The Existence Of Nonradial Solutions For Semilinear Elliptic Equation With Hardy Term

In this paper, we are concerned with the existence of positive radial and non-radial symmetric solutions for the following semilinear elliptic problem with Hardy term:whereΩ= {x | x∈R~n, n≥3, a < |x| < 1} is a annulus , and 0≤μ<μ|-, = ((n-2)/2)~2, f(u) is some given function.We shall study the problem according to f(0) > 0 and f(0) = 0, and obtain the positive radial or nonradial symmetric solutions by shooting method and bifurcation theorey when f satisfies different conditions.The organization of this paper is as follows:In section 1, as an introduction, we list some results relativing to the radial solution of semilinear elliptic equation and the major results of this paper.In section 2, firstly, we obtain the positive radial symmetric solution for problem (1) by backward shooting under the assumption (A-1) (A-2). Secondly, we discuss the propotions of positive radial solutions.In section 3, firstly, we prove the minimizer of the corresponding variational functional exists by variational method under the hypothese (H-1)-(H-4). Then to study the existence of nonradial solutions, we investigate the linearized eigenvalue problem corresponding to (1).In section 4, we get the positive nonradial symmetric solution by bifurcation theoery.