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

Abstract

For dimensions \(n\ge 8\), we are concerned with the quotient functional of the biharmonic Brézis-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 \(\varepsilon \rightarrow 0^+\), 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.