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Positive solutions of elliptic systems with bounded nonlinearities

Published online by Cambridge University Press:  14 November 2011

Ezzat S. Noussair  and
Charles A. Swanson
Ezzat S. Noussair
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW 2033, Australia
Charles A. Swanson
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Y4
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Synopsis

Semilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 3-4 , 1988 , pp. 321 - 332
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Allegretto, W.. On positive L ∞ solutions of a class of elliptic systems. Math. Z. 191 (1986), 479–484.Google Scholar
2Amann, H.. On the existence of positive solutions of nonlinear elliptic boundary value problems. Indiana Univ. Math. J. 21 (1971), 125–146.CrossRefGoogle Scholar
3Cantrell, R. S. and Cosner, C.. On the positone problem for elliptic systems. Indiana Univ. Math. J. 34 (1985), 517–532.Google Scholar
4Cohen, D. S. and Keller, H. B.. Some positone problems suggested by nonlinear heat conduction. J. Math. Mech. 16 (1967), 1361–1376.Google Scholar
5Friedman, A.. Bounded entire solutions of elliptic equations. Pacific J. Math. 44 (1973), 497–507.CrossRefGoogle Scholar
6Hess, P.. On the eigenvalue problem for weakly coupled elliptic systems. Arch. Rational Mech. Anal. 81 (1983), 151–159.Google Scholar
7Kawano, N.. On bounded entire solutions of semilinear elliptic equations. Hiroshima Math. J. 14 (1984), 125–158.Google Scholar
8Kawano, N. and Kusano, T.. On positive entire solutions of a class of second order semilinear elliptic systems. Math. Z. 186 (1984), 287–297.CrossRefGoogle Scholar
9Noussair, E. S. and Swanson, C. A.. Positive solutions of quasilinear elliptic equations in exterior domains. J. Math. Anal. Appl. 75 (1980), 121–133.Google Scholar
10Noussair, E. S. and Swanson, C. A.. Global positive solutions of semilinear elliptic problems. Pacific J. Math. 115 (1984), 177–192.Google Scholar
11Noussair, E. S. and Swanson, C. A.. Decaying entire solutions of quasilinear elliptic equations. Funkcial. Ekvac. (to appear).Google Scholar
12Nussbaum, R.. Positive solutions of nonlinear elliptic boundary problems. J. Math. Anal. Appl. 51 (1975), 461–482.CrossRefGoogle Scholar
13Rabinowitz, P. H.. Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23 (1973), 173–186.Google Scholar
14Sattinger, D. H.. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979–1000.Google Scholar
15Schmitt, K.. Boundary value problems for quasilinear second order elliptic equations. Nonlinear Anal. 2 (1978). 263–309.Google Scholar