Variable exponent $p(\cdot)$-Kirchhoff type problem with convection in variable exponent Sobolev spaces

We establish the existence of weak solution for a class of $p(x)$-Kirchhoff type problem for the $p(x)$-Laplacian-like operators with Dirichlet boundary condition and with gradient dependence (convection) in the reaction term. Our result is obtained using the topological degree for a class of demicontinuous operators of generalized $(S_{+})$ type and the theory of the variable exponent Sobolev spaces. Our results extend and generalize several corresponding results from the existing literature.