  An inverse problem for the two-dimensional Schrodinger equation with
$L^p_{com}$-potential, $p>1$, is considered. Using the
$\overline{\partial}$-method, the potential is recovered from the
Dirichlet-to-Neumann map on the boundary of a domain containing the support of
the potential. We do not assume that the potential is small or that the Faddeev
scattering problem does not have exceptional points. The paper contains a new
estimate on the Faddeev Green function that immediately implies the absence of
exceptional points near the origin and infinity when $v\in L^p_{com}$.
