Finding multiple solutions to elliptic systems with polynomial nonlinearity
Xuping Zhang, Jintao Zhang, Bo Yu · Numerical Methods for Partial Differential Equations · 2020
Abstract Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L2 norm and H1 norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D4 symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.