### Abstract

We consider the p-Laplacian boundary-value problem −φ
_{p}(u
^{′})
^{′}=λf(u),on (−1,1),u(±1)=0, where p>1 (p≠2), φ
_{p}(z):=|z|
^{p−1}sgnz, z∈R, λ⩾0, f:R→R is C
^{2} and f>0 on R. Under these conditions the set of solutions (λ,u) of (1)–(2) consists of the trivial solution (λ,u)=(0,0) together with a single (connected) C
^{2} curve S⊂R
_{+}×C
_{0}
^{1}[−1,1] (R
_{+}=(0,∞)). Under additional conditions on f the ‘shape’ of S can be determined. Solutions of (1)–(2) are equilibrium solutions of a related time-dependent, parabolic problem, and in this time-dependent setting the stability of these equilibria is of interest. It will be shown that the stability of solutions on S is determined by the shape of S. This will first be discussed in a general setting, and the results will then be applied to the specific case where S is ‘S-shaped’. Finally, similar results will be obtained, for ‘generic’ λ, without any additional conditions on f.

Original language | English |
---|---|

Article number | 111757 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 196 |

Early online date | 4 Feb 2020 |

DOIs | |

Publication status | Published - Jul 2020 |

### Keywords

- Bifurcation curve
- Stability
- p-Laplacian

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics