One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue

Abstract

In this paper, we prove the existence of eigenvalues for the problem { φ p ( u ′ ( t ) ) ′ + λ h ( t ) φ p ( u ( t ) ) = 0 , a.e. in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where φ p ( s ) = | s | p − 2 s , p > 1 , λ is a real parameter and the indefinite weight h is a nonnegative measurable function on ( 0 , 1 ) which may be singular at 0 and/or 1, and h ≢ 0 on any compact subinterval in ( 0 , 1 ) . We derive similar properties of eigenvalues as known in linear case ( p = 2 ) or continuous case ( h ∈ C [ 0 , 1 ] ) if h satisfies ∫ 0 1 t p − 1 ( 1 − t ) p − 1 h ( t ) d t < ∞ when 1 < p ⩽ 2 and ∫ 0 1 / 2 φ p −1 ( ∫ s 1 / 2 h ( τ ) d τ ) d s + ∫ 1 / 2 1 φ p −1 ( ∫ 1 / 2 s h ( τ ) d τ ) d s < ∞ when p ⩾ 2 , respectively. For the result, we establish the C 1 -regularity of all solutions at the boundary for the above problem as well as the following problem: { φ p ( u ′ ( t ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , a.e. in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where f ∈ C ( R , R ) , s f ( s ) > 0 for s ≠ 0 , f is odd and f ( s ) / φ p ( s ) is bounded above as s → 0 .