Blow-up of solutions of critical elliptic equations in three dimensions

Dear friends and colleagues,

on Monday, 10 May 2021 at 1 PM, Professor Rupert Frank (Caltech University, United States, and @ Ludwig Maximilian University of Munich, Germany) is giving a talk in our Asia-Pacific Analysis and PDE Seminar on

Blow-up of solutions of critical elliptic equations in three dimensions .

Abstract:

We describe the asymptotic behavior of positive solutions $u_{\epsilon }$ of the equation $$-\mathrm{\Delta }u+au=3u^{5-\epsilon }\text{in }\mathrm{\Omega }\subseteq {\mathbb{R}}^3$$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_{\epsilon }$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $$-\mathrm{\Delta }u+(a+\epsilon V)u=3u^5\text{ in }\mathrm{\Omega }\text{.}$$ For the variational problem corresponding to the latter problem we also obtain precise energy asymptotics and a detailed description of the blow-up behavior of almost minimizers (but not necessarily minimizers or solutions).

More information and how to attend this talk can be found at the seminar webpage .

Best wishes,

Daniel

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