Unique continuation for a class of degenerate elliptic operators
The paper is concerned with the strong unique continuation property for the system of degenerate elliptic equations \[ \text{div}(| u|^{p-2}\nabla u^{(k)})-f^{(k)}(x,u)=0\quad\text{for } x\in\Omega,\;k=1,\ldots,s. \tag{*} \] Here \(\Omega\subset\mathbb{R}^ n\) \((n\geq 3)\) is a domain, \(s\in\mathbb{N}\), \(p\geq 2\). On \(f:\Omega\times\mathbb{R}^ s\to\mathbb{R}^ s\) the growth condition \(| f(x,t)|\leq K| t|^{p-1}\) is imposed. The author presents the following result: Let \(u\in W^{2,2}_{\text{loc}}(\Omega,\mathbb{R}^ s)\) be a solution to (*). Then \(u\) cannot vanish of infinite order at some \(x_ 0\in\Omega\) unless \(u\) vanishes identically in \(\Omega\). Reviewer's remark: The solution class \(W^{2,2}_{\text{loc}}\) is not adequate for the degenerate system. The author should have given some information or reference concerning the natural solution class for (*). In the case of systems \((s>1)\) the proof contains some gaps. In the case of equations \((s=1)\) by means of the substitution \(w:=| u|^{p-2}u\) the problem essentially reduces to a linear one, which has been treated e.g. by \textit{N. Garofalo} and \textit{F.-H. Lin} [Indiana Univ. Math. J. 35, 245-268 (1986; Zbl 0678.35015)].
scientific article; zbMATH DE number 150612
scientific article