Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem−(u′/1+u′2)′=λf(t,u),u(0)=0,u(1)=0, depending on the behaviour at the origin and at infinity of the potential ∫0uf(t,s)ds. Besides solutions in W2,1(0,1)W2,1(0,1), we also consider solutions in Wloc2,1(0,1) which are possibly discontinuous at the endpoints of [0,1][0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.