A supercritical elliptic equation in the annulus

TL;DR

This paper develops a novel approach combining variational and topological methods to find positive, axially symmetric solutions for a supercritical elliptic equation in an annulus, including conditions for nonradial solutions.

Contribution

It introduces a new technique for detecting axially symmetric solutions of supercritical elliptic equations with nonradial coefficients in annular domains.

Findings

Existence of positive axially symmetric solutions in annuli.

Conditions for solutions to be nonradial depending on parameters.

Application of combined variational and topological methods.

Abstract

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