On the existence and non-existence of bounded solutions for a fourth order ODE

We prove that if F∈C1(R) is coercive and {F′=0} is discrete, then the EFK equation(1)u⁗−c2u″+F′(u)=0 possesses L∞(R) solutions if and only if F′ changes sign at least twice. As a corollary we prove that if un solvesun⁗+cn2un″+F′(un)=0, then ‖un‖∞→+∞ if cn→0, provided F has a unique local minimum, its only minimum is nondegenerate and int({F′=0})=∅. Finally we give criteria ensuring existence and non-existence of T-periodic solutions to (1) when F has multiple wells.