Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line
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 \[ u^{\prime\prime}=2\sinh u(x), \quad u(0)=c, \lim\limits_{x\to+\infty}u(x)=0=\lim\limits_{x\to+\infty} u^\prime(t). \]
scientific article; zbMATH DE number 2102513