Electron. J. Diff. Equ., Vol. 2016 (2016), No. 218, pp. 1-19.

Existence of solutions to nonlinear fractional Schrodinger equations with singular potentials

Qingxuan Wang, Dun Zhao, Kai Wang

Abstract:
We study the eigenvalue problem
$$ (-\Delta)^s u(x)+ V(x)u(x)-K(x)|u|^{p-2}u(x) =\lambda u(x) \quad \text{in } \mathbb{R}^N, $$
where $s\in(0,1)$, $N>2s$, $2<p<2^{*}=\frac{2N}{N-2s}$, V(x) is indefinite and allowed to be unbounded from below, and K(x) is nonnegative and allowed to be unbounded from above. When $\lambda <\lambda_0=\inf \sigma((-\Delta)^s +V(x))$ (the lowest spectrum of the operator $(-\Delta)^s +V(x))$, we obtain a positive ground state solution by using the constrained minimization method. Also we discuss the regularity of solutions.

Submitted November 22, 2015. Published August 16, 2016.
Math Subject Classifications: 35R11.
Key Words: Nonlinear fractional Schrodinger equation; ground state; positive solution; weakly continuous.

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Qingxuan Wang
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: wangqx12@lzu.edu.cn
Fax +86 931 8912481
Dun Zhao
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: zhaod@lzu.edu.cn
Kai Wang
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: wangk2010@lzu.edu.cn

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