Partially symmetric solutions of the generalized Hénon equation in symmetric domains

Abstract

We study the generalized H\'{e}non equation in a symmetric domain $\Omega$. Let $H$ and $G$ be closed subgroups of the orthogonal group such that $H \varsubsetneq G$ and $\Omega$ is $G$ invariant. Then we show the existence of a positive solution which is $H$ invariant but $G$ non-invariant under suitable assumptions of $H$, $G$ and the coefficient function of the equation.