On the Caffarelli–Kohn–Nirenberg inequalities

Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities

“…We start by recalling a reformulation of the inequalities (1) given in [4], [5]. The proofs of our main results will be based on this new formulation.…”

“…More precisely, they proved in [4], [5] that there is a function h(a) defined for a ≤ 0, satisfying h(0) = 0, a < h(a) < a + 1 for a < 0, and a+1−h(a) → 0 as −a → ∞, such that for any (a, b) satisfying a < 0 and a < b < h(a), the extremal function for S(a, b) is non-radial. In a recent preprint [9] of Felli and Schneider, it is observed that this curve h(a) can be sharpened to the following:…”

“…In Section 2 we first recall a new formulation of the inequalities (1) from [4], [5]. For this alternative formulation some new ideas developed in [7], [15] for proving symmetry properties via the maximum principle can be applied to derive the desired results.…”

On the Hénon equation: Asymptotic profile of ground states, II

“…This idea has been used in [2,7,8] to reformulate the problem. We use notation x = ( , t) ∈ C = S N−1 ×R.…”

“…Our method also allows extensions of the results to the cases of exterior domains and to the weighted versions of the equation (see Section 4). We have extensive study on the limiting problem (7) which may be of independent interest.…”