Existence and Multiplicity of Weak Positive Solutions to a Class of Fractional Laplacian with a Singular Nonlinearity
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This paper is devoted to the study of a class of fractional Laplacian with a singular nonlinearity. The purpose of this article is to give the existence and multiplicity of weak positive solutions by the combined effects of a superlinear and singular term. It is worth pointing out that the testing function in the definition of weak positive solutions does not need to have compact support in bounded domain. Hence the results of this paper are new even in the fractional Laplacian case.
KeywordsFractional Laplacian nondifferentiable functional existence and multiple
Mathematics Subject Classification35J25 47J30 46E35
The authors are very grateful to the referees for their helpful suggestions and comments which have improved the paper.
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Conflict of interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Nos. 11801038, 11626185) and Natural Science Foundation of Shaanxi Provincial Department of Education (No. 16KJ1558). This work is also supported by the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2017JQ1011, 2018JQ1023).
- 19.Mukherjee, T., Sreenadh, K.: Critical growth fractional elliptic problem with singular nonlinearities. Preprint (2017). http://arxiv.org/pdf/1602.07886.pdf
- 22.Fang, Y.: Existence, uniqueness of positive solution to a fractional Laplacians with singular nonlinearity. Mathematic. preprint (2014). http://arxiv.org/pdf/1403.3149.pdf
- 25.Mukherjee, T., Sreenadh, K.: On Dirichlet problem for fractional p-Laplacian with singular nonlinearity. Adv. Nonlinear Anal. (2016). http://arxiv.org/pdf/1602.00872.pdf (in press)
- 28.Mazya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 342, 2nd edn. Springer, Heidelberg (2011)Google Scholar
- 29.Musina, R., Nazarov, A.I.: Strong maximum principles for fractional Laplacians. Preprint (2017). http://arxiv.org/pdf/1612.01043.pdf