Boundary value problem for \(r^2d^2f/dr^2+f= f^3\). I: Existence and uniqueness (Q1599767)

The author studies a boundary value problem of the type: \[ r^{2}f'' + f = f^{3}, \qquad 0<r<\infty,\tag{1} \] \[ f(r) \to 0, \quad \text{ as}\quad r\to 0, \tag{2} \] \[ f(\infty) = 1. \tag{3} \] It is known that the solution to this boundary value problem has the asymptotics \[ f(r) \sim \alpha r^{\frac{1}{2}} \sin (\frac{\sqrt{3}}{2} \log r + \beta) \tag{4} \] as \(r\to 0\), and \(f(r) \sim 1 + \frac{\gamma}{r}\) if \(r\to \infty\) for some parameters \(\alpha, \beta\) and \(\gamma \). Here, formulas for the parameters \(\alpha, \beta\) and \(\gamma \), which are called connection formulas for this problem, are derived. At first, the author considers the existence and uniqueness of the solution for another boundary value problem \(f(1)=0\), \(f(\infty)=1\), and \(f(r)>0\) for \(r>1\). Using shooting arguments and variation methods, he proves that this BVP has a unique solution, where the solution has the asymptotics \(f(r) \sim a^{*} \log r\), as \(r\to 1\), for a positive constant \(a^{*}\). Finally, the uniqueness of the solution to problem (1)--(3) is proven by a variational method.