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

AbstractIn this work we study the range of the operatoru↦(|u′|p−2u′)′+λ1|u|p−2u,u(0)=u(T)=0,p>1. We prove that all functionsh∈C1[0,T] satisfying ∫T0h(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|u|p+∫T0huis unbounded from below if 1<p<2 and bounded from below (with a global minimizer) ifp>2, 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)