Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem

For dimensions $n\ge 8$, we are concerned with the quotient functional of the biharmonic Brézis-Nirenberg problem under the Navier boundary condition $$S(\epsilon V):=\underset{0\equiv ̸u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })}{\inf }\frac{{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^{2}dx+\epsilon {\int }_{\mathrm{\Omega }}V\mathrm{\mid }u{\mathrm{\mid }}^{2}dx}{({\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^{2^{\star }}dx{)}^{2\mathrm{/}{2}^{\star }}},$$ where $2^{\star }=\frac{2n}{n-4}$ is the critical Sobolev exponent of the embedding $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })\hookrightarrow L^{2^{\star }}(\mathrm{\Omega })$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded open set and $V:\bar{\mathrm{\Omega }}\to \mathbb{R}$ is a continuous function. Under certain assumptions on $V$, we establish sharp asymptotics for the energy difference $S(0)-S(\epsilon V)$, as $\epsilon \to 0^{+}$, by means of matching upper and lower bound estimates. Moreover, we give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the blow-up rate and the location of concentration points.