Abstract: Let G he a hounded domain in E Consider the following quasi-linear elliptic equation Although the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions:

Key words: quasi-linear elliptic equations, generalized solutions, maximum modulus, a priori estimate