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

Abstract

We consider a nonlinear degenerate elliptic partial differential equation - div(/del u/(p-2)del u) = H(x, u, del u) with the critical growth condition on H(x, u, del u) less than or equal to g(x) + /del u/(p), where g is sufficiently integrable and p is between 1 and infinity. Our first goal of this paper is to prove the existence of the solution in W-0(1,p) boolean AND L(infinity). The main idea is to obtain the uniform L(infinity)-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 of - div(/del u/(p-2)del u) = /del u/(p) for 1 < p less than or equal to n.