Existence and multiplicity results for a -Kirchhoff equation with a concave–convex term

The existence and multiplicity of solutions to the following p -Kirchhoff equation is considered:(0.1) { − M ( ‖ u ‖ p ) Δ p u = λ h ( x ) | u | q − 2 u + H ( x ) | u | r − 2 u , in Ω , u ( x ) = 0 , on ∂ Ω , where Ω is a bounded domain in R N , M ( s ) = a + b s k , Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian operator, and 1 < p < N .