Uniform bounds for solutions to elliptic problems on simply connected planar domains

TL;DR

This paper establishes uniform bounds for solutions to certain elliptic problems on simply connected planar domains, using complex analysis and integral identities, with implications for nonlinear PDEs.

Contribution

It provides new uniform bounds for solutions to the Liouville and Henon-Lane-Emden equations on simply connected domains, extending to systems and general nonlinearities.

Findings

Solutions satisfy a uniform mass bound

Results apply to systems and broader nonlinearities

Proofs utilize Riemann mapping theorem and Pohozaev identity

Abstract

We consider the singular Liouville equation and the Henon-Lane-Emden problem on simply connected planar domains. We show that any solution to each problem must satisfy a uniform bound on the mass. The same results applies to some systems and more general non-linearities. The proofs are based on the Riemann mapping theorem and a Pohozaev-type identity.