Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(R^ n\).

*(English)*Zbl 0676.35032We establish the uniqueness of the positive, radially symmetric solution to the differential equation \(\Delta u-u+u^ p=0\) (with \(p>1)\) in a bounded or unbounded annulus region in \(R^ n\) for all \(n\geq 1\), 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 is a well-known result of C. V. Coffman [Arch. Ration. Mech. Anal. 46, 81-95 (1972; Zbl 0249.35029)], which was later extended by K. McLeod and J. Serrin [ibid. 99, 115-145 (1987)] 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 an elaboration of a method due to Coffman. A survey of this method is given in a forthcoming paper entitled “On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP”, to appear in Z. Angew. Math. Phys. 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.

Reviewer: M.K.Kwong

##### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

##### Keywords:

uniqueness; positive, radially symmetric solution; annulus region; Neumann boundary condition; Dirichlet boundary condition; outer ball; decaying to zero; Kolodner-Coffman method; Sturm’s oscillation theory; second-order
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\textit{M. K. Kwong}, Arch. Ration. Mech. Anal. 105, No. 3, 243--266 (1989; Zbl 0676.35032)

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