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.

Classical and non-classical solutions of a prescribed curvature equation

OBERSNEL, Franco;OMARI, PIERPAOLO
2007-01-01

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/1697220
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