Electron. J. Diff. Equ., Vol. 2010(2010), No. 58, pp. 1-12.

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

Bryan P. Rynne

We consider the boundary-value problem
 - \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
 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 $(0,\lambda_{max}) \times C^0[0,1]$, with a parametrisation of the form $\lambda \to (\lambda,u(\lambda))$, where $u$ is a $C^1$ function defined on $(0,\lambda_{max})$, and $\lambda_{max}$ is a suitable weighted eigenvalue of the $p$-Laplacian ($\lambda_{max}$ may be finite or $\infty$), and $u$ satisfies
\lim_{\lambda\to 0} u(\lambda) = 0,
\lim_{\lambda \to \lambda_{max}} |u(\lambda)|_0 = \infty .
We also show that for each $\lambda \in (0,\lambda_{max})$ 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.

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Bryan P. Rynne
Department of Mathematics and the Maxwell Institute for Mathematical Sciences
Heriot-Watt University
Edinburgh EH14 4AS, Scotland
email: bryan@ma.hw.ac.uk

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