By Zhao W., Liu P.
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3. 5). Let (zi )i∈N ⊂ RN and consider the measures µi := |x − zi |−2a dx, then we have for 0 < r < 2 as r → 0 8 |y − zi |−bp sup x∈B2 (0), i∈N Br (x) |x−y| s ds dy → 0. µi (Bs (x)) Proof. We use as c a generic constant that may change its value from line to line. Fix x ∈ B2 (0). From the doubling property of the measure µi (see ) we find 8 Mi (x, |x − y|) := |x−y| ≤c s ds µi (Bs (x)) |x − y|−N +2a+2 , if |x − y| > 12 |x − zi | |x − y|−N +2 |x − zi |2a + |x − zi |−N +2a+2 , if |x − y| ≤ 21 |x − zi |.
Gi zK(0) (rui (xi )2/(N −2−2a) )− 2 + r( gi ) zK(0) DEGENERATE CRITICAL ELLIPTIC EQUATIONS 31 Since for Cui (xi )−2/(N −2−2a) ≤ r ≤ ri , there results C ≤ rui (xi )2/(N −2−2a) ≤ Ri , we have that d gi ≤ εi . gi ≤ εi , dr ∂B ∂B 2/(N −2−2a) (0) 2/(N −2−2a) (0) rui (xi ) rui (xi ) 1+δ K(0) Moreover for C = 2 (p−2)(N −2−2a) p(N −2−2a) − 2 Choosing εi = o Ri we have 1 − K(0)ui (xi )p−2 r (p−2)(N −2−2a) 2 ≤ −δ. ✷ the claim follows. 3. 5). Let (zi )i∈N ⊂ RN and consider the measures µi := |x − zi |−2a dx, then we have for 0 < r < 2 as r → 0 8 |y − zi |−bp sup x∈B2 (0), i∈N Br (x) |x−y| s ds dy → 0.
Abdellaoui, V. Felli and I. Peral. Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole RN (2003). Preprint.  A. Ambrosetti and M. Badiale. Homoclinics: Poincar´e-Melnikov type results via a variational approach. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 15 (1998), no. 2, 233–252.  A. Ambrosetti and M. Badiale. Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. Roy.
A Biplurisubharmonic Characterization of AUMD Spaces by Zhao W., Liu P.