The Nirenberg Problem on Half Spheres: A Bubbling-off Analysis

Abstract

Abstract In this paper, we perform a refined blow-up analysis of finite energy approximated solutions to a Nirenberg-type problem on half spheres. The latter consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular, we give a precise location of blow-up points and blow-up rates. Such an analysis shows that the blow-up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed, besides the combination of interior and boundary blow ups, there are nonsimple blow-up points for subcritical solutions having zero or nonzero weak limit. The formation of such nonsimple blowups is governed by a vortex problem, unveiling an unexpected connection with Euler equations in fluid dynamic and mean fields type equations in mathematical physics.