Global compactness properties of semilinear elliptic equations with critical exponential growth

Abstract

Sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the Sobolev H¹-topology or concentrate at isolated points of the domain. For energies allowing at most single-point blow-up, we establish a universal blow-up pattern near the concentration point and uniquely characterize the blow-up energy in terms of a geometric limiting problem.