Caffarelli–Kohn–Nirenberg type equations of fourth order with the critical exponent and Rellich potential

We study the existence/nonexistence of positive solutions ofΔ2u−μu|x|4=|u|qβ−2u|x|βin Ω, where Ω is a bounded domain and N≥5N≥5, qβ=2(N−β)N−4, 0≤β<40≤β<4 and 0≤μ<(N(N−4)4)2. We prove the nonexistence result when Ω is an open subset of RNRN, which is star-shaped with respect to the origin. We also study the existence of positive solutions when Ω is a smooth bounded domain with a nontrivial topology and β=0β=0, μ∈(0,μ0)μ∈(0,μ0), for certain μ0<(N(N−4)4)2 and N≥8N≥8. Different behaviors are obtained for Palais–Smale sequences depending on whether β=0β=0 or β>0β>0.