On some second order differential equations with convergent solutions (Q877185)

A solution \(x(t)\) of a differential equation is called convergent if it has the property that \[ x(t) =\gamma+o(1),\qquad\text{and}\quad x'(t) =o(t^{-1}) \] as \(t\to+\infty.\) The author studies existence of convergent solutions to a nonlinear differential equation \[ x''+a(t) f(x) =0,\quad t>0,\tag{a} \] where \(a\in C\left[ [0,+\infty),\mathbb{R}\right] \) and \(f\in C\left[ \mathbb{R},\mathbb{R}\right] .\) If (a) has a convergent solution, the change of variables \[ x(t)=\exp\left( y(u) \right) -\lambda,\qquad u(t)=\int_{c} ^{t}\frac{d\tau}{\lambda+x(\tau)}\tag{b} \] with the appropriate choice of \(c>0\) and \(\lambda> | \gamma| \) reduces (a) to the form \[ y''+b(u)g(e^{y})=0,\quad u\geq0,\tag{c} \] and equation (c) has a convergent solution \(y(u).\) Using the transformation (b), the author establishes existence of a solution \(u(x)\) of the elliptic differential equation \[ \Delta u+K\left(| x| \right) e^{u}=0,\qquad | x| >x_{0}>0 \] which has the asymptotic representation \[ u(x)=C_{1}\log | x| +C_{2}+o(1) \] as \(| x| \to+\infty,\) where \(C_{1}<0.\) This is an improvement of the result due to \textit{T. Kusano, M. Naito} and \textit{H. Usami} [Hiroshima Math. J. 16, 149--159 (1986; Zbl 0612.34052)].