A remark on unique continuation (Q1568966)

The following result is proved: There is a numerical constant \(\varepsilon > 0\) making the following true. Assume that \(u\) is a function defined on a (connected) neighborhood \(U\) of the origin in \(\mathbb R^n\) for some \(n \geq 2,\) which belongs to the Sobolev space \(W^{2,2}\) and satisfies the differential inequality \[ |\Delta u|\leq \frac{A}{|x|^2}u + \frac{\varepsilon}{|x|}|\nabla u|, \] for some constant \(A\). Assume also that \[ \lim_{r\to 0}\int_{|x|\leq r} |u|^2 =0 \quad \text{for all } k<\infty. \] Then \(u=0.\)