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 $$S(\epsilon V):=\underset{0\not\equiv u\in H^2(\Omega )\cap H_0^1(\Omega )}{\inf }\frac{{\int }_{\Omega }|\Delta u{|}^{2}dx+\epsilon {\int }_{\Omega }V|u{|}^{2}dx}{({\int }_{\Omega }|u{|}^{2^{\star }}dx{)}^{2/2^{\star }}},$$ 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 $V:\overline{\Omega }\to \mathbb{R}$ 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.

This paper has been accepted by Annali di Matematica Pura ed Applicata (1923 -)

References in corpus (2)

• On the sharp constant in the Bianchi-Egnell stability inequality
• Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem