We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$\left\{\begin{array}{cc}-\text{div}\left(\left(1+{\left|u\right|}^r\right)\nabla u\right)+{\left|u\right|}^{m-2}u{\left|\nabla u\right|}^2=f\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=0\hfill & \text{su}\partial \mathrm{\Omega }.\hfill \end{array}\right.$$

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...