Liouville type results for periodic and almost periodic linear operators

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1. [1] Agmon S., On positive solutions of elliptic equations with periodic coefficients in $R^n$, spectral results and extensions to elliptic operators on Riemannian manifolds, in: Differential Equations, Birmingham, Alabama, 1983, North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 7-17. Zbl0564.35033MR799327
2. [2] Amerio L., Prouse G., Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971. Zbl0215.15701MR275061
3. [3] Avellaneda M., Lin F.-H., Un théorème de Liouville pour des équations elliptiques à coefficients périodiques, C. R. Acad. Sci. Paris Sér. I Math.309 (5) (1989) 245-250. Zbl0691.35022MR1010728
4. [4] Berestycki H., Capuzzo-Dolcetta I., Nirenberg L., Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal.4 (1) (1994) 59-78. Zbl0816.35030MR1321809
5. [5] Bidaut-Véron M.-F., Initial blow-up for the solutions of a semilinear parabolic equation with source term, in: Équations aux dérivées partielles et applications, Gauthier–Villars/Éd. Sci. Méd. Elsevier, Paris, 1998, pp. 189-198. Zbl0914.35055MR1648222
6. [6] Bochner S., Beiträge zur Theorie der fastperiodischen Funktionen, Math. Ann.96 (1) (1927) 119-147. Zbl52.0261.01MR1512308JFM52.0261.01
7. [7] Bohr H., Zur Theorie der Fastperiodischen Funktionen II. Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen, Acta Math.46 (1–2) (1925) 101-214. Zbl51.0212.02MR1555201JFM51.0212.02
8. [8] Brezis H., Chipot M., Xie Y., Some remarks on Liouville type theorems, in: Recent Advances in Nonlinear Analysis, World Sci. Publ., Hackensack, NJ, 2008, pp. 43-65. Zbl05375300MR2410737
9. [9] Caffarelli L.A., Cabré X., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ., vol. 43, American Mathematical Society, Providence, RI, 1995. Zbl0834.35002MR1351007
10. [10] Capuzzo Dolcetta I., Cutrì A., Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Commun. Contemp. Math.5 (2003) 435-448. Zbl1040.35024MR1992357
11. [11] Cutrì A., Leoni F., On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire17 (2) (2000) 219-245. Zbl0956.35035MR1753094
12. [12] Dancer E.N., Du Y., Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc.131 (6) (2003) 1891-1899, (electronic). Zbl1076.35038MR1955278
13. [13] De Giorgi E., Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3)3 (1975) 25-43. Zbl0084.31901
14. [14] Fink A.M., Almost Periodic Differential Equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin, 1974. Zbl0325.34039MR460799
15. [15] Gidas B., Spruck J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations6 (8) (1981) 883-901. Zbl0462.35041MR619749
16. [16] Gilbarg D., Serrin J., On isolated singularities of solutions of second order elliptic differential equations, J. Anal. Math.4 (1955/1956) 309-340. Zbl0071.09701MR81416
17. [17] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224, second ed., Springer-Verlag, Berlin, 1983. Zbl0562.35001MR737190
18. [18] Hile G.N., Mawata C.P., Liouville theorems for nonlinear parabolic equations of second order, Differential Integral Equations9 (1) (1996) 149-172. Zbl0840.35043MR1364039
19. [19] Hu Z., Mingarelli A.B., On a question in the theory of almost periodic differential equations, Proc. Amer. Math. Soc.127 (9) (1999) 2665-2670. Zbl0924.34039MR1485481
20. [20] Hu Z., Mingarelli A.B., Almost periodicity of solutions for almost periodic evolution equations, Differential Integral Equations18 (4) (2005) 469-480. Zbl1212.34172MR2122710
21. [21] Kuchment P., Pinchover Y., Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal.181 (2) (2001) 402-446. Zbl0986.35028MR1821702
22. [22] Ladyženskaja O.A., Solonnikov V.A., Ural'ceva N.N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, American Mathematical Society, Providence, RI, 1967, translated from the Russian by S. Smith. Zbl0174.15403MR241822
23. [23] Li P., Wang J., Polynomial growth solutions of uniformly elliptic operators of non-divergence form, Proc. Amer. Math. Soc.129 (12) (2001) 3691-3699, (electronic). Zbl0989.35042MR1860504
24. [24] Li Y., Zhang L., Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math.90 (2003) 27-87. Zbl1173.35477MR2001065
25. [25] Lieberman G.M., Second Order Parabolic Differential Equations, World Scientific Publishing, River Edge, NJ, 1996. Zbl0884.35001MR1465184
26. [26] Mingarelli A.B., Pu F.Q., Zheng L., A counterexample in the theory of almost periodic differential equations, Rocky Mountain J. Math.25 (1) (1995) 437-440. Zbl0833.34041MR1340018
27. [27] Moser J., Struwe M., On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus, Bull. Braz. Math. Soc. (N.S.)23 (1–2) (1992) 1-20. Zbl0787.35028MR1203171
28. [28] Phóng V.Q., Stability and almost periodicity of trajectories of periodic processes, J. Differential Equations115 (2) (1995) 402-415. Zbl0815.34050MR1310938
29. [29] Pinsky R.G., Positive Harmonic Functions and Diffusion, Cambridge Stud. Adv. Math., vol. 45, Cambridge University Press, Cambridge, 1995. Zbl0858.31001MR1326606
30. [30] Pinsky R.G., Second order elliptic operators with periodic coefficients: Criticality theory, perturbations, and positive harmonic functions, J. Funct. Anal.129 (1) (1995) 80-107. Zbl0826.35030MR1322643
31. [31] Protter Murray H., Weinberger Hans F., Maximum Principles in Differential Equations, Prentice–Hall, Englewood Cliffs, NJ, 1967. Zbl0153.13602MR219861
32. [32] Rossi L., Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains, Commun. Pure Appl. Anal.7 (1) (2008) 125-141. Zbl1187.35076MR2358359
33. [33] Troianiello G.M., Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987. Zbl0655.35002MR1094820