Electronic Journal of Differential Equations,
Conference 05 (2000), pp. 1-12.
Title: Explicit construction, uniqueness, and bifurcation curves
of solutions for a nonlinear Dirichlet problem in a ball
Authors: Horacio Arango (Univ. Nacional de Colombia, Medellin, Colombia)
Jorge Cossio (Univ. Nacional de Colombia, Medellin, Colombia)
Abstract:
This paper presents a method for the explicit construction of
radially symmetric solutions to the semilinear elliptic problem
$$\displaylines{
\Delta v + f(v) = 0 \quad \hbox{in }B\cr v = 0
\quad \hbox{on }\partial B\,, }
$$
where $B$ is a ball in ${\mathbb R}^N$ and $f$ is a continuous
piecewise linear function. Our construction method is inspired
on a result by E. Deumens and H. Warchall [8], and uses spline
of Bessel's functions. We prove uniqueness of solutions for this
problem, with a given number of nodal regions and different
sign at the origin. In addition, we give a bifurcation diagram
when $f$ is multiplied by a parameter.
Published October 24, 2000.
Math Subject Classifications: 35B32, 35J60, 65D07, 65N99.
Key Words: Nonlinear Dirichlet problem; radially symmetric solutions;
bifurcation; explicit solutions; spline.