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

• Main Authors: Horacio Arango, Jorge Cossio
• Format: Article
• Language: English
• Published: Texas State University 2000-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Nonlinear Dirichlet problem; radially symmetric solutions; bifurcation; explicit solutions; spline.
• Online Access: http://ejde.math.txstate.edu/conf-proc/05/a1/abstr.html

This paper presents a method for the explicit construction of radially symmetric solutions to the semilinear elliptic problem $$displaylines{ Delta v + f(v) = 0 quad hbox{in }Bcr v = 0 quad hbox{on }partial B,, } $$ 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.