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Non-well-ordered lower and upper solutions for semilinear systems of PDEs

    We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

    AMSC: 35J57, 35J60, 35K50, 35K55


    • 1. H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972) 346–384. CrossrefGoogle Scholar
    • 2. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620–709. Crossref, ISIGoogle Scholar
    • 3. H. Amann, A. Ambrosetti and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978) 179–194. Crossref, ISIGoogle Scholar
    • 4. J. W. Bebernes and K. Schmitt, Periodic boundary value problems for systems of second order differential equations, J. Differential Equations 13 (1973) 32–47. Crossref, ISIGoogle Scholar
    • 5. J. W. Bebernes and K. Schmitt, Invariant sets and the Hukuhara–Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math. 7 (1977) 557–567. CrossrefGoogle Scholar
    • 6. C. De Coster and P. Habets, Two-Point Boundary Value Problems, Lower and Upper Solutions (Elsevier, Amsterdam, 2006). Google Scholar
    • 7. C. De Coster and M. Henrard, Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. Differential Equations 145 (1998) 420–452. Crossref, ISIGoogle Scholar
    • 8. C. De Coster, F. Obersnel and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, in Handbook of Differential Equations, ODE’s, eds. A. CanadaP. DrabekA. Fonda (Elsevier, Amsterdam, 2006), pp. 203–339. CrossrefGoogle Scholar
    • 9. C. De Coster and P. Omari, Stability and instability in periodic parabolic problems via lower and upper solutions, Quad. Mat. 539 (2003) 1–103. Google Scholar
    • 10. A. Fonda, G. Klun and A. Sfecci, Periodic solutions of second order differential equations in Hilbert spaces, Mediterr. J. Math., to appear. Google Scholar
    • 11. A. Fonda and R. Toader, Lower and upper solutions to semilinear boundary value problems: An abstract approach, Topol. Methods Nonlinear Anal. 38 (2011) 59–94. ISIGoogle Scholar
    • 12. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. (Springer, Berlin, 1983). CrossrefGoogle Scholar
    • 13. J.-P. Gossez and P. Omari, A necessary and sufficient condition of nonresonance for a semilinear Neumann problem, Proc. Amer. Math. Soc. 114 (1992) 433–442. Crossref, ISIGoogle Scholar
    • 14. P. Habets and P. Omari, Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order, Top. Methods Nonlinear Anal. 8 (1996) 25–56. CrossrefGoogle Scholar
    • 15. J. Hernández, Qualitative methods for nonlinear diffusion equations, in Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Mathematics, Vol. 1224 (Springer, Berlin, 1986), pp. 47–118. CrossrefGoogle Scholar
    • 16. P. Omari, Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations, Rend. Istit. Mat. Univ. Trieste 20 (1988) 54–64. Google Scholar
    • 17. C. V. Pao, Nonlinear Parabolic and Elliptic Equations (Plenum Press, New York, 1992). Google Scholar
    • 18. G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems (Plenum Press, New York, 1987). CrossrefGoogle Scholar
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