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Existence, Stability and Oscillation Properties of Slow-Decay Positive Solutions of Supercritical Elliptic Equations with Hardy Potential

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  16 April 2014

Vitaly Moroz  and
Jean van Schaftingen
Vitaly Moroz
Affiliation:
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK, (v.moroz@swansea.ac.uk)
Jean van Schaftingen
Affiliation:
Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium, (jean.vanschaftingen@uclouvain.be)
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Abstract

We prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potential

and study the stability and oscillation properties of these solutions. We also show that if the equation on ℝN has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝN a family of the exterior Dirichlet problems in ℝN \ K admits a continuum of stable slow-decay infinite-energy solutions.

Keywords

supercritical elliptic equationsHardy potentialslow-decay solutionsstable solutionsJoseph—Lundgren critical exponent

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B33: Critical exponents 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 255 - 271
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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