On a New Kato Class and Singular Solutions of a Nonlinear Elliptic Equation in Bounded Domains of $$\mathbb{R}^n$$

Abstract

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.

On the subordinate killed B.M in bounded domains and existence results for nonlinear fractional Dirichlet problems

“…These interesting inequalities extend those for the Green function G D of the killed Brownian motion χ D , in the case d ≥ 3 (see [18]) and consequently it was shown a 3G-inequality for G D (see [8]) allowing to introduce and study the Kato class of functions K (D) (see [13], for d ≥ 3 and [19] for d = 2). This class was extensively used in the study of various elliptic differential equations in bounded domains (see [2,13] and [19]).…”

“…While the classical formulation of the Dirichlet problem becomes impossible, the authors of [6] provide an appropriate reformulated Dirichlet problem associated to (− |D ) α 2 (see Proposition 3 and Remark 2 below). This approach allows us to study two different nonlinear Dirichlet problems associated to (− |D ) α 2 and to transfer existence results about nonlinear equations based on Brownian motion techniques, obtained in [12] and [13], into existence results in the new situation as it is stated in Theorems 3 and 4 below.…”

Positive solutions for some competitive elliptic systems

“…Ò Ø ÓÒ 1.3º (see [5] and [22] We remark that in the case where D is bounded and if d denotes its diameter,…”

“…The following compactness result will be used and it is proved in [22] for bounded domains and in [5] for unbounded ones.…”

“…where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and p, q are two nonnegative functions in the Kato class K(D) introduced and studied in [5] and [22].…”