Journal of the European Mathematical Society

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

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

Henry Berestycki[1] and Luca Rossi[2]

(1) CAMS - EHESS, Ecole des hautes études en sciences sociales, 190-198, avenue de France, 75244, Paris CEDEX 13, France
(2) Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale A. Moro 2, 00185, Roma, 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