Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials

Abstract

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials{−Δun=Wn(x)unpn,un>0,inΩ,un=0,on∂Ω,∫ΩpnWn(x)unpndx≤C, where Ω is a smooth bounded domain in R2, Wn(x)≥0 are bounded functions with zeros in Ω, and pn→∞ as n→∞. A typical example is Wn(x)=|x|2α with 0∈Ω, i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for α=0 has been well studied in the literature. While for α>0, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case α>0 and prove a quantization property (suppose 0 is a concentration point)pn|x|2αun(x)pn−1+t→8πet2∑i=1kδai+8π(1+α)et2ctδ0,t=0,1,2, for some k≥0, ai∈Ω∖{0} and some c≥1. Moreover, for α∉N, we show that the blow up must be simple, i.e. c=1. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation.