Fractional Kirchhoff problem with critical indefinite nonlinearity

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the formM∫ ∫R 2 N| u ( x ) − u ( y ) | 2| x − y | N + 2 sd x d y( − Δ ) s u= λ f ( x ) | u | q − 2 u + | u |2 s ∗ − 2 u in Ω , u = 0 inR N ∖ Ω ,where Ω ⊂ R Nis a smooth bounded domain, 0 < s < 1 and 1 < q < 2 . Here M is the Kirchhoff coefficient and2 s ∗ = 2 N / ( N − 2 s )is the fractional critical Sobolev exponent. The parameter λ is positive and the 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.