Multiple Solutions of Nonlinear Schrödinger Equations with the Fractional $p$-Laplacian

Abstract

We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation\[ (-\Delta)^\alpha_p u + V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.