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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

 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
 $\mathbb{R} \times \mathbb{R}^{2}$
.
Citation: Stefano Biagi, Teresa Isernia. On the solvability of singular boundary value problems on the real line in the critical growth case. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. Bobisud, Steady-state turbolent flow with reaction, Rocky Mountain J. Math., 21 (1991), 993-1007.  doi: 10.1216/rmjm/1181072925.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] L. Bobisud, Steady-state turbolent flow with reaction, Rocky Mountain J. Math., 21 (1991), 993-1007.  doi: 10.1216/rmjm/1181072925.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar
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