Let \(\Omega = \{x\in {\mathbb R}^n : | x| 0\) is a parameter and two typical examples of \(f\) are \(f(u) = u-| u| ^{p-1}u\), where \(p>1\), and \(f(u) = \sin u\). The main result of the paper asserts that \(G\) is non-transitive if and only if for each positive integer \(k\) there exists \(\lambda_{k}\) such that whenever \(\lambda > \lambda_{k}\), then the boundary value problem above has at least \(k\) non-radial \(G\)-invariant solutions. The proof uses Clark's theorem in critical point theory for even functionals. More precisely, solutions are found as critical points of the Euler-Lagrange functional in the Sobolev space \(H^1_{0,G}(\Omega)\) of \(G\)-invariant functions, and existence of non-radial solutions is established by comparing the total number of critical points with the number of critical points corresponding to the radial solutions. The paper is a continuation of the author's earlier work where a similar problem, but for a different class of nonlinearities, has been considered.