Some discontinuous variational problems (Q804818)

This paper studies elliptic boundary value problems of the type \[ -\Delta u=h(u-a)p(u)\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where h is the Heaviside function, \(a>0\) and \(\Omega\) is a smooth domain in \({\mathbb{R}}^ n\). The nonlinearity p is non-decreasing in u and may depend on x and other parameters, and in addition to the zero solution allows two additional solutions corresponding to a minimum and to a saddle point. The methods employed involve dual variational principles and symmetry arguments. An interesting application is considered in the final section to an electric arc.