Critical behavior for the polyharmonic operator with Hardy potential

Let us consider the Dirichlet problem {L-mu[u] := (-Delta)(m)u - mu u/vertical bar X vertical bar(2m) = u(2*-1) + lambda u, u > 0 in B D(beta)u vertical bar(partial derivative B) = 0 for vertical bar beta vertical bar <= m - 1 where B is the unit ball in R-n, n > 2m, 2* = 2n/(n - 2m). We find that, whatever n may be, this problem is critical (in the sense of Pucci-Serrin and Grunau) depending on the value of mu is an element of[0, (mu) over bar), (mu) over bar being the best constant in Rellich inequality. The present work extends to the perturbed operator (-Delta)(m) - mu vertical bar x vertical bar I-2m a well-known result by Grunau regarding the polyharmonic operator (see Grunau (1996)