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