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    188宝金博页面版: Existence of solutions for fractional p -Laplacian problems via Leray-Schauder’s nonlinear alternative

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    内容提示： Qiu and Xiang BoundaryValueProblems (2016) 2016:83 DOI 10.1186/s13661-016-0593-8RESEARCH Open AccessExistence of solutions for fractionalp-Laplacian problems via Leray-Schauder’snonlinear alternativeHong Qiu and Mingqi Xiang ** Correspondence:mqxiang@cauc.edu.cnCollege of Science, Civil AviationUniversity of China, Tianjin, 300300,P.R. ChinaAbstractIn this paper, we are concerned with the existence of solutions for a class ofquasilinear elliptic problems driven by a nonlocal integro-dif f erential operato...

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    Qiu and Xiang BoundaryValueProblems (2016) 2016:83 DOI 10.1186/s13661-016-0593-8RESEARCH Open AccessExistence of solutions for fractionalp-Laplacian problems via Leray-Schauder’snonlinear alternativeHong Qiu and Mingqi Xiang ** Correspondence:mqxiang@cauc.edu.cnCollege of Science, Civil AviationUniversity of China, Tianjin, 300300,P.R. ChinaAbstractIn this paper, we are concerned with the existence of solutions for a class ofquasilinear elliptic problems driven by a nonlocal integro-dif f erential operator withhomogeneous Dirichlet boundary data. As a particular case, we study the followingproblem:(– ? ) sp u = f(x,u)in ? ,u = 0 in R N \ ? ,where (– ? ) sp is the fractional p-Laplace operator, ? is an open bounded subset of RNwith Lipschitz boundary, and f : ? ×R → R is a Carathéodory function. The existenceof nonnegative solutions is obtained by using Leray-Schauder’s nonlinear alternative.MSC: 35R11; 35A15; 47G20Keywords: fractional p-Laplacian equation; integro-dif f erential operator;Leray-Schauder alternative theorem1 Introduction and main resultsRecently, a great deal of attention has been paid to the study of problems involvingfractional and nonlocal operators, both in pure mathematical research and in real-world applications, such as optimization, f i nance, continuum mechanics, phase transi-tion phenomena, population dynamics, minimal surfaces, and game theory, as they arethe typical outcome of stochastically stabilization of Lévy processes; see [?–?] and thereferences therein. Especially, the fractional Laplacian operators of the form (–?) s canbe viewed as the inf i nitesimal generators of stable Lévy processes; see for instance [?].Some interesting topics concerning the nonlocal fractional operators, such as the nonlin-ear fractional Schrödinger equation (see [?]), the fractional porous medium equation (see[?, ?]), and so on, have attracted considerable attention. There is no doubt that the liter-ature on fractional and nonlocal operators is quite large; see for example [?–??]. For thebasic properties of fractional Sobolev spaces and their applications to elliptic fractionalproblems, we refer the reader to [??, ??] and the references therein.© 2016 Qiu and Xiang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

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