A homotopy along p for systems with a vector p-laplace operator

Abstract

We extend to the vector p-Laplace operator (φp(u'))' = (|u'|p-2u')', p > 1, (|·| denotes the Euclidean norm in RN) 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 (φp(u'))' = f(t, u) a.e. in (0, T), u(0) = 0, u(T) = 0, where f: [0, T]×RN→ RN is a Carathéodory function, with f(t, 0) = 0.