Electron. J. Diff. Equ., Vol. 2012 (2012), No. 171, pp. 1-18.

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

Cristina Marcelli

Abstract:
This article concerns the existence and non-existence of solutions to the strongly nonlinear non-autonomous boundary-value problem
$$\displaylines{ (a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \quad \hbox{a.e. }t\in \mathbb{R}\cr x(-\infty)=\nu^- ,\quad x(+\infty)= \nu^+ }$$
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 4, 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

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Cristina Marcelli
Dipartimento di Scienze Matematiche
Università Politecnica delle Marche
Via Brecce Bianche, 60131 Ancona, Italy
email: marcelli@dipmat.univpm.it

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