A note on a non-linear Krein–Rutman theorem

By Rajesh Mahadevan. 2006

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Abstract In this note we will present an extension of the Krein–Rutman theorem [M.G. Kreĭn, M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. (26) (1950). [9] ] for an abstract non-linear, compact, positively 1-homogeneous operator on a Banach space having the properties of being increasing with respect to a convex cone K and such that there is a non-zero u ∈ K for which M T u ≽ u for some positive constant M . This will provide a uniform framework for recovering the Krein–Rutman-like theorems proved for many non-linear differential operators of elliptic type, like the p -Laplacian, cf. Anane [A. Anane, Simplicite et isolation de la premiere valeur propre du p -laplacien avec poids (Simplicity and isolation of the first eigenvalue of the p -Laplacian with weight), C. R. Acad. Sci. Paris 305 (16) (1987) 725–728 (in French)], the Hardy–Sobolev operator, cf. Sreenadh [K. Sreenadh, On the eigenvalue problem for the Hardy–Sobolev operator with indefinite weights, Electron. J. Differential Equations (33) (2002) 1–12], Pucci’s operator, cf. Felmer and Quaas [P. Felmer, A. Quaas, Positive radial solutions to a ‘semilinear’ equation involving the Pucci’s operator, J. Differential Equations 199 (2) (2004) 376–393]. Our proof follows the same lines as in the linear case, cf. Rabinowitz [P. Rabinowitz, Theorie du Degre Topologique et Applications a des Problemes aux Limites Non Lineaires, Lecture Notes Lab. Analyse Numerique, Universite Paris VI, 1975], and is based on a bifurcation theorem.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

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