Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 02, pp. 1-12.

Title: The nonlocal bistable equation: Stationary solutions
       on a bounded interval
   
Authors: Adam J. J. Chmaj (Heriot-Watt Univ., Edinburgh, Scotland, UK)
         Xiaofeng Ren (Utah State Univ., Logan, UT, USA)

Abstract:  
 We discuss instability and existence issues for the nonlocal
 bistable equation. This model arises as the Euler-Lagrange
 equation of a nonlocal, van der Waals type functional. Taking the
 viewpoint of the calculus of variations, we prove that for a class
 of nonlocalities this functional does not admit nonconstant $C^1$
 local minimizers. By taking variations along non-smooth paths, we
 give examples of nonlocalities for which the functional does not
 admit local minimizers having a finite number of discontinuities.
 We also construct monotone solutions and give a criterion for
 nonexistence of nonconstant solutions.

Submitted July 18, 2001. Published January 2, 2002.
Math Subject Classifications:   45G10.
Key Words: local minimizers; monotone solutions.