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We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O\left(N\right)$ such that $HG\subset O\left(N\right)$. We denote the orbit of $G$ through $x\in {ℝ}^N$ by $G\left(x\right)$, i.e., $G\left(x\right):=\{gx:g\in G\}$. We prove that if $H\left(x\right)G\left(x\right)$ for all $x\in \overline{\Omega }$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore...