Energy asymptotics in the three-dimensional Brezis-Nirenberg problem. (English) Zbl 1511.35123
Summary: For a bounded open set \(\Omega \subset \mathbb{R}^3\) we consider the minimization problem
\[
S(a+\epsilon V) = \inf_{0\not \equiv u\in H^1_0(\Omega)} \frac{\int_\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int_\Omega u^6\,dx)^{1/3}}
\]
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 \(\epsilon >0\).
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
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