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

Bryan P. Rynne

doi:10.1016/j.na.2004.03.021

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

Bryan P. Rynne

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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]; c_i0²+c_i1²0, 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 lim_t?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 language	English
Pages (from-to)	905-916
Number of pages	12
Journal	Nonlinear Analysis: Theory, Methods and Applications
Volume	57
Issue number	7-8
DOIs	https://doi.org/10.1016/j.na.2004.03.021
Publication status	Published - Jun 2004

Keywords

Asymmetric superlinearity
Nonlinear boundary value problems

Access to Document

10.1016/j.na.2004.03.021

Cite this

@article{a2a9b9091b4a464da18329eb22ba348c,

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

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. {\textcopyright} 2004 Elsevier Ltd. All rights reserved.",

keywords = "Asymmetric superlinearity, Nonlinear boundary value problems",

author = "Rynne, {Bryan P.}",

year = "2004",

month = jun,

doi = "10.1016/j.na.2004.03.021",

language = "English",

volume = "57",

pages = "905--916",

journal = "Nonlinear Analysis: Theory, Methods and Applications",

publisher = "Elsevier",

number = "7-8",

}

TY - JOUR

T1 - Second order, Sturm-Liouville problems with asymmetric, superlinear nonlinearities II

AU - Rynne, Bryan P.

PY - 2004/6

Y1 - 2004/6

N2 - 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.

AB - 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.

KW - Asymmetric superlinearity

KW - Nonlinear boundary value problems

UR - http://www.scopus.com/inward/record.url?scp=2942596538&partnerID=8YFLogxK

U2 - 10.1016/j.na.2004.03.021

DO - 10.1016/j.na.2004.03.021

M3 - Article

VL - 57

SP - 905

EP - 916

JO - Nonlinear Analysis: Theory, Methods and Applications

JF - Nonlinear Analysis: Theory, Methods and Applications

IS - 7-8

ER -

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this