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The Quarterly Journal of Mathematics Advance Access published online on December 12, 2007
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham047
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

Huan-Song Zhou{dagger} and Hongbo Zhu

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071, P. R. China

{dagger} Corresponding author. E-mail: hszhou{at}wipm.ac.cn

Received 21 February 2007; revised 9 August 2007
  Abstract

In this paper, we consider the following semilinear elliptic problem:

Formula

where f(x, t) tends to p(x) and q(x) isin L{infty}(RN), respectively, as t -> 0 and t -> +{infty}. We prove that there exist two numbers l and L with L < l such that problem (P) has at least one positive solution for {lambda} isin (–l, –L) and has no positive solution for all {lambda} {notin} [–l,–L]. The existence and non-existence of positive solutions for problem (P) at {lambda} = –l and {lambda} = –L are also discussed.


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