A GLOBAL CURVE OF STABLE, POSITIVE SOLUTIONS FOR A p-LAPLACIAN PROBLEM

作者：Rynne Bryan P^*

来源：Electronic Journal of Differential Equations, 2010, 58.

• Ordinary differential equations
• p-Laplacian
• nonlinear boundary value problems
• positive solutions
• stability

We consider the boundary-value problem
-phi(p)(u '(x))' - lambda f(x, u(x)), x is an element of (0, 1),
u(0) = u(1) = 0,
where p > 1 (p not equal 2), phi(p)(s) := vertical bar s vertical bar p(-1) sign s, s is an element of R, lambda >= 0, and the function f : [0, 1] x R -> R is C-1 and satisfies
f(x, xi) > 0, (x, xi) is an element of [0, 1] x R,
(p - 1) f(x, xi) >= f(xi)(x, xi)xi, (x, xi) is an element of [0, 1] x (0, infinity).
These assumptions on f imply that the trivial solution (lambda, u) = (0, 0) is the only solution with lambda = 0 or u = 0, and if lambda > 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, lambda(max)) x C-0[0, 1], with a parametrisation of the form lambda ! (lambda, u(lambda)), where u is a C-1 function defined on (0, lambda(max)), and lambda(max) is a suitable weighted eigenvalue of the p-Laplacian (lambda(max) may be finite or 1), and u satisfies
lim lambda -> 0 u(lambda) - 0, lim lambda ->lambda(max) vertical bar u(lambda)vertical bar(0-infinity).
We also show that for each lambda is an element of (0, lambda(max)) the solution u(lambda) is globally asymptotically stable, with respect to positive solutions (in a suitable sense).

• 出版日期2010-4-28