On the solvability of singular boundary value problems on the real line in the critical growth case

Stefano Biagi; Teresa Isernia

doi:10.3934/dcds.2020073

Article Contents

2020, Volume 40, Issue 2: 1131-1157. Doi: 10.3934/dcds.2020073

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On the solvability of singular boundary value problems on the real line in the critical growth case

Stefano Biagi and
Teresa Isernia

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy

Received: May 2019

Revised: August 2019

Published: November 2019

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Abstract

Combining fixed point techniques with the method of lower-upper solutions we prove the existence of at least one weak solution for the following boundary value problem

$ \begin{equation*} \left\{ \begin{array}{ll} \left( \, \Phi(a(t, x(t)) \, x'(t) )\, \right)' = f(t, x(t), x'(t)) &\mbox{ in } \mathbb{R}\\ x(-\infty) = \nu_{1}, \quad x(+\infty) = \nu_{2} \end{array} \right. \end{equation*} $

where $ \nu_{1}, \nu_{2}\in \mathbb{R} $, $ \Phi: \mathbb{R} \rightarrow \mathbb{R} $ is a strictly increasing homeomorphism extending the classical $ p $-Laplacian, $ a $ is a nonnegative continuous function on $ \mathbb{R} \times \mathbb{R} $ which can vanish on a set having zero Lebesgue measure and $ f $ is a Carathéodory function on $ \mathbb{R} \times \mathbb{R}^{2} $.

Keywords:

Heteroclinic solutions on $ \mathbb{R} $,
boundary value problems,
singular ODEs,
critical rate of decay,
$ \Phi $-Laplace operator.

Mathematics Subject Classification: Primary: 34B40, 34C37; Secondary: 34B16, 34L30.

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References

[1]	C. Bereanu and J. Mawhin, Boundary-value problems with non-surjective $\Phi$-Laplacian and one-side bounded nonlinearity, Adv. Differential Equations, 11 (2006), 35-60.
[2]	C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\Phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557. doi: 10.1016/j.jde.2007.05.014.
[3]	C. Bereanu and J. Mawhin, Periodic solutions of nonlinear perturbations of $\Phi$-Laplacians with possibly bounded $\Phi$, Nonlinear Anal., 68 (2008), 1668-1681. doi: 10.1016/j.na.2006.12.049.
[4]	C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\Phi$-Laplacian, J. Fixed Point Theory Appl., 4 (2008), 57-75. doi: 10.1007/s11784-008-0072-7.
[5]	S. Biagi, On the existence of weak solutions for singular strongly nonlinear boundary value problems on the half-line, Annali di Matematica, (2019), 1-30. doi: 10.1007/s10231-019-00893-2.
[6]	S. Biagi, A. Calamai and F. Papalini, Heteroclinic solutions for a class of boundary value problems associated with singular equations, Nonlinear Anal., 184 (2019), 44-68. doi: 10.1016/j.na.2019.01.030.
[7]	S. Biagi, A. Calamai and F. Papalini, Existence results for boundary value problems associated with singular strongly nonlinear equations, arXiv: 1910.10802 (preprint, 2018) doi: 10.1016/j.na.2019.01.030.
[8]	B. Bianconi and F. Papalini, Non-autonomous boundary value problems on the real line, Discrete Contin. Dyn. Syst., 15 (2006), 759-776. doi: 10.3934/dcds.2006.15.759.
[9]	L. Bobisud, Steady-state turbolent flow with reaction, Rocky Mountain J. Math., 21 (1991), 993-1007. doi: 10.1216/rmjm/1181072925.
[10]	A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl., 2011 (2011), Art. ID 893753, 18 pp. doi: 10.1155/2011/893753.
[11]	A. Cabada and R. L. Pouso, Existence results for the problem $(\phi(u'))' = f(t, u, u')$ with periodic and Neumann boundary conditions, Nonl. Anal. TMA, 30 (1997), 1733-1742. doi: 10.1016/S0362-546X(97)00249-6.
[12]	A. Cabada and R. L. Pouso, Existence results for the problem $(\phi(u'))' = f(t, u, u')$ with nonlinear boundary conditions, Nonl. Anal. TMA, 35 (1999), 221-231. doi: 10.1016/S0362-546X(98)00009-1.
[13]	A. Cabada, D. O'Regan and R. L. Pouso, Second order problems with functional conditions including Sturm-Liouville and multipoint conditions, Math. Nachr., 281 (2008), 1254-1263. doi: 10.1002/mana.200510675.
[14]	A. Calamai, Heteroclinic solutions of boundary value problems on the real line involving singular $\Phi$-Laplacian operators, J. Math. Anal. Appl., 378 (2011), 667-679. doi: 10.1016/j.jmaa.2011.01.056.
[15]	A. Calamai, C. Marcelli and F. Papalini, Boundary value problems for singular second order equations, Fixed Point Theory Appl., 2018 (2018), Paper No. 20, 22pp. doi: 10.1186/s13663-018-0645-0.
[16]	G. Cupini, C. Marcelli and F. Papalini, Heteroclinic solutions of boundary–value problems on the real line involving general nonlinear differential operators, Differential Integral Equations, 24 (2011), 619-644.
[17]	G. Cupini, C. Marcelli and F. Papalini, On the solvability of a boundary value problem on the real line, Bound. Value Probl., 2011 (2011), 17 pp. doi: 10.1186/1687-2770-2011-26.
[18]	N. El Khattabi, M. Frigon and N. Ayyadi, Multiple solutions of boundary value problems with $\phi$-Laplacian operators and under a Wintner-Nagumo growth condition, Bound. Value Probl., 2013 (2013), 1-21. doi: 10.1186/1687-2770-2013-236.
[19]	J. Esteban and J. Vazquez, On the equation of turbolent fitration in one-dimensional porus media, Nonlinear Anal., 10 (1986), 1303-1325. doi: 10.1016/0362-546X(86)90068-4.
[20]	L. Ferracuti and F. Papalini, Boundary-value problems for strongly non-linear multivalued equations involving different $\Phi$-Laplacians, Adv. Differential Equations, 14 (2009), 541-566.
[21]	C. Marcelli, Existence of solutions to boundary-value problems governed by general non-autonomous nonlinear differential operators, Electron. J. Differential Equations, 2012 (2012), No. 171, 18 pp.
[22]	C. Marcelli, The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators, Bound. Value Probl., 2013 (2013), 14 pp. doi: 10.1186/1687-2770-2013-252.