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 $N\geq 5$, $q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0<\beta<4$, $1<q\leq 2^{**}$ and $\mu<\mu_1:=\big(\frac{N(N-4)}{4}\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\leq\mu<\mu_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.