Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

Abstract

For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ 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 $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ 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 $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .

Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

“…The derivation is simpler for d ≥ 7 and becomes computationally challenging for 3 ≤ d ≤ 6 due to different leading order terms in the expansion of I ω and due to different regularity of the non-singular part H of Green's function. Some similar computations can be found in [5,6,15,16,32] for d ≥ 4 and in [12,13,16,20] for d = 3.…”

“…Since H is not C 1 , we need to expand H(x) as that in [16]. We define ψ = H(x) − 1 2 |x|, then by (1.11), ψ satisfies…”

“…By (2.23) and (2.24), it is standard (cf. [3,16,32]) to show that e(ω) = o ω (1) is attained by some ε ω satisfying ε ω → 0 as ω → ω * , which implies that (2.19) hold with ûω → 0 in X as ω → ω * . The orthogonality conditions in X for ûω ∈ M ⊥ ω are obtained from…”

“…However, the main difficulty in proving Theorem 1.1 is to obtain a good lower bound of I ω which will match the upper bound generated by the Aubin-Talanti bubbles up to the leading order terms. To achieve this, we need to further employ the ideas in literature [7,12,13,15,16,20,21,31,32], that is, splitting of u ω into two parts in X and estimating of these two parts precisely up to the leading order term. We remark that, due to the growth of the harmonic potential at infinity and the unboundedness of R d , the regular part of the Green function of the operator −∆ + |x| 2 − ω * is no longer bounded for all d ≥ 3.…”

Degenerate stability of some Sobolev inequalities

“…Namely, while Bianchi and Egnell project on the nearest optimizer, we do the same, but then zoom further in and project on the nearest zero-mode of the Hessian. This argument bears some vague resemblance to how in [29] we handled an asymptotic minimization situation where the expected leading term vanishes. We have not encountered this kind of argument in the context of stability of functional inequalities and we hope that it will be of use in related problems.…”