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

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February 2020, 40(2): 1131-1157. doi: 10.3934/dcds.2020073

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

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.

Citation: Stefano Biagi, Teresa Isernia. On the solvability of singular boundary value problems on the real line in the critical growth case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1131-1157. doi: 10.3934/dcds.2020073

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.

show all references

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.

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