Nonlinear Differential Equations,
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 1-12.

Explicit construction, uniqueness, and bifurcation curves of solutions for a nonlinear Dirichlet problem in a ball

Horacio Arango & Jorge Cossio

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 Ocotber 24, 2000.
Math Subject Classifications: 35B32, 35J60, 65D07, 65N99.
Key Words: Nonlinear Dirichlet problem, radially symmetric solutions, bifurcation, explicit solutions, spline.

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Horacio Arango
Departamento de Matematicas
Universidad Nacional de Colombia
Apartado Aereo 3840
Medellin, Colombia
Jorge Cossio
Departamento de Matematicas
Universidad Nacional de Colombia
Apartado Aereo 3840
Medellin, Colombia
e-mail: jcossio@perseus.unalmed.edu.co

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