Three Weak Solutions for Nonlocal Fractional Laplacian Equations

Abstract

The existence of three weak solutions for the following nonlocal fractional equation $(-\mathrm{\Delta }{)}^{s}u-\lambda u=\mu f(x,u)$ in $\mathrm{}\mathrm{}\mathrm{\Omega },\mathrm{}u=\mathrm{0}\mathrm{}$ in ${ℝ}^n\setminus \mathrm{\Omega },$ is investigated, where $s\in (\mathrm{0,1})$ is fixed, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator, $\lambda$ and $\mu$ are real parameters, $\mathrm{\Omega }$ is an open bounded subset of ${ℝ}^n$, $n>\mathrm{2}s$, and the function $f$ satisfies some regularity and natural growth conditions. The approach is based on a three-critical-point theorem for differential functionals.