Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation.

We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problem $$ -\left( {u'}/{\sqrt{1+{u'}^2}}\right)' = f(u), \quad u(0)=0,\,\,u(1)=0, $$ depending on the behaviour at the origin and at infinity of the function $f$. We consider solutions that are possibly discontinuous at the points where they attain the value zero.