2 0 Ju n 20 05 Solutions for the quasi - linear equations in multipliers spaces
نویسنده
چکیده
In this paper, we give a necessary and sufficient condition for the existence of a nonnegative solution for the equation u = W r u 2 + f , where f ∈ L 2 loc R d. (1) with W r σ = f is the Wolff's potential. Equations of this type arise in applications of Schrödinger operators and harmonic analysis (see [CWW], [?], [KS]) to the problem of the existence of nonnegative solutions for the nonlinear inhomogeneous integral equation u(x) = R d K(x, y)u q (y)dω(y) + f (y), x ∈ R d , (2) where q > 1 and f is a nonnegative measurable function on R d. This type equations has been treated by P. Barras and M. Pierre in [BP]. A necessary and sufficient condition for existence of solutions (in a weak sense) was given in terms of a certain nonlinear functional. Later, Adams and Pierre [AP] showed that (1) has a solution for sufficiently small λ > 0, if and only if, for all compact sets e ⊂ R d , σ(e) ≤ C.Cap e,. H 2 p , where p > 1. The proof is based on capacitary estimates and certain estimates weighted L p-estimates. The approach we present has several advantages of its own and is based on pointwise inequalities and dyadic version of W r .
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