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Multiple Positive Solutions of Semi‐Positone Sturm–Liouville Boundary Value Problems
Author(s) -
Lan K. Q.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018327
Subject(s) - sturm–liouville theory , mathematics , eigenvalues and eigenvectors , eigenfunction , boundary value problem , mathematics subject classification , value (mathematics) , combinatorics , boundary (topology) , pure mathematics , mathematical analysis , physics , statistics , quantum mechanics
This paper treats the existence of multiple positive solutions of the semi‐positone Sturm–Liouville boundary value problem. λ( p ( t ) y ′ ( t ) ) ′ + g ( t ) f ( t , y ( t ) ) = 0almost everywhere on [ R 0 , R 1 ],α z ( R 0 ) − rβ p ( R 0 ) z ′ ( R 0 ) = 0 ,γ z ( R 1 ) + δ p ( R 1 ) z ′ ( R 1 ) = 0 ,where g ∈ L + ∞ [ R 0 , R 1 ] and f is allowed to take negative values (that is, f is semi‐positone). When λ = 1, new results on the existence of one or two nonzero positive solutions are obtained. These results generalize previous results for positone cases (that is, f ⩾ 0) to the semi‐positone cases. We illustrate our results with an explicit example which has two nonzero positive solutions. These results are used to deduce results on intervals of eigenvalues for which there exist one or two nonzero positive eigenfunctions. Applications of these eigenvalue results are provided. 2000 Mathematics Subject Classification 34B18 (primary), 34B15, 34B16, 47H10, 47H30 (secondary).