Concentration and energy invariance for a class of fully nonlinear elliptic equations

TL;DR

This paper investigates the asymptotic behavior of positive solutions to a class of fully nonlinear elliptic equations in a ball, revealing concentration phenomena and energy invariance as the nonlinearity exponent approaches a critical value.

Contribution

It demonstrates the concentration and blow-up of solutions at the center, along with the invariance of a related energy, as the exponent nears the critical value.

Findings

Solutions concentrate at the center of the ball.

Solutions blow up as the exponent approaches the critical value.

A suitable energy remains invariant during this process.

Abstract

We study the asymptotic behaviour of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.