{"title":"BIFURCATION PROBLEM FOR A CLASS OF QUASILINEAR FRACTIONAL SCHRÖDINGER EQUATIONS","authors":"I. Abid","doi":"10.4134/JKMS.J190646","DOIUrl":null,"url":null,"abstract":"We study bifurcation for the following fractional Schrödinger equation (−∆)su+ V (x)u = λ f(u) in Ω u > 0 in Ω u = 0 inRn \\ Ω where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of Rn, (−∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is lim t→+∞ f(t) t = a ∈ (0,+∞). We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"1347-1372"},"PeriodicalIF":0.7000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Korean Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4134/JKMS.J190646","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
We study bifurcation for the following fractional Schrödinger equation (−∆)su+ V (x)u = λ f(u) in Ω u > 0 in Ω u = 0 inRn \ Ω where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of Rn, (−∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is lim t→+∞ f(t) t = a ∈ (0,+∞). We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.
我们研究了以下分数阶Schrödinger方程(−∆)su+ V (x)u = λ f(u)在Ω u > 0在Ω u = 0 inRn \ Ω中的分岔,其中0 < s < 1, n > 2s, Ω是Rn的有界光滑域,(−∆)s是s阶分数阶拉普拉斯算子,V是满足适当假设的势能,λ是一个正实参数。非线性项f是一个正的非降凸函数,其渐近线性为lim t→+∞f(t) t = a∈(0,+∞)。讨论了正解的存在性、唯一性和稳定性,证明了极值解的存在性和唯一性。考虑了一类拟线性分数阶Schrödinger方程的分岔问题的类型,并建立了解在分岔点附近的渐近性态。
期刊介绍:
This journal endeavors to publish significant research of broad interests in pure and applied mathematics. One volume is published each year, and each volume consists of six issues (January, March, May, July, September, November).