Existence of solution for a class of fractional problems with sign-changing functions

Abstract

Here we study the existence and multiplicity of solutions for the ‎following‎ ‎fractional‎‎ problem‎ ‎$$‎ ‎(-\Delta)_p^s u+a(x) |u|^‎{‎‎p‎-2} ‎u‎= f(x,u)‎, ‎$$‎ ‎with ‎the ‎Dirichlet‎ boundary condition $u=0$ on $\partial\Omega$‎ ‎where $\Omega$ is a bounded domain with smooth boundary‎, ‎$p\geq 2$‎,‎ $s\in(0,1)$ and ‎‎$‎a(x)‎‎$‎ ‎is a‎ sign-changing ‎function.‎ ‎Moreover, we consider two different assumptions on the ‎function‎ $‎f(x,u)‎$‎, ‎including the cases of nonnegative and sign-changing ‎function.‎‎