Existence and global behavior of positive solutions for some fourth-order boundary value problems (Q1724689)

Summary: We establish the existence and uniqueness of a positive solution to the following fourth-order value problem: \(u^{(4)}(x) = a(x) u^\sigma(x)\), \(x \in(0,1)\) with the boundary conditions \(u(0) = u(1) = u'(0) = u'(1) = 0\), where \(\sigma \in(- 1,1)\) 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.