We consider the Schrodinger equation $$ \Delta u + V (x)u = \lambda K(x)f(u) + \mu L(x)g(u) \mbox{ in } R^N; \ u\in H^1(R^N), \eqno{(P)} $$ where $N\ge  2$, $\lambda , \mu  \ge 0$ are parameters, $V,K,L : R^N\rightarrow R$ are radially symmetric potentials, $f : R\rightarrow R$ is a continuous function with sublinear growth at in finity, and $g : R\rightarrow R$ is a continuous sub-critical function. We first prove that for $\lambda $ small enough no non-zero solution exists for $(P)$, while for $\lambda $ large and $\mu $ small enough at least two distinct non-zero radially symmetric solutions do exist for $(P)$. By exploiting a Ricceri-type three-critical points theorem, the principle of symmetric criticality and a group-theoretical approach, the existence of at least $N-3$ ($N$ mod 2) distinct pairs of non-zero solutions is guaranteed for $(P)$ whenever $\lambda $ is large and  $\mu $ is small enough, $N\neq 3$, and $f, g$ are odd.