Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian

被引：38
|
作者：
Zhang Binlin [1 ]
Radulescu, Vicentiu D. [2 ,3 ]
Wang, Li [4 ]
机构：
[1] Heilongjiang Inst Technol, Dept Math, Harbin 150050, Heilongjiang, Peoples R China
[2] AGH Univ Sci & Technol, Fac Appl Math, Al Mickiewicza 30, PL-30059 Krakow, Poland
[3] Romanian Acad, Inst Math Simion Stoilow, POB 1-764, Bucharest 014700, Romania
[4] East China Jiaotong Univ, Coll Sci, Nanchang 330013, Jiangxi, Peoples R China
来源：
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS | 2019年 / 149卷 / 04期
基金：
黑龙江省自然科学基金; 中国国家自然科学基金;
关键词：
Fractional Laplacian; Kirchhoff-type problem; critical groups; Morse theory; NONTRIVIAL SOLUTIONS; MULTIPLICITY;
D O I：
10.1017/prm.2018.105
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:.{M(integral integral(R2N) vertical bar u(x) -u(y)vertical bar(2)/vertical bar x-y vertical bar(N vertical bar 2s) dxdy) (-Delta)(s)u = f(x,u) in Omega, , where (-.)s is the fractional Laplace operator, s. (0, 1), N > 2s, O is an open bounded subset of RN with smooth boundary.O, M : R+ 0. R+ is a continuous function satisfying certain assumptions, and f(x, u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff-type Laplacian problems.
引用
收藏
页码：1061 / 1081
页数：21
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