Solution curves for semilinear equations on a ball

Korman, Philip

doi:10.1090/S0002-9939-97-04119-1

Solution curves for semilinear equations on a ball
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by Philip Korman

Proc. Amer. Math. Soc. 125 (1997), 1997-2005

DOI: https://doi.org/10.1090/S0002-9939-97-04119-1

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Abstract:

We show that the set of positive solutions of semilinear Dirichlet problem on a ball of radius $R$ in $R^n$ \[ \Delta u+\lambda f(u)=0 \; \; \text {for} \; \; |x|<R, \; \; u=0 \; \; \text {on} \; \; |x|=R \] 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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