Abstract
We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville boundary value problem: { 1 p ( t ) ( p ( t ) u ‴ ( t ) ) ′ − g ( t ) F ( t , u ) = 0 , 0 < t < 1 , α 1 u ( 0 ) − β 1 u ′ ( 0 ) = 0 , γ 1 u ( 1 ) + δ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) − β 2 lim t → 0 + p ( t ) u ‴ ( t ) = 0 , γ 2 u ″ ( 1 ) + δ 2 lim t → 1 − p ( t ) u ‴ ( t ) = 0 , where g , p may be singular at t = 0 and/or 1. Moreover F ( t , x ) may also have singularity at x = 0 . The existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm–Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases.
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