On maximum principle and existence of solutions for elliptic systems on \(\mathbb{R}^ n\) (Q1842744)

This is a variant of the paper reviewed above. Now \[ {\mathcal L} U(x):= -\Delta U(x)- A(x) U(x),\tag{\(*\)} \] \(N= 2\) and \(A\) is a matrix-valued function decaying at infinity. Conditions \(C_1(A)\), \(C_2(A)\) on \(A\) are formulated which are claimed to be necessary resp. sufficient for the validity of a maximum principle. Furthermore, \(C_2(A)\) is shown to imply existence in similar cases as in the preceding review. Reviewer's remarks: i) The proof of necessity of \(C_1(A)\) contains the same gap as described in the preceding review. ii) In formula (7) it is asserted that \((a- \rho)u\) belongs to \(L^2_{1/\rho}\) which is not true in general.