  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 }
\partial \Omega, \end {array}\right. \qquad(1)\end{equation} where $\Omega$ is
a bounded annular domain (not necessarily an annulus) in $\mathbb{R}^N$ $(N
\ge3)$ and $ a(x)$ is a nonnegative smooth function. We show the existence of a
classical positive solution for all $p>2$ when the problem enjoys certain mild
symmetry conditions. As a consequence of our results, we shall show that (1)
has a nonradial solution when $ a(x)=1$ and $\Omega$ is an annulus with certain
assumptions on the radii. 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.
