Fractional Kirchhoff problem with critical indefinite nonlinearity

Abstract

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_Ω|(-Δ)^{\fracα{2}}u|^2dx\right)(-Δ)^α u= λf(x)|u|^{q-2}u+|u|^{2^*_α-2}u\;\; \text{in}\; Ω,\;u=0\;\textrm{in}\;\mathbb R^n\setminus Ω, \end{equation*} where $Ω\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<α<1, \; 2α<n<4α$ and $ \; 1<q<2$. Here $2^*_α={2n}/{(n-2α)}$ is the fractional critical Sobolev exponent, $λ$ is a positive parameter and the coefficient $f(x)$ is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.