Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-02-19 , DOI: 10.1007/s00526-021-01929-3 Rupert L. Frank , Tobias König , Hynek Kovařík
For a bounded open set \(\Omega \subset {\mathbb {R}}^3\) we consider the minimization problem
$$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of \(S(a+\epsilon V)-S\) as \(\epsilon \rightarrow 0+\), where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have \(S(a+\epsilon V)<S\) for all sufficiently small \(\epsilon >0\).
中文翻译：
三维Brezis-Nirenberg问题中的能量渐近性
对于有界开放集\（\ Omega \ subset {\ mathbb {R}} ^ 3 \），我们考虑最小化问题
$$ \ begin {aligned} S（a + \ epsilon V）= \ inf _ {0 \ not \ equiv u \ in H ^ 1_0（\ Omega}} \ frac {\ int _ \ Omega（| \ nabla u | ^ 2+（a + \ epsilon V）| u | ^ 2）\，dx} {（\ int _ \ Omega u ^ 6 \，dx）^ {1/3}} \ end {aligned} $$涉及关键的Sobolev指数。在赫比和沃贡的意义上，假设函数a是关键的。在a和V的某些假设下，我们将\（S（a + \ epsilon V）-S \）的渐近性计算为\（\ epsilon \ rightarrow 0+ \），其中S是Sobolev常数。（几乎）最小化器集中在Robin函数的零集中对应于a的点，我们确定集中点在该集中的位置。我们还表明，对于所有足够小的\（\ epsilon> 0 \），具有\（S（a + \ epsilon V）<S \）几乎是必要的。
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