Three solutions for a fractional Schrödinger equation with vanishing potentials

Appl. Math. Lett. 76, 90–95

In this paper, we study the following fractional Schrödinger equation

\begin{equation*} (-\Delta)^su+V(x)u=K(x)f(u)+\lambda W(x)|u|^{p-2}u,\ \ x\in\mathbb{R}^N, \end{equation*}

where $\lambda>0$ is a parameter, $(-\Delta)^s$ denotes the fractional Laplacian of order $s\in(0,1)$, $N>2s$, $W\in L^{\frac{2}{2-p}}(\mathbb{R}^N,\mathbb{R}^+)$, $1< p <2$, $V,K$ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Under some mild assumptions, we prove that the above equation has three solutions.