A generalized anti-maximum principle for the periodic one-dimensional pp-Laplacian with sign-changing potential

It is known that the anti-maximum principle holds for the quasilinear periodic problem (|u′|p−2u′)′+μ(t)(|u|p−2u)=h(t),u(0)=u(T),u′(0)=u′(T), if μ≥0μ≥0 in [0,T][0,T] and 0<‖μ‖∞≤(πp/T)p,whereπp=2(p−1)1/p∫01(1−sp)−1/pds, or p=2and0<‖μ‖α≤inf{‖u′‖22‖u‖α2:u∈W01,2[0,T]∖{0}}for some α,1≤α≤∞. In this paper we give sharp conditions on the LαLα-norm of the potential μ(t)μ(t) in order to ensure the validity of the anti-maximum principle even in the case where μ(t)μ(t) can change its sign in [0,T][0,T].