Fractional Schrodinger equations with new conditions

Abstract

In this article, we study the nonlinear fractional Schrodinger equation $$\displaylines{ (-\Delta)^{\alpha}u+ V(x)u= f(x,u)\cr u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), }$$ where $(-\Delta)^{\alpha}(\alpha \in (0, 1))$ stands for the fractional Laplacian of order $\alpha$, $x\in \mathbb{R}^{n}$, $V\in C(\mathbb{R}^{n},\mathbb{R})$ may change sign and f is only locally defined near the origin with respect to u. Under some new assumptions on V and f, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result.

Keywords

• Fractional Schrodinger equations
• critical point theory
• symmetric mountain pass theorem