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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 $O(N)$ such that $H \varsubsetneq G \subset O(N)$. We denote the orbit of $G$ through $x\in \mathbb {R}^N$ by $G(x)$, i.e., $G(x):=\{gx\colon g\in G \}$. We prove that if $H(x)\varsubsetneq G(x)$ 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 there exists an $H$ invariant $G$ non-invariant positive solution.

Positive solutions of the $p$-Laplace Emden-Fowler equation in hollow thin symmetric domains