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Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077-1092. [33] M. Musso, A. Pistoia, Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J. 51 (2002), 541-579. D. Murrey, Mathematical biology, Springer, Berlin, 1989. [35] K. Nagasaki, T. Suzuki, Asymptotic analysis for a two dimensional elliptic eigenvalue problem with exponentially dominated nonlinearity, Asymptotic Analysis 3 (1990), 173-188.

15) follows. 6. , k. If p ≥ 1 and i = h, then ρ2 eUi P ψhj Lp =O ρ 2(1−p) p as ρ → 0. Proof We have |eUi P ψhj |p = ρ2p ρ2p Ω |eUi P ψhj |p . 4 we get |eUi P ψhj |p = O(ρp ). 24) we obtain the desired estimate. 33 p Lp = O(ρ2−2p ). 7. Let j = 1, 2 and i = 1, . . , k. If p ≥ 1, then ρ2 eUi (P ψij − ψij ) Lp = O ρ2 1−p p as ρ → 0. 2 we get, ρ2 eUi (P ψij − ψij ) 10 Lp =O ρ2 eUi Lp = O(ρ2 1−p p ). Appendix D We are interested to prove a result concerning the structure of the solutions of the linearized problem “at infinity”.

J. Math. 25 (1995), 957-975. [19] W. Ding, J. Jost, J. Li, G. Wang, Existence results for mean field equations, Ann. IHP Analyse Non Lin´eaire 16 (1999), 653-666. [20] K. El Mehdi, M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, preprint. 36 [21] P. Esposito, Blow up solutions for a Liouville equation with singular data, preprint (2003). [22] P. Esposito, A class of Liouville-type equations arising in Chern-Simons vortex theory: asymptotics and construction of blowing up solutions, Thesis, Roma “Tor Vergata” (2003).

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