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Liouville theorems for elliptic systems and nonlinear equations of fourth order

Published online by Cambridge University Press:  14 November 2011

Vinod B. Goyal  and
Philip W. Schaefer
Vinod B. Goyal
Affiliation:
Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India
Philip W. Schaefer
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916, U.S.A.
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Synopsis

Liouville theorems are obtained for fourth order elliptic systems of the form Δ2U + B Δ U + AU = 0 and for fourth order nonlinear equations of the form Δ2u−q(x) Δu + p(x)f(u) = 0 as a consequence of two related subharmonic functions, the mean value property of subharmonic functions, and a basic Green identity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 91 , Issue 3-4 , 1982 , pp. 235 - 242
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Dunninger, D. R.. On Liouville-type theorems for elliptic and parabolic systems. J. Math. Anal. Appl. 72 (1979), 413–417.CrossRefGoogle Scholar
2Goyal, V. B. and Schaefer, P. W.. Liouville theorems for a class of fourth order elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 129–137.CrossRefGoogle Scholar
3Goyal, V. B. and Schaefer, P. W.. On entire solutions in some nonlinear fourth order elliptic equations. Proc. Conf. Nonlinear Phenomena Math. Sci., University of Texas, Arlington, 1980.Google Scholar
4Huilgol, R. R.. On Liouville's theorem for biharmonic functions. SIAM J. Appl. Math. 20 (1971), 37–39.CrossRefGoogle Scholar
5Nehari, Z.. A differential inequality. J. Analyse Math. 14 (1965), 297–302.CrossRefGoogle Scholar
6Redheffer, R.. On the inequality Δu≧f(u, |grad u|). J. Math. Anal. Appl. 1 (1960), 277–299.CrossRefGoogle Scholar
7Serrin, J.. Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc. 24 (1972), 348–366.CrossRefGoogle Scholar
8Serrin, J.. Liouville theorems and gradient bounds for quasilinear elliptic systems. Arch. Rational Mech. Anal. 66 (1977), 295–310.CrossRefGoogle Scholar