Electron. J. Diff. Eqns.,
Vol. 2002(2002), No. 02, pp. 1-12.
### The nonlocal bistable equation: Stationary solutions
on a bounded interval

Adam J. J. Chmaj & Xiaofeng Ren

**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.

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Adam J. J. Chmaj

Department of Mathematics,

Heriot-Watt University Riccarton,

Edinburgh EH14 4AS,
Scotland, UK

e-mail: A.J.Chmaj@ma.hw.ac.uk
Xiaofeng Ren

Department of Mathematics and Statistics

Utah State University

Logan, UT 84322, USA

e-mail: ren@math.usu.edu

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