On critical double phase Kirchhoff problems with singular nonlinearity

The paper deals with the following double phase problem -m[∫Ω(|∇u|pp+a(x)|∇u|qq)dx]div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=λu-γ+up∗-1inΩ,u>0inΩ,u=0on∂Ω,where Ω ⊂ RN is a bounded domain with Lipschitz boundary ∂Ω , N≥ 2 , m represents a Kirchhoff coefficient, 1 < p< q< p∗ with p∗= Np/ (N- p) being the critical Sobolev exponent to p, a bounded weight a(·) ≥ 0 , λ> 0 and γ∈ (0 , 1). By the Nehari manifold approach, we establish the existence of at least one weak solution.