A global curve of stable, positive solutions for a p-laplacian problem

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Abstract

We consider the boundary-value problem These assumptions on f imply that the trivial solution (?, u) = (0, 0) is the only solution with ? = 0 or u = 0, and if ? > 0 then any solution u is positive, that is, u > 0 on (0, 1).We prove that the set of nontrivial solutions consists of a C1 curve of positive solutions in (0, ?max) × C0[0, 1], with a parametrisation of the form ? ? (?, u(?)), where u is a C1 function defined on (0, ?max), and ?max is a suitable weighted eigenvalue of the p-Laplacian (?max may be finite or 1), and u satisfies We also show that for each ? ? (0, ?max) the solution u(?) is globally asymptotically stable, with respect to positive solutions (in a suitable sense). © 2010 Texas State University - San Marcos.

Original languageEnglish
Pages (from-to)1-12
Number of pages12
JournalElectronic Journal of Differential Equations
Volume2010
Issue number58
Publication statusPublished - 2010

Keywords

  • Nonlinear boundary value problems
  • Ordinary differential equations
  • P-laplacian
  • Positive solutions
  • Stability

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