A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities

We study the existence and non-existence of nontrivial weak solution of $$ {\Delta^2u-\mu\frac{u}{|x|^{4}} = \frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0<\beta<4$, $1Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.