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

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.