By Esposito P., Musso M., Pistoia A.

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Let ε > 0. The following expansion holds: F (ξ) = 4πmp 32π 2 4πm − 2 ϕm (ξ1 , . . , ξm ) + 2 2 γ γ γ m 8 1 + 2 ( v∞ − ∆w0 ) + O( 3 ) 2 2 2γ R2 (1 + |y| ) p uniformly for ξ ∈ Oε . Proof. 4). 6) we get that Ω (Uξ + φξ )p+1 = Ω |∇(Uξ + φξ )|2 + O( 1 ) p4 uniformly for ξ ∈ Oε . 5) Ω |∇φξ |2 + O( 1 ). 20) we have that B(0, δε ) 8 1 1 − ∆w0 − 2 ∆w1 + O(p2 e−p ) × 2 2 (1 + |y| ) p p j p p p 1 1 × p + v∞ + w0 + 2 w1 + O(e− 4 |y| + e− 4 ) + O(e− 2 ) p p m 1 8 1 = 8πp + ( v − ∆w0 ) + O( ) 4 2 )2 ∞ 2 (1 + |y| p R j=1 γ 2 µ p−1 j = 8πmp 32π − 2 γ2 γ − m log µj + j=1 m γ2 ( R2 8 1 v∞ − ∆w0 ) + O( 3 ) 2 2 (1 + |y| ) p 4 since µj p−1 = 1 − p4 log µj + O( p12 ).

Felmer, M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differential Equations 182 (2002), no. 2, 511–540. [10] M. del Pino, P. Felmer, M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries. Bull. London Math. Soc. 35 (2003), no. 4, 513–521. [11] M. del Pino, P. Felmer, J. Wei, On the role of distance function in some singular perturbation problems. Comm. Partial Differential Equations 25 (2000), no.

Calc. Var. Partial Differential Equations, to appear. [13] P. Esposito, Blow up solutions for a Liouville equation with singular data. Recent advances in elliptic and parabolic problems (Taiwan, 2004), 39–57, edited by Chiun-Chuan Chen, Michel Chipot and Chang-Shou Lin. [14] P. Esposito, M. Grossi, A. Pistoia, On the existence of blowing-up solutions for a mean field equation. Ann. IHP Analyse Non Lin´eaire 22, no. 2, 227–257. [15] P. Esposito, G. Mancini, A prescribed scalar curvature-type equation: almost critical manifolds and multiple solutions.

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Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent by Esposito P., Musso M., Pistoia A.


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