References
S. Agmon,On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, inMethods of Functional Analysis and Theory of Elliptic Equations, ed. D. Greco, Liguori, Naples, 1982, pp. 19–52.
W. Allegreto and A. B. Mingarelli,On the non-existence of positive solutions for a Schrödinger equation with an indefinite weight-function, C. R. Math. Rep. Acad. Sci. Canada8 (1986), 69–73.
M. BrelotOn Topologies and Boundaries in Potential Theory, Lecture Notes in Mathematics175, Springer-Verlag, Berlin, 1971.
R. S. Cantrell and C. Cosner,Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh112A (1989), 293–318.
R. S. Cantrell and C. Cosner,Diffusive logistic equations with indefinite weights: population models in disrupted environments II, Siam J. Math. Anal.22 (1991), 1043–1064.
M. D. Donsker and S. R. S. Varadhan,On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. U.S.A.72 (1975), 780–783.
M. D. Donsker and S. R. S. Varadhan,On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math.29 (1976), 595–621.
C. Holland,A minimum principle for the principal eigenvalue for second order linear elliptic equations with natural boundary conditions. Comm. Pure Appl. Math.31 (1978), 509–519.
Y. Kifer,Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Isr. J. Math.70 (1990), 1–47.
R. D. Nussbaum,Positive operators and elliptic eigenvalue problems, Math. Z.186 (1984), 247–264.
R. D. Nussbaum,Convexity and log convexity for the spectral radius, Linear Algebra Appl.73 (1986), 59–122.
R. D. Nussbaum and Y. Pinchover,On variational formulas for the generalized principal eigenvalue of second order elliptic equations with general boundary conditions, in preparation.
Y. Pinchover,On positive solutions of second-order elliptic equations, stability results and classification, Duke Math. J.57 (1988), 955–980.
Y. Pinchover,On criticality and ground states of second-order elliptic equations II, J. Differential Equations87 (1990), 353–364.
R. G. Pinsky,A generalized Dirichlet principle for second order nonselfadjoint elliptic operators, SIAM J. Math. Anal.19 (1988), 204–213.
M. H. Protter and H. F. Weinberger,On the spectrum of general second order operators, Bull. Am. Math. Soc.72 (1966), 251–255.
M. Reed and B. Simon,Methods of Modern Mathematical Physics. 1. Functional Analysis, revised edition, Academic Press, New York, 1980.
M. Schechter,Hamiltonians for singular potentials, Indiana Univ. Math. J.22 (1972), 483–503.
M. Schechter,Exact estimates for potentials, SIAM J. Math. Anal.19 (1988), 1324–1328.
M. Schechter,The spectrum of the Schrödinger operator, Trans. Am. Math., Soc.312 (1989) 115–128.
M. Schechter,Multiplication operators, Can. J. Math.61 (1989), 234–249.
M. Sion,On general minimax theorems, Pacific. J. Math.8 (1958), 171–176.
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Dedicated to Professor Shmuel Agmon
Partially supported by NSF DMS 89-03018.
Partially supported by Technion VPR-Fund-K. and M. Bank Mathematics R. Fund.
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Nussbaum, R.D., Pinchover, Y. On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. J. Anal. Math. 59, 161–177 (1992). https://doi.org/10.1007/BF02790223
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DOI: https://doi.org/10.1007/BF02790223