Stability of solutions of a 1-dimensional, p-Laplacian problem and the shape of the bifurcation curve

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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−1sgnz, 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 languageEnglish
Article number111757
JournalNonlinear Analysis: Theory, Methods and Applications
Volume196
Early online date4 Feb 2020
DOIs
Publication statusPublished - Jul 2020

Keywords

  • Bifurcation curve
  • Stability
  • p-Laplacian

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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