We consider the equation (Formula presented.) in the sense of distribution in (Formula presented.) where (Formula presented.) and (Formula presented.). Then it is known that (Formula presented.) solves (Formula presented.) , for some nonnegative constants (Formula presented.) and (Formula presented.). In this paper, we study the existence of singular solutions to (Formula presented.) in a domain (Formula presented.) , (Formula presented.) is a nonnegative measurable function in some Lebesgue space. If (Formula presented.) in (Formula presented.) , then we find the growth of the nonlinearity (Formula presented.) that determines (Formula presented.) and (Formula presented.) to be (Formula presented.). In case when (Formula presented.) , we will establish regularity results when (Formula presented.) , for some (Formula presented.). This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (Formula presented.) with a specific weight function (Formula presented.). Later, we discuss its analogous generalization for the polyharmonic operator. © 2015 Taylor & Francis.