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arXiv · 2102.10525

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

Rupert L. Frank·Tobias König·Hynek Kovařík
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Abstract

We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ 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 $-Δu + (a+εV) u = 3\,u^5$ in $Ω$.

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Rupert L. Frank, Tobias König, Hynek Kovařík. 2022-09-27. Blow-up of solutions of critical elliptic equations in three dimensions. https://doi.org/10.2140/apde.2024.17.1633

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