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

We consider the Dirichlet problem-Delta(Kp)(p(x))u(x) - Delta(Kq)(q(x))u(x) = f(x, u(x), del u(x)) in Omega, u vertical bar(partial derivative Omega) = 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.