On The Number of Interior Peak Solutions for A Singularly Perturbed Neumann Problem Fang-Hua Lin Courant Institute of Mathematical Sciences Wei-Ming Ni University of Minnesota AND Jun-Cheng Wei Chinese University of Hong Kong Abstract We consider the following singularly perturbed Neumann problem: in in and on where is the Laplace operator, is a constant, is a bounded smooth domain in with its unit outward normal , and is super- linear and subcritical. A typical is where when and when . We show that there exists an such that for and for each integer bounded by where is a constant depending on and only, there exists a solution with interior peaks. (An explicit formula for is also given.) As a con- sequence, we obtain that for sufficiently small, there exists at least number of solutions. Moreover, for each there exist solutions with energies in the order of . c 2000 Wiley Periodicals, Inc. 1 Introduction The main theme of this paper is the concentration phenomena of the following singularly perturbed elliptic problem (1.1) in in on Communications on Pure and Applied Mathematics, Vol. 000, 0001–0027 (2000) c 2000 Wiley Periodicals, Inc.