Unique continuation for non-negative solutions of quasilinear elliptic equations (Q2748275)

The unique continuation problem was formulated by B. Simon for Schrödinger equation on bounded domain \(\Omega\in \mathbb{R}^n\). The property means that the nonnegative solution of the homogeneous equation which vanishes of infinite order at one point is identically zero in the whole domain. The solution \(u>0\) has zero of infinite order at \(x_0\in \Omega\) if NEWLINE\[NEWLINE\lim_{\sigma\to 0} \int_{B(x_0,\sigma)} u(x)/|B(x_0, \sigma)|^k dx= 0,NEWLINE\]NEWLINE for any positive \(k\). It is proved that a nonnegative solution of a nonnegative solution of a quasilinear elliptic equations NEWLINE\[NEWLINE\text{div }A(x, u,\nabla u)= B(x,u,\nabla u)NEWLINE\]NEWLINE has no zero of infinite order in \(\Omega\) if \(A\) and \(B\) are continuous functions satisfying the generalized Kato conditions.