Blow-up solutions of the self-dual Chern–Simons–Higgs vortex equation

Kwangseok Choe; Namkwon Kim

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We investigate the spectral properties of the differential operator $- r^{s} Δ$ , $s \geq 0$ with the Dirichlet boundary condition in unbounded domains whose boundaries satisfy some geometrical condition. Considering this operator as a self-adjoint operator in the space with the norm ${∥ u ∥}_{L_{2, s} (Ω)}^{2} = \int_{Ω} r^{- s} {| u |}^{2} d x$ , we study the structure of the spectrum with respect to the parameter $s$ . Further we give an estimate of the rate of condensation of discrete spectra when it changes to continuous.

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