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

We study the existence/nonexistence of positive solution of $${\mathrm{\Delta }}^{2}u-\mu \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^4}=\frac{\mathrm{\mid }u{\mathrm{\mid }}^{q_{\beta }-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{\beta }}\text{in }\mathrm{\Omega }\text{,}$$ when $\mathrm{\Omega }$ is a bounded domain and $N\ge 5$, $q_{\beta }=\frac{2(N-\beta )}{N-4}$, $0\le \beta <4$ and $0\le \mu <(\frac{N(N-4)}{4}{)}^2$. We prove the nonexistence result when $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ when $\mathrm{\Omega }$ is a bounded domain with non trivial topology and $\beta =0$, $\mu \in (0,{\mu }_0)$, for certain ${\mu }_0<(\frac{N(N-4)}{4}{)}^2$ and $N\geq 8$\mathrm{.}DifferentbehaviorofPSsequenceshavebeenobtaineddependingon$\beta=0$or$\beta>0$.