Critical behavior for the polyharmonic operator with Hardy potential

Let us consider the Dirichlet problem {Lμ[u]≔(−Δ)mu−μu|x|2m=u2∗−1+λu,u>0inBDβu|∂B=0for|β|≤m−1 where BB is the unit ball in RnRn, n>2mn>2m, 2∗=2n/(n−2m)2∗=2n/(n−2m). We find that, whatever nn may be, this problem is critical (in the sense of Pucci–Serrin and Grunau) depending on the value of μ∈[0,μ¯), μ¯ being the best constant in Rellich inequality. The present work extends to the perturbed operator (−Δ)m−μ|x|−2mI(−Δ)m−μ|x|−2mI a well-known result by Grunau regarding the polyharmonic operator (see Grunau (1996)).