Nonlinear singular Navier problem of fourth order

Abstract

We present an existence result for a nonlinear singular differential equation of fourth order with Navier boundary conditions. Under appropriate conditions on the nonlinearity ƒ(t, x, y), we prove that the problem L2u = L(Lu) = ƒ(., u, Lu) a.e. in (0, 1), u'(0) = 0, (Lu)' (0) = 0, u(1) = 0, Lu(1) = 0. has a positive solution behaving like (1 - t) on [0, 1]. Here L is a differential operator of second order, Lu = 1/A(au')'. For f(t, x, y) = f(t, x), we prove a uniqueness result. Our approach is based on estimates for Green functions and on Schauder's fixed point theorem.