## Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line.(English)Zbl 1053.34028

The authors consider the following singular boundary value problem $\begin{cases} \frac{\mathstrut 1}{p(t)}(p(t)x^\prime(t))^\prime+q(t)f(t,x(t),p(t)x^\prime(t))=0,\\ x(0)=a\geq 0,\\ \lim\limits_{t\to\infty}{p(t)x^\prime(t)}=b\geq 0, \end{cases}$ with $$f:[0,+ \infty)^2\times \mathbb{R}\to [0,+ \infty)$$, $$p\in C([0,+ \infty),\mathbb{R})\cap C^1(0,+ \infty)$$, $$p(t)>0$$ and $$q(t)\geq 0$$ for $$t\in (0,+\infty)$$. The next two conditions are assumed throughout $\int_{0}^{1}\frac{\mathstrut ds}{p(s)}<\infty \text{ and }\int_{0}^{t}p(s)q(s)ds<\infty \text{ for any }t\geq 0.$ The existence of a global, monotone, positive and unbounded solution is proved. The approach is based on a corresponding vector field investigation.
The article also provides two applications of the main result which are related to superconductivity theory and the theory of colloids correspondingly. The first application is the proof of the existence of strictly positive solutions of the following boundary value problems for Painlevé-type equations $x^{\prime\prime}=x^4-tx^2, \quad x^\prime(0)=0, \lim\limits_{t\to+\infty} \displaystyle\frac{\mathstrut x(t)}{\sqrt t}=1,$ and $x^{\prime\prime}=x^3-tx, \quad x(0)=a, \lim\limits_{t\to-\infty}x(t)=0=\lim\limits_{t\to-\infty} x^\prime(t).$ The second application is the proof of the existence of a strictly positive and decreasing solution of the problem $u^{\prime\prime}=2\sinh u(x), \quad u(0)=c, \lim\limits_{x\to+\infty}u(x)=0=\lim\limits_{x\to+\infty} u^\prime(t).$

### MSC:

 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B60 Applications of boundary value problems involving ordinary differential equations 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
Full Text:

### References:

  Agarwal, R.; O’Regan, D., Twin solutions to singular Dirichlet problems, J. math. anal. appl., 240, 433-445, (1999) · Zbl 0946.34022  Agarwal, R.; O’Regan, D., Boundary value problems on the half line in the theory of colloids, Math. prob. eng., 8, 2, 143-150, (2002) · Zbl 1049.34032  Baxley, J.V., Existence and uniqueness for nonlinear boundary value problems on infinite intervals, J. math. anal. appl., 147, 127-133, (1990) · Zbl 0719.34037  Charman, S.J., Superheating field of type II superconductors, SIAM J. appl. math., 55, 5, 1233-1258, (1995) · Zbl 0839.35129  Chen, S.; Zhang, Y., Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019  Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Heath & Co Boston · Zbl 0154.09301  Guedda, M., Note on the uniqueness of a global positive solution to the second Painlevé equation, Electron. J. differential equations, 173, 318-324, (1993)  Guo, D., Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications, J. math. anal. appl., 195, 449-468, (1995)  Hasting, S.P.; McLeod, J.B., A boundary value problem associated with the second Painlevé equation transcendent and the korteweg – de Vries equation, Arch. ration. mech. anal., 73, 1, 31-51, (1980) · Zbl 0426.34019  Helffer, B.; Weissler, F.B., On a family of solutions of the second Painlevé equation related to superconductivity, Eur. J. appl. math., 9, 3, 223-243, (1998) · Zbl 0920.34051  Kuratowski, K., Topology II, (1968), Academic Press New York  D. Levi, P. Winternitz, Painlevé Transcendents: Their Asymptotics and Physical Applications, NATO ASI Series, Series B: Physics, Vol. 278, 1990.  Liu, X., Solutions of impulsive boundary value problems on the half space, J. math. anal. appl., 222, 411-430, (1998) · Zbl 0912.34021  Meehan, M.; Regan, D.O’, Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals, Nonlinear anal., 35, 355-387, (1999) · Zbl 0920.45006  Ntouyas, S.K.; Palamides, P.K., On sturm – liouville and thomas – fermi singular boundary value problems, Nonlinear oscillations, 5, 326-344, (2001) · Zbl 1049.34023  Palamides, P., Singular points of the consequent mapping, Ann. math. pura appl., CXXIX, 383-395, (1981) · Zbl 0502.34035  Palamides, P.; Sficas, Y.; Staikos, V., Wazewski’s topological method for caratheodory systems, Math. systems theory, 7, 243-261, (1984) · Zbl 0631.34019  Palamides, P.K., Conjugate boundary value problems, via Sperner’s lemma, J. math. anal. appl., 46, 299-308, (2001) · Zbl 0995.34010  Tzanetis, D., Blow-up of radially symmetric solutions of a non-local problem modelling ohmic heating, Electron. J. differential equations, 11, 1-26, (2002) · Zbl 0993.35018  Yan, B., Multiple unbounded solutions of boundary value problems for second-order differential equations on the half space, Nonlinear anal. TMA, 51, 1031-1044, (2002) · Zbl 1021.34021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.