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

Abstract

We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional Laplacian (− |D ) α 2 associated to the subordinate killed Brownian motion process Z D α in a bounded C 1,1 domain D. Our arguments are based on potential theory tools on Z D α and properties of an appropriate Kato class of functions K α (D).

Nonhomogeneous boundary conditions for the spectral fractional Laplacian

“…One aspect of the theory is however left unanswered: the formulation of natural nonhomogeneous boundary conditions. A first attempt can be found in the work of Dhifli, Mâagli and Zribi [11]. The investigations that have resulted in the present paper turn out, we hope, to shed some further light on this question.…”

“…By Lemma 27, ψ ≥ 0 in Ω and by Lemma 26 − ∂ψ ∂ν ≥ 0 on ∂Ω. Thus, by (11), Ω uf ≥ 0. Since this is true for every f ∈ C ∞ c (Ω), the result follows.…”

Positive Solutions for Some Competitive Fractional Systems in Bounded Domains

“…For more examples of functions belonging to ( ), we refer to [14]. Note that for the classical case (i.e., = 2), the class of functions 2 ( ) was introduced and studied in [15].…”

“…Finally, let us recall some potential theory tools that are needed, and we refer to [14,16,17] for more details. For ∈ + ( ), we define the kernel on + ( ) by…”

“…Lemma 7 (see [14,Proposition 6]). Let ℎ ∈ + ( ) and be a nonnegative excessive function on with respect to .…”