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

Let ϵ>0ϵ>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⩾1N⩾1, 22p>2 if N=1N=1 or N=2N=2, a∈C(RN)a∈C(RN), a(x)>0a(x)>0 for x∈RNx∈RN, and lim|x|→∞a(x)=0lim|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⩾3N⩾3, K∈C([0,∞))K∈C([0,∞)), K(r)>0K(r)>0 for r>0r>0, limr→0K(r)=0limr→0K(r)=0, and limr→∞K(r)=0limr→∞K(r)=0.