Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions

Abstract

In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu= a_λ(x)|u|^{q-2}u+b(x)|u|^{2^*_α-1}u &{\rm in}\,\,Ω, u=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $0<α<1$, $ Ω$ is a bounded domain with smooth boundary in $ \R^N $ with $N>2α$ and $ 2^*_α=2N/(N-2α)$ is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Ljusternik-Schnirelmann category.