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Acnowledgements: I express my deep gratitude to Gabriella Tarantello, I am indebted to her for the stimulating discussions and the helpful comments about the manuscript. References [1] A. Bahri, Y. Y. Li, O. Rey, On a variational problem with lack of compactness: the topological effect of critical points at infinity, Calc. Var. Partial Differential Equations, 3(1995), pp. 67-93. [2] S. Baraket, Construction de limites singuli`eres pour des probl`emes elliptiques non lin´eaires en dimension deux, C.

Thus, for any f ∈ Y there exists an unique (h, λ, a) ∈ E solving Λ ρ w = f in Ω with ||(h, λ, a)||E ≤ C||L−1 ρ f ||E . We rewrite the solution w(z) in the form w(z) = h (z) + i λi ∂λi v(ρ, 0, 0)(z) + 2(−a) · ∂a v(ρ, 0, 0)(z), with h (z) = h(z) + χ(z − p)λ1 (∂τ vρ,τ1 (z − p) − ∂λ1 v(ρ, 0, 0)(z)) +χ(z − q)λ2 (∂τ vρ,τ2 ,γ (z − q) − ∂λ2 v(ρ, 0, 0)(z)) + 2 (1 − χ(z − q)) a · ∂a v(ρ, 0, 0)(z) +2χ(z − q) a · (∂z¯vρ,τ2 ,γ (z − q) + ∂a v(ρ, 0, 0)(z)) , where we have taken into account that (1 − χ(z − p)) ∂ λ1 v(ρ, 0, 0)(z) and (1 − χ(z − q)) ∂λ2 v(ρ, 0, 0)(z) are identically zero.

Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations, 9(1999), pp. 31-91. [31] J. Prajapat, G. Tarantello, On a class of elliptic problem in R 2 : symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131(2001), pp. 967-985. [32] O. Rey, The role of Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. , 89(1990), pp. 1-52. [33] T. Ricciardi, G. Tarantello, Vortices in the Maxwell-Chern-Simons theory, Comm.