In this paper we prove the existence of infinitely many radial solutions of $ \Delta u + K(r)f(u) = 0 $ on the exterior of the ball of radius $ R>0 $, $ B_{R} $, centered at the origin in $ {\mathbb R}^{N} $ with $ u = 0 $ on $ \partial B_{R} $ and $ \lim_{r \to \infty} u(r) = 0 $ where $ N>2 $, $ f $ is odd with $ f<0 $ on $ (0, \beta) $, $ f>0 $ on $ (\beta, \infty), $ $ f $ superlinear for large $ u $ and $ 0< K(r) \leq \frac{K_{1}}{r^{\alpha}} $ with $ 2<\alpha <2(N-1) $ for large $ r $.
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