Heriot-Watt Mathematics Report Series
HWM11-14, 8 Jun 2011
A global curve of stable, positive solutions for a $p$-Laplacian problem
B P Rynne
Abstract
We consider the boundary-value problem
\begin{gather*}
- \phi_p (u'(x))' = \lambda f(x,u(x)) , \quad x \in (0,1),\\
u(0) = u(1) = 0,
\end{gather*}
where
$p>1$ ($p \ne 2$), $\phi_p(s) := |s|^{p-1} \mathop{\rm sign} s$, $s
\in \mathbb{R}$,
$\lambda \ge 0$,
and the function
$f : [0,1] \times \mathbb{R} \to \mathbb{R}$ is $C^1$ and
satisfies
\begin{gather*} %\tag{3}
f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\\
(p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi ,
\quad (x,\xi) \in [0,1] \times (0,\infty) .
\end{gather*}
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 $C^1$ 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).
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