Eigenfunction Expansion Method for Multiple Solutions of Semilinear Elliptic Equations with Polynomial Nonlinearity

Abstract

Eigenfunction expansion discretization is considered for finding multiple solutions of semilinear elliptic equations with polynomial nonlinearity. Error estimates of the discretization are derived. An efficient polynomial homotopy is constructed for computing all solutions of the resulting polynomial system on coarse level. A filter strategy on successively finer levels is designed to remove spurious solutions and to refine nonspurious solutions, simultaneously. The filtered solutions are further refined by a finite element Newton method. Numerical results are included to verify the error estimates derived, the efficiency of the homotopy, and the effectiveness of the filter strategy. A specific case of the Lazer--McKenna conjecture is numerically verified.