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

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.