Abstract
This article is concerned with boundary value problems of the type (BVP) u″ + g(r)u′ + f(u) = 0, r > 0; u′(0) = 0, limr → ∞ u(r) = 0; where f(0) = 0. Such problems arise in the study of semi-linear elliptic differential equations in ℝn. It is shown that (BVP) has at most one non-negative non-trivial solution under appropriate conditions on f and g. The conditions are weaker than those given by Peletier and Serrin [6], who considered the special case g(r) = (n - 1)/r, n = 2, 3,⋯.
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References
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), pp. 209–243.
H. Berestycki and P.-L. Lions, Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon. C. R. Acad. Sci. Paris, Série A 287 (1978), pp. 503–506.
H. Berestycki, P. L. Lions and L. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in IRN. Indiana Univ. Math. J. 30 (1981), pp. 141–157.
K. McLeod and J. Serrin, Uniqueness of the ground state solution for Δu + f(u) = 0. Proc. Nat Acad. Sci. USA 78 (1981), pp. 6592–6595.
L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in IRn. Arch. Rat. Mech. Anal. 81 (1983), pp. 181–197.
L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in IRn. J. Diff. Eq. 61 (1986), pp. 380–397.
H. G. Kaper and M.-K. Kwong, Uniqueness for a class of non-linear initial value problems. J. Math. Anal. Applic. (to appear).
B. Franchi, E. Lanconelli, and J. Serrin, Existence and uniqueness of ground state solutions of quasilinear elliptic equations. Preprint (1986).
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© 1988 Springer-Verlag New York Inc.
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Kaper, H.G., Kwong, M.K. (1988). Uniqueness of non-negative solutions of a class of semi-linear elliptic equations. In: Ni, WM., Peletier, L.A., Serrin, J. (eds) Nonlinear Diffusion Equations and Their Equilibrium States II. Mathematical Sciences Research Institute Publications, vol 13. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9608-6_1
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DOI: https://doi.org/10.1007/978-1-4613-9608-6_1
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