Boundary blow-up solutions in the unit ball : asymptotics, uniqueness and symmetry

TL;DR

This paper provides a comprehensive asymptotic analysis of boundary blow-up solutions in the unit ball, establishing conditions for uniqueness, symmetry, and universality of blow-up rates across various nonlinearities.

Contribution

It introduces a full asymptotic expansion method applicable to any nonlinearity and characterizes when blow-up rates are universal, advancing understanding of boundary blow-up phenomena.

Findings

Asymptotic expansions for boundary blow-up solutions are derived.

Conditions for uniqueness and radial symmetry are established.

The paper characterizes nonlinearities with universal blow-up rates.

Abstract

We calculate the full asymptotic expansion of boundary blow-up so- lutions, for any nonlinearity f . Our approach enables us to state sharp qualitative results regarding uniqueness and ra- dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. At last, we study in more detail the standard nonlinearities f (u) = u^p, p > 1