Far field refraction problem with loss of energy in negative refractive index material

This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $\kappa$, that is, $\kappa<-1$ and $-1<\kappa<0$. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Amp\`ere type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.