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October  2018, 23(8): 3297-3308. doi: 10.3934/dcdsb.2018252

## Global Kneser solutions to nonlinear equations with indefinite weight

 1 Department of Mathematics and Statistics, Masaryk University, CZ-61137 Brno, Czech Republic 2 Department of Mathematics and Computer Sciences "Ulisse Dini", University of Florence, I-50139 Florence, Italy

* Corresponding author: Serena Matucci

Received  March 2018 Published  October 2018 Early access  August 2018

Fund Project: The first author is supported by the grant GA 17-03224S of the Czech Grant Agency. The third author is partially supported by Gnampa, National Institute for Advanced Mathematics (INdAM).

The paper deals with the nonlinear differential equation
 $\bigl(a(t)\Phi(x^{\prime})\bigr)^{\prime}+b(t)F(x)=0,\ \ \ t\in\lbrack1,\infty),$
in the case when the weight
 $b$
has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions
 $x$
such that
 $x(t)>0, {{x}'}(t)<0$
on the whole closed interval
 $[1,\infty )$
, is considered. Moreover, conditions assuring that these solutions tend to zero as
 $t\rightarrow\infty$
are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight
 $b$
is a periodic function or it is unbounded from below.
Citation: Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252
##### References:
 [1] J. Andres, G. Gabor and L. Górniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc., 351 (1999), 4861-4903.  doi: 10.1090/S0002-9947-99-02297-7. [2] A. Boscaggin and F. Zanolin, Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl., 194 (2015), 451-478.  doi: 10.1007/s10231-013-0384-0. [3] M. Cecchi, Z. Došlá and M. Marini, Principal solutions and minimal sets of quasilinear differential equations, Dynam. Systems Appl., 13 (2004), 221-232. [4] M. Cecchi, Z. Došlá and M. Marini, Half-linear differential equations with oscillating coefficient, Differential Integral Equations, 18 (2005), 1243-1256. [5] M. Cecchi, Z. Došlá, I. Kiguradze and M. Marini, On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems, Differential Integral Equations, 20 (2007), 1081-1106. [6] M. Cecchi, M. Furi and M. Marini, On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals, Nonlinear Anal., 9 (1985), 171-180.  doi: 10.1016/0362-546X(85)90070-7. [7] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic equations, Research Notes in Mathematics, 106, Pitman (Advanced Publishing Program), Boston, MA, 1985. [8] Z. Došlá, M. Marini and S. Matucci, On some boundary value problems for second order nonlinear differential equations, Math. Bohem., 137 (2012), 113-122. [9] Z. Došlá, M. Marini and S. Matucci, A boundary value problem on a half-line for differential equations with indefinite weight, Commun. Appl. Anal., 15 (2011), 341-352. [10] Z. Došlá, M. Marini and S. Matucci, Positive solutions of nonlocal continuous second order BVP's, Dynam. Systems Appl., 23 (2014), 431-446. [11] Z. Došlá, M. Marini and S. Matucci, A Dirichlet problem on the half-line for nonlinear equations with indefinite weight, Ann. Mat. Pura Appl., 196 (2017), 51-64.  doi: 10.1007/s10231-016-0562-y. [12] O. Došlý and A. Elbert, Integral characterization of principal solution of half-linear differential equations, Studia Sci. Math. Hungar., 36 (2000), 455-469.  doi: 10.1556/SScMath.36.2000.3-4.16. [13] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier Sci. B. V., Amsterdam, 2005. [14] P. Drábek and A. Kufner, Discreteness and symplicity of the spectrum of a quasilinear Sturm-Liouville-type problem on an infinite interval, Proc. Amer. Math. Soc., 134 (2006), 235-242.  doi: 10.1090/S0002-9939-05-07958-X. [15] P. Drábek, A. Kufner and K. Kuliev, Half-linear Sturm Liouville problem with weights: Asymptotic behavior of eigenfunctions, Proc. Steklov Inst. Math., 284 (2014), 148-154.  doi: 10.1134/S008154381401009X. [16] A. Elbert and T. Kusano, Principal solutions of non-oscillatory half-linear differential equations, Adv. Math. Sci. Appl., 8 (1998), 745-759. [17] P. Hartman, Ordinary Differential Equations, Reprint of the second edition, Birkäuser, Boston, Mass., 1982. [18] J. Jaroš and T. Kusano, Decreasing regularly varying solutions of sublinearly perturbed superlinear Thomas-Fermi equation, Results Math., 66 (2014), 273-289.  doi: 10.1007/s00025-014-0376-4. [19] K. Kamo, Asymptotic equivalence for positive decaying solutions of the generalized Emden-Fowler equations and its application to elliptic problems, Arch. Math. (Brno), 40 (2004), 209-217. [20] I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Publ. G., Dordrecht, 1993. doi: 10.1007/978-94-011-1808-8. [21] T. Kusano, V. Marić and T. Tanigawa, Regularly varying solutions of generalized Thomas-Fermi equations, Bull. Cl. Sci. Math. Nat. Sci. Math., 34 (2009), 43-73. [22] M. Marini and S. Matucci, A boundary value problem on the half-line for superlinear differential equations with changing sign weight, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 117–132. [23] S. Matucci, A new approach for solving nonlinear BVP's on the half-line for second order equations and applications, Mathematica Bohemica, 140 (2015), 153-169. [24] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952. [25] D. D. Mirzov, Principal and nonprincipal solutions of a nonlinear system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 31 (1988), 100-117. [26] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179.

