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

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials {-Delta u(n) = W-n(x)u(n)(p)n, u(n)>0, in Omega, u(n)=0, on partial derivative Omega, integral(Omega)p(n)W(n)(x)u(n)p(n) dx <= C, where Omega is a smooth bounded domain in R-2, W-n(x)>= 0 are bounded functions with zeros in Omega, and p(n) -> infinity as n -> infinity. A typical example is W-n(x)=|x|(2 alpha) with 0 is an element of Omega, i.e. the equation turns to be the well-known Henon equation. The asymptotic behavior for alpha=0 has been well studied in the literature. While for alpha>0, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case alpha>0 and prove a quantization property (suppose 0 is a concentration point) p(n)|x|(2 alpha)u(n)(x)(pn-1)+t -> 8 pi e (t/2)Sigma(k)(i=1)delta(ai)+8 pi(1+alpha)e(t/2)c(t)delta(0), t=0,1,2, for some k >= 0, a(i) is an element of Omega\{0} and some c >= 1. Moreover, for alpha is not an element of 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 Henon equation.