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On the solvability of singular boundary value problems on the real line in the critical growth case
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy |
$ \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*} $ |
$ \nu_{1}, \nu_{2}\in \mathbb{R} $ |
$ \Phi: \mathbb{R} \rightarrow \mathbb{R} $ |
$ p $ |
$ a $ |
$ \mathbb{R} \times \mathbb{R} $ |
$ f $ |
$ \mathbb{R} \times \mathbb{R}^{2} $ |
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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