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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.

MSC:

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
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