Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set$$\Omega \subset {\mathbb {R}}^3$$Ω⊂R3we consider the minimization problem$$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$S(a+ϵV)=inf0≢u∈H01(Ω)∫Ω(|∇u|2+(a+ϵV)|u|2)dx(∫Ωu6dx)1/3involving the critical Sobolev exponent. The functionais assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions onaandVwe compute the asymptotics of$$S(a+\epsilon V)-S$$S(a+ϵV)-Sas$$\epsilon \rightarrow 0+$$ϵ→0+, whereSis the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding toaand we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have$$S(a+\epsilon V)S(a+ϵV)<Sfor all sufficiently small$$\epsilon >0$$ϵ>0.