An Elliptic Equation With No Monotonicity Condition on the Nonlinearity

Gregory S. Spradlin

doi:10.1051/cocv:2006022

An Elliptic Equation With No Monotonicity Condition on the Nonlinearity

Gregory S. Spradlin

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	American English
Journal	ESAIM: Control, Optimisation and Calculus of Variations
Volume	12
DOIs	https://doi.org/10.1051/cocv:2006022
State	Published - Oct 2006

Keywords

Mountain Pass Theorem
variational methods
Nehari manifold
Brouwer degree
concentration-compactness

Disciplines

Partial Differential Equations

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Cite this

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title = "An Elliptic Equation With No Monotonicity Condition on the Nonlinearity",

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.",

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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.

KW - Mountain Pass Theorem

KW - variational methods

KW - Nehari manifold

KW - Brouwer degree

KW - concentration-compactness

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DO - 10.1051/cocv:2006022

M3 - Article

SN - 1262-3377

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JO - ESAIM: Control, Optimisation and Calculus of Variations

JF - ESAIM: Control, Optimisation and Calculus of Variations

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