Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells

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• 作者：MiaoMiao Niu ; ZhongWei Tang
• 关键词：nonlinear Schrödinger equation ; least energy solution ; fractional Laplacian ; variational methods
• 刊名：Science China Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：60
• 期：2
• 页码：261-276
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics;
• 出版者：Science China Press
• ISSN：1869-1862
• 卷排序：60

文摘

We are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian $$\begin{array}{*{20}c} {( - \Delta )^s u(x) + \lambda V(x)u(x) = u(x)^{p - 1} ,} & {u(x) \geqslant 0,} & {x \in \mathbb{R}^N ,} \\ \end{array} $$ for sufficiently large λ, 2 < p < \(\frac{{2N}}{{N - 2s}}\) for N ≥ 2. V (x) is a real continuous function on RN. Using variational methods we prove the existence of least energy solution u_λ(x) which localizes near the potential well int V−1(0) for λ large. Moreover, if the zero sets int V −1(0) of V (x) include more than one isolated component, then u_λ(x) will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter λ is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V−1(0). 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J Funct Anal, 2015, 269: 47–79MathSciNetCrossRefMATHGoogle ScholarCopyright information© Science China Press and Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsMiaoMiao Niu1ZhongWei Tang1Email author1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina About this article CrossMark Publisher Name Science China Press Print ISSN 1674-7283 Online ISSN 1869-1862 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; 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