Journal of the European Mathematical Society

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Volume 8, Issue 2, 2006, pp. 195–215
DOI: 10.4171/JEMS/47

Published online: 2006-06-30

On the principal eigenvalue of elliptic operators in $\R^N$ and applications

Henry Berestycki[1] and Luca Rossi[2]

(1) Ecole des hautes études en sciences sociales, Paris, France
(2) Università di Roma La Sapienza, Italy

Two generalizations of the notion of principal eigenvalue for elliptic operators in $\R^N$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and ``limit periodic'' operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.

Keywords: Elliptic operators, principal eigenvalue, generalized principal eigenvalue in $\R^N$, limit periodic operators

Berestycki Henry, Rossi Luca: On the principal eigenvalue of elliptic operators in $\R^N$ and applications. J. Eur. Math. Soc. 8 (2006), 195-215. doi: 10.4171/JEMS/47