On the growth of positive entire solutions of elliptic PDEs and their gradients

Antonio Vitolo

doi:10.3934/dcdss.2014.7.1335

Article Contents

2014, Volume 7, Issue 6: 1335-1346. Doi: 10.3934/dcdss.2014.7.1335

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On the growth of positive entire solutions of elliptic PDEs and their gradients

Antonio Vitolo^1,

1.
Department of Mathematics, University of Salerno, via Giovanni Paolo II n.132, 84084 Fisciano (SA)

Received: December 31, 2012

Revised: June 30, 2013

Published: November 2014

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Abstract

We investigate the growth of entire positive functions $u(x)$ and their gradients $Du$ in Sobolev spaces when a polynomial growth is assumed for their image $Lu$ through a linear second-order uniform elliptic operator $L$. In particular, under suitable assumptions on the coefficients, we show that if $Lu$ is bounded, then $u(x)$ may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations.

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Mathematics Subject Classification: Primary: 35B08, 35B09; Secondary: 35J60.

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