Second order, Sturm-Liouville problems with asymmetric, superlinear nonlinearities II

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Abstract

We consider the nonlinear Sturm-Liouville problem -(p(x)u'(x)) '+q(x)u(x)=f(x,u(x))+h(x),in (0,p),c00u(0)+c 01u'(0)=0,c10u(p)+c11u'(p)=0, wherep?C1[0,p],q?C0[0,p],with p(x)>0 for all x?[0,p]; ci02+ci120, i=0,1; h?L2(0,p). We suppose that f:[0,p]×R?R is continuous and there exist increasing functions ?l, ?u:[0,8)?R, and positive constants A, B, such that limt?8 ?l(t)=8 and -A+?l(?)?=f(x,?)=A+?u(?)?, ?=0,|f(x,?)|=A+B|?|,?=0,for all x?[0,p] (thus the nonlinearity is superlinear as u(x)?8, but linearly bounded as u(x)?-8). Existence and non-existence results are obtained for the above problem. Similar results have been obtained before for problems in which f is linearly bounded as |?|?8, and these results have been expressed in terms of 'half-eigenvalues' of the problem. The results obtained here for the superlinear case are expressed in terms of certain asymptotes of these half-eigenvalues. © 2004 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)905-916
Number of pages12
JournalNonlinear Analysis: Theory, Methods and Applications
Volume57
Issue number7-8
DOIs
Publication statusPublished - Jun 2004

Keywords

  • Asymmetric superlinearity
  • Nonlinear boundary value problems

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