Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators

Abstract

AbstractThe paper deals with the existence and non-existence of solutions of the following nonlinear non-autonomous boundary value problem governed by the p-Laplacian operator: $$(P) \quad \textstyle\begin{cases} (h(t,x(t)) \vert x'(t) \vert ^{p-2} x'(t))' = g(t,x(t),x'(t)) \quad \text{a.e. } t\in \mathbb{R}, \\ x(-\infty )=a, \qquad x(+\infty )= b \end{cases} $$(P){(h(t,x(t))|x′(t)|p−2x′(t))′=g(t,x(t),x′(t))a.e. t∈R,x(−∞)=a,x(+∞)=b with $a< b$a<b, where a is a positive, continuous function and g is a Caratheódory nonlinear function.We prove an existence result, underlying the relationship between the behavior of $g(t,x,\cdot )$g(t,x,⋅) as $y\to 0$y→0 related to that of $g(\cdot ,x,y)$g(⋅,x,y) and $h(\cdot ,x) $h(⋅,x) as $|t|\to +\infty $|t|→+∞.