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Using a new form of the 3G-Theorem for the Green function of a bounded domain in R n , we introduce a new Kato class K( ) which contains properly the classical Kato class K n ( ). Next, we exploit the properties of this new class, to extend some results about the existence of positive singular solutions of nonlinear differential equations.Mathematics Subject Classifications (1991): 34B15, 34B27.

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of R n (n ≥ 2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions K m,n containing properly the classical Kato class K n . We exploit properties of functions belonging to K m,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in R n (n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class K ∞ n (D). Next, we study the existence and the uniqueness of a positive continuous solution u inD of the following nonlinear singular elliptic problem      ∆u + ϕ(., u) = 0 , in D(in the sense of distributions)where ϕ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions ϕ(x, t) = q(x)t −σ , σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

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