Existence of multiple positive solutions of singular nonlinear boundary value problems

For a given positive integer N , we provide conditions on the nonlinear function f which guarantee that the boundary value problem y ″ + f ( t , y ) = 0 , 0 < x ≤ r , y ( 0 ) = 0 , y ′ ( r ) = 0 , has N positive solutions. The nonlinear function f is allowed to be singular at y = 0 and t = 0 but is required to satisfy an integrability condition reminiscent of a condition first used by S. Taliaferro in 1979 and growth conditions as y increases similar to those assumed in the nonsingular case by Henderson and Thompson in 2000. Our main results depend on shooting methods.