Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonlinear perturbations of linear elliptic systems at resonance

Author: Philip Korman
Journal: Proc. Amer. Math. Soc. 140 (2012), 2447-2451
MSC (2010): Primary 35J60
Published electronically: November 21, 2011
MathSciNet review: 2898707
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a semilinear system

$\displaystyle \Delta u$ $\displaystyle + \lambda v+b_1(v)=f(x),\;\; x \in \Omega ,\quad \;\;\; u=0$$\displaystyle \mbox {\ \ \ \,\,for $x \in \partial \Omega $} ,$    
$\displaystyle \Delta v$ $\displaystyle +\frac {\lambda ^2 _1}{\lambda } u+b_2(u) =g(x),\;\; x \in \Omega ,\quad v=0$$\displaystyle \mbox {\ \ \ \, for $x \in \partial \Omega $},$    

whose linear part is at resonance. Here $ \lambda >0$ and the functions $ b_1(t)$ and $ b_2(t)$ are bounded and continuous. Assuming that $ tb_i(t)>0 $ for all $ t \in R$, $ i=1,2$, and that the first harmonics of $ f(x)$ and $ g(x)$ lie on a certain straight line, we prove the existence of solutions. This extends a similar result for one equation, due to D.G. de Figueiredo and W.-M. Ni.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J60

Retrieve articles in all journals with MSC (2010): 35J60

Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025

Keywords: Elliptic system at resonance, existence of solutions
Received by editor(s): February 25, 2011
Published electronically: November 21, 2011
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia