Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

Abstract

For a bounded open set Ω⊂ℝ³ we consider the minimization problem S(a + V) = inf|[0≡u∈H¹₀(Ω) [∫_Ω(|∇u|² + (a + ϵV)|u|²) dx]/(∫_Ω u⁶ 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+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 + ϵV) < S for all sufficiently small ϵ > 0.

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