Geodesic
The nonlocal bistable equation: Stationary solutions on a bounded interval
Electronic journal of differential equations, Tome 2002 (2002)
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.
Classification : 45G10
Keywords: local minimizers, monotone solutions 
@article{EJDE_2002__2002__a247,
     author = {Chmaj,  Adam J.J. and Ren,  Xiaofeng},
     title = {The nonlocal bistable equation: {Stationary} solutions on a bounded interval},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {0992.45002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a247/}
}
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Chmaj,  Adam J.J.; Ren,  Xiaofeng. The nonlocal bistable equation: Stationary solutions on a bounded interval. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a247/