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 − u = λeu in Ω, u =0 on∂Ω in R2, where λ>0 is a parameter, and the almost critical problem − u = u N+2 N−2−ε, u>0 inΩ, u =0 on∂Ω in higher dimensions, where ε >0. © 2010 Elsevier Inc. All rights reserved