Electron. J. Differential Equations, Vol. 2018 (2018), No. 126, pp. 1-21.

Multiplicity of solutions for a perturbed fractional Schrodinger equation involving oscillatory terms

Chao Ji, Fei Fang

Abstract:
In this article we study the perturbed fractional Schrodinger equation involving oscillatory terms
$$\displaylines{ (-\Delta)^{\alpha} u+u =Q(x)\Big(f(u)+\epsilon g(u)\Big), \quad x\in \mathbb{R}^N\cr u\geq 0, }$$
where $\alpha\in (0, 1)$ and $N> 2\alpha$, $(-\Delta)^{\alpha}$ stands for the fractional Laplacian, $Q: \mathbb{R}^N\to \mathbb{R}^N$ is a radial, positive potential, $f\in C([0, \infty), \mathbb{R})$ oscillates near the origin or at infinity and $g\in C([0, \infty), \mathbb{R})$ with $g(0)=0$. By using the variational method and the principle of symmetric criticality for non-smooth Szulkin-type functionals, we establish that: (1) the unperturbed problem, i.e. with $\epsilon=0$ has infinitely many solutions; (2) the number of distinct solutions becomes greater and greater when $| \epsilon|$ is smaller and smaller. Moreover, various properties of the solutions are also described in terms of the $L^{\infty}$- and $H^{\alpha}(\mathbb{R}^N)$-norms.

Submitted January 7, 2018. Published June 18, 2018.
Math Subject Classifications: 35J60, 47J30.
Key Words: Fractional Schrodinger equation; multiple solutions; oscillatory terms.

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Chao Ji
Department of Mathematics
East China University of Science and Technology
200237 Shanghai, China
email: jichao@ecust.edu.cn
Fei Fang
Department of Mathematics
Beijing Technology and Business University
100048 Beijing, China
email: fangfei68@163.com

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