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

Abstract

We study the existence/nonexistence of positive solution of $$ {Δ^2u-μ\frac{u}{|x|^4}=\frac{|u|^{q_β-2}u}{|x|^β}\quad\textrm{in $Ω$,}} $$ when $Ω$ is a bounded domain and $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0\leq β<4$ and $0\leqμ<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when $Ω$ is an open subset of $\mathbf R^N$ which is star shaped with respect to the origin. We also study the existence of positive solution in $Ω$ when $Ω$ is a bounded domain with non trivial topology and $β=0$, $μ\in(0,μ_0)$, for certain $μ_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq 8$. Different behavior of PS sequences have been obtained depending on $β=0$ or $β>0$.