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Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity

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Abstract

In this paper, we investigate the following critical fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^su+V(x)u=|u|^{2_s^*-2}u+\lambda K(x)f(u), \ x \in \mathbb {R}^N, \end{aligned}$$

where \(\lambda >0\), \(0<s<1\), \((-\Delta )^s\) denotes the fractional Laplacian of order s, \(V, \ K\) are nonnegative continuous functions satisfying some conditions and f is a continuous function, \(N>2s\) and \(2_s^*=\frac{2N}{N-2s}\). We prove that the equation has a positive solution for large \(\lambda \) by the variational method.

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Authors and Affiliations

  1. Department of Mathematics, Honghe University, Mengzi, 661100, Yunnan, People’s Republic of China

    Quanqing Li

  2. Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, People’s Republic of China

    Kaimin Teng

  3. Department of Mathematics, Yunnan Normal University, Kunming, 650092, Yunnan, People’s Republic of China

    Xian Wu

Authors
  1. Quanqing Li
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  2. Kaimin Teng
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  3. Xian Wu
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Correspondence to Quanqing Li.

Additional information

This work is supported in part by the National Natural Science Foundation of China (11261070; 11501403; 11461023) and the Shanxi Province Science Foundation for Youths under Grant 2013021001-3.

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Li, Q., Teng, K. & Wu, X. Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity. Mediterr. J. Math. 14, 80 (2017). https://doi.org/10.1007/s00009-017-0846-5

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  • DOI: https://doi.org/10.1007/s00009-017-0846-5

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