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

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.

Keywords

• Fractional Schrodinger equation
• multiple solutions
• oscillatory terms