Infinitely many solutions for fractional Schrödinger equation with potential vanishing at infinity

Abstract

Abstract The paper investigates the following fractional Schrödinger equation: (−Δ)su+V(x)u=K(x)f(u),x∈RN, $$\begin{aligned} (-\Delta )^{s}u+V(x)u=K(x)f(u), \quad x\in \mathbb{R}^{N}, \end{aligned}$$ where 0<s<1 $0< s<1$, 2s<N $2s< N$, (−Δ)s $(-\Delta )^{s}$ is the fractional Laplacian operator of order s. V(x) $V(x)$, K(x) $K(x)$ are nonnegative continuous functions and f(x) $f(x)$ is a continuous function satisfying some conditions. The existence of infinitely many solutions for the above equation is presented by using a variant fountain theorem, which improves the related conclusions on this topic. The interesting result of this paper is the potential V(x) $V(x)$ vanishing at infinity, i.e., lim|x|→+∞V(x)=0 $\lim_{|x|\rightarrow +\infty }V(x)=0$.

Keywords

• Fractional Laplacian
• Fountain theorem
• Fractional Schrödinger equation
• Weak solution