Two generalizations of the notion of principal eigenvalue for elliptic operators in ^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.
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Berestycki et al. (2006) studied this question.