Hidden Symmetry and Bifurcation Analysis of the Emden-Fowler Equation

Abstract

The Emden-Fowler equation yields bifurcations at the same two critical exponents β ₁and β ₂already occuring in [20]. This coincidence allows to construct counterexamples confirming that the uniform a priori estimates for the Dirichlet boundary value problem of the nonlinear Poisson equation obtained there are rather optimal. As shown there, for β < β₁, finite L₁-norm of the source term implies a uniform bound. On the other hand, for ,β [d] β ₁, the phase plane analysis of the Emden-Fowler equation yields unbounded weak solutions of finite L₁-norm. Analogously, we have shown in [20] that for ,β < β ₂finite Dirichlet integral implies a uniform bound. On the other hand, for β > β ₂the phase plane analysis of the Emden-Fowler equation yields a sequence of classical solutions with bounded Dirichlet integrals hut maximum norms growing to infinity. The bifurcation analysis of the Emden-Fowler equation is based on a hidden symmetry, which allows to reduce the dimension of phase space from three to two. For the resulting flow in the phase plane, the bifurcations can be discussed completely. At β ₁we get a pitchfork bifurcation of equilibria and at β ₂a degenerate Hopf-and homoclinic bifurcation.

Keywords:

• Emden-Fowler equation
• strong and distributional solutions
• homology classes
• phase-plane analysis
• negative Bendixon criteriurn
• Hopf-and saddle-node bifurcation