Spectral properties and nodal solutions for second-order, $m$-point, $p$-Laplacian boundary value problems

Niall Dodds, Bryan P. Rynne

DOI: http://dx.doi.org/10.12775/TMNA.2008.033

Abstract


We consider the $m$-point boundary value problem consisting
of the equation
$$
-\phi_p (u')'=f(u), \quad \text{on $(0,1)$},\tag 1
$$
together with the boundary conditions
$$
u(0) = 0,\quad u(1) = \sum^{m-2}_{i=1}\alpha_i u(\eta_i) ,\tag 2
$$
where
$p> 1$, $\phi_p(s) := |s|^{p-1} \text{\rm sgn} s$, $s \in {\mathbb R}$,
$m \ge 3$,
$\alpha_i , \eta_i \in (0,1)$,
for $i=1,\dots,m-2$,
and $\sum^{m-2}_{i=1} \alpha_i < 1$.
We assume that the function
$f \colon {\mathbb R} \to{\mathbb R}$ is continuous, satisfies
$sf(s) > 0$ for $s \in {\mathbb R} \setminus \{0\}$,
and that
$f_0 := \lim_{\xi \rightarrow 0}{f(\xi)}/{\phi_p(\xi)} > 0$.
%(we assume that the limit exists and is finite).

Closely related to the problem (1), (2), is the spectral problem
consisting of the equation
$$
-\phi_p (u')' = \la \phi_p(u) , \tag 3
$$
together with the boundary conditions (2).
It will be shown that
the spectral properties of (2), (3),
are similar to those of the standard
Sturm-Liouville problem with separated (2-point) boundary conditions
(with a minor modification to deal with the multi-point boundary
condition).
The topological degree of a related operator is also obtained.
These spectral and degree theoretic results are then used to prove
a Rabinowitz-type global bifurcation theorem for a bifurcation problem
related to the problem (1), (2).
Finally, we use the global bifurcation theorem to obtain nodal solutions
%(that is, sign-changing solutions with a specified number of zeros)
of (1), (2), under various conditions on the asymptotic behaviour
of $f$.

Keywords


$m$-point; $p$-Laplacian; spectral properties; nodal solutions

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