Critical blowup exponent to a class of semilinear elliptic equations with constraints in higher dimension - local properties

We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup +\inf$ type inequality and the quantized blowup mechanism.