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

AbstractLet ϵ>0 be a small parameter. In this paper, we study existence of multiple multi-bump positive solutions for the semilinear Schrödinger equation−Δu+u=(1−ϵa(x))|u|p−2u,u∈H1(RN), where N⩾1, 2<p<2N/(N−2) if N⩾3, p>2 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−Δu=(1−ϵK(|x|))uN+2N−2,u∈D1,2(RN), where N⩾3, K∈C([0,∞)), K(r)>0 for r>0, limr→0K(r)=0, and limr→∞K(r)=0