Existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli

Abstract

We consider the nonlinear elliptic problem−Δu=upin ΩR,u>0in ΩR,u=0in ΩR where p>1 and ΩR={x∈RN:R<|x|<R+1} with N⩾3. It is known that as R→∞, the number of nonequivalent solutions of the above problem goes to ∞ when p∈(1,(N+2)/(N−2)), N⩾3. Here we prove the same phenomenon for any p>1 by finding O(N−1)-symmetric clustering bump solutions which concentrate near the set {(x1,…,xN)∈ΩR:xN=0} for large R>0.