Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem: u ( 4 ) ( x ) = a ( x ) u σ ( x ) , x ∈ ( 0,1 ) with the boundary conditions u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = 0 , where σ ∈ ( - 1,1 ) and a is a nonnegative continuous function on (0, 1) that may be singular at x = 0 or x = 1 . We also give the global behavior of such a solution.