Energy asymptotics and blow-up phenomena for biharmonic Br\'{e}zis-Nirenberg problem

Abstract

For dimensions $n\geq8$, we are concerned with the quotient functional of the biharmonic Br\'{e}zis-Nirenberg problem under the Navier boundary condition $$ S(\varepsilon V):=\inf_{0\not\equiv u\in H^2(\Omega)\cap H_0^1(\Omega)}\frac{\int_{\Omega}|\Delta u|^2dx+\varepsilon\int_{\Omega}V|u|^2dx}{\big(\int_{\Omega}|u|^{2^\star}dx\big)^{2/2^\star}}, $$ where $2^\star=\frac{2n}{n-4}$ is the critical Sobolev exponent of the embedding $H^2(\Omega)\cap H_0^1(\Omega)\hookrightarrow L^{2^\star}(\Omega)$, $\Omega\subset\mathbb{R}^n$ is a bounded open set and $V:\overline{\Omega}\rightarrow\mathbb{R}$ is a continuous function. Under certain assumptions on $V$, we establish sharp asymptotics for the energy difference $S(0)-S(\varepsilon V)$, as $\varepsilon\rightarrow0^+$, 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.