### Eigenvalue criteria for existence of positive solutions of second-order, multi-point, $p$-Laplacian boundary value problems

#### Abstract

In this paper we consider the existence and uniqueness of positive

solutions of the multi-point boundary value problem

$$

\gather

- (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 ,

\quad\text{a.e. on $(-1,1)$},

\tag{1}

\\

u(\pm 1) = \sum^{m^\pm}_{i=1}\al^\pm_i u(\eta^\pm_i) ,

\tag{2}

\endgather

$$

where

$p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$,

$m^\pm \ge 1$ are integers,

and

$$

\eta_i^\pm \in (-1,1),\quad

\al_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad

\sum^{m^\pm}_{i=1} \al_i^\pm < 1 .

$$

Also,

$a \in L^1(-1,1),$

and

$g \colon [-1,1] \X \mathbb R^2 \to \mathbb R$ is Carathéodory,

with

$$

g(x,0,0) = 0, \quad x \in [-1,1].

\tag{3}

$$

Our criteria for existence of positive solutions of

(1), (2)

will be expressed in terms of the asymptotic behaviour of

$g(x,s,t)$, as $s \to \infty$,

and the principal eigenvalues of the

multi-point boundary value problem consisting of the equation

$$

-\phi_p (u')' + a \phi_p (u) = \la \phi_p (u) ,

\quad \text{on $(-1,1)$},

\tag{4}

$$

solutions of the multi-point boundary value problem

$$

\gather

- (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 ,

\quad\text{a.e. on $(-1,1)$},

\tag{1}

\\

u(\pm 1) = \sum^{m^\pm}_{i=1}\al^\pm_i u(\eta^\pm_i) ,

\tag{2}

\endgather

$$

where

$p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$,

$m^\pm \ge 1$ are integers,

and

$$

\eta_i^\pm \in (-1,1),\quad

\al_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad

\sum^{m^\pm}_{i=1} \al_i^\pm < 1 .

$$

Also,

$a \in L^1(-1,1),$

and

$g \colon [-1,1] \X \mathbb R^2 \to \mathbb R$ is Carathéodory,

with

$$

g(x,0,0) = 0, \quad x \in [-1,1].

\tag{3}

$$

Our criteria for existence of positive solutions of

(1), (2)

will be expressed in terms of the asymptotic behaviour of

$g(x,s,t)$, as $s \to \infty$,

and the principal eigenvalues of the

multi-point boundary value problem consisting of the equation

$$

-\phi_p (u')' + a \phi_p (u) = \la \phi_p (u) ,

\quad \text{on $(-1,1)$},

\tag{4}

$$

#### Keywords

Positive solutions of nonlinear boundary value problems; principal eigenvalues

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