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

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 $f$.

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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