Asymptotic behavior and existence results for the biharmonic problems involving Rellich potentials

In this paper, we first investigate the following limit problem:Lμ[u]:=Δ2u−μu|x|4=u2⁎(s)−1|x|s,u>0,x∈Rn∖{0}, where 0≤s<4, 2⁎(s):=2(n−s)n−4, 0≤μ<μ¯:=116n2(n−4)2. The asymptotic properties at the origin and infinity of its ground state solutions are obtained, which minimize the related best Rellich-Sobolev constant. Secondly, we study the following critical biharmonic problem:{Lμ[u]=|u|2⁎(s)−2u|x|s+λ|u|q−2u,x∈Ω,u=∂u∂n=0,x∈∂Ω, where Ω⊂Rn(n≥5) is a smooth bounded domain containing the origin, 0≤s<4, 2≤q<2⁎:=2nn−4, λ>0. By variational argument the existence of nontrivial solutions to the problem is established.