An elliptic equation with no monotonicity condition on the nonlinearity

Gregory S. Spradlin

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 786-794
  • ISSN: 1292-8119

Abstract

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An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.

How to cite

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Spradlin, Gregory S.. "An elliptic equation with no monotonicity condition on the nonlinearity." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 786-794. <http://eudml.org/doc/249614>.

@article{Spradlin2006,
abstract = { An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used. },
author = {Spradlin, Gregory S.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Mountain-pass theorem; variational methods; Nehari manifold; Brouwer degree; concentration-compactness.; Brouwer degree; concentration-compactness},
language = {eng},
month = {10},
number = {4},
pages = {786-794},
publisher = {EDP Sciences},
title = {An elliptic equation with no monotonicity condition on the nonlinearity},
url = {http://eudml.org/doc/249614},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Spradlin, Gregory S.
TI - An elliptic equation with no monotonicity condition on the nonlinearity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 786
EP - 794
AB - An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.
LA - eng
KW - Mountain-pass theorem; variational methods; Nehari manifold; Brouwer degree; concentration-compactness.; Brouwer degree; concentration-compactness
UR - http://eudml.org/doc/249614
ER -

References

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