A generalized anti-maximum principle for the periodic one-dimensional [formula omitted]-Laplacian with sign-changing potential

Abstract

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