Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

John Villavert

doi:10.3934/cpaa.2015.14.493

Article Contents

2015, Volume 14, Issue 2: 493-515. Doi: 10.3934/cpaa.2015.14.493

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Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

John Villavert^1,

1.
Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103

Received: January 31, 2014

Revised: August 31, 2014

Published: February 2015

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Abstract

This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. Then a sharp criterion for the existence and non-existence of positive solutions is determined for both differential and integral versions of a Hardy--Sobolev type system with variable coefficients. In the constant coefficient case, Liouville type theorems for positive radial solutions are also established using radial decay estimates and Pohozaev type identities in integral form.

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Mathematics Subject Classification: Primary: 35B53, 45G05, 45G15; Secondary: 35J48, 35J91.

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