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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
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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.

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Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: kormanp@math.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11288-7
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.