Article PDF
Avoid common mistakes on your manuscript.
References
S. Bernstein,Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differential gleichungen vom Elliptisches Typus, Math. Z.26 (1927), 551–558.
L. Bers and L. Nirenberg,On linear and nonlinear boundary value problems in the plane, inAtti Convegno Internazionale Sulle Equazioni Derivate Partiali, Trieste, Edizioni Cremonese, Rome, 1955, pp. 141–167.
E. Bohn and L. K. Jackson,The Liouville theorem for a quasilinear elliptic partial differential equation, Trans. Amer. Math. Soc.104 (1962), 392–397.
R. Finn,Sur quelques généralisations du théorème de Picard, C. R. Acad. Sci. Paris235 (1952), 596–598.
R. Finn and D. Gilbarg,Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math.10 (1957), 23–63.
D. Gilbarg,Some local properties of elliptic equations, Proc. Symp. Pure Math., Vol. IV, Providence, RI, Amer. Math. Soc., 1961, 127–141.
D. Gilbarg and J. Serrin,On isolated singularities of solutions of second order elliptic differential equations, J. Analyse. Math.4 (1955–1956), 309–340.
S. Hildebrandt and K. O. Widman,Sätze vom Liouvillschen Typ für quasilineare elliptische Gleichungen und systeme, Nachr. Akad. Wiss. Göttingen, No. 4 (1979), 1–19.
E. Hopf,On S. Bernstein’s theorem on surfaces z(x, y) of nonpositive curvature, Proc. Amer. Math. Soc.1 (1950), 80–85.
E. Hopf,Bemerkungen zu einem Satze vom S. Bernstein aus der theorie der elliptischen Differentialgleichungen, Math. Z.29 (1928), 744–745.
A. V. Ivanov,Local estimates for the first derivatives of solutions of quasilinear second order elliptic equations and their application to Liouville type theorems, Sem. Math. V. A. Steklov Math. Inst. Leningrad30 (1972) 40–50 (translated as J. Soviet Math.4 (1975), 335–344).
L. Karp,Subharmonic functions on real and complex manifolds, Math. Z., to appear.
L. Karp,Asymptotic behavior of solutions of elliptic equations II, J. Analyse Math.39 (1981), 103–115.
M. Meier,Liouville theorems for nonlinear elliptic equations and systems, Manuscripta Math.29 (1979), 207–228.
E. J. Mickle,A remark on a theorem by S. Bernstein, Proc. Amer. Math. Soc.1 (1950), 86–89.
J. Moser,On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math.14 (1961), 577–591.
L. A. Peletier and J. Serrin,Gradient bounds and Liouville theorems for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, IV,5 (1978), 65–104.
M. H. Protter and H. Weinberger,Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1967.
R. M. Redheffer,On the inequality Δu ≧f (u, |gradu|), J. Math. Anal. Appl.1 (1960), 277–299.
J. Serrin,On the Harnack inequality for linear elliptic equations, J. Analyse Math.4 (1955/56), 297–308.
J. Serrin,A Harnack inequality for nonlinear equations, Bull. Amer. Math. Soc.69 (1963), 481–486.
J. Serrin,Local behavior of solutions of quasilinear elliptic equations, Acta Math.111 (1964), 247–302.
J. Serrin,Singularities of solutions of quasilinear elliptic equations, Proc. Symp. Appl. Math., Vol. 17, Amer. Math. Soc., 1965, pp. 68–88.
J. Serrin,Isolated singularities of solutions of quasilinear equations, Acta Math.113 (1965), 219–240.
J. Serrin,Entire solutions of nonlinear Poisson equations, Proc. London Math. Soc.24 (1972), 348–366.
I. N. Tavgelidze,Liouville theorems for second order elliptic and parabolic equations, Moscow Univ. Math. Bull.31 (1976), 70–76.
N. S. Trudinger,On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math.20 (1967), 721–747.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Karp, L. Asymptotic behavior of solutions of elliptic equations I: Liouville-type theorems for linear and nonlinear equations onR n . J. Anal. Math. 39, 75–102 (1981). https://doi.org/10.1007/BF02803331
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02803331