Open AccessOn the growth of positive entire solutions of elliptic PDEs and their gradients
Author(s) -
Antonio Vitolo
Publication year - 2014
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2014.7.1335
Subject(s) - quadratic growth , sobolev space , mathematics , bounded function , polynomial , infinity , mathematical analysis , order (exchange) , counterexample , operator (biology) , nonlinear system , pure mathematics , discrete mathematics , physics , biochemistry , chemistry , finance , repressor , quantum mechanics , transcription factor , economics , gene
We investigate the growth of entire positive functions u(x) and\udtheir gradients Du in Sobolev spaces when a polynomial growth is assumed for their image Lu through a linear second-order uniform elliptic operator L.\udIn 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\udresults to viscosity solutions of fully nonlinear equations