Existence and asymptotics of solutions for a class of second-order quasilinear ordinary differential equations (Q2567819)

The author considers the boundary value problem for the second-order ordinary differential equation \[ v''(x)-x^2v'(x)-\mu F(v(x))=0,\tag{1} \] \[ v(0)=1,\qquad v(x)=O(1)\quad\text{as}\quad x\to +\infty.\tag{2} \] Here, \(\mu>0\) is a constant, and \(F:[0,1]\to [0,1]\) is a monotone increasing and continuous function, satisfying a Lipschitz condition on the interval \([0,1]\), and \(F(0)=0\). The author proves existence and uniqueness of a solution of (1), (2). Moreover, the asymptotic representations of the solution of (1), (2), as \(\mu\to 0\) and also as \(x\to +\infty\), are established.