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We study the existence/nonexistence of positive solution of $${\Delta^2u-\mu\frac{u}{|x|^4}=\frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}\quad\textrm{in$\Omega$,}} $$ when $\Omega$ is a bounded domain and $N\geq 5$,$q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0\leq \beta<4$ and$0\leq\mu<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when$\Omega$ is an open subset of $\mathbf R^N$ which is star shaped with respectto the origin. We also study the existence of positive solution in $\Omega$when $\Omega$ is a bounded domain with non trivial topology and $\beta=0$,$\mu\in(0,\mu_0)$, for certain $\mu_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq8$. Different behavior of PS sequences have been obtained depending on$\beta=0$ or $\beta>0$.

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