Search extension method for multiple solutions of a nonlinear problem

Based on the eigensystem {λj, φj} of −Δ, multiple solutions of nonlinear problem Δu + u3 = 0 min Ω, u = 0 on Ω are approximated. A new method called the search extension method (SEM) composed of three steps is proposed. Numerical examples in a square and the unit circle are presented to show the efficiency of our method.