Chern-Simons solitons and a nonlinear elliptic equation

We prove an existence theorem for the following quasilinear elliptic equation (1 - e^u)Δu = |∇u|²e^u - λ(1 - e^u)²e^u + 4πΣ_j=1^Nδ_pj over the full plane subject to the boundary condition that u → 0 as |x| → ∞, where λ > 0 is a physical parameter and δ is the Dirac distribution concentrated at the point p. The solutions of the equation are vortex-like multi-solitons arising in a unified relativistic self-dual Chern-Simons theory.