Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

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 -pagination

Keywords:

• Fractional Laplacian
• Radon measure
• Dirac mass
• Green kernel
• Schauder’s fixed point theorem

AMS Subject Classifications:

• 35R11
• 35J61
• 35R06

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.

Additional information

Funding

H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].