The existence of solutions for the modified ( p ( x ) , q ( x ) ) -Kirchhoff equation

Abstract

We consider the Dirichlet problem − Δ p ( x ) K p u ( x ) − Δ q ( x ) K q u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) in Ω , u | ∂ Ω = 0 , driven by the sum of a p ( x ) -Laplacian operator and of a q ( x ) -Laplacian operator, both of them weighted by indefinite (sign-changing) Kirchhoff type terms. We establish the existence of weak solution and strong generalized solution, using topological tools (properties of Galerkin basis and of Nemitsky map). In the particular case of a positive Kirchhoff term, we obtain the existence of weak solution ( = strong generalized solution), using the properties of pseudomonotone operators.