Kun Cheng, Li Wang
In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent
where is a smooth bounded domain, , , and are two parameters, and , where . and in . For some and N, by the barrier method and mountain pass lemma, we prove that there exists such that there are exactly two positive solutions if and no positive solutions for . Moreover, if , there is a unique solution (), which means that ( ) is a turning point for the above problem. Furthermore, in case and if is a ball in and f satisfies some additional conditions, then a uniqueness existence result is obtained for small enough.
Submitted September 23, 2017. Published December 11, 2017.
Math Subject Classifications: 35A15,35J60, 46E35.
Key Words: Non-homogeneous; fractional Laplacian; critical Sobolev exponent; variational method.
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| Kun Cheng |
Department of Information Engineering
Jingdezhen Ceramic Institute
Jingdezhen 333403, China
| Li Wang|
College of Science
East China Jiaotong University
Nanchang 330013, China
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