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On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. (English) Zbl 1320.34029

This is an interesting paper which deals with the investigation of a class of two-point boundary value problems concerning the Ermakov-Painlevé II type equation \[ \mu_{xx}=a\mu^3+bx\mu+\frac{c}{\mu^3},\quad 0<x<1, \] associated with the Dirichlet boundary conditions \[ \mu(0)=\mu_0,\quad \mu(1)=\mu_1, \] where \(\mu_0, \mu_1\) are real non-negative reals. The authors present conditions for the existence of at least one solution, when \(a, c>0\), \(\mu_0\), \(\mu_1>0\), (Theorem 1.1), and \(a\leq 0\), \(c<0\) \(\mu_0\), \(\mu_1>0\), (Theorem 1.4), of infinity many positive solutions, when \(a<0<c\), \(\mu_0\), \(\mu_1>0\), (Theorem 1.2), of at least one positive solution, when \(a>0>c\), \(\mu_0\), \(\mu_1>0\), (Theorem 1.3) and of at least one solution \(\mu\) such that \(\mu(x)>0\), \(0<x<1\), when \(a>0>c\), \(\mu_0=\mu_1=0\), (Theorem 1.5). In an appendix, the authors discuss the case of a particular three-component Ermakov-Ray-Reid system, which can be examined by using Ermakov invariants.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B60 Applications of boundary value problems involving ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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