  We provide a-priori $L^\infty$ bounds for positive solutions to a class of
subcritical elliptic problems in bounded $C^2$ domains. Our arguments rely on
the moving planes method applied on the Kelvin transform of solutions. We prove
that locally the image through the inversion map of a neighborhood of the
boundary contains a convex neighborhood; applying the moving planes method, we
prove that the transformed functions have no extremal point in a neighborhood
of the boundary of the inverted domain. Retrieving the original solution $u$,
the maximum of any positive solution in the domain $\Om,$ is bounded above by a
constant multiplied by the maximum on an open subset strongly contained in
$\Om.$ The constant and the open subset depend only on geometric properties of
$\Om,$ and are independent of the non-linearity and on the solution $u$. Our
analysis answers a longstanding open problem.
