Let A be a symmetric N × N real-matrix-valued function on a connected region in Rn, with A positive definite a.e. and A, A−1 locally integrable. Let b and c be locally integrable, non-negative, real-valued functions on Ω, with c positive a.e. Put a(u, v) = =\(\mathop \smallint \limits_\Omega \)((A∇u, ∇v) + buv) dx. We consider in X the weak boundary value problem a(u, v) = =\(\mathop \smallint \limits_\Omega \)fvcdx, all v ε X; where X is a suitable Hilbert space contained in Hloc1,1(Ω). Criteria are given in order that the Green's operator for this problem have an integral representation and bounded eigenfunctions; in addition, criteria for compactness are given.