COMMUNICATIONS ON doi:10.3934/cpaa.2018098 PURE AND APPLIED ANALYSIS Volume 17, Number 5, September 2018 pp. 2063–2084 ON SPIKE SOLUTIONS FOR A SINGULARLY PERTURBED PROBLEM IN A COMPACT RIEMANNIAN MANIFOLD Shengbing Deng School of Mathematics and Statistics Southwest University, Chongqing 400715, China Zied Khemiri University of Tunis El Manar D´ epartement de Math´ ematiques Facult´ e des Sciences de Tunis, Campus Universitaire 2092 Tunis El Manar, Tunisia Fethi Mahmoudi ∗ Centro de Modelamiento Matem´atico, Universidad de Chile Beauchef 851, Edificio Norte–Piso 7, Santiago de Chile (Communicated by Jaeyoung Byeon) Abstract. Let (M,g) be a smooth compact riemannian manifold of dimension N ≥ 2 with constant scalar curvature. We are concerned with the following elliptic problem -ε 2 Δg u + u = u p-1 , u> 0, in M. where Δg is the Laplace-Beltrami operator on M, p> 2 if N = 2 and 2 <p< 2N N-2 if N ≥ 3, ε is a small real parameter. We prove that there exist a function Ξ such that if ξ 0 is a stable critical point of Ξ(ξ) there exists ε 0 > 0 such that for any ε ∈ (0,ε 0 ), problem (1) has a solution uε which concentrates near ξ 0 as ε tends to zero. This result generalizes previous works which handle the case where the scalar curvature function of (M,g) has non-degenerate critical points. 1. Introduction. We consider the following problem − ε 2 Δ g u + u − u p−1 =0 in M (1) where (M,g) is a smooth compact Riemannian manifold without boundary of di- mension N ≥ 2, ε> 0 is a small parameter and p> 2 if N = 2, 2 <p< 2 ∗ = 2N N−2 if N ≥ 3. The energy functional J ε associated to (1) is defined by J ε [u]=  M ( ε 2 2 |∇ g u| 2 + 1 2 u 2 − 1 p u p )dµ g for u ∈ H 1 (M ). 2000 Mathematics Subject Classification. Primary: 35J20, 35J60, 35B33; Secondary: 35B40. Key words and phrases. Singular perturbation problems, concentration phenomena, finite di- mensional reduction. S. Deng has been partly supported by National Natural Science Foundation of China 11501469 and the Basic Science and Advanced Technology Research of Chongqing cstc2016jcyA0032 and XDJK2017B014. F. Mahmoudi has been supported by Fondecyt Grant 1140311, fondo Basal PFB03 C.C. 2420 CMM and “Millennium Nucleus Center for Analysis of PDE NC130017”. * Corresponding author. 2063