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Abstract.We consider the forced Liénard equationtogether with the boundary conditionswhere g is continuous on R x (0, +oo) and becomes infinite at u = 0. We consider classical solutions as well as generalized solutions that can go into the singularity u = 0 . The method of approach uses upper and lower solutions and degree theory.

We discuss existence and multiplicity of positive solutions of the \ud one-dimensional prescribed curvature problem \ud $$\ud -\left(\ud {u'}/{\sqrt{1+{u'}^2}}\right)' = \lambda f(t,u),\ud \quad\ud u(0)=0,\,\,u(1)=0, \ud $$\ud 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