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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
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