Abstract
We use cone methods combined with distribution theory and blow ups to find the asymptotic limit of the principal eigenvalue of a cooperative elliptic linear system when the diffusion is small.
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Bachman G.: Elements of abstract harmonic analysis. Academic Press, New York (1964)
H. Berestycki and P. L. Lions, Some applications of the methods of sub and supersolutions, pp. 16–41 in Lecture Notes in Mathematics 782, Springer-Verlag, Berlin, 1980.
Dancer E.N.: On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large. Proc. London Math. Soc. (3) 53, 429–452 (1986)
Dancer E.N.: Moving plane methods for systems on half spaces. Math. Ann. 342, 245–254 (2008)
E. N. Dancer and P. Hess, Behaviour of a semilinear periodic-parabolic problem when a parameter is small, pp. 12–19 in Functional analytic methods for partial differential equations, Lecture Notes in Mathematics 1450, Springer-Verlag, Berlin, 1990.
Folland G.: Introduction to partial differential equations. Princeton University Press, Princeton (1976)
I. M. Gelfand and N. Vilenkin, Generalized functions Vol. 4. Applications of harmonic analysis. Translated from the Russian by Amiel Feinstein. Academic Press, New York, 1964.
D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd edition. Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983.
Hess P.: On the eigenvalue problem for weakly coupled elliptic systems. Arch. Rational Mech. Anal. 81, 151–159 (1983)
Hess P., Kato T.: On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. Partial Differential Equations 5, 999–1030 (1980)
L. Hörmander, The analysis of linear differential operators I. Distribution theory and Fourier analysis, 2nd edition. Grundlehren der Mathematischen Wissenschaften 256, Springer-Verlag, Berlin, 1990.
Kavian O.: Introduction à la théorie des points critiques. Springer-Verlag, Paris (1993)
Krasnosel’skiĭ M.A.: Positive solutions of operator equations. Noordhoff, Groningen (1964)
Schaefer H.H.: Banach lattices and positive operators. Springer-Verlag, Berlin (1974)
Sweers G.: Strong positivity in \({C(\overline\Omega)}\) for elliptic systems. Math. Z. 209, 251–271 (1992)
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Dancer, E.N. On the principal eigenvalue of linear cooperating elliptic systems with small diffusion. J. Evol. Equ. 9, 419–428 (2009). https://doi.org/10.1007/s00028-009-0011-0
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DOI: https://doi.org/10.1007/s00028-009-0011-0