Two dimensional elliptic equation with critical nonlinear growth

Abstract

Abstract. We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions.where Ω is a unit disk in R 2 and λ denotes a positive parameter.We show that for a radially symmetric solution of (1.1) satisfiesMoreover, 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 asWe also show some extensions of the above results for general two dimensional domains.

Blow-up analysis for nodal radial solutions in Moser-Trudinger critical equations in $R^2$

“…In [1] it was proved the existence of solutions to (1.4) for any λ ∈ (0, λ 1 ) where λ 1 is the first eigenvalue of −∆ with Dirichlet boundary conditions (see also [13]). As λ → 0, the corresponding solution u λ concentrates around the origin and its asymptotic behavior was studied in [22] and [3]. These results hold also for more general problems like (1.5) −∆u = λf (u)e u 2 in B u = 0 on ∂B.…”