A homotopy along $p$ for systems with a vector $p$-Laplace operator

Abstract

We extend to the vector $p-$Laplace operator $(\phi_p(u'))'=(|u'|^{p-2}u')',$ $p>1,$ ($|\cdot|$ denotes the Euclidean norm in $\mathbb R^N$) a method that uses a suitable homotopy along $p$ to evaluate at the level $p=2$ a Leray-Schauder degree for an associated operator depending on $p.$ We apply this result to prove existence of nontrivial solutions to the system $$(\phi_p(u'))'=f(t,u)\quad\text{a.e.}\quad \mbox{in}\quad (0,T),\quad u(0)=0,\quad u(T)=0, $$ where $f:[0,T]\times\mathbb R^N\mapsto \mathbb R^N$ is a Carathéodory function, with $f(t,0)=0.$