Duke Mathematical Journal

Structure of positive solutions to $(-\Delta+V)u=0$ in $R^n$

Minoru Murata

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 53, Number 4 (1986), 869-943.

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
Secondary: 31B35: Connections with differential equations


Murata, Minoru. Structure of positive solutions to ( − Δ + V ) u = 0 in R n . Duke Math. J. 53 (1986), no. 4, 869--943. doi:10.1215/S0012-7094-86-05347-0. http://projecteuclid.org/euclid.dmj/1077305358.

Export citation


  • [1] S. Agmon, On positive solutions of elliptic equations with periodic coefficients in ${\bf R}\sp n$, spectral results and extensions to elliptic operators on Riemannian manifolds, Differential equations (Birmingham, Ala., 1983) eds. I. W. Knowles and R. T. Lewis, North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 7–17.
  • [2] W. Allegretto, On the equivalence of two types of oscillation for elliptic operators, Pacific J. Math. 55 (1974), 319–328.
  • [3] W. Allegretto, Positive solutions and spectral properties of second order elliptic operators, Pacific J. Math. 92 (1981), no. 1, 15–25.
  • [4] M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429–461.
  • [5] L. A. Caffarelli and W. Littman, Representation formulas for solutions to $\Delta u-u=0$ in ${\bf R}\sp{n}$, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, D.C., 1982, pp. 249–263.
  • [6] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
  • [7] M. Cranston, S. Orey, and U. Rösler, The Martin boundary of two-dimensional Ornstein-Uhlenbeck processes, Probability, statistics and analysis, London Math. Soc. Lecture Note Ser., vol. 79, Cambridge Univ. Press, Cambridge, 1983, pp. 63–78.
  • [8] P. Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964.
  • [9] F. I. Karpelevic, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Trudy Moskow Math. Obsc. 14 (1965), 48–185, Trans. Moscow Math. Soc. 1965, 51–199.
  • [10] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.
  • [11] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172.
  • [12] W. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), no. 1, 219–226.
  • [13] M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal. 56 (1984), no. 3, 300–310.
  • [14] M. Murata, Isomorphism theorems for elliptic operators in ${\bf R}\sp{n}$, Comm. Partial Differential Equations 9 (1984), no. 11, 1085–1105.
  • [15] M. Murata, Large time asymptotics for fundamental solutions of diffusion equations, Tohoku Math. J. (2) 37 (1985), no. 2, 151–195.
  • [16] M. Murata, Positive solutions of Schrödinger equations, IMA Preprint Ser. No. 124, 1985. Inst. Math. Applicat., Univ. Minnesota, Minneapolis.
  • [17] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.
  • [18] B. Simon, Large time behavior of the $L\sp{p}$ norm of Schrödinger semigroups, J. Funct. Anal. 40 (1981), no. 1, 66–83.
  • [19] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.
  • [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258.
  • [21] M. G. Šur, The Martin boundary for a linear, elliptic, second-order operator, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 45–60, Amer. Math. Translations Ser. 2, 56, 19–35.