Smoothness of certain degenerate elliptic equations

Abstract

Given p > 1 , p ≠ 2 p > 1,p \ne 2 , let u be a solution to div ( | grad u | P − 2 grad u ) = 0 {\text {div}}(|{\text {grad}}\;u{|^{P - 2}}{\text {grad}}\;u) = 0 on a domain D in Euclidean two space. We prove that if u is nonconstant and real analytic in D, then the gradient of u does not vanish in D. Some examples of Krol’ are used to show this result and a related result of Ural’tseva are nearly best possible.