Abstract
Let us consider the Dirichlet problem {Lμ[u]≔(−Δ)mu−μu|x|2m=u2∗−1+λu,u>0inBDβu|∂B=0for|β|≤m−1 where B is the unit ball in Rn, 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 μ∈[0,μ¯), μ¯ being the best constant in Rellich inequality. The present work extends to the perturbed operator (−Δ)m−μ|x|−2mI a well-known result by Grunau regarding the polyharmonic operator (see Grunau (1996)).
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