Some discontinuous variational problems. (English) Zbl 0728.35037

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.


35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces