Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

The existence and location of a positive bounded weak solution is established for a class of parametric Dirichlet problems with convection term provided the parameter is suitably restricted. The nonlinearity involves variable exponents with no upper limitation neither in the solution nor in its gradient, which is a novelty. Our approach is based on a version of sub-supersolution method that fits the specific character of the considered problem. We pass through auxiliary contracted and truncated problems allowing to handle the unbounded variable exponents. A priori estimates for the solution are also obtained.