Optimal regularity of positive solutions of the Henon-Hardy equation and related equations

We present a new method to determine the optimal regularity of positive solutions u is an element of C-4(Omega\{0}) boolean AND C-0((Omega) over bar) of the Henon-Hardy equation, i.e., Delta(2)u = vertical bar x vertical bar(alpha)u(p) in Omega, (0.1) where Omega subset of R-N (N >= 4) is a bounded smooth domain with 0 is an element of Omega, alpha > -4, and p is an element of R. It is clear that 0 is an isolated singular point of solutions of (0.1) and the optimal regularity of u in Omega relies on the parameter alpha. It is also important to see that the regularity of u at x = 0 determines the regularity of u in Omega. We first establish asymptotic expansions up to arbitrary orders at x = 0 of prescribed positive solutions u is an element of C-4(Omega\{0}) boolean AND C-0((Omega) over bar) of (0.1). Then we show that the regularity at x = 0 of each positive solution u of (0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation (0.1) with Omega = B, where B is the unit ball of R-N. The main idea works for more general equations with singular weights.