Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. (English) Zbl 0688.35025
Summary: We prove that nonnegative solutions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular this answers a question of B. Gidas, W. M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)], where positive solutions were proven to be radially symmetric. In Section 4 we provide a sufficient condition on the geometry of the domain which ensures that nonnegative solutions are positive in the interior.
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35B99 | Qualitative properties of solutions to partial differential equations |
Citations:
Zbl 0425.35020References:
| [1] | Castro A., Proc. Royal Soc. of Edinburgh 108 pp 291– (1988) · Zbl 0659.34018 · doi:10.1017/S0308210500014670 |
| [2] | Castro, A., and Shivaji, R., Nonnegative solutions for a class,of radially symmetric nonpositone problems, Proc. Amer. Math. Soc. (to appear). |
| [3] | Gidas B., Comun. Math. Phys. 68 pp 209– (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 |
| [4] | Guillemin V., Differential Topology (1974) |
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