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−Δu=λeuinΩ,u=0on∂Ω in R2R2, where λ>0λ>0 is a parameter, and the almost critical problem−Δu=uN+2N−2−ε,u>0inΩ,u=0on∂Ω in higher dimensions, where ε>0ε>0.