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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?

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You need to be more specific if you expect a useful answer. –  Liviu Nicolaescu Feb 15 '12 at 17:40

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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 $u(x) = u(\|x\|)$.

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