Advances in Differential Equations Volume 8, Number 9, September 2003, Pages 1107–1124 MULTIPLE SOLUTIONS FOR PERTURBED INDEFINITE SEMILINEAR ELLIPTIC EQUATIONS Paola Magrone Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy Silvia Mataloni Dipartimento di Matematica, Universit` a di Roma Tre Largo San L. Murialdo, 00100 Roma, Italy (Submitted by: Antonio Ambrosetti) Abstract. We are looking for infinitely many weak solutions for a semilinear elliptic equation with indefinite nonlinearity. The presence of an L 2 function perturbs the symmetry of the problem. The result is obtained using the approach introduced by Rabinowitz for positive nonlinearities. 1. Introduction We are interested in the following problem: (P f )  −Δu − λu = W (x)p(u)+ f (x) in Ω u =0 on ∂ Ω, where Ω is a bounded open subset of R N (N ≥ 3) with a smooth boundary. The parameter λ varies in the whole real line R and f is an L 2 (Ω) func- tion. The function W (x) is bounded in Ω and different from zero almost everywhere (“thin” zero set; see (W )), while p(u) is a continuous, subcriti- cal (with respect to the Sobolev embedding), and superlinear function (see (p1)–(p3)). Moreover, we ask that p is an odd function ((p4)). Therefore, for f ≡ 0, problem (P f ) becomes symmetric and thus problems like (P f ) are called perturbed while the indefiniteness of the problem is due to the change of sign of W (x). Accepted for publication: April 2003. AMS Subject Classifications: 35J25, 35J60. The authors are supported by MURST, Project “Variational Methods and Nonlinear Differential Equations.”. 1107