Sharp counterexamples in unique continuation for second order elliptic equations

We construct nontrivial solutions with compact support for the elliptic equation Deltau = Vu with V epsilon L(P), p < n/2 or V epsilon L(w)(n/2) for n greater than or equal to 3 and with V epsilon L(1) for n = 2. The same method also yields nontrivial solutions with compact support for the elliptic equation Deltau = Wdelu with W epsilon L(q), p < n or W epsilon L(w)(n) for n greater than or equal to 2.