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Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities

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Abstract

In this paper we consider the nonlinear fractional Schrödinger equations in presence of a critical power nonlinearity and a subcritical term

$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^s u+V(x)u=f(u)+u^{2^*_s-1}\quad x\in \mathbb R^N, \end{aligned}$$

where \(s\in (0,1), 2^*_s=2N/(N-2s), N>2s, (-\Delta )^s\) is the fractional laplacian and f(u) is of subcritical nonlinearity, and \(\varepsilon \) is a positive parameter. Under a local condition imposed on the potential V, we relate the number of positive solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.

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Acknowledgments

The authors would like to express their sincere gratitude to the referee for careful reading the manuscript and valuable comments and suggestions.

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Authors and Affiliations

  1. College of Science, Minzu University of China, Beijing, 100081, China

    Xiaoming He

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wenming Zou

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  1. Xiaoming He
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  2. Wenming Zou
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Correspondence to Xiaoming He.

Additional information

Communicated by P. Rabinowitz.

X. He is supported by NSFC (11371212, 10601063, 11271386) while W. Zou by NSFC (11371212, 11271386).

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He, X., Zou, W. Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. 55, 91 (2016). https://doi.org/10.1007/s00526-016-1045-0

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