I am not getting the most of the article concern with the existence and uniqueness of positive solution and radial solution. I just want to know how a positive solution and radial solution of nonlinear elliptic BVP can be described?

Without any further context, I suppose a positive solution is one that is greater than zero everywhere in the domain, i.e. for all $x$ it holds that $u(x) > 0$. A radial solution might mean one that is only a function of the norm of the argument, i.e. for all $x$ it holds that $u(x) = u(\x\)$. 

