Upper and lower solutions for a homogeneous Dirichlet problem with nonlinear diffusion and the principle of linearized stability

We consider a class of quasilinear elliptic equations on a bounded domain subject to homogeneous Dirichlet boundary data. We establish a means of constructing upper and lower solutions in a neighborhood of a given solution to the quasilinear boundary value problem, leading to a principle of linearized stability-instability for the solution viewed as an equilibrium to the corresponding parabolic problem.