We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius R in Rⁿ \[ Δ u+λ f(u)=0 \; \; for \; \; |x|<R, \; \; u=0 \; \; on \; \; |x|=R \] 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).
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Philip Korman (1997) studied this question.