Non-autonomous boundary value problems on the real line (Q874384)

The general non-autonomous boundary value problem \[ (\Phi(x'(t)))' = f(t,x(t),x'(t)), \qquad x(-\infty)=0 , \quad x(+\infty)=1 \] is investigated, where \(\Phi\) is a monotone functional, including the classical \(p\)-Laplacian operator as a particular case. The authors obtain both existence and non-existence results under mild assumptions, which become very efficient criteria when the right-hand side has the product structure \(f(t,x,x')= a(t,x)b(x,x')\), based on the behavior of \(a(t,x)\) as \(| t| \to +\infty\) and \(b(x,y)/\Phi(y)\) as \(y\to 0\). Many examples are presented, both for general functionals \(\Phi\) and for the \(p\)-Laplacian operator.