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 $$ -\left( {u'}/{\sqrt{1+{u'}^2}}\right)' = \lambda f(t,u), \quad u(0)=0,\,\,u(1)=0, $$ depending on the behaviour at the origin and at infinity of the potential $\int_0^u f(t,s)\,ds$. Besides solutions in $W^{2,1}(0,1)$, we also consider solutions in $W_{loc}^{2,1}(0,1)$ which are possibly discontinuos at the endpoints of $[0,1]$. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.