Nonlinear Analysis 54 (2003) 445–456 www.elsevier.com/locate/na Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities Norimichi Hirano a ; * , Wan Se Kim b; 1 a Department of Mathematics, Faculty of Engineering, Yokohama National University, 156 Tokiwadai, Hodogaya-ku, Yokohama, Japan b Department of Mathematics, College of Natural Sciences, Hanyang University, Seoul 133-791, South Korea Received 20 April 2002; accepted 3 December 2002 Abstract In this paper, we establish a multiple existence result of periodic solutions for a nonlinear parabolic equations with singular nonlinearity at zero. To establish our result, We use an ap- proximating process by smooth nonlinear functions to avoid a singularity in nonlinear term. And we adapt topological argument based on the variational structure of functionals corresponding to approximating equations. ? 2003 Elsevier Science Ltd. All rights reserved. MSC: 35k20; 35k55; 35k60 Keywords: Multiplicity; Sublinear grow; Parabolic equation; Stability; Unstability 1. Introduction Let N ¿ 2 and  be a bounded domain in R N with a smooth boundary 9 and h : R ×   → R be a continuous function such that for each x ∈ ; h(·;x) is T -periodic and continuous, and h¿ 0 on R × . In this paper, we are concerned with multiple * Corresponding author. Tel.: +81-45-339-4212. E-mail addresses: hirano@math.sci.ynu.ac.jp (N. Hirano), wanskim@hanyang.ac.kr (W.S. Kim). 1 This work was supported by Korea Research Foundation Grant KRF-2000-15-DP0027. 0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00101-9