Smoothness of certain degenerate elliptic equations

Abstract:

Given $p > 1,p \ne 2$, let u be a solution to ${\text {div}}(|{\text {grad}}\;u{|^{P - 2}}{\text {grad}}\;u) = 0$ 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.