Nonlinear degenerate elliptic partial differential equations with critical growth conditions on the gradient

We consider a nonlinear degenerate elliptic partial differential equation with the critical growth condition on where g is sufficiently integrable and p is between 1 and ∞. Our first goal of this paper is to prove the existence of the solution.The main idea is to obtain the uniform L^∞-estimate of suitable approximate solutions, employing a truncation technique and radially decreasing symmetrization techniques based on rearrangements. We also find an example of unbounded weak solution.