# Blow up solutions for a Liouville equation with singular by Esposito P.

By Esposito P.

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493-512. [37] G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. , 37(1996), pp. 3769-3796. [38] V. H. Weston, On the asymptotic solution of a partial differential equation with exponential nonlinearity, SIAM J. Math. , 9(1978), pp. 1030-1053. [39] Y. Yang, Solitons in field theory and nonlinear analysis, Springer Verlag, 2001.

Let Φ ∈ H 1 (∂B) such that (S0 − T0 )Φ = 0 ∈ L2 (∂B). 9 there exists a solution w 0 for the problem    w0 = 0 in Ω \ S w0 = 0 on ∂Ω   w0 = Φ on ∂B such that w(z) = s1 + 2q1 · z − p + O(|z − p|2 ) s2 + 2q2 · z−q |z−q|2 − γ¯ z − q + O(|z − q|2 ) as z → p as z → q for some si ∈ R, qi ∈ C. The assumption (S0 − T0 )Φ = 0 ensures that we are gluing harmonic ˜ and B which coincide together with their normal derivative on ∂B. In this functions in Ω 30 way the resulting function is harmonic in Ω \ S.

1-52. [33] T. Ricciardi, G. Tarantello, Vortices in the Maxwell-Chern-Simons theory, Comm. Pure Appl. , 53(2000), pp. 811-851. [34] J. Spruck, Y. Yang, On multivortices in the electroweak theory I: existence of periodic solutions, Comm. Math. , 144(1992), pp. 1-16. [35] M. Struwe, G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. , 8(1998), pp. 109-121. 39 [36] T. Suzuki, Two dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear Diffusion Equations and their equilibrium states, 3(1992), pp.