Existence and nonexistence of positive solutions of -Kolmogorov equations perturbed by a Hardy potential

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• 作者：Jerome A. Goldstein^a ; ^{jgoldste@memphis.edu" class="auth_mail" title="E-mail the corresponding author} ; Daniel Hauer^b ; ^{daniel.hauer@sydney.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Abdelaziz Rhandi^c ; ^{arhandi@unisa.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35A01 ; 35B09 ; 35B25 ; 35D30 ; 35D35 ; 35K92
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：131
• 期：Complete
• 页码：121-154
• 全文大小：842 K

文摘

In this article, we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi-linear equations perturbed by a singular Hardy potential on the whole Euclidean space depending on the controllability of the given singular potential. To control the singular potential we use a weighted Hardy inequality with an optimal constant, which was recently discovered in Hauer and Rhandi (2013). Our results in this paper extend the ones in Goldstein et al. (2012) concerning a linear Kolmogorov operator significantly in several ways: firstly, by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15002539&_mathId=si1.gif&_user=111111111&_pii=S0362546X15002539&_rdoc=1&_issn=0362546X&md5=47c8d579246aa09e2384638e30d8f9d6" title="Click to view the MathML source">pclass="mathContainer hidden">class="mathCode">$p$-Laplace type with a nonlinear convection term for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X15002539&_mathId=si3.gif&_user=111111111&_pii=S0362546X15002539&_rdoc=1&_issn=0362546X&md5=2748d864dcfca9e9bba396ff5180ee0e" title="Click to view the MathML source">1<p<∞class="mathContainer hidden">class="mathCode">$1<p<\infty$, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions.