Affiliation:
1. Departamento de Análisis Matemático y Matemática Aplicada , Instituto de Matemática Interdisciplinar (IMI) , Universidad Complutense de Madrid , Plaza de Ciencias 3, 28040 Madrid , Spain
2. Sezione di Matematica e Informatica , Dipartimento di Matematica e Geoscienze , Università degli Studi di Trieste , Via A. Valerio 12/1, 34127 Trieste , Italy
Abstract
Abstract
This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem
{
-
(
u
′
1
+
u
′
2
)
′
=
λ
a
(
x
)
f
(
u
)
in
(
0
,
1
)
,
u
′
(
0
)
=
0
,
u
′
(
1
)
=
0
,
\begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{%
\prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\
u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases}
where
λ
∈
ℝ
{\lambda\in\mathbb{R}}
is a parameter,
a
∈
L
∞
(
0
,
1
)
{a\in L^{\infty}(0,1)}
changes sign, and
f
∈
C
1
(
ℝ
)
{f\in C^{1}(\mathbb{R})}
is positive in
(
0
,
+
∞
)
{(0,+\infty)}
. The attention is focused on the case
f
(
0
)
=
0
{f(0)=0}
and
f
′
(
0
)
=
1
{f^{\prime}(0)=1}
, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions.
Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established.
The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
13 articles.
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