Computation And Research For Multiple Solutions Of Nonlinear Elliptic Problem

Based on the Search-Extend-Method (SEM) that proposed by C.M.Chen and Z.Q.Xie for computing multiple solutions of nonlinear elliptic equation, this paper introduces the FEM and the interpolated coefficient finite element method into the SEM and calls this improved version as ISEM. ISEM not only can reduce our expensive computation greatly but also can be generalized to non-odd nonlinearity cases as well as many different domains (including the concave domains).Main results follows:(1) The computational result of this paper verifies (though only in numerical sense) the assumption about the structure and distribution of the multiple solutions of odd nonlinear equations [4], and we can obtain the corresponding solution from any given initial guess [4] More solutions can be obtained by ISEM than by any other existed methods (such as MPA, HLA) in some domains (such as square, triangle and L domain).(2) Further, the result of this paper verifies the assumption about the amount of the solutions is at least 3k -1[4] (k is the multiplity of the eigenvalue of the operator in equation), when the nonlinear term is f(u) = u3.