We consider the existence of radially symmetric non-negative solutions for the boundary value problem \[ − Δ u ( x ) = λ f ( u ( x ) ) ‖ x ‖ ≤ 1 , x ∈ R N ( N ≥ 2 ) u ( x ) = 0 ‖ x ‖ = 1 \begin {array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\quad \left \| x \right \| \leq 1,x \in {R^N}(N \geq 2)} \\ {u(x) = 0\quad \left \| x \right \| = 1} \\ \end {array} \] where λ > 0 , f ( 0 ) > 0 \lambda > 0,f(0) > 0 (non-positone), f ′ ≥ 0 f’ \geq 0 and f f is superlinear. We establish existence of non-negative solutions for λ \lambda small which extends some work of our previous paper on non-positone problems, where we considered the case N = 1 N = 1 . Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.