The Asymptotic Behavior Of The Least-Energy Solutions For Hénon Equation In Dimension 2

We consider the Henon equation where Ω is the unit ball in R~N, a > 0 is aj constant, and p is superlinear and subcritical, that is When the dimension N > 3, it is known that the least-energy solutions cannot be, as p approaches the critical exponent, radially symmetric, and the maximum point will converge to the boundary.In this paper, the case N = 2 is considered, which implies the critical exponent is the infinity, i.e we want to know! the asymptotic behavior of the least-energy solutions when p approaches the infinity. We first use the Co-Area formula to prove the boundedness of thje least-energy solutions at least for large p, and we also can use the iteration mathod to get the same result. Furthermore, by blow-up analysis, we know that when p large enough, the least-energy solutions cannot be radially symmetric.