Solutions of Semilinear Problems in Symmetric Planar Domains — ODE Behavior and Uniqueness of Branches

Abstract

We prove an existence and uniqueness theorem for an “Initial Value Problem” in the plane, related to the semilinear elliptic equation

$$ - \Delta u = f(u) $$

in the case f is a C¹-convex function. This result is applied to show the uniqueness of a global bifurcation branch for the problem

$$ \begin{gathered} - \Delta u = {u^p} + \lambda u in \Omega \hfill \\ u > 0 in \Omega \hfill \\ u = 0 on \partial \Omega , \hfill \\ \end{gathered} $$

whereΩis a symmetric bounded domain inℝ²