Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions

We investigate the stability properties of positive steady-state solutions of semilinear initial-boundary-value problems with nonlinear boundary conditions. In particular, we employ a principle of linearized stability for this class of problems to prove sufficient conditions for the stability and instability of such solutions. These results shed some light on the combined effects of the reaction term and the boundary nonlinearity on stability properties. We also discuss various examples satisfying our hypotheses for stability results in dimension 1. In particular, we provide complete bifurcation curves for positive solutions for these examples.