Abstract
We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in R n for all n≧1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.
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Communicated by James Serrin
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Kwong, M.K. Uniqueness of positive solutions of Δu−u+up=0 in Rn . Arch. Rational Mech. Anal. 105, 243–266 (1989). https://doi.org/10.1007/BF00251502
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DOI: https://doi.org/10.1007/BF00251502