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Abstract
The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display)
where ,
or
,
with
is the fractional Laplacian defined in the principle value sense,
is a bounded
open set in
with
,
is a bounded Radon measure supported in
and
is defined in the distribution sense, i.e.
here denotes the unit inward normal vector at
. In this paper, we prove that (0.1) with
admits a unique weak solution when g is a continuous nondecreasing function satisfying
Our interest then is to analyse the properties of weak solution when with
, including the asymptotic behaviour near
and the limit of weak solutions as
. Furthermore, we show the optimality of the critical value
in a certain sense, by proving the non-existence of weak solutions when
. The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when
and
is a bounded nonnegative Radon measure supported in
. We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying
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Acknowledgements
Suad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047).
Notes
No potential conflict of interest was reported by the authors.