Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set Ω ⊂ R 3 we consider the minimization problem S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 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 + ϵ V ) - S as ϵ → 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 + ϵ V ) < S for all sufficiently small ϵ > 0 .