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

被引：40

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：

中国国家自然科学基金; 黑龙江省自然科学基金;

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

页码：1061 / 1081

页数：21

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