Electron. J. Diff. Equ., Vol. 2015 (2015), No. 25, pp. 1-11.

Multiple solutions for fractional Schrodinger equations

Hongxia Shi, Haibo Chen

Abstract:
In this article we study the fractional Schr\"odinger equations
$$ (-\Delta)^{\alpha}u+V(x)u=f(x,u) \quad\text{in }\mathbb{R}^{N}, $$
where $0<\alpha<1$, $N\geq2$, $(-\Delta)^{\alpha}$ stands for the fractional Laplacian of order $\alpha$. First by using Morse theory in combination with local linking arguments, we prove the existence of at least two nontrivial solutions. Next we prove that the problem has k distinct pairs of solutions by using the Clark theorem.

Submitted November 16, 2014. Published January 27, 2015.
Math Subject Classifications: 35B38, 35G99.
Key Words: Fractional Schrodinger equations; variational methods; Morse theory; local linking.

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Hongxia Shi
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: shihongxia5617@163.com
Haibo Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: math_chb@163.com

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