A non-autonomous second order boundary value problem on the half-line (Q601750)

The following boundary value problem \[ \begin{cases} x'' = a(t)V'(x), \\ x(0) = 0, \quad x(t) \to 1 \quad \text{as} \;t \to \infty \end{cases} \] is considered, where \(V \in C^1({\mathbb R})\), \(V(x)>0\) on \((0,1)\), \(V(0)=V(1)=0\), \(a \in L^\infty[0,\infty)\), \(a(t) \geq 0\), and \(a(t) \to l>0\) as \(t\to\infty\). The existence of a solution is established. The proof is based on variational methods.