The existence of solutions for the modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/1998/Math/MathML"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math>-Kirchhoff equation

We consider the Dirichlet problem − Δ p ( x stretchy="false">) K u q = f , mathvariant="normal">∇<!-- ∇ <mml:mspace width="1em" /> in mathvariant="normal">Ω<!-- Ω maxsize="1.2em" minsize="1.2em">| mathvariant="normal">∂<!-- ∂ <mml:mn>0 driven by sum of a xmlns="http://www.w3.org/1998/Math/MathML"> -Laplacian operator and operator, both them weighted indefinite (sign-changing) Kirchhoff type terms. establish existence weak solution strong generalized solution, using topological tools (properties Galerkin basis Nemitsky map). In particular case positive term, we obtain (<mml:math solution), properties pseudomonotone operators.