Quasilinear Systems with Unbounded Variable Exponents and Convection Terms

We prove the existence of positive solutions for a system of quasilinear equations driven by \(p_i\) $p_i$ -Laplacian operators, with Dirichlet boundary conditions, variable exponents, convection terms, and depending on two parameters. No upper bound on the variable exponents, neither in u nor in \(\nabla u\) $\mathrm{\nabla }u$ , is imposed. It is for the first time when such systems are investigated. The range of solvability for the involved parameters is explicitly determined. Our approach relies on a version of sub-supersolution method for systems that is adapted to the specific character of our problem.