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Symmetry and Monotonicity of a Nonlinear Schrödinger Equation Involving the Fractional Laplacian

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Abstract

In this paper, we consider a nonlinear Schrödinger equation involving the fractional Laplacian with Dirichlet condition:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{\frac{\alpha }{2}}u+A(x)u=f(x,u,\nabla u) \ \text{ in } \ \Omega ,\\ u>0, \ \text{ in }\ \Omega ; \ u\equiv 0, \ \text{ in }\ \mathbb R^n\backslash \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a domain (bounded or unbounded) in \(\mathbb R^n\) which is convex in \(x_1\)-direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in \(x_1\)-direction in the left half domain of \(\Omega \). Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space \(\mathbb R^n_+\).

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Acknowledgements

The authors would like to express their gratitude to referees for many valuable suggestions which help to improve the presentation of this paper and provide good directions for further research.

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Correspondence to Ping Li.

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Communicated by Maria Alessandra Ragusa.

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The first author was supported by the Science and Technology Research Program for the Education Department of Hubei province of China under Grant No. D20163101. The Second author was supported by the NNSF of China under Grant No. 11871096.

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Yuan, L., Li, P. Symmetry and Monotonicity of a Nonlinear Schrödinger Equation Involving the Fractional Laplacian. Bull. Malays. Math. Sci. Soc. 44, 4109–4125 (2021). https://doi.org/10.1007/s40840-021-01158-z

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  • DOI: https://doi.org/10.1007/s40840-021-01158-z

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