An approximate bifurcation function and existence of solutions for semilinear elliptic PDEs

The bifurcation function associated with an elliptic boundary value problem Au + g[u] = 0 is a vector field B(omega) on R(d) with the property that the solutions of the boundary value problem are in a one-to-one correspondence with the zeros of B. An approximation B(h) of B using a finite element approach is formulated and optimal order error estimates are given. After the zeros of B(h), i.e., approximations for the zeros of B, have been given, the existence of the zeros of B close to the approximations is shown. (C) 2004 Elsevier Ltd. All rights reserved.