Nonexistence of multi-bubble solutions to some elliptic equations on convex domains

We prove the nonexistence of multi-bubble solutions for several types of problems on smooth-bounded convex domains. Problems we study include the Liouville equation -Delta u = lambda e(u) in Omega, u = 0 on partial derivative Omega in R(2), where lambda > 0 is a parameter, and the almost critical problem -Delta u = u(N+2/N-2 - epsilon), u > 0 in Omega, u = 0 on partial derivative Omega in higher dimensions, where epsilon > 0. (C) 2010 Elsevier Inc. All rights reserved.