Multiplicity Of Solutions For Perturbed Symmetric Nonlinear Elliptic Equations

Variational problems which are invariant under the actions of symmetric group often possess multiple solutions (e.g. see [1-6] and the reference therein). But when the symme- try is destroyed by a perturbation which is not small, does the same result still survive? Such kind of problems have been studied by many authors during the last two decades and some interesting results are also obtained (see [7-14]). In this paper, the multiplicity of solutions to the following elliptic problem ?2~,u )~.a(x)IuI~2u + g(x, u) + f(x, 慳), x C ?, (Er) { ~: 0, 2~C8fl) is considered, where f~ C R~ is a smooth bounded domain, g € x H), gQr,t) is odd with respect tot, and f € C(i~x H). According to 22?equivariant Ljusternik-Sehnirelman theory [13] and the method in [11], this paper follows, generalizes the previous results and obtains the existence of solutions in the following cases: (1) when fQr,抲) fQr), equation 梸u = AaQv)u+g(x,抋)+f(r) or ?jxu g(x,抋) + btf(x) has infinitely solutions (in the weak sense). (2) While aQv) 0, (Es,) has infinitely solutions (in the weak sense). Here 慳 C is said to be a weak solution of (Er) we t黣an that ~ 桳N~u.抳dx )~J~ a(r)luI~2uvd~x+ f~ g(x, u)vdrc + J拁2 f(x, u)vdx holds for any v C In ?, we give an introduction of this paper; In ?, some preliminaries and well- known notions are given; In ?, a variant of 22 equivariant Ljusternik-Schnirelman theory proposed by Ekeland and Ghoussoub [13] is used to prove case (1); Following the ideas in [11], ? is concerned with the proof of case (2).