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Existence of positive entire solutions for weakly coupled semilinear elliptic systems

Published online by Cambridge University Press:  14 November 2011

Yasuhiro Furusho
Yasuhiro Furusho
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840, Japan
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Synopsis

Weakly coupled semilinear elliptic systems of the form

are considered in RN, N≧2, where k = 1, 2, …, M, u = (u1, …, uM) and λ is a real constant. The aim of this paper is to give sufficient conditions for (*) to have entire solutions whose components are positive in RN and converge to non-negative constants as |x| tends to ∞. For this purpose a new supersolution-subsolution method is developed for the system (*) without any hypotheses on the monotonicity of the non-linear terms fk with respect to u.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 79 - 91
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Allegretto, W.. On positive L∞ solutions of a class of elliptic systems. Math. Z. 191 (1986), 479–484.CrossRefGoogle Scholar
2Cantrell, R. S. and Cosner, C.. On the positone problem for elliptic systems. Indiana Univ. Math. J. 34 (1985), 517–532.Google Scholar
3Fukagai, N.. On decaying entire solution of second order sublinear elliptic equations. Hiroshima Math. J. 14 (1984), 551–562.Google Scholar
4Furusho, Y.. On decaying entire positive solutions of semilinear elliptic equations. Japan. J. Math. 14 (1988), 97–118.Google Scholar
5Furusho, Y.. Existence of global positive solutions of quasilinear elliptic equations in unbounded domains. Funkcial. Ekvac. 32 (1989), 227–242.Google Scholar
6Furusho, Y. and Kusano, T.. On the existence of positive decaying entire solutions for a class of sublinear elliptic equations. Canad. J. Math. 40 (1988), 1156–1173.Google Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn, (Berlin: Springer, 1983).Google Scholar
8Kawano, N.. On bounded entire solutions of semilinear elliptic equations. Hiroshima Math. J. 14 (1984), 125–158.Google Scholar
9Kawano, N. and Kusano, T.. On positive solutions of a class of second order semilinear elliptic systems. Math. Z. 186 (1984), 287–297.Google Scholar
10Kawano, N. and Kusano, T.. Positive solutions of a class of second order semilinear elliptic systems in exterior domains. Math. Nach. 121 (1985), 11–23.CrossRefGoogle Scholar
11Kusano, T. and Swanson, C. A.. A general method for quasilinear elliptic problems in RN. J. Math. Anal. Appl. (to appear).Google Scholar
12Ladde, G. S., Lakshmikantham, V. and Vatsala, A. S.. Monotone iterative techniques for nonlinear differential equations (Boston: Pitman, 1985).Google Scholar
13Ladyzenskaya, O. A. and Ural'tseva, N. N.. Linear and Quasilinear Elliptic Equations, English translation, (New York: Academic Press, 1968).Google Scholar
14Leung, A. W.. Systems of nonlinear partial differential equations, Applications to Biology and Engineering (Dordrecht: Kluwer Academic Publishers, 1989).CrossRefGoogle Scholar
15Ni, W. M.. On the elliptic equation, its generalizations, and applications in geometry. Indiana Univ. Math. J. 31 (1982), 493–529.CrossRefGoogle Scholar
16Noussair, E. S. and Swanson, C. A.. Decaying entire solutions of quasilinear elliptic equations. Funkcial. Ekvac. 31 (1988), 839–861.Google Scholar
17Noussair, E. S. and Swanson, C. A.. Positive solutions of elliptic systems with bounded nonlinearities. Proc. Roy. Soc. Edinburgh. Sect. A 108 (1988), 321–332.Google Scholar
18Sattinger, D.. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979–1000.Google Scholar
19Schmitt, K.. Boundary value problems for quasilinear second order elliptic equations. Nonlinear Anal. 2 (1978), 263–309.Google Scholar
20Tsai, L. Y.. Nonlinear boundary value problems for systems of second-order elliptic differential equations. Bull. Inst. Math. Acad. Sinica 5 (1977), 157–165.Google Scholar
21Tsai, L. Y.. Existence of solutions of nonlinear elliptic systems. Bull. Inst. Math. Acad. Sinica 8 (1980), 111–127.Google Scholar