A New Solution Operator of One-Dimensional p-Laplacian with a Sign-Changing Weight and Its Application. (8th August 2012) - Sim, Inbo; Lee, Yong Hoon
Abstract : We establish a new solution operator for the following problem - φ p ( u ′ ) ′ = g, t ∈ ( 0, 1 ), u ( 0 ) = 0 = u ( 1 ), where φ p ( x ) = | x | p - 2 x, p > 1 . g may be singular at the boundary or change signs or may not be in L 1 ( 0, 1 ) so that this solution operator can cover larger class of g than previously known ones. As an application, by checking complete continuity of the solution operator, we show the existence of nontrivial solutions for p -Laplacian φ p ( u ′ ) ′ + λ h ( t ) f ( u ( t ) ) = 0, t ∈ ( 0, 1 ), u ( 0 ) = 0 = u ( 1 ), where λ > 0 a parameter and f ∈ C ( ℝ, ℝ ) and f ( 0 ) > 0 and h may change signs or may be beyond of L 1 ( 0, 1 ) .