A supercritical elliptic equation in the annulus

By a combination of variational and topological techniques in the presence invariant cones, we detect new type positive axially symmetric solutions Dirichlet problem for elliptic equation $$ -\Delta u + = a(x)|u|^{p-2}u an annulus $A \subset \mathbb{R}^N$ ($N\ge3$). Here $p>2$ is allowed to be supercritical $a(x)$ but possibly nonradial function with additional symmetry monotonicity properties, which are shared by solution $u$ construct. In case where $a$ equals constant, obtain exponent $p$ large or when $A$ fixed width.