Quasilinear Dirichlet problems with competing operators and convection

The paper deals with a quasilinear Dirichlet problem involving a competing ( )-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing ( )-Laplacian we consider the usual ( )-Laplacian, our results ensure the existence of weak solutions.