The Fredholm Alternative at the First Eigenvalue for the One Dimensionalp-Laplacian

In this work we study the range of the operatoru (u′p−2 u′)′+λ1 up−2 u,u(0)=u(T)=0,p>1. We prove that all functionsh C1[0, T] satisfying ∫T0 h(t) sinp(πpt/T) dt=0 lie in the range, but that ifp≠2 andh 0 the solution set is bounded. Here sin(πpt/T) is a first eigenfunction associated toλ1. We also show that in this case the associated energy functionalu (1/p) ∫T0 u′p−(λ1/p) ∫T0 up+∫T0 huis unbounded from below if 12, onW1, p0(0, T) (λ1corresponds precisely to the best constant in theLp-Poincaré inequality). Moreover, we show that unlike the linear casep=2, forp≠2 the range contains a nonempty open set inL∞(0, T).