## 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),c_{00}u(0)+c _{01}u'(0)=0,c_{10}u(p)+c_{11}u'(p)=0, wherep?C^{1}[0,p],q?C^{0}[0,p],with p(x)>0 for all x?[0,p]; c_{i0}^{2}+c_{i1}^{2}0, i=0,1; h?L^{2}(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 |
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Pages (from-to) | 905-916 |

Number of pages | 12 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 57 |

Issue number | 7-8 |

DOIs | |

Publication status | Published - Jun 2004 |

## Keywords

- Asymmetric superlinearity
- Nonlinear boundary value problems