Existence and multiplicity for radially symmetric solutions to Hamilton-Jacobi-Bellman equations

Abstract

This article concerns the existence and multiplicity of radially symmetricnodal solutions to the nonlinear equation $$\displaylines{ -\mathcal{M}_\mathcal{C}^{\pm}(D^2u)=\mu f(u) \quad \text{in } \mathcal{B},\cru=0 \quad \text{on } \partial\mathcal{B}, }$$ where \(\mathcal{M}_\mathcal{C}^{\pm}\) are general Hamilton-Jacobi-Bellman operators,  (\mu\) is a real parameter and \(\mathcal{B}\) is the unit ball. By using bifurcation theory, we determine the range of parameter \(\mu\)  in which the above problem has one or multiple nodal solutions according to the behavior of \(f\) at 0 and infinity, and whether \(f\) satisfies the signum condition \(f(s)s>0 \) for \(s\neq 0\) or not. For more information see https://ejde.math.txstate.edu/Volumes/2021/31/abstr.html