Adam J. J. Chmaj & Xiaofeng Ren
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 C1 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.
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Department of Mathematics and Statistics
Utah State University
Logan, UT 84322, USA