Multi-bump solutions and multi-tower solutions for equations on RN

Let >0 be a small parameter. In this paper, we study existence ofmultiple multi-bump positive solutions for the semilinear Schrödinger equation − u+ u = 1− a(x) |u|p−2u, u ∈ H1 RN , where N 1, 22 if N = 1 or N = 2, a ∈ C(RN), a(x) > 0 for x ∈ RN, and lim|x|→∞ a(x) = 0. We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equation − u = 1− K |x| u N+2 N−2, u∈ D1,2 RN , where N 3, K ∈ C([0,∞)), K(r)>0 for r >0, limr→0 K(r) = 0, and limr→∞K(r) = 0. © 2009 Elsevier Inc. All rights reserved.