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_\Omega|(-\Delta)^{\frac{\alpha}{2}}u|^2dx\right)(-\Delta)^{\alpha} u= \lambda f(x)|u|^{q-2}u+|u|^{2^*_\alpha-2}u\;\; \text{in}\; \Omega,\;u=0\;\textrm{in}\;\mathbb R^n\setminus \Omega, \end{equation*} where $\Omega\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<\alpha<1, \; 2\alpha<n<4\alpha$ and $ \; 1<q<2$. Here $2^*_\alpha={2n}/{(n-2\alpha)}$ is the fractional critical Sobolev exponent, $\lambda$ 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.