Elliptic Kirchhoff-type system with two convections terms and under Dirichlet boundary conditions

Abstract

This work discusses the existence  of "weak solutions" for a system of Kirchhoff-type involving variable exponent $(\alpha_1(m),\alpha_2(m))$-Laplacian operators and under the Dirichlet boundary conditions. Under appropriate hypotheses on the nonlinear terms and the Kirchhoff functions, the existence of weak solutions is obtained on the spaces $W_0^{1, \alpha_{1}(m)}(\mathcal{D}) \times W_0^{1, \alpha_{2}(m)}(\mathcal{D})$. The proof of the main result is based on a  topological degree argument for a class of demicontinuous operators of $(S_+)$-type.