“…The main difference with is that they used the ‘polar’ representation of u (in analogy to the second branch in ) in the whole domain, even when . This point, however, demanded a rather involved justification, which our argument bypasses (a related observation was made independently in very recently).Remark It is worth mentioning that the passage from to if n = 1 or m = 1 can be shown by a similar ‘local replacement’ argument, which remarkably does not require condition , see and respectively.Remark A careful examination of the proof of Theorem reveals that the assumption of the boundedness of u , as well as that of the finiteness of the number of wells, can be dropped at the expense of imposing some natural uniformity conditions on W and compromising with the softer estimate with n = 2 as the conclusion. More precisely, we further assume that holds with the same r 0 for all i = 1,⋯, and that the set shrinks uniformly to { a 1 , a 2 ,⋯} as ρ →0.…”