Multiple solutions to the nonhomogeneous -Kirchhoff elliptic equation with concave–convex nonlinearities

In this paper, we study the multiplicity of solutions for the nonhomogeneous p -Kirchhoff elliptic equation (0.1) − M ( ‖ ∇ u ‖ p p ) Δ p u = λ h 1 ( x ) | u | q − 2 u + h 2 ( x ) | u | r − 2 u + h 3 ( x ) , x ∈ Ω , with zero Dirichlet boundary condition on ∂ Ω , where Ω is the complement of a smooth bounded domain D in R N ( N ≥ 3 ) . λ > 0 , M ( s ) = a + b s k , a , b > 0 , k ≥ 0 , h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are continuous functions which may change sign on Ω . The parameters p , q , r satisfy 1 < q < p ( k + 1 ) < r < p ∗ = N p N − p . A new existence result for multiple solutions is obtained by the Mountain Pass Theorem and Ekeland’s variational principle.