show all references

##### References:
 [1] J. Andres, G. Gabor and L. Górniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc., 351 (1999), 4861-4903.  doi: 10.1090/S0002-9947-99-02297-7. [2] A. Boscaggin and F. Zanolin, Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl., 194 (2015), 451-478.  doi: 10.1007/s10231-013-0384-0. [3] M. Cecchi, Z. Došlá and M. Marini, Principal solutions and minimal sets of quasilinear differential equations, Dynam. Systems Appl., 13 (2004), 221-232. [4] M. Cecchi, Z. Došlá and M. Marini, Half-linear differential equations with oscillating coefficient, Differential Integral Equations, 18 (2005), 1243-1256. [5] M. Cecchi, Z. Došlá, I. Kiguradze and M. Marini, On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems, Differential Integral Equations, 20 (2007), 1081-1106. [6] M. Cecchi, M. Furi and M. Marini, On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals, Nonlinear Anal., 9 (1985), 171-180.  doi: 10.1016/0362-546X(85)90070-7. [7] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic equations, Research Notes in Mathematics, 106, Pitman (Advanced Publishing Program), Boston, MA, 1985. [8] Z. Došlá, M. Marini and S. Matucci, On some boundary value problems for second order nonlinear differential equations, Math. Bohem., 137 (2012), 113-122. [9] Z. Došlá, M. Marini and S. Matucci, A boundary value problem on a half-line for differential equations with indefinite weight, Commun. Appl. Anal., 15 (2011), 341-352. [10] Z. Došlá, M. Marini and S. Matucci, Positive solutions of nonlocal continuous second order BVP's, Dynam. Systems Appl., 23 (2014), 431-446. [11] Z. Došlá, M. Marini and S. Matucci, A Dirichlet problem on the half-line for nonlinear equations with indefinite weight, Ann. Mat. Pura Appl., 196 (2017), 51-64.  doi: 10.1007/s10231-016-0562-y. [12] O. Došlý and A. Elbert, Integral characterization of principal solution of half-linear differential equations, Studia Sci. Math. Hungar., 36 (2000), 455-469.  doi: 10.1556/SScMath.36.2000.3-4.16. [13] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier Sci. B. V., Amsterdam, 2005. [14] P. Drábek and A. Kufner, Discreteness and symplicity of the spectrum of a quasilinear Sturm-Liouville-type problem on an infinite interval, Proc. Amer. Math. Soc., 134 (2006), 235-242.  doi: 10.1090/S0002-9939-05-07958-X. [15] P. Drábek, A. Kufner and K. Kuliev, Half-linear Sturm Liouville problem with weights: Asymptotic behavior of eigenfunctions, Proc. Steklov Inst. Math., 284 (2014), 148-154.  doi: 10.1134/S008154381401009X. [16] A. Elbert and T. Kusano, Principal solutions of non-oscillatory half-linear differential equations, Adv. Math. Sci. Appl., 8 (1998), 745-759. [17] P. Hartman, Ordinary Differential Equations, Reprint of the second edition, Birkäuser, Boston, Mass., 1982. [18] J. Jaroš and T. Kusano, Decreasing regularly varying solutions of sublinearly perturbed superlinear Thomas-Fermi equation, Results Math., 66 (2014), 273-289.  doi: 10.1007/s00025-014-0376-4. [19] K. Kamo, Asymptotic equivalence for positive decaying solutions of the generalized Emden-Fowler equations and its application to elliptic problems, Arch. Math. (Brno), 40 (2004), 209-217. [20] I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Publ. G., Dordrecht, 1993. doi: 10.1007/978-94-011-1808-8. [21] T. Kusano, V. Marić and T. Tanigawa, Regularly varying solutions of generalized Thomas-Fermi equations, Bull. Cl. Sci. Math. Nat. Sci. Math., 34 (2009), 43-73. [22] M. Marini and S. Matucci, A boundary value problem on the half-line for superlinear differential equations with changing sign weight, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 117–132. [23] S. Matucci, A new approach for solving nonlinear BVP's on the half-line for second order equations and applications, Mathematica Bohemica, 140 (2015), 153-169. [24] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952. [25] D. D. Mirzov, Principal and nonprincipal solutions of a nonlinear system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 31 (1988), 100-117. [26] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179.
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