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We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius R in Rⁿ \ u+ f (u) =0 \; \; for \; \; |x|<R, \; \; u=0 \; \; on \; \; |x|=R \ consists of smooth curves. Our results can be applied to compute the direction of bifurcation. We also give an easy proof of a uniqueness theorem due to Smoller and Wasserman (1984).
Philip Korman (Wed,) studied this question.