure of solutions to Δu=1/u

Abstract

We study properties of the semilinear elliptic equation $\Delta u = 1/u$ on domains in $R^n$, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low dimensions when we also require them to be stable for the corresponding variational problem. The problem of finding singular solutions is related to the general study of singularities of minimal hypersurfaces in Euclidean space.