×

Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(R^ n\). (English) Zbl 0676.35032

We 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

Citations:

Zbl 0249.35029
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berestycki, H., & Lions, P.-L., Non-linear scalar field equations I, existence of a ground slate; II existence of infinitely many solutions, Arch. Rational Mech. Analysis 82 (1983), 313–375. · Zbl 0533.35029
[2] Berestycki, H., Lions, P.-L., & Peletier, L. A., An ODE approach to the existence of positive solutions for · Zbl 0522.35036
[3] Coffman, C. V., Uniqueness of the ground stat · Zbl 0249.35029
[4] Gidas, B., Ni, W. M., & Nirenberg, L.. Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243. · Zbl 0425.35020
[5] Gidas, B., Ni, W. M., & Nirenberg, L., Symmetry of positive solutions of nonlinear · Zbl 0469.35052
[6] Kaper, H. G., & Kwong, Man Kam, Uniqueness of non-negative solutions of a class of semi-linear elliptic equations. Nonlinear Diffusion Equations and Their Equilibrium States II (Ed. by W.-M. Ni, L. A. Peletier, J. Serrin), Mathematical Sciences Research Institute Publications, Springer-Verlag (1988), 1–18. · Zbl 0662.35037
[7] Kaper, H. G., & Kwong, Man Kam, Concavity and monotonicity properties of solutions of Emden-Fowler equations, Differential and Integral Equations (1988), 327–340. · Zbl 0723.34026
[8] Kolodner, I. I., Heavy rotating string–a nonlinear eigenvalue problem, Comm. Pure and Appl. Math. 8 (1955), 395–408. · Zbl 0065.17202
[9] Kwong, Man Kam, On Kolodner’s method for the uniqueness of Emden-Fowler boundary value problems (preprint). · Zbl 0744.34023
[10] McLeod, K., & Serrin, J., Uniqueness of positive radial solu · Zbl 0667.35023
[11] Ni, W. M., Uniqueness of solutions of nonlinear Dirichlet problems, J. Diff. Eq. 50 (1983), 289–304. · Zbl 0476.35033
[12] Ni, W. M., & Nussbaum, R., Uniqueness and nonuniqueness for positive radial solutions of {\(\Delta\)}u+f(u,r)=0, Comm. Pure and Appl. Math. 38 (1985), 69–108. · Zbl 0581.35021
[13] Peletier, L. A., & Serrin, J., Uniqueness of solutions of s · Zbl 0577.35035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.