A Principle of Reduced Stability for Reaction–Diffusion Equations

A semilinear elliptic boundary value problem,Au+f(x, u, λ)=0 (withfu(x, u, λ) bounded below) can be shown to be equivalent to a finite-dimensional problem,B(c, λ)=0 d(c d), in the sense that their solution sets, which are not necessarily singletons, are in a one–to–one correspondence (c(u)↔u(c)). The functionB(c, λ) is called the bifurcation function. It is shown that, for any solutionu(c), the number of negative (resp. zero) eigenvalues of the matrixBc(c, λ) is identical to the number of negative (resp. zero) eigenvalues of the linearized elliptic operatorAv+fu(x, u(c), λ) v. This results in a version of the principle of reduced stability for the problemut+Au+f(x, u, λ)=0 and its reductionc′+B(c, λ)=0.