  In this paper we are interested in positive classical solutions of
\begin{equation} \label{eqx} \left\{\begin{array}{ll} -\Delta u = a(x) u^{p-1}
& \mbox{ in } \Omega, \\ u>0 & \mbox{ in } \Omega, \\ u= 0 & \mbox{ on } \pOm,
\end {array}\right. \end{equation} where $\Omega$ is a bounded annular domain
(not necessarily an annulus) in $\IR^N$ $(N \ge3)$
  and $ a(x)$ is a nonnegative continuous function. We show the existence of a
classical positive solution for a range of supercritical values of $p$ when the
problem enjoys certain mild symmetry and monotonicity conditions. As a
consequence of our results, we shall show that (\ref{eqx}) has
$\Bigl\lfloor\frac{N}{2} \Bigr\rfloor$ (the floor of $\frac{N}{2}$) positive
nonradial solutions when $ a(x)=1$ and $\Omega$ is an annulus with certain
assumptions on the radii. We also obtain the existence of positive solutions in
the case of toroidal domains. Our approach is based on a new variational
principle that allows one to deal with supercritical problems variationally by
limiting the corresponding functional on a proper convex subset instead of the
whole space at the expense of a mild invariance property.
