Some Research About Hénon Equation

This thesis considers two problems. Firstly, we consider a semilinear elliptic equation on an annulus in R2 with large exponent in the nonlinear term. whereΩ≤R2,Ω=BR2(0)\BR1(0), R2> R1. The equation is (?) We investigate positive solutions obtained by the variational method. It turns out that the contrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that Iα,p2=(?)∫Ω|▽u|2dχ/(∫Ω|χ|α|u|p+1dχdχ)2/p+1 the minimum of energy functional with the nonlinear exponent equal to p, p> 0, is Iα,P= (8πe)1/2p-1/2, as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded uniformly in p. As p tends to infinity, the solutions develop one peak. Precisely the solutions approach zero except at one point where they stay away form zero and bounded from above. For this domain, the solutions enlarged by a suitable quantity behave like a Green function of-△. In this case we shall also prove that the peak must appear at a critical point of the Robin function of the annulus. The second problem is a related fourth order equation in a unit ball of RN. The equation is (?) where p> 1. We analyse symmetry breaking for ground states of the fourth order equation when pis close to 2*, where 2*= 2N/N-4.