Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary

Abstract: We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial \eta} = \lambda f(u)$ on $\partial\Omega$ as well as an extension to a corresponding system of equations. While existence, uniqueness, nonexistence, and multiplicity results for such problems are well-established, their numerical treatment presents computational challenges due to the absence of comparison principles and complex bifurcation phenomena. We present finite difference formulations for both single equations and coupled systems with cross-coupling boundary conditions, establishing admissibility results for the finite difference method. We derive principal eigenvalue analysis for the linearized problems to determine unique bifurcation points from trivial solutions. The eigenvalue analysis provides additional insight into the theoretical properties of the problem while also providing intuition for computing approximate solutions based on the proposed finite difference formulation. We combine our finite difference methods with continuation methods to trace complete bifurcation curves, validating established existence and uniqueness results and consistent with the results of the principle eigenvalue analysis.