Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian

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• 作者：Yuanhong Wei ; Xifeng Su
• 关键词：35J60 ; 47J30
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：52
• 期：1-2
• 页码：95-124
• 全文大小：367 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We consider the semi-linear elliptic PDE driven by the fractional Laplacian: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} (-\Delta )^s u=f(x,u) &{} \hbox {in }\Omega , \\ u=0 &{} \hbox {in }\mathbb {R}^n\backslash \Omega .\\ \end{array} \right. \end{aligned}$$ An \(L^{\infty }\) regularity result is given, using De Giorgi–Stampacchia iteration method. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais–Smale condition without Ambrosetti–Rabinowitz condition for non-local elliptic equations is proved. Two non-trivial solutions are given under some weak hypotheses. Non-local elliptic equations with concave–convex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of Ambrosetti–Brezis–Cerami type is given, which shows that the effect of the parameter \(\lambda \) in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.