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

Bryan P. Rynne

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
$$
-\phi_p (u')' + a \phi_p (u) = \la \phi_p (u) ,
\quad \text{on $(-1,1)$},
\tag{4}
$$

Keywords


Positive solutions of nonlinear boundary value problems; principal eigenvalues

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