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Nontrivial solution for Schrödinger–Poisson equations involving the fractional Laplacian with critical exponent

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Abstract

This paper deals with a class of nonlocal Schrödinger equations with critical exponent

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s} u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=f(x,u),&{}\quad \mathrm{in}\ \mathbb {R}^3,\\ (-\Delta )^s \phi =K(x)|u|^{2^*_s-1},&{}\quad \mathrm{in}\ \mathbb {R}^3. \end{array}\right. \end{aligned}$$

By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of nontrivial solution is obtained under appropriate assumptions on V, K and f.

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Acknowledgements

The author would like to express their sincere gratitude to anonymous referees for his/her constructive comments for improving the quality of this paper.

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  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Xiaojing Feng

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  1. Xiaojing Feng
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Projects supported by National Natural Science Foundation of China (Grant Nos. 11671239, 12071266), Natural Science Foundation of Shanxi Province (201801D211001, 201801D121002, 201801D221012) and Shanxi Scholarship Council of China (Grant No. 2020-005).

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Feng, X. Nontrivial solution for Schrödinger–Poisson equations involving the fractional Laplacian with critical exponent. RACSAM 115, 10 (2021). https://doi.org/10.1007/s13398-020-00953-w

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