Two Dimensional Elliptic Equation with Critical Nonlinear Growth

We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. { −∆u = λueu , u > 0 in Ω, u = 0 on ∂Ω, (1.1) where Ω is a unit disk in R2 and λ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies ∫ D |∇u| dx → 4π, λ ↘ 0. Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as um(γmx)− um(γm) → 2 log 2 1 + |x|2 as λm ↘ 0 locally uniformly in x ∈ R2. We also show some extensions of the above results for general two dimensional domains.