By Nolasco M., Tarantello G.
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Extra resources for Vortex condensates for the SU(3) Chern-Simons theory
13) is a novelty of the SU (3)-theory and it is certainly worthwhile investigating. Acknowledgments: The authors wish to express their gratitude to G. Dunne for useful comments. 34 M. NOLASCO AND G. TARANTELLO References  A. Abrikosov, On the magnetic properties of superconductors of the second group, Soviet Phys. JETP 5 (1957), 1174–1182.  A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381.  T. Aubin, Nonlinear analysis on manifolds.
Lett. 66 (1991), 553–555.  H. Nielsen and P. Olesen, Vortex-line models for dual strings, Nucl. Phys. B 61 (1973), 45–61.  M. Nolasco and G. Tarantello, On a sharp Sobolev type inequality on two dimensional compact manifolds, Arch. Rational Mech. Anal. 145 (1998), 161–195.  , Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. and PDE. 9 (1999), 31–94.  T. Ricciardi and G. Tarantello, Self-dual vortices in the Maxwell- Chern-Simons-Higgs theory, Preprint, 1999.
Recall that in this case N1 = 0, and hence h1 = 1, N2 = 1. 42) 2 2 ≤ C, for suitable C > 0 (independent of λ) and i = 1, 2. 43) (i = 1, 2) Ω with C > 0 independent of λ. 45) c± 1,λ = 1 2 ± Ω Ω ew1,λ e ± 2w1,λ + 4π 1 (N1 + N2 )(N2 − 9N1 ) + o( ) 9λ λ as λ → +∞. 43) we derive immediately that c± 2,λ → −∞, as λ → +∞. e. n in Ω. e. in Ω. Note in particular that, ew1,n → 1, ew2,n → ew± in Lp (Ω), ∀ p ≥ 1. 45), we find: ± ± ± −∆(w1,n + 2w2,n ) =3λn h2 ec2,n ew2,n (1 + ec1,n ew1,n − 2ec2,n ew2,n ) − 4π(2N2 + N1 ) =4π(2N2 + N1 ) h2 ew2,n 1 − w2,n |Ω| h e Ω 2 + φn in Ω ± p with c± i,n := ci,λn (i = 1, 2) and φn → 0 strongly in L (Ω), ∀ p ≥ 1.
Vortex condensates for the SU(3) Chern-Simons theory by Nolasco M., Tarantello G.