On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in \(\mathbb {R}^N\)

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• 作者：Sihua Liang ; Shaoyun Shi
• 关键词：Mathematics Subject Classification35J60 ; 35Q55
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

In this paper, we study a class of nonlinear Schrödinger equations with electromagnetic fields and critical nonlinearity in \(\mathbb {R}^N: -\Delta _Au + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2^*-2}u\), where f is a continuous function, \(V, Z: \mathbb {R}^N \rightarrow \mathbb {R}\) are continuous functions verifying suitable hypotheses. We show that if the zero set of V(x) has several isolated connected components \(\Omega _1,\ldots ,\Omega _k\) such that the interior of \(\Omega _i\) is not empty and \(\partial \Omega _i\) is smooth, then for \(\lambda > 0\) large there exists, for any non-empty subset \(\Gamma \subset \{1,\ldots ,k\}\), a bump solution trapped in a neighborhood of \(\cup _{j\in \Gamma }\Omega _j\).