Two generalizations of the notion of principal eigenvalue for elliptic operators in 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.
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Henry Berestycki, Luca Rossi, On the principal eigenvalue of elliptic operators in and applications. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 195–215DOI 10.4171/JEMS/47