Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 58, pp. 1-12.

Title: A global curve of stable, positive
       solutions for a p-Laplacian problem

Author: Bryan P. Rynne (Heriot-Watt Univ., Edinburgh, Scotland)

Abstract: 
 We consider the boundary-value problem
 $$\displaylines{
 - \phi_p (u'(x))' = \lambda f(x,u(x))  , \quad  x \in (0,1),\cr
 u(0) = u(1) = 0,
 }$$
 where $p>1$ ($p \ne 2$), $\phi_p(s)  :=  |s|^{p-1} \hbox{\rm sign} s$, 
 $s \in \mathbb{R}$, $\lambda \ge 0$,
 and the function
 $f : [0,1] \times \mathbb{R} \to \mathbb{R}$ is $C^1$ and
 satisfies
 $$\displaylines{ 
 f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\cr
 (p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi ,
 \quad  (x,\xi) \in [0,1] \times (0,\infty) .
 }$$
 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 {\em positive},
 that is, $u > 0$ on $(0,1)$.

We prove that  the set of nontrivial solutions
consists of a $C^1$ curve of positive solutions in
$(0,\lambda_{\rm max}) \times C^0[0,1]$,
with a parametrisation of the form
$\lambda \to (\lambda,u(\lambda))$,
where $u$ is a $C^1$ function defined on $(0,\lambda_{\rm max})$,
and $\lambda_{\rm max}$ is a suitable weighted eigenvalue of the
$p$-Laplacian
($\lambda_{\rm max}$ may be finite or $\infty$),
and $u$ satisfies
$$
\lim_{\lambda\to 0} u(\lambda) = 0,
\quad
\lim_{\lambda \to \lambda_{\rm max}} |u(\lambda)|_0 = \infty .
$$
We also show that for each $\lambda \in (0,\lambda_{\rm max})$
the solution $u(\lambda)$ is globally asymptotically stable,
with respect to positive solutions
(in a suitable sense).

Submitted August 13, 2009. Published April 28, 2010.
Math Subject Classifications: 34B15.
Key Words: Ordinary differential equations; p-Laplacian; 
           nonlinear boundary value problems;  positive solutions; 
	   stability.