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

Abstract

We consider the nonlinear elliptic problem-Delta u = u(p) in Omega(R), u > 0 in Omega(R), u = 0 in Omega(R)where p > 1 and Omega(R) = {x is an element of R-N: R < vertical bar x vertical bar < R + 1} with N >= 3. It is known that as R -> infinity, the number of nonequivalent solutions of the above problem goes to infinity when p is an element of (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 {(x(1), ... , x(N)) is an element of Omega(R): x(N) = 0} for large R > 0.