Boundedness for solutions of equations of the Chern-Simons-Higgs type

This paper is concerned with a priori estimate of an equation of the Chern-Simons-Higgs type. We study the uniform bound of classical solutions u of the semilinear equation and integrable solutions u of the fractional order equation. We prove |u|≤1 in Rn, which comes into play in studying the quantization effects of those equations.