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In this paper, we extend our earlier unique continuation results PZ2 for the Schr\"odinger-type inequality | u| V|u| on a domain in Cⁿ by removing the smoothness assumption on solutions u = (u₁, , uN). More specifically, we establish the unique continuation property for W₋₎₂^1, 1 solutions when the potential V L₋₎₂ᵖ, p>2n; and for W₋₎₂^1, 2n+ solutions when V L₋₎₂^2n with N=1 or n = 2. Although the unique continuation property fails in general if V L₋₎₂^p, p<2n, we show that the property still holds for W₋₎₂^1, 1 solutions when V is a small constant multiple of 1|z|.
Pan et al. (Sat,) studied this question.