Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 171, pp. 1-18.

Title: Existence of solutions to boundary-value problems governed 
       by general non-autonomous nonlinear differential operators

Author: Cristina Marcelli (Univ. Politecnica delle Marche, Ancona, Italy)

Abstract:
 This article concerns the existence and non-existence of solutions to
 the  strongly nonlinear non-autonomous boundary-value problem
 $$\displaylines{
 (a(t,x(t))\Phi(x'(t)))' =  f(t,x(t),x'(t)) \quad \hbox{a.e. } t\in \mathbb{R}  \\
 x(-\infty)=\nu^- ,\quad  x(+\infty)= \nu^+ 
 }$$
 with $\nu^-<\nu^+$, where $\Phi:\mathbb{R} \to \mathbb{R}$ is a general
 increasing homeomorphism, with $\Phi(0)=0$, a is a positive,
 continuous function and f is a Caratheodory nonlinear function.
 We provide sufficient conditions for the solvability which result 
 to be optimal  for a wide class of problems. In particular, 
 we focus on the role played  by the behaviors of $f(t,x,\cdot)$ 
 and $\Phi(\cdot)$ as $y\to 0$ related  to that of 
 $f(\cdot,x,y)$ and $a(\cdot,x) $ as $|t|\to +\infty$.

Submitted September 3, 2012. Published October 04, 2012.
Math Subject Classifications: 34B40, 34C37, 34B15, 34L30.
Key Words: Boundary value problems; unbounded domains;
           heteroclinic solutions; nonlinear differential operators; 
           p-Laplacian operator;  Phi-Laplacian operator