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Liouville theorems on some indefinite equations

Published online by Cambridge University Press:  14 November 2011

Meijun Zhu
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver BC, Canada V6T 1Z2

Abstract

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Berestycki, H., Dolcetta, I. Capuzzo and Nirenberg, L.. Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Meth. Nonlinear Analysis 4 (1994), 5978.CrossRefGoogle Scholar
2Bianchi, G.. Non-existence of positive solutions to semilinear elliptic equations on ℝn or ℝn+ through the method of moving planes. Commun. PDE 22 (1997), 16711690.Google Scholar
3Caffarelli, L., Gidas, B. and Spruck, J.. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42 (1989), 271297.CrossRefGoogle Scholar
4Chen, W. and Li, C.. Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), 615623.CrossRefGoogle Scholar
5Chen, W. and Li, C.. A priori estimates for prescribing scalar curvature equation. Ann. Math. 145 (1997), 547564.CrossRefGoogle Scholar
6Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry of positive solutions of nonlinear elliptic equations in Rn. In Mathematical analysis and applications (ed. Nachbin, L.), part A, pp. 369402. Adv. Math. Suppl. Stud. 7 (New York: Academic Press, 1981).Google Scholar
7Gidas, B. and Spruck, J.. A priori bounds for positive solutions of nonlinear elliptic equations. Commun. PDE 8 (1981), 883901.CrossRefGoogle Scholar
8Hu, B.. Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition. Diff. Int. Eqns 7 (1994), 301313.Google Scholar
9Li, Y. Y. and Zhu, M.. Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), 383417.CrossRefGoogle Scholar
10Lou, Y. and Zhu, M.. Classifications of non-negative solutions to some elliptic problems. Diff. Int. Eqns. (In the press.)Google Scholar
11Ou, B.. Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition. Diff. Int. Eqns 9 (1996), 11571164.Google Scholar
12Terracini, S.. Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions. Diff. Int. Eqns 8 (1995), 19111922.Google Scholar