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

Abstract:

We consider a nonlinear degenerate elliptic partial differential equation $- {\operatorname {div}}(|\nabla u{|^{p - 2}}\nabla u) = H(x,u,\nabla u)$ with the critical growth condition on $H(x,u,\nabla u) \leq g(x) + |\nabla u{|^p}$, where g is sufficiently integrable and p is between 1 and $\infty$. Our first goal of this paper is to prove the existence of the solution in $W_0^{1,p} \cap {L^\infty }$. The main idea is to obtain the uniform ${L^\infty }$-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 $- {\operatorname {div}}(|\nabla u{|^{p - 2}}\nabla u) = |\nabla u{|^p}$ for $1 < p \leq n$.